Methods of soil analysis. Part 4, Physical methods [1 ed.] 9780891188933, 0891188932, 9780891188414
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English Pages 866 [1711] Year 2002
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CONTENTS
FOREWORD
PREFACE
CONTRIBUTORS
Conversion Factors for SI and non-SI Units
1
1.2 Soil Variability
1.2.1 Sources and Structure of Variability
1.2.1.1 Introduction
1.2.1.2 Properties and Processes
1.2.1.3 Sources of Variability
1.2.1.4 Structure of Variability
1.2.2 Variability and Scale
1.2.2.1 Scale of Research Domain
1.2.2.2 Scale of Observation
1.2.3 References
2
1.3 Errors, Variability, and Precision
1.3.1 Introduction
1.3.2 Classification of Measurement Errors
1.3.3 Scientific Validity of Measurements
1.3.4 Characterization of Variability
1.3.5 Skewed Frequency Distributions
1.3.5.1 Impact of Mathematical Distribution on Imprecision
1.3.6 Transformations of a Random Variable or Functions of a Random (Explanatory) Variable
1.3.7 The Estimation of Precision
1.3.8 Precision of Derived Observations
1.3.8.1 Case 1: A Single-Valued Function of an Observation
1.3.8.2 Case 2: A Number Derived from Measurements of More Than One Attribute on the Same Sample
1.3.8.3 Case 3: A Single Function of Numerous Measurements with Same Attribute
1.3.9 Error Propagation in Modeling
1.3.10 The Roles of Bias and Precision
1.3.11 How to Study Errors of Observation
1.3.12 Role of Errors of Observation in the Study of Relationships
1.3.13 A Note on Terminology
1.3.14 Statistical Problems and Techniques in General
1.3.15 References
01
3
1.4 Sampling
1.4.1 Designing a Sampling Scheme
1.4.1.1 Towards Better Planning
1.4.1.2 A Guiding Principle in Designing Sampling Schemes
1.4.1.3 Practical Issues
1.4.1.4 Scientific Issues
1.4.1.5 Statistical Issues
1.4.2 Design-Based and Model-Based Approach
1.4.3 Design-Based Strategies
1.4.3.1 Scope of Design-Based Strategies
1.4.3.2 Simple Random Sampling
1.4.3.3 Stratified Sampling
1.4.3.4 Two-Stage Sampling
1.4.3.5 Cluster Sampling
1.4.3.6 Systematic Sampling
1.4.3.7 Advanced Design-Based Strategies
1.4.4 Model-Based Strategies
1.4.5 Composite Sampling
1.4.6 Sampling in Dimensions Other than Two-Dimensional Space
1.4.6.1 Sampling in Three-Dimensional Space and at Depth
1.4.6.2 Sampling in Time
1.4.6.3 Sampling in Space–Time
1.4.7 References
4
5
1.6 Time and Space Series
1.6.1 General Information
1.6.2 Auto- and Cross-Correlation
1.6.3 Spectral Analysis
1.6.4 State–Space Analysis
1.6.4.1 Autoregressive State–Space Model for Spatial Processes
1.6.4.2 State–Space Analysis for Time Series
1.6.5 References
6
1.7 Parameter Optimization and Nonlinear Fitting
1.7.1 Introduction
1.7.2 Maximum-Likelihood and Weighted Least-Squares Estimator
1.7.3 Methods of Solution
1.7.4 Correlation and Confidence Intervals
1.7.5 Goodness of Fit
1.7.6 Examples and Optimization Programs
1.7.7 Discussion
1.7.8 References
02
7
1.8 Newer Application Techniques
1.8.1 Fractal Dimensions
1.8.1.1 Theory
1.8.1.2 Quantifying Soil Structure Using Fractal Geometry
1.8.1.3 Applications of Fractal Geometry to Soil Physics
1.8.1.4 Characterizing Soil Spatial Variability Using Fractal Geometry
1.8.1.5 Future Prospects
1.8.2 Fuzzy Sets
1.8.2.1 The Concept of Fuzzy Sets
1.8.2.2 Some Definitions and Examples Related to Fuzzy Sets
1.8.2.3 Fuzzy Models of Soil Physical Processes: Example
1.8.2.4 Application to Soil Classification, Mapping, and Land Evaluation
1.8.3 Wavelet Analysis
1.8.3.1 The Continuous Wavelet Transform
1.8.3.2 The Discrete Wavelet Transform
1.8.3.3 Prospects for Wavelet Analysis
1.8.4 References
8
Chapter 2: The Solid Phase
2.1 Bulk Density and Linear Extensibility
2.1.1 Introduction
2.1.1.1 Principles
2.1.1.2 Variability
2.1.1.3 Application
2.1.1.4 Dealing with Rock Fragments
2.1.2 Core Method
2.1.2.1 Introduction
2.1.2.2 Method
2.1.2.3 Comments
2.1.3 Excavation Method
2.1.3.1 Introduction
2.1.3.2 Method
2.1.3.3 Comments
2.1.4 Clod Method
2.1.4.1 Introduction
2.1.4.2 Equipment and Supplies
2.1.4.3 Procedure
2.1.4.4 Comments
2.1.5 Radiation Methods
2.1.5.1 Introduction
2.1.5.2 Equipment and Procedures
2.1.5.3 Comments
2.1.6 Linear Extensibility
2.1.6.1 Introduction
2.1.6.2 Calculations
2.1.6.3 Comments
2.1.7 References
9
2.2 Particle Density
2.2.1 Introduction
2.2.2 Principles
2.2.3 Methods
2.2.3.1 Calculation from Porosity and Bulk Density
2.2.3.2 Liquid Displacement
2.2.3.3 Gas Displacement
2.2.3.4. Estimation from Constituent Properties
2.2.4 Comments
2.2.5 References
10
2.3 Porosity
2.3.1 Introduction
2.3.2 Total Porosity
2.3.2.1 Calculation from Particle and Bulk Densities
2.3.2.2 Gravimetric Method with Water Saturation
2.3.2.3 Volumetric Method with Gas Pycnometry
2.3.3 Pore-Size Distribution
2.3.3.1 Water-Desorption Method
2.3.3.2 Visualization Method Using Impregnation
2.3.3.3 Mercury-Porosimetry Method
2.3.4 References
11
2.4 Particle-Size Analysis
2.4.1 Introduction
2.4.2 Pretreatment and Dispersion Techniques
2.4.2.1 General Principles
2.4.2.2 Organic Matter Removal
2.4.2.3 Iron Oxide Removal
2.4.2.4 Carbonate Removal
2.4.2.5 Soluble Salts Removal
2.4.2.6 Sample Dispersion
2.4.3 Specific Methods of Particle-Size Analysis
2.4.3.1 Introduction
2.4.3.2 Analysis by Sieving
2.4.3.3 Analysis by Gravitational Sedimentation
2.4.3.4 Pipette Method
2.4.3.5 Hydrometer Method
2.4.3.6 Modern Methods for Particle-Size Measurement
2.4.4 References
12
2.5 Specific Surface Area
2.5.1 Introduction
2.5.2 Liquid-Phase Adsorption Methods
2.5.2.1 Principles
2.5.2.2 Equipment and Supplies
2.5.2.3 Experimental Procedure
2.5.2.4 Comments
2.5.3 Gas-Phase Adsorption Methods
2.5.3.1 Principles
2.5.3.2 Equipment and Supplies
2.5.3.3 Experimental Procedure
2.5.3.4 Comments
2.5.4 Retention of Polar Liquids
2.5.4.1 Principles
2.5.4.2 Equipment and Supplies
2.5.4.3 Experimental Procedure
2.5.4.4 Comments
2.5.5 Comparison of Surface Area Methods
2.5.6 References
13
2.6 Aggregate Stability and Size Distribution
2.6.1 Principles
2.6.2 Apparatus and Procedures
2.6.2.1 General Sample Preparation
2.6.2.2 Modifications of Informally Standardized Methods and Apparatus
2.6.2.3 Representation of Data
2.6.3 Comments
2.6.4 References
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2.7 Shear Strength of Unsaturated Soils
2.7.1 Introduction
2.7.2 Shear Strength Equation for Unsaturated Soils
2.7.3 Triaxial Shear Tests for Unsaturated Soils
2.7.3.1 Test Procedures for Triaxial Tests
2.7.4 Direct Shear Tests for Unsaturated Soils
2.7.5 Failure Criteria for Unsaturated Soils
2.7.5.1 Strain Rates for Triaxial and Direct Shear Tests
2.7.6 Interpretation of Drained Test Results Using Multistage Testing Procedures
2.7.7 Nonlinearity of Failure Envelope
2.7.8 Interpretation of Undrained Test Results
2.7.8.1 Confined Compression Tests
2.7.8.2 Unconfined Compression Tests
2.7.9 Relationship Between the Soil Water Characteristic Curve and the Shear Strength of Unsaturated Soils
2.7.10 Procedure for Predicting the Shear Strength of Unsaturated Soils
2.7.11 Summary
2.7.12 References
03
2.8
2.9
2.9 Atterberg Limits
2.9.1 Introduction
2.9.2 Liquid Limit
2.9.2.1 Casagrande Method
2.9.2.2 One-Point Casagrande Method
2.9.2.3 Drop-Cone Penetrometer Method
2.9.3 Plastic Limit
2.9.3.1 Casagrande Method
2.9.4 References
2.10
2.10 Soil Compressibility
2.10.1 Introduction
2.10.2 Principles
2.10.3 Methods
2.10.3.1 Apparatus
2.10.3.2 Procedure
2.10.4 Comments
2.10.5 References
3.1
Chapter 3: The Soil Solution Phase
3.1 Water Content
3.1.1 General Information
3.1.2 Scope of Methods and Brief Description
3.1.2.1 Thermogravimetric Method Using Convective Oven-Drying
3.1.2.2 Gravimetric Method Using Microwave Oven-Drying
3.1.2.3 Time Domain Reflectometry
3.1.2.4 Ground Penetrating Radar
3.1.2.5 Capacitance Devices
3.1.2.6 Radar Scatterometry or Active Microwave
3.1.2.7 Passive Microwave
3.1.2.8 Electromagnetic Induction
3.1.2.9 Neutron Thermalization
3.1.2.10 Nuclear Magnetic Resonance
3.1.2.11 Gamma Ray Attenuation
3.1.3 Methods for Measurement of Soil Water Content
3.1.3.1 Thermogravimetric Using Convective Oven-Drying
3.1.3.2 Gravimetric Using Microwave Oven-Drying
3.1.3.3 The Basis of Electromagnetic Methods: A Wave Equation Framework
3.1.3.4 Time Domain Reflectometry
3.1.3.5 Ground Penetrating Radar to Measure Soil Water Content
3.1.3.6 Capacitance Devices
3.1.3.7 Active Microwave Remote Sensing Methods
3.1.3.8 Passive Microwave Remote Sensing Methods
3.1.3.9 Electromagnetic Induction
3.1.3.10 Neutron Thermalization
3.1.3.11 Nuclear Magnetic Resonance
3.1.4 References
3.2
3.2 Water Potential
3.2.1 Piezometry
3.2.1.1 Introduction
3.2.1.2 Principles
3.2.1.3 Drilling Methods
3.2.1.4 Installation
3.2.1.5 Monitoring
3.2.1.6 Technical Considerations
3.2.1.7 References
3.2.2
3.2.2 Tensiometry
3.2.2.1 Introduction
3.2.2.2 Soil Water Matric Potential
3.2.2.3 Principles
3.2.2.3.a Gravitational Potential
3.2.2.3.b Matric Potential
3.2.2.3.c Pneumatic Potential
3.2.2.3.d Osmotic Potential
3.2.2.3.e Soil Overburden and Swelling Potential
3.2.2.3.f Hydraulic Head
3.2.2.4 Essential Components of Tensiometers
3.2.2.4.a Porous Cup
3.2.2.4.b Water Reservoir
3.2.2.4.c Measurement Gauge
3.2.2.5 Alternative Types of Tensiometers
3.2.2.5.a Osmotic Tensiometer
3.2.2.5.b Methanol Tensiometers
3.2.2.6 Field and Laboratory Applications
3.2.2.6.a Field Applications
3.2.2.6.b Laboratory Applications
3.2.2.7 Field Measurements
3.2.2.7.a Equipment Suppliers
3.2.2.7.b Fabrication and Assembly
3.2.2.7.c Testing Tensiometer Units
3.2.2.7.d Tensiometer Installation
3.2.2.7.e Data Collection
3.2.2.7.f Maintenance and Servicing
3.2.2.8 Laboratory Measurements
3.2.2.8.a Equipment Suppliers
3.2.2.8.b Fabrication and Assembly
3.2.2.8.c Testing Tensiometer Units
3.2.2.8.d Data Collection
3.2.2.8.e Maintenance and Servicing
3.2.2.9 Interpretation of Tensiometric Readings
3.2.2.9.a Calculating Matric Potential from Tensiometric Measurements
3.2.2.9.b Calculation of Hydraulic Head Gradient
3.2.2.9.c Bias, Precision, and Accuracy
3.2.2.10 Gauge Calibration
3.2.2.11 References
3.2.3
3.2.3 Thermocouple Psychrometry
3.2.3.1 Principles
3.2.3.2 Equipment
3.2.3.2.a Sample-Chamber Sensors
3.2.3.2.b In Situ Sensors
3.2.3.2.c Electronics
3.2.3.3 Procedures
3.2.3.3.a Cleaning and Handling
3.2.3.3.b Calibration, Signal Generation, and Signal Interpretation
3.2.3.3.c Sample Handling and Laboratory Measurement
3.2.3.3.d In Situ Sensor Installation and Measurement
3.2.3.3.e Separation of Matric and Osmotic Water-Potential Components
3.2.3.4 Comments
3.2.3.5 Commercial Sources
3.2.3.5.a Laboratory Sample Chambers
3.2.3.5.b In Situ Sensors
3.2.3.5.c Electronic Measurement and Control Equipment
3.2.3.6 References
3.24
3.2.4 Miscellaneous Methods for Measuring Matric or Water Potential
3.2.4.1 Introduction
3.2.4.2 Heat Dissipation Sensors
3.2.4.2.a Principles
3.2.4.2.b Equipment
3.2.4.2.c Calibration and Measurement
3.2.4.2.d Errors
3.2.4.3 Electrical Resistance Sensors
3.2.4.3.a Principles
3.2.4.3.b Equipment
3.2.4.3.c Calibration and Measurement
3.2.4.3.d Errors
3.2.4.4 Frequency Domain and Time Domain Matric Potential Sensors
3.2.4.4.a Principles
3.2.4.4.b Equipment
3.2.4.4.c Calibration
3.2.4.4.d Errors
3.2.4.5 Electro-Optical Switches
3.2.4.6 Dew Point Potentiameter
3.2.4.6.a Principles
3.2.4.6.b Equipment
3.2.4.6.c Measurement
3.2.4.6.d Errors
3.2.4.7 Filter Paper Technique
3.2.4.7.a Principles
3.2.4.7.b Calibration
3.2.4.7.c Measurement
3.2.4.7.d Errors
3.2.4.8 Vapor Equilibration
3.2.4.8.a Principles
3.2.4.8.b Measurement
3.2.4.8.c Errors
3.2.4.9 References
3.3
3.3 Water Retention and Storage
3.3.1 Introduction
3.3.1.1 References
3.32
3.3.2 Laboratory
3.3.2.1 Introduction
3.3.2.1.a General
3.3.2.1.b Samples
3.3.2.1.c Sample Preparation
3.3.2.1.d Wetting Solution
3.3.2.1.e Sample Wetting
3.3.2.1.f Temperature Effects
3.3.2.2 Hanging Water Column
3.3.2.2.a Principles
3.3.2.2.b Equipment and Supplies
3.3.2.2.c Procedure
3.3.2.2.d Comments
3.3.2.3 Pressure Cell
3.3.2.3.a Principles
3.3.2.3.b Equipment and Supplies
3.3.2.3.c Procedure
3.3.2.3.d Comments
3.3.2.4 Pressure Plate Extractor
3.3.2.4.a Principles
3.3.2.4.b Equipment and Supplies
3.3.2.4.c Procedure
3.3.2.4.d Comments
3.3.2.5 Long Column
3.3.2.5.a Principles
3.3.2.5.b Equipment and Supplies
3.3.2.5.c Procedure
3.3.2.5.d Comments
3.3.2.6 Suction Table
3.3.2.6.a Principles
3.3.2.6.b Equipment and Supplies
3.3.2.6.c Procedure
3.3.2.6.d Comments
3.3.2.7 Controlled Liquid Volume
3.3.2.7.a Principles
3.3.2.7.b Equipment and Supplies
3.3.2.7.c Procedure
3.3.2.7.d Comments
3.3.2.8 Determination of Soil Water Characteristic by Freezing Method
3.3.2.8.a. Introduction
3.3.2.8.b Matric Potential in Frozen Soil
3.3.2.8.c Liquid Water Content in Frozen Soil
3.3.2.8.d Similarity Between Soil Freezing and Soil Moisture Characteristic
3.3.2.8.e Comments
3.3.2.9 Miscellaneous Methods
3.3.2.9.a Controlled Vapor Pressure-Description and Principles
3.3.2.9.b Measured Vapor Pressure-Description and Principles
3.3.2.9.c Controlled Osmotic Pressure-Description and Principles
3.3.2.9.d Comments on Vapor-and Osmotic-Based Methods
3.3.2.9.e Transient Liquid-Phase Methods
3.3.2.10 Computational Corrections
3.3.2.11 Guidelines For Method Selection
3.3.2.12 References
3.3.3
3.3.3 Field
3.3.3.1 Introduction
3.3.3.2 Field Water Capacity
3.3.3.2.a Field Test
3.3.3.2.b Laboratory Approximation
3.3.3.2.c Analytical Determination of Field Water Capacity
3.3.3.2.d Comments
3.3.3.3 Permanent Wilting Point
3.3.3.4 Available Water
3.3.3.5 Specific Yield
3.3.3.6 References
3.3.4
3.3.4 Parametric Models
3.3.4.1 Introduction
3.3.4.2 General Characteristics of Water Retention Curves and Important Parameters
3.3.4.3 Brooks and Corey Type Power Function
3.3.4.4 van Genuchten Type Power Function
3.3.4.5 Exponential Function
3.3.4.6 Lognormal Distribution Function
3.3.4.7 Water Capacity Functions
3.3.4.8 Unsaturated Hydraulic Conductivity Functions
3.3.4.9 Multimodal Retention Functions
3.3.4.10 Soil Water Hysteresis
3.3.4.11 Model Fitting
3.3.4.12 Comments
3.3.4.13 References
3.3.5
3.3.5 Property-Transfer Models
3.3.5.1 Physically Based Water Retention Prediction Models
3.3.5.1.a Introduction
3.3.5.1.b Arya-Paris Model: Principles
3.3.5.1.c Haverkamp-Parlange Model: Principles
3.3.5.1.d Comments
3.3.5.2 Property Transfer from Particle and Aggregate Size to Water Retention
3.3.5.2.a Principles
3.3.5.2.b Procedure
3.3.5.2.c Comments
3.3.5.3 References
3.3.6
3.3.6 Air-Water Interfacial Area
3.3.6.1 Introduction
3.3.6.2 Principles
3.3.6.3 Aqueous-Static Method
3.3.6.3.a Apparatus
3.3.6.3.b Procedure
3.3.6.3.c Calculation
3.3.6.4 Aqueous-Dynamic Method
3.3.6.4.a Apparatus
3.3.6.4.b Procedure
3.3.6.4.c Calculations
3.3.6.5 Gaseous-Dynamic Method
3.3.6.5.a Apparatus
3.3.6.5.b Procedure
3.3.6.5.c Calculation
3.3.6.6 Comments
3.3.6.7 References
3.4
3.4 Saturated and Field-Saturated Water Flow Parameters
3.4.1 Introduction
3.4.1.1 Principles and Parameter Definitions
3.4.1.2 References
3.4.2
3.4.2 Laboratory Methods
3.4.2.1 Introduction
3.4.2.2 Constant Head Soil Core (Tank) Method
3.4.2.3 Falling Head Soil Core (Tank) Method
3.4.2.4 Steady Flow Soil Column Method
3.4.2.5 Other Laboratory Methods
3.4.2.6 References
3.4.3
3.4.3 Field Methods (Vadose and Saturated Zone Techniques)
3.4.3.1 Introduction
3.4.3.2 Ring or Cylinder Infiltrometers (Vadose Zone)
3.4.3.3
3.4.3.3 Constant Head Well Permeameter (Vadose Zone)
3.4.3.3.a Introduction
3.4.3.3.b Apparatus and Procedures
3.4.3.3.c Analyses
3.4.3.3.d Example Calculations
3.4.3.3.e Comments
3.4.3.3.f References
04
3.4.3.3
3.4.3.4 Auger-Hole Method (Saturated Zone)
3.4.3.4.a Introduction
3.4.3.4.b Analysis
3.4.3.4.c Equipment and Supplies
3.4.3.4.d Procedure
3.4.3.4.e Layered Soils
3.4.3.4.f Example Calculations
3.4.3.4.g Comments
3.4.3.4.h References
3.4.3.5
3.4.3.5 Piezometer Method (Saturated Zone)
3.4.3.5.a Introduction
3.4.3.5.b Analysis
3.4.3.5.c Equipment and Supplies
3.4.3.5.d Procedure
3.4.3.5.e Layered Soils
3.4.3.5.f Example Calculations
3.4.3.5.g Comments
3.4.3.5.h References
3.4.3.6
3.4.3.6 Other Saturated Zone Methods
3.4.3.6.a Introduction
3.4.3.6.b References
3.5
3.5 Unsaturated Water Transmission Parameters Obtained from Infiltration
3.5.1 Basic Theory
3.5.2 One-Dimensional Infiltration Equations and Their Use
3.5.3 Horizontal Absorption-The Bruce and Klute Experiment
3.5.4 Three-Dimensional Infiltration Using Disk Permeameters
3.5.4.1 Early-Time Observations
3.5.4.2 Steady-State Observations
3.5.5 Conclusions
3.5.6 References
3.6.11
3.6 Simultaneous Determination of Water Transmission and Retention Properties
3.6.1. Direct Methods
3.6.1.1 Laboratory
3.6.1.2
3.6.1.2 Field
3.6.1.2.a Instantaneous Profile
3.6.1.2.b Plane of Zero Flux
3.6.1.2.c Constant Flux Vertical Time Domain Reflectometry
3.6.1.2.d References
3.6.2
3.6.2. Inverse Methods
3.6.2.1 Introduction
3.6.2.2 Theory of Flow and Optimization
3.6.2.2.a Water Flow Modeling
3.6.2.2.b Parameter Optimization
3.6.2.3 Multistep Outflow Method
3.6.2.3.a Introduction
3.6.2.3.b Experimental Procedures
3.6.2.3.c Simulations and Optimization
3.6.2.4 Evaporation Method
3.6.2.4.a Introduction
3.6.2.4.b Experimental Procedures
3.6.2.4.c Simulation and Optimization
3.6.2.5 Tension Disc Infiltrometer
3.6.2.5.a Introduction
3.6.2.5.b Experimental Procedures
3.6.2.5.c Simulation and Optimization
3.6.2.6 Field Drainage
3.6.2.6.a Introduction
3.6.2.6.b Experimental Procedures
3.6.2.6.c Simulation and Optimization
3.6.2.7 Additional Applications
3.6.2.8 Example
3.6.2.9 Discussion
3.6.2.10 References
05
3.6.3
3.6.3 Indirect Methods
3.6.3.1 Introduction
3.6.3.2 Semiempirical Approaches
3.6.3.2.a Pore-Size Distribution
3.6.3.2.b Particle-Size Distribution
3.6.3.3 Empirical Approaches
3.6.3.3.a Regression Analysis
3.6.3.3.b Neural Network Analysis
3.6.3.3.c Databases
3.6.3.4 References
3.7
3.7 Evaporation from Natural Surfaces
3.7.1 Soil-Based Methods
3.7.1.1 Lysimetry
3.7.1.2 Soil Water Balance
3.7.2 Plant-Based Methods
3.7.2.1 Sap Flow Measurement
3.7.2.2 Cuvettes
3.7.3 Micrometeorological Methods
3.7.3.1 Bowen Ratio Energy Balance Method
3.7.3.2 Eddy Covariance
3.7.3.3 Conditional Sampling
3.7.4 Remote Sensing Methods
3.7.5 References
4.1
Chapter 4 The Soil Gas Phase
4.1 Introduction
4.1.1 References
4.2
4.2 Gas Sampling and Analysis
4.2.1 Principles
4.2.2 Sampling
4.2.2.1 Air-Filled Pores Above the Water Table
4.2.2.2 Soil Water Sampling
4.2.3 Gas Analysis
4.2.3.1 Principles of Gas Chromatography
4.2.3.2 Laboratory Analysis by Gas Chromatography
4.2.3.3 Field-Based Gas Chromatography Systems
4.2.3.4 Alternative Gas Analysis Systems
4.2.4 In Situ Analyses at the Gas-Liquid Interface
4.2.4.1 Platinum Microelectrode Methods
4.2.4.2 Membrane-Covered Electrode Methods
4.2.4.3 Miscellaneous Gas-Sensing Probes
4.2.5 References
4.3
4.3 Gas Diffusivity
4.3.1 Introduction
4.3.2 Laboratory Methods
4.3.2.1 The Currie Method
4.3.2.2 The Two-Chamber Method
4.3.3 Field Method
4.3.3.1 The MacIntyre and Philip Method
4.3.4 Predicting Gas Diffusivity
4.3.4.1 Undisturbed Soil
4.3.4.2 Sieved, Repacked Soil
4.3.4.3 Values of D0
4.3.5 References
4.4
4.4 Air Permeability
4.4.1 Introduction
4.4.2 Laboratory Methods
4.4.3 Field Methods
4.4.3.1 Acoustic Method
4.4.3.2 Buried Probe and Well Techniques
4.4.4 Recommended Laboratory Method
4.4.4.1 Apparatus and Materials
4.4.4.2 Procedures
4.4.4.3 Comments
4.4.5 Recommended Field Method
4.4.5.1 Apparatus and Materials
4.4.5.2 Procedures
4.4.5.3 Comments
4.4.6 Choice of Method, Including Scaling and Variability Aspects
4.4.7 Predicting Air Permeability
4.4.8 Conclusions and Summary
4.4.9 References
4.5
4.5 Soil–Atmosphere Gas Exchange
4.5.1 Introduction
4.5.2 Computation from Fick’s Law
4.5.2.1 Principles
4.5.2.2 Apparatus, Materials, and Procedure
4.5.2.3 Comments and Cautions
4.5.3 Chamber Systems
4.5.3.1 Principles
4.5.3.2 Apparatus and Materials
4.5.3.3 Procedure
4.5.3.4 Comments and Cautions
4.5.4 Sampling Design, Data Analyses, and Data Summaries
4.5.5 Concluding Remarks
4.5.6 References
5.1
Chapter 5: Soil Heat
5.1 Temperature
5.1.1 Thermocouple Thermometry
5.1.2 Integrated Circuit Thermometers
5.1.3 Resistance Thermometers
5.1.3.1 Platinum Resistance Thermometers
5.1.3.2 Resistance Temperature Detectors
5.1.3.3 Thermistors
5.1.4 Nonelectric Thermometers
5.1.5 Infrared Radiation Thermometers
5.1.6 Installation and Operation
5.1.7 References
5.2
5.2 Heat Capacity and Specific Heat
5.2.1 General Introduction
5.2.2 General Principles
5.2.2.1 Basic Definitions
5.2.2.2 Relationship Between Volumetric Heat Capacity and Specific Heat
5.2.3 Methods
5.2.3.1 De Vries Approximation
5.2.3.2 Dual-Probe Heat-Pulse Method
5.2.4 References
5.3
5.3 Thermal Conductivity
5.3.1 Introduction
5.3.2 Predictive Methods
5.3.2.1 Predicting Soil Thermal Conductivity, IncludingTemperature Effects
5.3.2.2 Predicting Soil Thermal Conductivity from Readily AvailableSoils Data
5.3.3 Steady-State Methods
5.3.3.1 Guarded Hot Plate Method
5.3.4 Transient-State Methods
5.3.4.1 Theory
5.3.4.2 Probe Design and Construction
5.3.4.3 Single Heat Probe Method
5.3.4.4 Dual-Probe Heat-Pulse Method
5.3.5 Comments Concerning Thermal Conductivity
5.3.6 References
5.4
5.4 Soil Thermal Diffusivity
5.4.1 Laboratory Method for Determining Soil Thermal Diffusivity
5.4.1.1 Principles
5.4.1.2 Measurements
5.4.1.3 Analysis of Soil Column Temperature Observations
5.4.2 Field Method for Determining Soil Thermal Diffusivity
5.4.3 References
5.5
5.5 Heat Flux Density
5.5.1 Calorimetric
5.5.1.1 Principles
5.5.1.2 Equipment
5.5.1.3 Procedures
5.5.1.4 Commentary on Advantages and Limitations
5.5.2 Gradient
5.5.2.1 Principles
5.5.2.2 Equipment
5.5.2.3 Procedures
5.5.2.4 Commentary on Advantages and Limitations
5.5.3 Combination
5.5.3.1 Principles
5.5.3.2 Equipment
5.5.3.3 Procedures
5.5.3.4 Commentary on Advantages and Limitations
5.5.4 Soil Heat Flux Plate
5.5.4.1 Principles
5.5.4.2 Equipment
5.5.4.3 Procedures
5.5.4.4 Commentary on Advantages and Limitations
5.5.5 References
5.6
5.6 Coupled Heat and Water Transfer
5.6.1 Introduction
5.6.2 Soil Thermal Water Diffusivity
5.6.3 References
6.1
Chapter 6: Miscible Solute Transport
6.1 Solute Content and Concentration
6.1.1 Introduction
6.1.2 Measurement of Solute Content Using Soil Extraction
6.1.2.1 General Principles
6.1.2.2 Equipment
6.1.2.3 Procedure
6.1.2.4 Comments
6.1.3 Measurement of Solute Concentration Using Soil Water Extraction
6.1.3.1 Suction Cups
6.1.3.2 Passive Capillary Samplers
6.1.3.3 Porous Matrix Sensors
6.1.4 Indirect Measurement of Solute Concentration
6.1.4.1 Relationship between Soil Water Solute Concentration and Apparent Soil Electrical Conductivity
6.1.4.2 Electrical Resistivity: Wenner Array
6.1.4.3 Electrical Resistivity: Four-Electrode Probe
6.1.4.4 Time Domain Reflectometry
6.1.4.5 Nonintrusive Electromagnetic Induction
6.1.5 Emerging Methods
6.1.5.1 Fiber Optic Sensors
6.1.5.2 Capillary Absorbers
6.1.6 References
6.2
6.2 Solute Diffusion
6.2.1 Introduction
6.2.2 Theory of Diffusion
6.2.2.1 Fick’s Laws of Diffusion
6.2.2.2 Estimation of Diffusion Coefficients in Liquids
6.2.3.3 Temperature Dependence of Diffusion Coefficients
6.2.2.4 Type of Diffusion Coefficients
6.2.2.5 Diffusion of Nonreactive Solutes in Porous Media
6.2.2.6 Estimation of Diffusion Coefficients for Nonreactive Solutesin Porous Media
6.2.2.7 Diffusion and Convection
6.2.2.8 Diffusion and Reactions
6.2.3 Methods
6.2.3.1 Steady-State Methods
6.2.3.2 Transient Methods
6.2.3.3 Other Methods
6.2.4 Conclusions
6.2.5 References
6.3
6.3 Solute Transport: Theoretical Background
6.3.1 Elementary Concepts
6.3.1.1 Solute Transport Experiments
6.3.1.2 Breakthrough Curves
6.3.1.3 Moments
6.3.1.4 Mass Balance
6.3.1.5 Flux and Resident Concentrations
6.3.2 Convection–Dispersion Model
6.3.2.1 Dimensionless Parameters
6.3.2.2 Flux Concentrations
6.3.2.3 Boundary Conditions
6.3.2.4 Analytical Solutions
6.3.2.5 Moments
6.3.3 Nonequilibrium Models
6.3.3.1 Two-Region Model
6.3.3.2 Two-Site Model
6.3.3.3 General Nonequilibrium Formulation
6.3.3.4 Analytical Solutions and Moments
6.3.3.5 Additional Nonequilibrium Formulations
6.3.4 Stochastic–Convective Model
6.3.4.1 Transfer Function Modeling
6.3.4.2 Stochastic–Convective Transfer Function Model
6.3.4.3 Convective Lognormal Transfer Function Model
6.3.4.4 Field Applications
6.3.4.5 Comments
6.3.5 Transport Equation Generalizations
6.3.6 References
6.4
6.4 Solute Transport: Experimental Methods
6.4.1 Laboratory
6.4.1.1 Introduction
6.4.1.2 Sample Collection and Preparation
6.4.1.3 Apparatus
6.4.1.4 Displacing and Resident Solutions
6.4.1.5 Procedures
6.4.1.6 Calculations and Data Presentation
6.4.1.7 Additional Apparatuses
6.4.1.8 Additional Procedures
6.4.1.9 Comments
6.4.2 Field
6.4.2.1 Introduction
6.4.2.2 Water and Solute Application
6.4.2.3 Solute Monitoring and Measurement
6.4.3 Tracers
6.4.4 Equipment Sources
6.4.5 References
6.5
6.5 Solute Transport: Data Analysis and Parameter Estimation
6.5.1 Least-Squares Fitting
6.5.1.1 Principles
6.5.1.2 Software
6.5.1.3 Convection–Dispersion Model
6.5.1.4 Nonequilibrium Models
6.5.1.5 Transient Water Flow and Depth-Dependent Water Content
6.5.2 Method of Moments
6.5.2.1 Principles and Procedures
6.5.2.2 Example
6.5.2.3 Recommendations
6.5.3 Determination of θIM and α Using a Disk Permeameter and Multiple Solutes
6.5.3.1 Principles
6.5.3.2 Procedure
6.5.3.3 Example
6.5.4 Determination of Travel-Time Probability Density Function from Concentration Data
6.5.5 Determination of Field Solute Mass Flux (Travel Time Probability Density Function) Using Time Domain Reflectometry
6.5.5.1 Principles
6.5.5.2 Vertically Installed Probes
6.5.5.3 Equipment and Methodology
6.5.6 References
6.6
6.6 Solute Transport During Variably Saturated Flow—Inverse Methods
6.6.1 Introduction
6.6.2 Theory of Flow, Transport, and Optimization
6.6.3 Examples
6.6.3.1 Steady-State Laboratory Flow Experiment with Nonlinear Transport
6.6.3.2 Laboratory Transport Subject to Flow Interruption
6.6.3.3 Transient Laboratory Experiment with Equilibrium Solute Transport
6.6.3.4 Field Experiment with Nonequilibrium Solute Transport
6.6.4 Conclusions
6.6.5 References
6.7
6.7 Processes Governing Transport of Organic Solutes
6.7.1 Phase Transfer and Distribution Coefficients
6.7.1.1 Air–Water Distribution
6.7.1.2 Soil–Water Distribution
6.7.1.3 Soil–Air Distribution
6.7.1.4. Implications of Soil–Water–Air Distribution on Solute Transport
6.7.2 Transformation
6.7.2.1 Chemical Transformation
6.7.2.2 Biological Degradation
6.7.2.3 Photodegradation
6.7.2.4 Implications of Transformation on Solute Transport
6.7.3 References
6.8
6.8 Microbial Transport
6.8.1 Introduction
6.8.1.1 Parasites
6.8.1.2 Bacteria
6.8.1.3 Viruses
6.8.2 Laboratory Methods
6.8.2.1 Batch Equilibration Method
6.8.2.2 Flowthrough Columns
6.8.3 Field Studies
6.8.4 Indicators of Human Enteroviruses
6.8.5 Microbial Transport Modeling
6.8.6 References
6.9
6.9 Geochemical Transport
6.9.1 Introduction
6.9.2 Geochemical Reaction Equations
6.9.2.1 Complexation
6.9.2.2 Cation Exchange Reactions
6.9.2.3 Adsorption Reactions
6.9.2.4 Precipitation–Dissolution
6.9.2.5 Reactions with Organic Matter and Effects of Bacteria
6.9.2.6 Activity Coefficients and Thermodynamic Equilibrium Constants
6.9.3 Mass-Balance Transport Equations
6.9.4 Numerical Implementation
6.9.5 Effects of Solution Composition on Hydraulic Properties and Reclamation Models
6.9.6 Applications
6.9.7 Use of Geochemical Transport Models
6.9.8 References
7.1
Chapter 7: Multi-Fluid Flow
7.1 Introduction
7.2 Fluid Contents
7.2.1 Introduction
7.2.2 Nondestructive Measurements
7.2.2.1 Gamma Radiation
7.2.2.2 X-ray Radiation
7.2.2.3 Other Methods
7.2.3 Destructive Measurements
7.3 Saturation-Pressure Relationships
7.3.1 Introduction
7.3.2 Air-Nonaqueous Phase Liquid Systems
7.3.3 Two Immiscible Liquids
7.3.4 Air and Two Immiscible Liquids
7.4 Relative Permeability Measurements
7.4.1 Introduction
7.4.2 Nonaqueous Phase Liquid-Gas Systems
7.4.3 Nonaqueous Phase Liquid-Water Systems
7.4.3.1 Theory
7.4.3.2 Equipment and Measurements
7.4.3.3 Summary and Comments
7.5 Prediction of Capillary Pressure-Relative Permeability Relations
7.5.1 Introduction
7.5.2 Extending Two-Phase Saturation-Pressure Relations to Air-Nonaqueous Phase Liquid-Water Systems: Nonhysteretic
7.5.3 Extending Two-Phase Saturation-Pressure Relations to Air-Nonaqueous Phase Liquid-Water Systems: Hysteretic
7.6 Measuring Interfacial Areas of Immiscible Fluids
7.6.1 Introduction
7.6.2 Experimental Techniques for the Measurement of Interfacial Areas
7.6.2.1 Trapped Nonwetting Phase
7.6.2.2 Continuous Nonwetting Phase
7.6.3 Water Saturation-Interfacial Area Relationship
7.6.4 Comments
7.7 References
8.1
Chapter 8: Soil Erosion by Water and Tillage
8.1 Introduction
8.2 Soil Erosion by Water
8.2.1 Soil Erosion Measurements
8.2.1.1 Monitoring Erosion during Natural Storm Events
8.2.1.2 Erosion Measurements during Simulated Rain Storms
8.2.1.3 Experimental Area
8.2.2 Soil Erosion Prediction
8.2.2.1 USLE-RUSLE Relationships
8.2.2.2 Water Erosion Protection Project Soil Erosion Prediction Relationship
8.3 Tillage Erosion
8.3.1 Experimental Measurement of Tillage Erosion using Tracers
8.3.1.1 Principle
8.3.1.2 Equipment, Software, and Supplies
8.3.1.3 Procedural Steps
8.3.2 Measurement of Tillage Erosion by Volumetric Assessment of Soil Translocation
8.3.2.1 Principle
8.3.2.2 Equipment, Software, and Supplies
8.3.2.3 Procedural Steps
8.3.3 Estimation of Tillage Erosion from Cesium-137 Inventories
8.3.3.1 Principle
8.3.3.2 Equipment, Software, and Supplies
8.3.3.3 Procedural Steps
8.4 References
8.2
SUBJECT INDEX
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Methods of Soil Analysis Part 4 Physical Methods
Soil Science Society of America Book Series Books in the series are available from the Soil Science Society of America, 677 South Segoe Road, Madison, WI 53711 USA. 1. MINERALS IN SOIL ENVIRONMENTS. Second Edition. 1989. J. B. Dixon and S. B. Weed, editors R. C. Dinauer, managing editor 2. PESTICIDES IN THE SOIL ENVIRONMENT: PROCESSES, IMPACTS, AND MODELING. 1990. H. H. Cheng, editor S. H. Mickelson, managing editor 3. SOIL TESTING AND PLANT ANALYSIS. Third Edition. 1990. R. L. Westerman, editor S. H. Mickelson, managing editor 4. MICRONUTRIENTS IN AGRICULTURE. Second Edition. 1991. J. J. Mortvedt et al., editors S. H. Mickelson, managing editor 5. METHODS OF SOIL ANALYSIS: PHYSICAL AND MINERALOGICAL METHODS. Part 1. Second Edition. 1986. Arnold Klute, editor R. C. Dinauer, managing editor METHODS OF SOIL ANALYSIS: MICROBIOLOGICAL AND BIOCHEMICAL PROPERTIES. Part 2. 1994. R. W. Weaver et al., editor S. H. Mickelson, managing editor METHODS OF SOIL ANALYSIS: CHEMICAL METHODS. Part 3. 1996. D. L. Sparks, editor J. M. Bartels, managing editor METHODS OF SOIL ANALYSIS: PHYSICAL METHODS. Part 4. 2002. J. H. Dane and G. C. Topp, Co-editors L. K. Al-Amoodi, managing editor 6. LAND APPLICATION OF AGRICULTURAL, INDUSTRIAL, AND MUNICIPAL BY-PRODUCTS. 2000. J. F. Power and W. A. Dick, editors J. M. Bartels, managing editor 7. SOIL MINERALOGY WITH ENVIRONMENTAL APPLICATIONS J. B. Dixon and D. G. Schulze, editors L. K. Al-Amoodi, managing editor
Methods of Soil Analysis Part 4 Physical Methods Jacob H. Dane and G. Clarke Topp, Co-editors
Editorial Committee Gaylon S. Campbell Robert Horton William A. Jury Donald R. Nielsen Harold M. van Es Peter J. Wierenga Jacob H. Dane (Co-editor) G. Clarke Topp (Co-editor)
Managing Editor: Lisa Al-Amoodi Editor-in-Chief SSSA: Warren A. Dick
Number 5 in the Soil Science Society of America Book Series
Published by: Soil Science Society of America, Inc. Madison, Wisconsin, USA 2002
Copyright© 2002 by the Soil Science Society of America, Inc. ALL RIGHTS RESERVED UNDER THE U.S. COPYRIGHT ACT OF 1976 (PL. 94-533). Any and all uses beyond the limitations of the “fair use” provision of the law require written permission from the publisher(s) and/or the author(s); not applicable to contributions prepared by officers or employees of the U.S. Government as part of their official duties. The views expressed in this publication represent those of the individual Editors and Authors. These views do not necessarily reflect endorsement by the Publisher(s). In addition, trade names are sometimes mentioned in this publication. No endorsement of these products by the Publisher is intended, nor is any criticism implied of similar products not mentioned.
Soil Science Society of America, Inc. 677 South Segoe Road, Madison, WI 53711-1086 USA
Library of Congress Control Number: 2002109389
Printed in the United States of America.
CONTENTS FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi CONVERSION FACTORS FOR SI AND NON-SI UNITS . . . . . . . . . . . . xxxvii
Chapter 1 Soil Sampling and Statistical Procedures 1.1 Introduction
A.W. WARRICK AND H.M. VAN ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Soil Variability H.M. VAN ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Sources and Structure of Variability . . . . . . . . . . . . . . . . . . . . . 1.2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1.2 Properties and Processes . . . . . . . . . . . . . . . . . . . . . . . 1.2.1.3 Sources of Variability . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1.4 Structure of Variability . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Variability and Scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2.1 Scale of Research Domain . . . . . . . . . . . . . . . . . . . . . 1.2.2.2 Scale of Observation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Errors, Variability, and Precision R.R. ALLMARAS AND OSCAR KEMPTHORNE . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Classification of Measurement Errors . . . . . . . . . . . . . . . . . . . . 1.3.3 Scientific Validity of Measurements . . . . . . . . . . . . . . . . . . . . . 1.3.4 Characterization of Variability . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Skewed Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5.1 Impact of Mathematical Distribution on Imprecision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Transformations of a Random Variable or Functions of a Random (Explanatory) Variable . . . . . . . . . . . . . . . . . . . . . 1.3.7 The Estimation of Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.8 Precision of Derived Observations. . . . . . . . . . . . . . . . . . . . . . . 1.3.8.1 Case 1: A Single-Valued Function of an Observation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.8.2 Case 2: A Number Derived from Measurements of More Than One Attribute on the Same Sample . . . 1.3.8.3 Case 3: A Single Function of Numerous Measurements with Same Attribute . . . . . . . . . . . . . . 1.3.9 Error Propagation in Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.10 The Roles of Bias and Precision . . . . . . . . . . . . . . . . . . . . . . . . 1.3.11 How to Study Errors of Observation . . . . . . . . . . . . . . . . . . . . . 1.3.12 Role of Errors of Observation in the Study of Relationships . . 1.3.13 A Note on Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
1 1 1 1 2 3 4 8 8 9 12 15 15 16 17 18 22 27 28 29 31 31 33 34 35 36 39 40 41
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CONTENTS
1.3.14 Statistical Problems and Techniques in General . . . . . . . . . . . . 1.3.15 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Sampling J.J. DE GRUIJTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Designing a Sampling Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.1 Towards Better Planning . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.2 A Guiding Principle in Designing Sampling Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.3 Practical Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.4 Scientific Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.5 Statistical Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Design-Based and Model-Based Approach. . . . . . . . . . . . . . . . 1.4.3 Design-Based Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.1 Scope of Design-Based Strategies . . . . . . . . . . . . . . . 1.4.3.2 Simple Random Sampling. . . . . . . . . . . . . . . . . . . . . . 1.4.3.3 Stratified Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.4 Two-Stage Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.5 Cluster Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.6 Systematic Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.7 Advanced Design-Based Strategies . . . . . . . . . . . . . . 1.4.4 Model-Based Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Composite Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.6 Sampling in Dimensions Other Than Two-Dimensional Space 1.4.6.1 Sampling in Three-Dimensional Space and at Depth. 1.4.6.2 Sampling in Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.6.3 Sampling in Space–Time. . . . . . . . . . . . . . . . . . . . . . . 1.4.7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Geostatistics S.R. YATES AND A.W. WARRICK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1.1 Geostatistical Investigations . . . . . . . . . . . . . . . . . . . . 1.5.2 Using Geostatistical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2.2 Spatial Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2.3 Geostatistics and Estimation . . . . . . . . . . . . . . . . . . . . 1.5.2.4 Geostatistics and Uncertainty . . . . . . . . . . . . . . . . . . . 1.5.3 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3.1 Selected Windows-Based Geostatistical Software . . . 1.5.3.2 Simulating Random Fields . . . . . . . . . . . . . . . . . . . . . 1.5.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Time and Space Series O. WENDROTH AND D.R. NIELSEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Auto- and Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 State–Space Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4.1 Autoregressive State–Space Model for Spatial Processes . . . . . . . . . . . . . . . . . . . . . . . . . .
43 43 45 45 45 46 47 48 48 50 52 53 54 58 60 62 63 65 67 70 71 72 72 74 77 81 81 82 83 83 84 94 107 114 114 115 116 119 119 119 123 128 131
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1.6.4.2 State–Space Analysis for Time Series . . . . . . . . . . . . 1.6.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Parameter Optimization and Nonlinear Fitting JIÌÍ ŠIMæNEK AND JAN W. HOPMANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Maximum-Likelihood and Weighted Least-Squares Estimator 1.7.3 Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.4 Correlation and Confidence Intervals . . . . . . . . . . . . . . . . . . . . 1.7.5 Goodness of Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.6 Examples and Optimization Programs. . . . . . . . . . . . . . . . . . . . 1.7.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.8 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Newer Application Techniques ALEX. B. MCBRATNEY, ALISON N. ANDERSON, R. MURRAY LARK, AND INAKWU O. ODEH . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.1 Fractal Dimensions
..................... Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantifying Soil Structure Using Fractal Geometry . Applications of Fractal Geometry to Soil Physics . . . Characterizing Soil Spatial Variability Using Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1.5 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Fuzzy Sets I.O. ODEH AND A.B. MCBRATNEY . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2.1 The Concept of Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 1.8.2.2 Some Definitions and Examples Related to Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2.3 Fuzzy Models of Soil Physical Processes: Example . 1.8.2.4 Application to Soil Classification, Mapping, and Land Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Wavelet Analysis R.M. LARK AND A.B. MCBRATNEY . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3.1 The Continuous Wavelet Transform . . . . . . . . . . . . . . 1.8.3.2 The Discrete Wavelet Transform. . . . . . . . . . . . . . . . . 1.8.3.3 Prospects for Wavelet Analysis. . . . . . . . . . . . . . . . . . 1.8.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.N. ANDERSON AND A.B. MCBRATNEY
1.8.1.1 1.8.1.2 1.8.1.3 1.8.1.4
133 136 139 139 140 143 146 148 149 155 156 159 159 160 163 168 170 171 171 172 172 178 181 184 187 189 192 195
Chapter 2 The Solid Phase 2.1 Bulk Density and Linear Extensibility R.B. GROSSMAN AND T.G. REINSCH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.2 Variability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.3 Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.4 Dealing with Rock Fragments. . . . . . . . . . . . . . . . . . .
201 201 201 202 203 205
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2.1.2 Core Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Excavation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Clod Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4.2 Equipment and Supplies . . . . . . . . . . . . . . . . . . . . . . . 2.1.4.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Radiation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5.2 Equipment and Procedures . . . . . . . . . . . . . . . . . . . . . 2.1.5.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Linear Extensibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6.2 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Particle Density ALAN L. FLINT AND LORRAINE E. FLINT . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.1 Calculation from Porosity and Bulk Density . . . . . . . 2.2.3.2 Liquid Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.3 Gas Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3.4 Estimation from Constituent Properties . . . . . . . . . . . 2.2.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Porosity LORRAINE E. FLINT AND ALAN L. FLINT . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Total Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Calculation from Particle and Bulk Densities . . . . . . 2.3.2.2 Gravimetric Method with Water Saturation . . . . . . . . 2.3.2.3 Volumetric Method with Gas Pycnometry . . . . . . . . . 2.3.3 Pore-Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3.1 Water-Desorption Method . . . . . . . . . . . . . . . . . . . . . . 2.3.3.2 Visualization Method Using Impregnation. . . . . . . . . 2.3.3.3 Mercury-Porosimetry Method. . . . . . . . . . . . . . . . . . . 2.3.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Particle-Size Analysis GLENDON W. GEE AND DANI OR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
207 207 207 210 211 211 211 216 218 218 218 219 222 222 222 223 223 223 223 224 225 225 229 229 229 230 230 230 235 237 239 240 241 241 242 242 243 245 246 246 249 251 253 255 255
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2.4.2 Pretreatment and Dispersion Techniques. . . . . . . . . . . . . . . . . . 2.4.2.1 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2.2 Organic Matter Removal . . . . . . . . . . . . . . . . . . . . . . . 2.4.2.3 Iron Oxide Removal . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2.4 Carbonate Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2.5 Soluble Salts Removal. . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2.6 Sample Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Specific Methods of Particle-Size Analysis. . . . . . . . . . . . . . . . 2.4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.2 Analysis by Sieving . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.3 Analysis by Gravitational Sedimentation . . . . . . . . . . 2.4.3.4 Pipette Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.5 Hydrometer Method . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.6 Modern Methods for Particle-Size Measurement. . . . 2.4.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Specific Surface Area K.D. PENNELL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Liquid-Phase Adsorption Methods . . . . . . . . . . . . . . . . . . . . . . 2.5.2.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2.2 Equipment and Supplies . . . . . . . . . . . . . . . . . . . . . . . 2.5.2.3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . 2.5.2.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Gas-Phase Adsorption Methods. . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3.2 Equipment and Supplies . . . . . . . . . . . . . . . . . . . . . . . 2.5.3.3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . 2.5.3.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Retention of Polar Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4.2 Equipment and Supplies . . . . . . . . . . . . . . . . . . . . . . . 2.5.4.3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . 2.5.4.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Comparison of Surface Area Methods. . . . . . . . . . . . . . . . . . . . 2.5.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Aggregate Stability and Size Distribution JOHN R. NIMMO AND KIM S. PERKINS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Apparatus and Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2.1 General Sample Preparation . . . . . . . . . . . . . . . . . . . . 2.6.2.2 Modifications of Informally Standardized Methods and Apparatus. . . . . . . . . . . . . . . . . . . . . . . . 2.6.2.3 Representation of Data . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Shear Strength of Unsaturated Soils D.G. FREDLUND AND S.K. VANAPALLI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
257 257 259 261 262 262 262 264 264 265 269 272 278 283 289 295 295 298 298 299 300 302 303 303 305 306 307 308 308 309 309 311 312 313 317 317 320 320 323 324 326 327 329 329
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2.7.2 Shear Strength Equation for Unsaturated Soils . . . . . . . . . . . . . 2.7.3 Triaxial Shear Tests for Unsaturated Soils. . . . . . . . . . . . . . . . . 2.7.3.1 Test Procedures for Triaxial Tests . . . . . . . . . . . . . . . . 2.7.4 Direct Shear Tests for Unsaturated Soils . . . . . . . . . . . . . . . . . . 2.7.5 Failure Criteria for Unsaturated Soils . . . . . . . . . . . . . . . . . . . . 2.7.5.1 Strain Rates for Triaxial and Direct Shear Tests. . . . . 2.7.6 Interpretation of Drained Test Results Using Multistage Testing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.7 Nonlinearity of Failure Envelope . . . . . . . . . . . . . . . . . . . . . . . . 2.7.8 Interpretation of Undrained Test Results . . . . . . . . . . . . . . . . . . 2.7.8.1 Confined Compression Tests. . . . . . . . . . . . . . . . . . . . 2.7.8.2 Unconfined Compression Tests. . . . . . . . . . . . . . . . . . 2.7.9 Relationship Between the Soil Water Characteristic Curve and the Shear Strength of Unsaturated Soils. . . . . . . . . . 2.7.10 Procedure for Predicting the Shear Strength of Unsaturated Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.12 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Soil Penetrometers and Penetrability BIRL LOWERY AND JOHN E. MORRISON, JR. . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2.1 Soil Mechanics Associated with Penetrometers. . . . . 2.8.3 Equipment, Software, and Supplies . . . . . . . . . . . . . . . . . . . . . . 2.8.3.1 Penetrometer Rod or Shaft and Active Element. . . . . 2.8.3.2 Force-Sensing Apparatus . . . . . . . . . . . . . . . . . . . . . . 2.8.3.3 Depth-Sensing Apparatus . . . . . . . . . . . . . . . . . . . . . . 2.8.3.4 Structure of Penetrometer . . . . . . . . . . . . . . . . . . . . . . 2.8.3.5 Data Logging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Data Logging Operational Details for Specific Penetrometers. 2.8.4.1 Pocket Penetrometer . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4.2 Cone Penetrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4.3 Friction-Sleeve Cone Penetrometer . . . . . . . . . . . . . . 2.8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Atterberg Limits R.A. MCBRIDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Liquid Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2.1 Casagrande Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2.2 One-Point Casagrande Method . . . . . . . . . . . . . . . . . . 2.9.2.3 Drop-Cone Penetrometer Method. . . . . . . . . . . . . . . . 2.9.3 Plastic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3.1 Casagrande Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Soil Compressibility S.C. GUPTA, J.M. BRADFORD, AND A. DRESCHER . . . . . . . . . . . . . . . . . . . . . 2.10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
331 332 333 343 347 347 349 351 351 353 354 354 357 360 360 363 363 366 367 375 375 375 376 377 378 378 378 379 383 385 385 389 389 390 391 393 394 395 396 398 399 399
CONTENTS
2.10.2 Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.3 Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.3.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.3.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
400 409 409 410 413 414
Chapter 3 The Soil Solution Phase 3.1 Water Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Information G. CLARKE TOPP AND P.A. (TY) FERRÉ . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Scope of Methods and Brief Description G. CLARKE TOPP AND P.A. (TY) FERRÉ . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1 Thermogravimetric Method Using Convective Oven-Drying. . . . . . . . . . . . . . . . . . . . . . . 3.1.2.2 Gravimetric Method Using Microwave Oven-Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.3 Time Domain Reflectometry. . . . . . . . . . . . . . . . . . . . 3.1.2.4 Ground Penetrating Radar . . . . . . . . . . . . . . . . . . . . . . 3.1.2.5 Capacitance Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.6 Radar Scatterometry or Active Microwave. . . . . . . . . 3.1.2.7 Passive Microwave . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.8 Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . 3.1.2.9 Neutron Thermalization . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.10 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . 3.1.2.11 Gamma Ray Attenuation . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Methods for Measurement of Soil Water Content . . . . . . . . . . . 3.1.3.1 Thermogravimetric Method Using Convective Oven-Drying G. CLARKE TOPP AND P.A. (TY) FERRÉ . . . . . . . . . . . . . . . 3.1.3.2 Gravimetric Method Using Microwave Oven-Drying G. CLARKE TOPP AND P.A. (TY) FERRÉ . . . . . . . . . . . . . . . 3.1.3.3 The Basis of Electromagnetic Methods: A Wave Equation Framework . . . . . . . . . . . . . . . . . . . 3.1.3.4 Time Domain Reflectometry P.A. (TY) FERRÉ AND G. CLARKE TOPP . . . . . . . . . . . . . . . 3.1.3.5 Ground Penetrating Radar to Measure Soil Water Content J.L. DAVIS AND A.P. ANNAN . . . . . . . . . . . . . . . . . . . . . . . 3.1.3.6 Capacitance Devices JAMES L. STARR AND IOAN C. PALTINEANU . . . . . . . . . . . 3.1.3.7 Active Microwave Remote Sensing Methods H. MCNAIRN, T.J. PULTZ, AND J.B. BOISVERT . . . . . . . . . . . 3.1.3.8 Passive Microwave Remote Sensing Methods THOMAS J. JACKSON . . . . . . . . . . . . . . . . . . . . . . . . . . .
417 417 419 419 419 419 420 420 420 420 421 421 421 421 422 422 425 428 434 446 463 475 488
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CONTENTS
3.1.3.9 Electromagnetic Induction R.G. KACHANOSKI, J.M.H. HENDRICKX, AND E. DE JONG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3.10 Neutron Thermalization
CLIFF HIGNETT AND STEVEN R. EVETT . . . . . . . . . . . . . . 3.1.3.11 Nuclear Magnetic Resonance CAROLINE M. PRESTON . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Water Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Piezometry MICHAEL H. YOUNG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.2 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.3 Drilling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.4 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.5 Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.6 Technical Considerations . . . . . . . . . . . . . . . . . . . . . . 3.2.1.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Tensiometry MICHAEL H. YOUNG AND JAMES B. SISSON. . . . . . . . . . . . . . . . . . . 3.2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.2 Soil Water Matric Potential . . . . . . . . . . . . . . . . . . . . . 3.2.2.3 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.4 Essential Components of Tensiometers . . . . . . . . . . . 3.2.2.5 Alternative Types of Tensiometers . . . . . . . . . . . . . . . 3.2.2.6 Field and Laboratory Applications . . . . . . . . . . . . . . . 3.2.2.7 Field Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.8 Laboratory Measurements. . . . . . . . . . . . . . . . . . . . . . 3.2.2.9 Interpretation of Tensiometric Readings. . . . . . . . . . . 3.2.2.10 Gauge Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Thermocouple Psychrometry BRIAN J. ANDRASKI AND BRIDGET R. SCANLON . . . . . . . . . . . . . . . 3.2.3.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.5 Commercial Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Miscellaneous Methods for Measuring Matric or Water Potential BRIDGET R. SCANLON, BRIAN J. ANDRASKI, AND JIM BILSKIE . . . . 3.2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.2 Heat Dissipation Sensors. . . . . . . . . . . . . . . . . . . . . . . 3.2.4.3 Electrical Resistance Sensors . . . . . . . . . . . . . . . . . . . 3.2.4.4 Frequency Domain and Time Domain Matric Potential Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.5 Electro-Optical Switches. . . . . . . . . . . . . . . . . . . . . . .
497 501 521 534 547 547 547 548 552 555 562 567 571 575 575 575 577 579 586 587 589 599 602 605 606 609 609 612 620 637 638 639 643 643 644 654 660 661
CONTENTS
3.2.4.6 Dew Point Potentiameter. . . . . . . . . . . . . . . . . . . . . . . 3.2.4.7 Filter Paper Technique. . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.8 Vapor Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Water Retention and Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Introduction J.H. DANE AND JAN W. HOPMANS . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Laboratory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1 Introduction J.H. DANE AND JAN W. HOPMANS . . . . . . . . . . . . . . . . . . . 3.3.2.2 Hanging Water Column J.H. DANE AND JAN W. HOPMANS . . . . . . . . . . . . . . . . . . . 3.3.2.3 Pressure Cell J.H. DANE AND JAN W. HOPMANS . . . . . . . . . . . . . . . . . . . 3.3.2.4 Pressure Plate Extractor J.H. DANE AND JAN W. HOPMANS . . . . . . . . . . . . . . . . . . . 3.3.2.5 Long Column J.H. DANE AND JAN W. HOPMANS . . . . . . . . . . . . . . . . . . . 3.3.2.6 Suction Table N. ROMANO, JAN W. HOPMANS, AND J.H. DANE . . . . . . . . 3.3.2.7 Controlled Liquid Volume K.A. WINFIELD AND J.R. NIMMO . . . . . . . . . . . . . . . . . . . 3.3.2.8 Determination of Soil Water Characteristic by Freezing Method E.J.A. SPAANS AND J.M. BAKER . . . . . . . . . . . . . . . . . . . . 3.3.2.9 Miscellaneous Methods J.R. NIMMO AND K.A. WINFIELD . . . . . . . . . . . . . . . . . . . 3.3.2.10 Computational Corrections J.H. DANE AND J.W. HOPMANS . . . . . . . . . . . . . . . . . . . . . 3.3.2.11 Guidelines for Method Selection J.R. NIMMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Field NUNZIO ROMANO AND ALESSANDRO SANTINI . . . . . . . . . . . . . . . . 3.3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.2 Field Water Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.3 Permanent Wilting Point . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.4 Available Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.5 Specific Yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Parametric Models KEN’ICHIROU KOSUGI, JAN W. HOPMANS, AND JACOB H. DANE . . . 3.3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2 General Characteristics of Water Retention Curves and Important Parameters . . . . . . . . . . . . . . . . . . . . . . 3.3.4.3 Brooks and Corey Type Power Function . . . . . . . . . . 3.3.4.4 van Genuchten Type Power Function . . . . . . . . . . . . . 3.3.4.5 Exponential Function. . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
662 665 668 669 671 671 673 675 675 680 684 688 690 692 698 704 710 714 717 717 721 721 723 729 731 734 736 739 739 740 741 742 745
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3.3.4.6 Lognormal Distribution Function . . . . . . . . . . . . . . . . 3.3.4.7 Water Capacity Functions . . . . . . . . . . . . . . . . . . . . . . 3.3.4.8 Unsaturated Hydraulic Conductivity Functions . . . . . 3.3.4.9 Multimodal Retention Functions . . . . . . . . . . . . . . . . 3.3.4.10 Soil Water Hysteresis. . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.11 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.12 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Property-Transfer Models RANDEL HAVERKAMP, PAOLO REGGIANI, AND JOHN R. NIMMO . . . 3.3.5.1 Physically Based Water Retention Prediction Models RANDEL HAVERKAMP AND PAOLO REGGIANI . . . . . . . . . 3.3.5.2 Property Transfer from Particle and Aggregate Size to Water Retention JOHN R. NIMMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Air–Water Interfacial Area P.S.C. RAO, HEONKI KIM, AND MICHAEL D. ANNABLE . . . . . . . . . . . 3.3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6.2 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6.3 Aqueous-Static Method. . . . . . . . . . . . . . . . . . . . . . . . 3.3.6.4 Aqueous-Dynamic Method . . . . . . . . . . . . . . . . . . . . . 3.3.6.5 Gaseous-Dynamic Method . . . . . . . . . . . . . . . . . . . . . 3.3.6.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Saturated and Field-Saturated Water Flow Parameters W.D. REYNOLDS, D.E. ELRICK, E.G. YOUNGS, A. AMOOZEGAR, H.W.G. BOOLTINK, AND J. BOUMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1.1 Principles and Parameter Definitions W.D. REYNOLDS AND D.E. ELRICK . . . . . . . . . . . . . . . . . . 3.4.1.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Laboratory Methods W.D. REYNOLDS, D.E. ELRICK, E.G. YOUNGS, H.W.G. BOOLTINK, AND J. BOUMA . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2.2 Constant Head Soil Core (Tank) Method W.D. REYNOLDS AND D.E. ELRICK . . . . . . . . . . . . . . . . . . 3.4.2.3 Falling Head Soil Core (Tank) Method W.D. REYNOLDS AND D.E. ELRICK . . . . . . . . . . . . . . . . . . 3.4.2.4 Steady Flow Soil Column Method H.W.G. BOOLTINK AND J. BOUMA . . . . . . . . . . . . . . . . . . . 3.4.2.5 Other Laboratory Methods W.D. REYNOLDS AND D.E. ELRICK . . . . . . . . . . . . . . . . . . 3.4.2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Field Methods (Vadose and Saturated Zone Techniques) W.D. REYNOLDS, D.E. ELRICK, E.G. YOUNGS, AND A. AMOOZEGAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.2 Ring or Cylinder Infiltrometers (Vadose Zone) W.D. REYNOLDS, D.E. ELRICK, AND E.G. YOUNGS . . . . . .
745 747 747 750 751 753 754 755 759 762 777 781 783 783 784 786 789 792 794 795 797 797 797 801 802 802 804 809 812 815 816 817 817 818
CONTENTS
xv
3.4.3.2.a Single-Ring and Double- or Concentric-Ring Infiltrometers W.D. REYNOLDS, D.E. ELRICK, AND E.G. YOUNGS . . . . . . . . . . . . . . . . . . . . . .
3.4.3.2.b Pressure Infiltrometer W.D. REYNOLDS AND D.E. ELRICK
..........
821 826
3.4.3.2.c Twin- or Dual-Ring and Multiple-Ring Infiltrometers W.D. REYNOLDS, D.E. ELRICK, AND E.G. YOUNGS . . . . . . . . . . . . . . . . . . . . . .
3.4.3.2.d References . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.3 Constant Head Well Permeameter (Vadose Zone) W.D. REYNOLDS AND D.E. ELRICK . . . . . . . . . . . . . . . . . . 3.4.3.4 Auger-Hole Method (Saturated Zone) AZIZ AMOOZEGAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.5 Piezometer Method (Saturated Zone) AZIZ AMOOZEGAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.6 Other Saturated Zone Methods . . . . . . . . . . . . . . . . . . 3.5 Unsaturated Water Transmission Parameters Obtained from Infiltration BRENT CLOTHIER AND DAVID SCOTTER . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 One-Dimensional Infiltration Equations and Their Use . . . . . . 3.5.3 Horizontal Absorption—The Bruce and Klute Experiment . . . 3.5.4 Three-Dimensional Infiltration Using Disk Permeameters. . . . 3.5.4.1 Early-Time Observations J.-P. VANDERVAERE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4.2 Steady-State Observations BRENT CLOTHIER AND DAVID SCOTTER . . . . . . . . . . . . . 3.5.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Simultaneous Determination of Water Transmission and Retention Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.1.a Long Column ARTHUR T. COREY . . . . . . . . . . . . . . . . . . . . . 3.6.1.1.b Steady-State Centrifuge JOHN R. NIMMO, KIM S. PERKINS, AND ANGUS M. LEWIS . . . . . . . . . . . . . . . . . . .
3.6.1.1.c Wind and Hot-Air Methods LALIT M. ARYA . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.1.d Suction Crust Infiltrometer H.W.G. BOOLTINK AND J. BOUMA . . . . . . . . . . 3.6.1.1.e Bypass Flow H.W.G. BOOLTINK AND J. BOUMA . . . . . . . . . . 3.6.1.1.f References . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.2 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.2.a Instantaneous Profile GEORGES VACHAUD AND J.H. DANE . . . . . . . .
836 842 844 859 870 877 879 879 881 884 888 889 894 896 896 899 899 899 899 903 916 926 930 933 937 937
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CONTENTS
3.6.1.2.b Plane of Zero Flux LALIT M. ARYA . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.2.c Constant Flux Vertical Time Domain Reflectometry GARY W. PARKIN AND R. GARY KACHANOSKI . 3.6.1.2.d References . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Inverse Methods JAN W. HOPMANS, JIÌÍ ŠIMæNEK, NUNZIO ROMANO, AND WOLFGANG DURNER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.2 Theory of Flow and Optimization. . . . . . . . . . . . . . . . 3.6.2.3 Multistep Outflow Method . . . . . . . . . . . . . . . . . . . . . 3.6.2.4 Evaporation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.5 Tension Disc Infiltrometer. . . . . . . . . . . . . . . . . . . . . . 3.6.2.6 Field Drainage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.7 Additional Applications . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Indirect Methods FEIKE J. LEIJ, MARCEL G. SCHAAP, AND LALIT M. ARYA . . . . . . . . . 3.6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3.2 Semiempirical Approaches . . . . . . . . . . . . . . . . . . . . . 3.6.3.3 Empirical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Evaporation from Natural Surfaces JOHN M. BAKER AND JOHN M. NORMAN . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Soil-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1.1 Lysimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1.2 Soil Water Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Plant-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2.1 Sap Flow Measurement . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2.2 Cuvettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Micrometeorological Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3.1 Bowen Ratio Energy Balance Method . . . . . . . . . . . . 3.7.3.2 Eddy Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3.3 Conditional Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Remote Sensing Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
945 954 960 963 963 965 971 978 981 983 986 987 1002 1004 1009 1009 1010 1025 1041 1047 1048 1048 1051 1051 1052 1053 1058 1061 1062 1065 1066 1071
Chapter 4 The Soil Gas Phase 4.1 Introduction
DENNIS E. ROLSTON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Gas Sampling and Analysis RICHARD E. FARRELL, EELTJE DE JONG, AND JANE A. ELLIOTT . . . . . . . . . . 4.2.1 Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1075 1075 1076 1076
CONTENTS
4.2.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.1 Air-Filled Pores Above the Water Table . . . . . . . . . . . 4.2.2.2 Soil Water Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Gas Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3.1 Principles of Gas Chromatography . . . . . . . . . . . . . . . 4.2.3.2 Laboratory Analysis by Gas Chromatography . . . . . . 4.2.3.3 Field-Based Gas Chromatography Systems . . . . . . . . 4.2.3.4 Alternative Gas Analysis Systems . . . . . . . . . . . . . . . 4.2.4 In Situ Analyses at the Gas–Liquid Interface . . . . . . . . . . . . . . 4.2.4.1 Platinum Microelectrode Methods . . . . . . . . . . . . . . . 4.2.4.2 Membrane-Covered Electrode Methods . . . . . . . . . . . 4.2.4.3 Miscellaneous Gas-Sensing Probes . . . . . . . . . . . . . . 4.2.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Gas Diffusivity DENNIS E. ROLSTON AND PER MOLDRUP . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Laboratory Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.1 The Currie Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.2 The Two-Chamber Method . . . . . . . . . . . . . . . . . . . . . 4.3.3 Field Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3.1 The MacIntyre and Philip Method . . . . . . . . . . . . . . . 4.3.4 Predicting Gas Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4.1 Undisturbed Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4.2 Sieved, Repacked Soil . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4.3 Values of D0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Air Permeability BRUCE C. BALL AND PER SCHJØNNING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Laboratory Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Field Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.1 Acoustic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.2 Buried Probe and Well Techniques . . . . . . . . . . . . . . . 4.4.4 Recommended Laboratory Method . . . . . . . . . . . . . . . . . . . . . . 4.4.4.1 Apparatus and Materials . . . . . . . . . . . . . . . . . . . . . . . 4.4.4.2 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Recommended Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5.1 Apparatus and Materials . . . . . . . . . . . . . . . . . . . . . . . 4.4.5.2 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Choice of Method, Including Scaling and Variability Aspects . 4.4.7 Predicting Air Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.8 Conclusions and Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Soil–Atmosphere Gas Exchange GORDON L. HUTCHINSON AND GERALD P. LIVINGSTON . . . . . . . . . . . . . . . 4.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
1077 1078 1081 1083 1084 1089 1091 1092 1096 1097 1103 1106 1106 1113 1113 1114 1114 1122 1126 1126 1131 1131 1134 1136 1137 1141 1141 1142 1143 1145 1146 1146 1146 1147 1149 1149 1150 1150 1151 1152 1153 1156 1157 1159 1159
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CONTENTS
4.5.2 Computation from Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2.2 Apparatus, Materials, and Procedure . . . . . . . . . . . . . 4.5.2.3 Comments and Cautions . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Chamber Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3.2 Apparatus and Materials . . . . . . . . . . . . . . . . . . . . . . . 4.5.3.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3.4 Comments and Cautions . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Sampling Design, Data Analyses, and Data Summaries. . . . . . 4.5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1160 1160 1162 1162 1163 1163 1165 1170 1175 1176 1179 1180
Chapter 5 Soil Heat 5.1 Temperature
KEVIN J. MCINNES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Thermocouple Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Integrated Circuit Thermometers . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Resistance Thermometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.1 Platinum Resistance Thermometers . . . . . . . . . . . . . . 5.1.3.2 Resistance Temperature Detectors . . . . . . . . . . . . . . . 5.1.3.3 Thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Nonelectric Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Infrared Radiation Thermometers . . . . . . . . . . . . . . . . . . . . . . . 5.1.6 Installation and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Heat Capacity and Specific Heat G.J. KLUITENBERG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 General Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.2 Relationship Between Volumetric Heat Capacity and Specific Heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3.1 De Vries Approximation . . . . . . . . . . . . . . . . . . . . . . . 5.2.3.2 Dual-Probe Heat-Pulse Method . . . . . . . . . . . . . . . . . 5.2.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Thermal Conductivity KEITH L. BRISTOW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Predictive Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2.1 Predicting Soil Thermal Conductivity, Including Temperature Effects . . . . . . . . . . . . . . . . . . 5.3.2.2 Predicting Soil Thermal Conductivity from Readily Available Soils Data. . . . . . . . . . . . . . . . . . . .
1183 1183 1190 1192 1192 1192 1193 1195 1195 1197 1199 1201 1201 1201 1201 1202 1203 1203 1204 1208 1209 1209 1210 1211 1215
CONTENTS
5.3.3 Steady-State Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3.1 Guarded Hot Plate Method . . . . . . . . . . . . . . . . . . . . . 5.3.4 Transient-State Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4.2 Probe Design and Construction. . . . . . . . . . . . . . . . . . 5.3.4.3 Single Heat Probe Method . . . . . . . . . . . . . . . . . . . . . 5.3.4.4 Dual-Probe Heat-Pulse Method . . . . . . . . . . . . . . . . . 5.3.5 Comments Concerning Thermal Conductivity . . . . . . . . . . . . . 5.3.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Soil Thermal Diffusivity ROBERT HORTON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Laboratory Method for Determining Soil Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.3 Analysis of Soil Column Temperature Observations . 5.4.2 Field Method for Determining Soil Thermal Diffusivity . . . . . 5.4.3 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Heat Flux Density THOMAS J. SAUER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Calorimetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1.4 Commentary on Advantages and Limitations . . . . . . 5.5.2 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2.4 Commentary on Advantages and Limitations . . . . . . 5.5.3 Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3.4 Commentary on Advantages and Limitations . . . . . . 5.5.4 Soil Heat Flux Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4.4 Commentary on Advantages and Limitations . . . . . . 5.5.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Coupled Heat and Water Transfer IBRAHIM N. NASSAR AND ROBERT HORTON . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Soil Thermal Water Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xix
1216 1216 1217 1217 1218 1220 1221 1223 1224 1227 1228 1228 1228 1229 1229 1231 1233 1235 1235 1236 1236 1237 1238 1238 1238 1239 1239 1240 1240 1240 1240 1241 1242 1242 1243 1244 1247 1247 1249 1249 1250 1251
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CONTENTS
Chapter 6 Miscible Solute Transport 6.1 Solute Content and Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253 6.1.1 Introduction JAN M.H. HENDRICKX, JON M. WRAITH, DENNIS L. CORWIN, AND R. GARY KACHANOSKI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Measurement of Solute Content Using Soil Extraction JAN M.H. HENDRICKX AND LOUIS W. DEKKER . . . . . . . . . . . . . . . . 6.1.2.1 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Measurement of Solute Concentration Using Soil Water Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3.1 Suction Cups DENNIS L. CORWIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3.2 Passive Capillary Samplers JOHN S. SELKER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3.3 Porous Matrix Sensors DENNIS L. CORWIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Indirect Measurement of Solute Concentration JAN M.H. HENDRICKX, B. DAS, DENNIS L. CORWIN, JON M. WRAITH, AND R. GARY KACHANOSKI . . . . . . . . . . . . . . . . .
1253 1255 1255 1256 1256 1260 1261 1261 1266 1269 1274
6.1.4.1 Relationship between Soil Water Solute Concentration and Apparent Soil Electrical Conductivity JAN M.H. HENDRICKX, B. DAS, DENNIS L. CORWIN, JON M. WRAITH AND R. GARY KACHANOSKI . . . . . . . . . .
6.1.4.2 Electrical Resistivity: Wenner Array
DENNIS L. CORWIN AND JAN M.H. HENDRICKX . . . . . . . . 6.1.4.3 Electrical Resistivity: Four-Electrode Probe DENNIS L. CORWIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4.4 Time Domain Reflectometry JON M. WRAITH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4.5 Nonintrusive Electromagnetic Induction JAN M.H. HENDRICKX AND R. GARY KACHANOSKI . . . . . 6.1.5 Emerging Methods JAN M.H. HENDRICKX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5.1 Fiber Optic Sensors JAN M.H. HENDRICKX . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5.2 Capillary Absorbers CARL KELLER AND JAN M.H. HENDRICKX . . . . . . . . . . . . 6.1.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Solute Diffusion MARKUS FLURY AND THOMAS F. GIMMI. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Theory of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2.1 Fick’s Laws of Diffusion . . . . . . . . . . . . . . . . . . . . . . . 6.2.2.2 Estimation of Diffusion Coefficients in Liquids. . . . . 6.2.2.3 Temperature Dependence of Diffusion Coefficients . 6.2.2.4 Type of Diffusion Coefficients . . . . . . . . . . . . . . . . . .
1275 1282 1287 1289 1297 1306 1307 1308 1311 1323 1323 1324 1324 1324 1327 1328
CONTENTS
6.2.2.5 Diffusion of Nonreactive Solutes in Porous Media . . 6.2.2.6 Estimation of Diffusion Coefficients for Nonreactive Solutes in Porous Media . . . . . . . . . . . . . 6.2.2.7 Diffusion and Convection . . . . . . . . . . . . . . . . . . . . . . 6.2.2.8 Diffusion and Reactions . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3.1 Steady-State Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3.2 Transient Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3.3 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Solute Transport: Theoretical Background TODD H. SKAGGS AND FEIKE J. LEIJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Elementary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.1 Solute Transport Experiments . . . . . . . . . . . . . . . . . . . 6.3.1.2 Breakthrough Curves. . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.3 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.4 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.5 Flux and Resident Concentrations . . . . . . . . . . . . . . . 6.3.2 Convection–Dispersion Model. . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2.1 Dimensionless Parameters. . . . . . . . . . . . . . . . . . . . . . 6.3.2.2 Flux Concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2.3 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2.4 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2.5 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Nonequilibrium Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3.1 Two-Region Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3.2 Two-Site Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3.3 General Nonequilibrium Formulation. . . . . . . . . . . . . 6.3.3.4 Analytical Solutions and Moments . . . . . . . . . . . . . . . 6.3.3.5 Additional Nonequilibrium Formulations . . . . . . . . . 6.3.4 Stochastic–Convective Model . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4.1 Transfer Function Modeling . . . . . . . . . . . . . . . . . . . . 6.3.4.2 Stochastic–Convective Transfer Function Model. . . . 6.3.4.3 Convective Lognormal Transfer Function Model . . . 6.3.4.4 Field Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Transport Equation Generalizations. . . . . . . . . . . . . . . . . . . . . . 6.3.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Solute Transport: Experimental Methods TODD H. SKAGGS, G.V. WILSON, PETER J. SHOUSE, AND FEIKE J. LEIJ . . . . . 6.4.1 Laboratory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1.2 Sample Collection and Preparation. . . . . . . . . . . . . . . 6.4.1.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1.4 Displacing and Resident Solutions . . . . . . . . . . . . . . . 6.4.1.5 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxi
1328 1330 1332 1332 1333 1333 1334 1347 1348 1348 1353 1353 1353 1354 1356 1358 1359 1360 1361 1361 1362 1364 1365 1368 1369 1370 1371 1372 1372 1372 1372 1374 1375 1375 1377 1377 1378 1381 1381 1381 1381 1382 1385 1386
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6.4.1.6 Calculations and Data Presentation. . . . . . . . . . . . . . . 6.4.1.7 Additional Apparatuses . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1.8 Additional Procedures . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1.9 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2.2 Water and Solute Application . . . . . . . . . . . . . . . . . . . 6.4.2.3 Solute Monitoring and Measurement . . . . . . . . . . . . . 6.4.3 Tracers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Equipment Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Solute Transport: Data Analysis and Parameter Estimation TODD H. SKAGGS, D.B. JAYNES, R. GARY KACHANOSKI, PETER J. SHOUSE, AND A.L. WARD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Least-Squares Fitting
TODD H. SKAGGS AND PETER J. SHOUSE . . . . . . . . . . . . . . . . . . . . .
6.5.1.1 6.5.1.2 6.5.1.3 6.5.1.4 6.5.1.5
Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convection–Dispersion Model . . . . . . . . . . . . . . . . . . Nonequilibrium Models . . . . . . . . . . . . . . . . . . . . . . . Transient Water Flow and Depth-Dependent Water Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Method of Moments TODD H. SKAGGS AND PETER J. SHOUSE . . . . . . . . . . . . . . . . . . . . . 6.5.2.1 Principles and Procedures . . . . . . . . . . . . . . . . . . . . . . 6.5.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Determination of θIM and α Using a Disk Permeameter and Multiple Solutes D.B. JAYNES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Determination of Travel-Time Probability Density Function from Concentration Data TODD H. SKAGGS ................................... 6.5.5 Determination of Field Solute Mass Flux (Travel Time Probability Density Function) Using Time Domain Reflectometry R. GARY KACHANOSKI AND A.L. WARD . . . . . . . . . . . . . . . . . . . . . 6.5.5.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5.2 Vertically Installed Probes. . . . . . . . . . . . . . . . . . . . . . 6.5.5.3 Equipment and Methodology . . . . . . . . . . . . . . . . . . . 6.5.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Solute Transport During Variably Saturated Flow—Inverse Methods JIÌÍ ŠIMæNEK, MARTINUS TH. VAN GENUCHTEN, DIEDERIK JACQUES, JAN W. HOPMANS, MITSUHIRO INOUE, AND MARKUS FLURY . . . . . . . . . .
1389 1389 1390 1392 1392 1392 1392 1393 1396 1397 1398 1403 1403 1403 1405 1406 1412 1414 1416 1416 1418 1418 1419 1419 1420 1421 1422
1423 1425 1426 1428 1431
1435 6.6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435 6.6.2 Theory of Flow, Transport, and Optimization . . . . . . . . . . . . . . 1436
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6.6.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3.1 Steady-State Laboratory Flow Experiment with Nonlinear Transport . . . . . . . . . . . . . . . . . . . . . . 6.6.3.2 Laboratory Transport Subject to Flow Interruption . . 6.6.3.3 Transient Laboratory Experiment with Equilibrium Solute Transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3.4 Field Experiment with Nonequilibrium Solute Transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Processes Governing Transport of Organic Solutes SHARON K. PAPIERNIK, J. GAN, AND S.R. YATES . . . . . . . . . . . . . . . . . . . . . 6.7.1 Phase Transfer and Distribution Coefficients . . . . . . . . . . . . . . 6.7.1.1 Air–Water Distribution . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1.2 Soil–Water Distribution. . . . . . . . . . . . . . . . . . . . . . . . 6.7.1.3 Soil–Air Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1.4 Implications of Soil–Water–Air Distribution on Solute Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2.1 Chemical Transformation . . . . . . . . . . . . . . . . . . . . . . 6.7.2.2 Biological Degradation . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2.3 Photodegradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2.4 Implications of Transformation on Solute Transport . 6.7.3 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Microbial Transport YAN JIN, MARYLYNN V. YATES, AND SCOTT R. YATES . . . . . . . . . . . . . . . . . 6.8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1.1 Parasites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1.2 Bacteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1.3 Viruses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Laboratory Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2.1 Batch Equilibration Method . . . . . . . . . . . . . . . . . . . . 6.8.2.2 Flowthrough Columns . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.3 Field Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.4 Indicators of Human Enteroviruses . . . . . . . . . . . . . . . . . . . . . . 6.8.5 Microbial Transport Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Geochemical Transport JIÌÍ ŠIMæNEK AND ALBERT J. VALOCCHI . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.2 Geochemical Reaction Equations . . . . . . . . . . . . . . . . . . . . . . . 6.9.2.1 Complexation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.2.2 Cation Exchange Reactions. . . . . . . . . . . . . . . . . . . . . 6.9.2.3 Adsorption Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.2.4 Precipitation–Dissolution . . . . . . . . . . . . . . . . . . . . . . 6.9.2.5 Reactions with Organic Matter and Effects of Bacteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxiii
1439 1439 1440 1442 1444 1448 1448 1451 1451 1451 1454 1461 1464 1465 1465 1471 1474 1476 1476 1481 1481 1482 1483 1484 1485 1485 1492 1499 1500 1500 1505 1511 1511 1513 1515 1515 1517 1519 1519
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6.9.3 6.9.4 6.9.5 6.9.6 6.9.7 6.9.8
6.9.2.6 Activity Coefficients and Thermodynamic Equilibrium Constants. . . . . . . . . . . . . . . . . . . . . . . . . Mass-Balance Transport Equations . . . . . . . . . . . . . . . . . . . . . . Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Solution Composition on Hydraulic Properties and Reclamation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Geochemical Transport Models . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1521 1522 1524 1525 1526 1529 1531
Chapter 7 Multi-Fluid Flow 7.1 Introduction
R.J. LENHARD, M. OOSTROM, AND J.H. DANE . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Fluid Contents
M. OOSTROM, J.H. DANE, AND R.J. LENHARD . . . . . . . . . . . . . . . . . . . . . . . .
7.3
7.4
7.5
7.6
7.2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Nondestructive Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.1 Gamma Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.2 X-ray Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.3 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Destructive Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturation–Pressure Relationships R.J. LENHARD, J.H. DANE, AND M. OOSTROM . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Air–Nonaqueous Phase Liquid Systems . . . . . . . . . . . . . . . . . . 7.3.3 Two Immiscible Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Air and Two Immiscible Liquids . . . . . . . . . . . . . . . . . . . . . . . . Relative Permeability Measurements J.H. DANE, R.J. LENHARD, AND M. OOSTROM . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Nonaqueous Phase Liquid–Gas Systems . . . . . . . . . . . . . . . . . . 7.4.3 Nonaqueous Phase Liquid–Water Systems . . . . . . . . . . . . . . . . 7.4.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3.2 Equipment and Measurements . . . . . . . . . . . . . . . . . . 7.4.3.3 Summary and Comments . . . . . . . . . . . . . . . . . . . . . . Prediction of Capillary Pressure–Relative Permeability Relations R.J. LENHARD, M. OOSTROM, AND J.H. DANE . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Extending Two-Phase Saturation–Pressure Relations to Air–Nonaqueous Phase Liquid–Water Systems: Nonhysteretic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Extending Two-Phase Saturation–Pressure Relations to Air–Nonaqueous Phase Liquid–Water Systems: Hysteretic . . . Measuring Interfacial Areas of Immiscible Fluids K. PRASAD SARIPALLI, P.S.C. RAO, AND MICHAEL D. ANNABLE . . . . . . . . . . 7.6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1537 1539 1539 1540 1540 1559 1561 1563 1565 1565 1567 1569 1573 1581 1581 1582 1584 1584 1585 1589 1591 1591
1592 1598 1609 1609
CONTENTS
xxv
7.6.2 Experimental Techniques for the Measurement of Interfacial Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2.1 Trapped Nonwetting Phase . . . . . . . . . . . . . . . . . . . . . 7.6.2.2 Continuous Nonwetting Phase . . . . . . . . . . . . . . . . . . 7.6.3 Water Saturation–Interfacial Area Relationship . . . . . . . . . . . . 7.6.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 8
1609 1610 1611 1613 1614 1615
Soil Erosion by Water and Tillage M.J.M. RÖMKENS, SETH M. DABNEY, GERARD GOVERS, AND J.M. BRADFORD
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Soil Erosion by Water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Soil Erosion Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1.1 Monitoring Erosion during Natural Storm Events . . . 8.2.1.2 Erosion Measurements during Simulated Rain Storms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1.3 Experimental Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Soil Erosion Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2.1 USLE–RUSLE Relationships . . . . . . . . . . . . . . . . . . . 8.2.2.2 Water Erosion Protection Project Soil Erosion Prediction Relationship . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Tillage Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Experimental Measurement of Tillage Erosion Using Tracers . 8.3.1.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1.2 Equipment, Software, and Supplies . . . . . . . . . . . . . . 8.3.1.3 Procedural Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Measurement of Tillage Erosion by Volumetric Assessment of Soil Translocation . . . . . . . . . . . . . . . . . . . . . . . 8.3.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2.2 Equipment, Software, and Supplies . . . . . . . . . . . . . . 8.3.2.3 Procedural Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Estimation of Tillage Erosion from Cesium-137 Inventories . . 8.3.3.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3.2 Equipment, Software, and Supplies . . . . . . . . . . . . . . 8.3.3.3 Procedural Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1621 1621 1622 1622 1624 1625 1626 1626 1639 1643 1646 1646 1648 1649 1654 1654 1654 1654 1655 1655 1656 1657 1660
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663
FOREWORD As the primary source of food and fiber and major interface with the environment, soil is the reservoir on which most life on earth depends. Soil science has provided us with a basic understanding of the physical, chemical, and biological properties and processes essential to ecosystem integrity and function. This knowledge has promoted our understanding of the importance of soils in enhancing human and ecosystem health through production of sufficient food and controlling the transport of soil and potentially toxic substances in the environment. Never before have we had such a technologically advanced set of tools for addressing the needs of humanity and the earth. The editors and 125 contributors to Methods of Soil Analysis. Part 4. Physical Methods have done an excellent job in expanding the scope and comprehensive nature of the original Methods of Soil Analysis. Part 1. Physical and Mineralogical Methods. Consultation with 19 resource experts in development of the book has ensured a high level of quality and inclusion of state-of-the-art methods. Newer methods have been added and the scope of the book expanded to keep pace with a broader evolving group of hydrologists, geologists, environmental scientists, soil chemists, and microbiologists interested in transport of pesticides, chemicals, and microorganisms and use of remote sensing procedures to assess large-scale geochemical processes. The first chapter in the book deals with soil sampling and innovative statistical procedures, highlighting the fact that this book includes more emphasis on handling data once it has been collected and inclusion of the implications of different methods. This work will undoubtedly join the ranks of the previously internationally successful soil methods monographs and become a standard item on the bookshelves of most soil and earth scientists. This work moves us another step forward in our to journey to “Sustain Earth and Its People” by providing a basic foundation upon which we can translate our science into practice. JOHN W. DORAN President Soil Science Society of America
xxvii
PREFACE Because of the rapid and numerous changes in measurement methods associated with soil physical and mineralogical properties, the Soil Science Society of America Editorial Board decided not to print a third edition of Methods of Soil Analysis. Part 1. Physical and Mineralogical Properties, which had been so ably completed under the editorial leadership of Professor Arnold Klute. Instead, the Editorial Board decided to divide the one volume into two parts. The part containing soil physical measurements has consequently been renamed as Methods of Soil Analysis. Part 4. Physical Properties. The planning of this book and its development have been overseen by an editorial committee consisting of: Jacob H. Dane (Co-editor), Auburn University, Auburn, AL G. Clarke Topp (Co-editor), Agriculture and Agri-Food Canada, Ottawa, Canada Gaylon S. Campbell, Washington State University, Pullman, WA Harold M. van Es, Cornell University, Ithaca, NY Robert Horton, Iowa State University, Ames, IA William A. Jury, University of California, Riverside, CA Donald R. Nielsen, University of California, Riverside, CA Peter J. Wierenga, University of Arizona, Tucson, AZ The approach in Part 4 differs substantially from that in Part 1 in that the new book uses a more hierarchical approach. As such it is divided into eight chapters, with each chapter covering a major aspect of soil physical properties. Following the table of contents, the reader can then refine the search until the specific topic or measurement of interest is indicated. Compared with Part 1, new methods have been added and some of the older methods have been updated or deleted. Although most of the methods have been presented in a cook-book type format, some of the sections provide less detail because the methods in question are still under development. In those cases an adequate literature review has been provided to bring the reader up to speed on the particular subject. Examples of the latter are the sections discussing diffusion, dispersion, solute content, multifluid flow, and remote sensing methods. Additionally, over the past ten years or so, a number of topics have received interest by people other than soil physicists. Consequently, we felt it was important to at least touch upon such areas as pesticide, geochemical, and virus transport; the increasing use of inverse methods involving transient experiments; laboratory measurements in multifluid flow experiments; and both remote and ground penetrating radar. Ignoring these less developed methods would have made this book less useful to hydrologists, geologists, environmental scientists, soil chemists, and microbiologists. To assure that this book documents state-of-the art methods, 19 resource people were consulted. They provided valuable advice and participated closely in the development of each chapter and section. Listed by chapter, these resource people were:
xxix
xxx
PREFACE
Chapter 1: Harold M. van Es, Cornell University, Ithaca, NY Art W. Warrick, University of Arizona, Tucson, AZ Chapter 2: Glendon W. Gee, Battelle, Pacific National Northwest Laboratory, Richland, WA Chapter 3: John Baker, USDA-ARS, St. Paul, MN Brent Clothier, HortResearch, Palmerston North, New Zealand Jacob H. Dane, Auburn University, Auburn, AL David Elrick, University of Guelph, Guelph, Ontario, Canada P.A. (Ty) Ferré, University of Arizona, Tucson, AZ John M. Norman, University of Wisconsin, Madison, WI. W. Dan Reynolds, Agriculture and Agri-Food Canada, Harrow, Ontario, Canada David R. Scotter, Massey University, Palmerston North, New Zealand G. Clarke Topp, Agriculture and Agri-Food Canada, Ottawa, Ontario, Canada Peter J. Wierenga, University of Arizona, Tucson, AZ Chapter 4: Dennis E. Rolston, University of California, Davis, CA Chapter 5: Robert Horton, Iowa State University, Ames, IA Chapter 6: M.Th. (Rien) van Genuchten, USDA-ARS, Riverside, CA Jan M.H. Hendrickx, New Mexico Tech, Socorro, NM Chapter 7: Robert J. Lenhard, Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID Chapter 8: Matt J. M. Römkens, USDA-ARS, Oxford, MS We would like to extend our thanks to the members of the editorial committee, to the resource people, to the 125 authors and co-authors who have contributed manuscripts, and to all the anonymous reviewers of this first edition of Methods of Soil Analysis. Part 4. Physical Properties. Without their help the production of this book would not have been possible. Thanks also to Lisa Al-Amoodi for serving as Managing Editor at SSSA headquarters. We hope that this new book will meet your expectations and will be as useful as the two editions of Part 1. JACOB H. DANE Co-editor Auburn University, Auburn, AL USA
G. CLARKE TOPP Co-editor Agriculture and Agri-Food Canada Ottawa, Ontario, Canada
CONTRIBUTORS R.R. Allmaras
USDA-ARS, Department of Soil, Water, and Climate, University of Minnesota, St. Paul, MN 55108
Aziz Amoozegar
Soil Science Department, Box 7619, North Carolina State University, Raleigh, NC 27695-7619
Alison N. Anderson
New South Wales Farmers’ Association, GPO Box 1068, Sydney NSW 1041, Australia
Brian J. Andraski
U.S. Geological Survey, 333 West Nye Lane, Rm. 203, Carson City, NV 89706
Michael D. Annable
Department of Environmental Engineering Sciences, 217 Black Hall, University of Florida, P.O. Box 116450, Gainesville, FL 32611-6450
A. Peter Annan
Sensors & Software Inc., 1091 Brevik Place, Mississauga, ON L4W 3R7 Canada
L.M. Arya
770 El Caballo Drive, Oceanside, CA 92507
John Baker
USDA-ARS, Department of Soil, Water, and Climate, 439 Borlaug Hall, University of Minnesota, 1991 Upper Buford Circle, St. Paul, MN 55108
Bruce C. Ball
Scottish Agricultural College, Environment Division, Bush Estate, Penicuik, Midlothian, UK EH26 0PH
Jim Bilskie
Campbell Scientific, Inc., 815 West 1800 North, Logan, UT 843211784
J.B. Boisvert
Agriculture and Agri-Food Canada, 930 Carling Avenue, Sir John Carling Building, Rm. 761, Ottawa, ON K1A 0C5, Canada
H.W.G. Booltink
Netherlands Organization for Energy and Environment, Elzenpas 13, 6666 HD Meteren, the Netherlands
J. Bouma
Department of Environmental Sciences, Laboratory of Soil Science and Geology, Wageningen Agricultural University, 6700 AA Wageningen, The Netherlands
J.M. Bradford
USDA-ARS, Subtropical Agricultural Research Laboratory, Weslaco, TX 78596-8344
Keith L. Bristow
CSIRO Land and Water, PMB Aitkenvale, Townsville, QLD 4814, Australia
Brent Clothier
Environment and Risk Management Group, HortResearch, Private Bag 11-030, Palmerston North, New Zealand
Arthur T. Corey
1309 Kirkwood, Apt. 601, Fort Collins, CO 80525
Dennis L. Corwin
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507-4617
Seth M. Dabney
USDA-ARS, National Sedimentation Laboratory, P.O. Box 1157, Oxford, MS 38655
J.H. Dane
Department of Agronomy and Soils, 202 Funchess Hall, Auburn University, AL 36849-5412
Bhabani Sankar Das
Agricultural and Food Engineering Department, Aquaculture Building, Indian Institute of Technology, Kharagpur, West Bengal
J. Les Davis
TERAD, 3509 Mississauga Road, Mississauga, ON L5L 2R9, Canada
J.J. de Gruijter
Alterra–Green World Research, P.O. Box 47, NL-6700 AA Wageningen, The Netherlands
Eeltje de Jong
Department of Soil Science, University of Saskatchewan, 51 Campus Drive, Saskatoon, SK S7N 5A8, Canada
xxxi
xxxii
CONTRIBUTORS
Louis W. Dekker
Alterra–Green World Research, P.O. Box 47, 6700 AA Wageningen, The Netherlands
A. Drescher
Department of Civil Engineering, 500 Pillsbury Drive SE, University of Minnesota, MN 55455
Wolfgang Durner
Institute of Geoecology, Technical University of Braunschweig, Langerkamp 19c, D-36106 Braunschweig, Germany
Jane A. Elliott
Environment Canada, National Water Research Institute, 11 Innovation Boulevard, Saskatoon, SK S7N 3H5, Canada
David E. Elrick
Department of Land Resource Science, University of Guelph, Guelph, ON N1G 2W1, Canada
Steven R. Evett
USDA-ARS-SPA-CPRL Water Management Research Unit, 2300 Experiment Station Road, Bushland, TX 79012
Richard E. Farrell
Department of Soil Science, University of Saskatchewan, 51 Campus Drive, Saskatoon, SK S7N 5A8, Canada
Ty Ferré
Department of Hydrology and Water Resources, University of Arizona, Tucson, AZ 85721-0011
Alan L. Flint
U.S. Geological Survey, Placer Hall, 6000 J Street, Sacramento, CA 95819-6129
Lorraine E. Flint
U.S. Geological Survey, Placer Hall, 6000 J Street, Sacramento, CA 95819-6129
Markus Flury
Department of Crop and Soil Sciences, Washington State University, Pullman, WA 99164
Delwyn G. Fredlund
Department of Civil Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada
J. Gan
Department of Environmental Sciences, University of California, Riverside, CA 92521
Glendon W. Gee
Battelle, Richland, WA 99352
Thomas F. Gimmi
Institute of Geology and Paul Scherrer Institute, University of Bern, Baltzerstrasse 1, CH-3012 Bern, Switzerland
Gerard Govers
K.U. Leuven, Laboratory for Experimental Geomorphology, Redingenstraat 16, B-3000 Leuven, Belgium
Robert B. Grossman
USDA-NRCS, National Soil Survey Center, Lincoln, NE 685083866
Satish Gupta
Department of Soil, Water, and Climate, University of Minnesota, 1991 Upper Buford Circle, St. Paul, MN 55108
Randel Haverkamp
Laboratoire d’etude des Transferts en Hydrologie et Environnement (LTHE), 1023 rue de la Piscine, BP 53, Domaine Universitaire, 38041 Grenoble, France
Jan M.H. Hendrickx
Hydrology Program, Department of Earth and Environmental Science, New Mexico Tech, 801 Leroy Place, Socorro, NM 87801
Cliff Hignett
Soil Water Solutions, 45A Ormond Avenue, Daw Park, South Australia 5041, Australia
Jan W. Hopmans
Hydrology Program, Department of Land, Air and Water Resources, 123 Veihmeyer Hall, University of California, Davis, CA 95616
Robert Horton
Agronomy Department, Iowa State University, Ames, IA 50011
Gordon L. Hutchinson
USDA-ARS, 301 South Howes Street, Rm. 435, P.O. Box E, Fort Collins, CO 80522
Mitsuhiro Inoue
Arid Land Research Center, Tottori University, 1390 Hamasaka, Tottori, 680-0001, Japan
CONTRIBUTORS
xxxiii
Thomas J. Jackson
USDA-ARS, Hydrology and Remote Sensing Laboratory, 104 Bldg. 007 BARC-West, Beltsville, MD 20705
Diederik Jacques
Belgina Nuclear Research Centre (SCK-CEN), Boeretang 2000, B2400 MOL, Belgium
Dan Jaynes
USDA-ARS, National Soil Tilth Laboratory, 2150 Pammel Drive, Ames, IA 50010
Yan Jin
Department of Plant and Soil Sciences, 152 Townsend Hall, University of Delaware, Newark, DE 19717-1303
R. Gary Kachanoski
University of Alberta, Edmonton, AB T6G 2J9, Canada
Carl Keller
Flexible Liner Underground Technology, Limited, 6 Easy Street, Santa Fe, New Mexico
O. Kempthorne
Department of Statistics, Iowa State University, Ames, IA 50011 (Deceased)
Heonki Kim
Department of Environmental Science, Hallyum University, 1 Okchon-Dong, Chunchon Kangwon, South Korea 200-702
Gerard Kluitenberg
Department of Agronomy, 2004 Throckmorton Hall, Kansas State University, Manhattan, KS 66506-5501
Ken’ichirou Kosugi
Laboratory of Erosion Control, Department of Forest Science, Graduate School of Agriculture, Kyoto University, Kyoto 606-8502, Japan
R. Murray Lark
Mathematics and Decision Systems Group, Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK45 4HS, UK
Feike J. Leij
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507-4617
R.J. Lenhard
Geosciences Research, INEEL, P.O. Box 1625, Idaho Falls, ID 834152025
Angus M. Lewis
Central Coast Regional Water Quality Control Board, 81 South Higuera Street, Suite 200, San Luis Obispo, CA 93401
Gerald P. Livingston
School of Natural Resources, University of Vermont, Burlington, VT 05405
Birl Lowery
Department of Soil Science, University of Wisconsin, 1525 Observatory Drive, Madison, WI 53706-1299
Alex. B. McBratney
School of Land, Water, and Crop Sciences, The University of Sydney, NSW 2006, Australia
R.A. McBride
Department of Land Resource Science, University of Guelph, Guelph, ON N1G 2W1, Canada
Kevin McInnes
Department of Soil and Crop Sciences, Heep Center, 370 Olsen Boulevard, Texas A&M University, TAMU 2474, College Station, TX 77843
Heather McNairn
Canada Centre for Remote Sensing, 588 Booth Street, Ottawa, ON K1A 0Y7, Canada
Per Moldrup
Department of Environmental Engineering, Aalborg University, Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark
John E. Morrison, Jr.
USDA-ARS, 808 East Blackland Road, Temple, TX 76502
Ibrahim N. Nassar
Faculty of Agriculture—Damnahour, Alexandria University, Behera, Egypt
Donald R. Nielsen
University of California, 1004 Pine Lane, Davis, CA 95616
John R. Nimmo
Unsaturated-Zone Flow Project, U.S. Geological Survey, 345 Middlefield Road, MS-421, Menlo Park, CA 94025
xxxiv
CONTRIBUTORS
John M. Norman
Department of Soil Science, University of Wisconsin, 1525 Observatory Drive, Madison, WI 53706
Inakwu Ominyi Odeh
School of Land, Water, and Crop Sciences, The University of Sydney, NSW 2006, Australia
M. Oostrom
Pacific Northwest National Laboratory, Environmental Technology Division, Richland, WA 99352
Dani Or
Department of Plants, Soils, and Biometeorology, Utah State University, Logan, UT 84322-4820
Ioan C. Paltineanu
Paltin International Inc., Laurel, MD 20707
Sharon K. Papiernik
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507
Gary W. Parkin
Department of Land Resource Science, University of Guelph, Guelph, ON N1G 2W1, Canada
Kurt D. Pennell
School of Civil and Environmental Engineering, Georgia Institute of Technology, 200 Bobby Dodd Way, Atlanta, GA 30332-0512
Kim S. Perkins
U.S. Geological Survey, 345 Middlefield Road, MS-421, Menlo Park, CA 94025
Caroline M. Preston
Pacific Forestry Centre, Natural Resources Canada, 506 West Burnside Road, Victoria, BC V8Z 1M5, Canada
Terry J. Pultz
Canada Centre for Remote Sensing, 588 Booth Street, Ottawa, ON K1A 0Y7, Canada
P.S.C. Rao
School of Civil Engineering, Purdue University, West Lafayette, IN 47907-1284
Paolo Reggiani
Laboratoire d’etude des Transferts en Hydrologie et Environnement (LTHE), 1023 Rue de la Piscine, BP 53, 38041 Grenoble, France
Thomas G. Reinsch
USDA-NRCS, National Soil Survey Center, Lincoln, NE 685083866
W. Daniel Reynolds
Greenhouse and Processing Crops Research Centre, Agriculture and Agri-Food Canada, 2585 County Road 20, Harrow, ON N0R 1G0, Canada
Dennis E. Rolston
Land, Air and Water Resources, University of California, One Shields Avenue, Davis, CA 95616-8627
Nunzio Romano
Department of Agricultural Engineering and Agronomy, University of Naples “Federico II”, Via Universitá 100, 80055 Portici (Napoli), Italy
M.J.M. Römkens
USDA-ARS National Sedimentation Laboratory, P.O. Box 1157, Oxford, MS 38655
Alessandro Santini
Department of Agricultural Engineering, University of Naples “Federico II”, Via Universitá 100, 80055 Portici (Napoli), Italy
Prasad Saripalli
Battelle National Northwest Laboratories, 1318 Sigma V Complex (K6-81), Richland, WA 99352
Thomas J. Sauer
USDA-ARS, National Soil Tilth Laboratory, 2150 Pammel Drive, Ames, IA 50011-4420
Bridget R. Scanlon
Bureau of Economic Geology, The University of Texas at Austin, 10100 Burnet Road, Austin, TX 78758
Marcel Schaap
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, Riverside, CA 92507-4617
Per Schjønning
Department of Crop Physiology and Soil Science, Research Center Foulum, P.O. Box 50, DK-8830 Tjele, Denmark
CONTRIBUTORS
xxxv
David R. Scotter
Institute of Natural Resources, Massey University, Palmerston North, New Zealand
John S. Selker
Room 240 Gilmore Hall, Oregon State University, Corvallis, OR 97331-3906
Peter J. Shouse
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507
JiÍÍí Šimççnek
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507
James B. Sisson
Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID 83415-2107
Todd H. Skaggs
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507
Egbert J.A. Spaans
EARTH University, Guácimo, Apt. Postal 4442-1000, San José, Costa Rica
James L. Starr
USDA-ARS, ANRI, EQL, BARC-West, Beltsville, MD 20705-2350
G. Clarke Topp
Land and Agronomy Program, Eastern Cereal and Oilseed Research Centre, Agriculture and Agri-Food Canada, 960 Carling Avenue, Ottawa, ON K1A 0C6, Canada
Georges Vachaud
Laboratoire d’etude des Transferts en Hydrologie et Environnement (LTHE), BP 53, 38041 Grenoble, France
Albert J. Valocchi
Department of Civil and Environmental Engineering, University of Illinois, 205 North Mathews, Urbana, IL 61801
Sai K. Vanapalli
Civil Engineering Department, Lakehead University, Thunder Bay, ON P7B 5E1, Canada
Jean-Pierre Vandervaere
Laboratoire d’etude des Transferts en Hydrologie et Environnement (LTHE), Université Joseph Fourier/UFR Mécanique, 38041 Grenoble Cedex 9, France
Harold M. van Es
Department of Crop and Soil Sciences, 1005 Bradfield Hall, Cornell University, Ithaca, NY 14853-1901
Martinus Th. van Genuchten
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507
Andy L. Ward
Pacific Northwest National Laboratories, Richland, WA 99352
Art W. Warrick
Department of Soil, Water and Environmental Sciences, University of Arizona, P.O. Box 210038, Tucson, AZ 85721-0038
Ole Wendroth
ZALF, Institut für Bodenlandschaftsforschung, D-15374 Müncheberg, Germany
Peter J. Wierenga
Department of Soil, Water and Environmental Sciences, University of Arizona, Tucson, AZ 85721
G.V. Wilson
USDA-ARS National Sedimentation Laboratory, 598 McElroy Drive, Oxford, MS 38655
Kari A. Winfield
U.S. Geological Survey, 345 Middlefield Road, MS-421, Menlo Park, CA 94025
Jon M. Wraith
Land Resources and Environmental Sciences Department, Montana State University, P.O. Box 173120, Bozeman, MT 59717-3120
Marylynn V. Yates
Department of Environmental Sciences, 4108 Hinderaker Hall, University of California, Riverside, CA 92521
Scott R. Yates
USDA-ARS, George E. Brown, Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507
xxxvi
CONTRIBUTORS
Michael H. Young
Desert Research Institute, University and Community College System of Nevada, 755 East Flamingo Road, Las Vegas, NV 89119
Edward G. Youngs
Institute of Water and Environment, Cranfield University, Silsoe, Bedfordshire MK45 4DT, UK
Conversion Factors for SI and non-SI Units
xxxvii
9.73 × 35.3 6.10 × 104 2.84 × 10−2 1.057 3.53 × 10−2 0.265 33.78 2.11
cubic meter, cubic meter, m3 cubic meter, m3 liter, L (10−3 m3) liter, L (10−3 m3) liter, L (10−3 m3) liter, L (10−3 m3) liter, L (10−3 m3) liter, L (10−3 m3)
m3
hectare, ha square kilometer, km2 (103 m)2 square kilometer, km2 (103 m)2 square meter, m2 square meter, m2 square millimeter, mm2 (10−3 m)2
2.47 247 0.386 2.47 × 10−4 10.76 1.55 × 10−3
10−3
m) kilometer, km meter, m meter, m micrometer, µm (10−6 m) millimeter, mm (10−3 m) nanometer, nm (10−9 m)
0.621 1.094 3.28 1.0 3.94 × 10−2 10
Column 1 SI Unit
(103
To convert Column 1 into Column 2, multiply by
Volume
Area
Length
acre-inch cubic foot, ft3 cubic inch, in3 bushel, bu quart (liquid), qt cubic foot, ft3 gallon ounce (fluid), oz pint (fluid), pt
acre acre square mile, mi2 acre square foot, ft2 square inch, in2
mile, mi yard, yd foot, ft micron, µ inch, in Angstrom, Å
Column 2 non-SI Units
Conversion Factors for SI and non-SI Units
102.8 2.83 × 10−2 1.64 × 10−5 35.24 0.946 28.3 3.78 2.96 × 10−2 0.473
0.405 4.05 × 10−3 2.590 4.05 × 103 9.29 × 10−2 645
1.609 0.914 0.304 1.0 25.4 0.1
To convert Column 2 into Column 1, multiply by
xxxviii CONVERSION FACTORS FOR SI AND NON-SI UNITS
megagram per cubic meter, Mg
megapascal, MPa (106 Pa) megapascal, MPa (106 Pa) pascal, Pa pascal, Pa
10 1000
1.00
9.90 10 2.09 × 10−2 1.45 × 10−4
pound, lb ounce (avdp), oz pound, lb quintal (metric), q ton (2000 lb), ton ton (U.S.), ton ton (U.S.), ton
Pressure
Density
atmosphere bar pound per square foot, lb ft−2 pound per square inch, lb in−2
gram per cubic centimeter, g cm−3
square centimeter per gram, cm2 g−1 square millimeter per gram, mm2 g−1
Specific Surface
pound per acre, lb acre−1 pound per bushel, lb bu−1 bushel per acre, 60 lb bushel per acre, 56 lb bushel per acre, 48 lb gallon per acre pound per acre, lb acre−1 pound per acre, lb acre−1 ton (2000 lb) per acre, ton acre−1 mile per hour
Yield and Rate
Mass
(continued on next page)
square meter per kilogram, square meter per kilogram, m2 kg−1
m−3
m2
kg−1
kilogram per hectare, kg kilogram per cubic meter, kg m−3 kilogram per hectare, kg ha−1 kilogram per hectare, kg ha−1 kilogram per hectare, kg ha−1 liter per hectare, L ha−1 tonne per hectare, t ha−1 megagram per hectare, Mg ha−1 megagram per hectare, Mg ha−1 meter per second, m s−1
0.893 7.77 × 10−2 1.49 × 10−2 1.59 × 10−2 1.86 × 10−2 0.107 893 893 0.446 2.24
ha−1
gram, g (10−3 kg) gram, g (10−3 kg) kilogram, kg kilogram, kg kilogram, kg megagram, Mg (tonne) tonne, t
2.20 × 10−3 3.52 × 10−2 2.205 0.01 1.10 × 10−3 1.102 1.102
0.101 0.1 47.9 6.90 × 103
1.00
0.1 0.001
1.12 12.87 67.19 62.71 53.75 9.35 1.12 × 10−3 1.12 × 10−3 2.24 0.447
454 28.4 0.454 100 907 0.907 0.907
CONVERSION FACTORS FOR SI AND NON-SI UNITS xxxix
57.3
35.97
10−4
5.56 × 10−3
3.60 ×
milligram per square meter second, mg m−2 s−1 milligram (H2O) per square meter second, mg m−2 s−1 milligram per square meter second, mg m−2 s−1 milligram per square meter second, mg m−2 s−1
radian, rad
British thermal unit, Btu calorie, cal erg foot-pound calorie per square centimeter (langley) dyne calorie per square centimeter minute (irradiance), cal cm−2 min−1
Energy, Work, Quantity of Heat
Celsius, °C Fahrenheit, °F
Temperature
Column 2 non-SI Units
degrees (angle), °
Plane Angle
gram per square decimeter hour, g dm−2 h−1 micromole (H2O) per square centimeter second, µmol cm−2 s−1 milligram per square centimeter second, mg cm−2 s−1 milligram per square decimeter hour, mg dm−2 h−1
Transpiration and Photosynthesis
joule, J joule, J joule, J joule, J joule per square meter, J m−2 newton, N watt per square meter, W m−2
9.52 × 10−4 0.239 107 0.735 2.387 × 10−5 105 1.43 × 10−3
10−2
kelvin, K Celsius, °C
Column 1 SI Unit
1.00 (K − 273) (9/5 °C) + 32
To convert Column 1 into Column 2, multiply by
Conversion Factors for SI and non-SI Units
1.75 × 10−2
2.78 × 10−2
104
180
27.8
1.05 × 103 4.19 10−7 1.36 4.19 × 104 10−5 698
1.00 (°C + 273) 5/9 (°F − 32)
To convert Column 2 into Column 1, multiply by
xl CONVERSION FACTORS FOR SI AND NON-SI UNITS
10−3
2.29 1.20 1.39 1.66
2.7 × 2.7 × 10−2 100 100
Elemental P K Ca Mg
becquerel, Bq becquerel per kilogram, Bq kg−1 gray, Gy (absorbed dose) sievert, Sv (equivalent dose)
Oxide P2O5 K2O CaO MgO
curie, Ci picocurie per gram, pCi g−1 rad, rd rem (roentgen equivalent man)
Radioactivity
milliequivalent per 100 grams, meq 100 g−1 percent, % parts per million, ppm
Concentrations
acre-inch, acre-in cubic foot per second, ft3 s−1 U.S. gallon per minute, gal min−1 acre-foot, acre-ft acre-inch, acre-in acre-foot, acre-ft
Water Measurement
millimho per centimeter, mmho cm−1 gauss, G
Plant Nutrient Conversion
gram per kilogram, g kg−1 milligram per kilogram, mg kg−1
0.1 1
10−11
centimole per kilogram, cmol kg−1
cubic meter, cubic meter per hour, m3 h−1 cubic meter per hour, m3 h−1 hectare meter, ha m hectare meter, ha m hectare centimeter, ha cm
m3
siemen per meter, S tesla, T
1
9.73 × 9.81 × 10−3 4.40 8.11 97.28 8.1 × 10−2
10 104
Electrical Conductivity, Electricity, and Magnetism m−1
0.437 0.830 0.715 0.602
3.7 × 1010 37 0.01 0.01
10 1
1
102.8 101.9 0.227 0.123 1.03 × 10−2 12.33
0.1 10−4
CONVERSION FACTORS FOR SI AND NON-SI UNITS xli
Published 2002
Chapter 1 Soil Sampling and Statistical Procedures 1.1 Introduction A. W. WARRICK, University of Arizona, Tucson, Arizona H. M. VAN ES, Cornell University, Ithaca, New York
Consideration of sampling and analyses are common to all natural sciences. In this book, methods of analyses are considered for a wide range of physical properties of soils. Each measurement procedure has associated problems dealing with analyses and reliability of results. This chapter addresses how some of these questions can be addressed. It is indisputable that soils vary across landscapes and change with time. This natural or induced variation is discussed, as well as how it affects sampling, experimental design, and statistical procedures. Variation and sources of differences are discussed in terms of random and systematic processes. Recent developments in geostatistics and in time and space analyses are addressed, as are some of the newer applications of fractals, fuzzy sets, and wavelets. Also, a section is devoted to parameter optimization and nonlinear fitting—a popular choice of deducing model parameters from increasingly complex experiments and measurements. Each of the sections contains examples drawn from studies on soil physical measurements. The discussion of each topic cannot cover all aspects encountered and do not serve as a substitute for numerous sources of information available. Nevertheless, principles are presented and methodologies suggested that are particularly relevant to soils research and the specific techniques discussed in this book.
1.2 Soil Variability H. M. VAN ES, Cornell University, Ithaca, New York
1.2.1 Sources and Structure of Variability 1.2.1.1 Introduction Dealing with variability is an integral component of the measurement of soil attributes. Experimental design, sampling protocols, measurement techniques, and parameterization methods need to be selected with conscious recognition that soil attributes may vary in space and time and are affected by extrinsic factors such as land use type and management practices. The structure of variability may be random, correlated, periodic, or any combination of these, and may also be scale dependent. Copyright 2002 © Soil Science Society of America, 677 S. Segoe Road, Madison, WI 53711, USA. Methods of Soil Analysis. Part 4. Physical Methods. SSSA Book Series, no. 5. 1
2
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The pervasiveness of variability, and therefore uncertainty, of soil properties demands an upfront treatise. This is especially the case for methods that focus on soil physical behavior, as such properties are often extremely variable and typically do not conform to conventional statistical assumptions. Bulking (compositing) is often inappropriate to increase precision, because many observations need to be made on undisturbed material. It should be recognized that variability is inherently subjective and a result of the choices that are made by the researcher, including the scale of the domain of observation, the implicit assumptions on the processes, and last but not least, the specifics of the measurement techniques (Goovaerts, 1997). Appropriate methodologies will in part be the result of the measurement choices of the researcher. For example, when measuring saturated water transmission (Section 3.4), mean values and variability will be affected by choices on laboratory vs. field methods, the type of permeameter, and even the size of soil columns. This section provides background information on these variability issues, which are further discussed in the following sections on errors, and methods of sampling and analysis. 1.2.1.2 Properties and Processes In many cases, soil physical properties are measured to obtain a better understanding or estimation of soil processes, often through the use of simulation models. It should be recognized that variability of properties may or may not affect the relevant processes in a significant manner, as these processes are often highly nonlinear. This issue of the functional sensitivity has been studied by Boesten (1991), Kickert (1984), Persicani (1996), and Holden et al. (1996), and results generally vary depending on the processes and boundary conditions that are simulated. A researcher should consider the relative significance in variability of a soil property relative to the process of interest. For example, an order of magnitude difference in soil hydraulic conductivity may only result in a minimal change in water percolation if the latter process is primarily defined by precipitation inputs. Conversely, small changes in soil bulk density may dramatically alter the soil’s mechanical behavior. Therefore, the former case may require a different level of precision and sampling effort than the latter to provide similar precision on the processes of interest. Efficiently designed experiments incorporate an understanding of the variability of the properties as well as their impact on the processes. State variables represent the condition of the soil at any given time and location as a result of one or multiple soil processes, which in turn are governed by soil properties (Table 1.2–1). Each may exhibit different variability structures. For example, soil temperature variations within a field generally show high temporal variability with diurnal and annual periodicity, and generally low spatial variabilTable 1.2–1. Examples of soil processes and associated state variables and properties. Process
State variable
Relevant properties
Heat transfer
Soil temperature
Water movement
Soil water content, soil water potential
Thermal conductivity, thermal diffusivity, heat capacity, specific heat Hydraulic conductivity, soil water diffusivity, water retentivity
SOIL SAMPLING AND STATISTICAL PROCEDURES
3
ity. However, soil thermal conductivity and heat capacity, the main soil properties affecting heat flux and consequently soil temperature, generally show low variability and are not affected by periodic patterns. Therefore, the appropriate characterization of the state variable soil temperature requires sampling and parameterization efforts that are very different from the properties that affect it. Another dimension of the variability issue relates to the fact that soil properties and state variables are often estimated rather than directly measured, as is done when using pedotransfer functions that relate easily measured properties (e.g., texture, bulk density, and organic matter content) to difficult to measure properties such as hydraulic conductivity functions (Bouma & van Lanen, 1987; Wösten & van Genuchten, 1988), or when using simulation models (Addiscott & Wagenet, 1985). The errors associated with the estimation procedures are then also propagated (Wösten et al., 1990; Heuvelink & Burrough, 1993; Schaap & Leij, 1998; see also Section 1.3). 1.2.1.3 Sources of Variability The sources of variability for soil physical properties are spatial and temporal. Each may be the result of intrinsic (natural) or extrinsic (cultural or management related) processes. Intrinsic soil variability is the results of the geological, hydrological, and biological factors that affect pedogenesis. They have a distinct spatial component that can be said to be regionalized; that is, it varies in space, with nearby areas tending to be more alike. However, this is strongly scale dependent, as these processes operate from the continental scale to the submeter level. This supports the notion of soil mapping, which implies that areas can be identified that are relatively uniform; that is, the variability among mapping units is larger than variability within them. Although this approach has greatly facilitated the use of soils information, alternative strategies have been developed that give greater recognition to the fact that soils generally constitute a continuum with variability at different scales, giving rise to the application of geostatistics (Section 1.5), and fractal and fuzzy set theories (Section 1.8). Soil maps nevertheless still provide an important framework for soil sampling and parameterization efforts, but should not be overly relied upon because the basic assumption that they represent the single most significant source of spatial variability in a domain does not always hold, especially when extrinsic sources of variability are present (van Es et al., 1999). Jenny’s time factor of soil formation relates mostly to long-term effects on static soil properties. Shorter-term temporal variability of soil processes and properties may also be significant and need to be explicitly accounted for in soil studies. Physical processes in soils such as water movement and heat diffusion are generally driven and dominated by weather related processes (at least for rainfed systems), and their study needs to explicitly consider weather-related variability and potential confounding effects. Soil physical properties are also affected by weather and climate, although often in more subtle ways. Notably, soil properties that affect hydraulic behavior, gas flow, erosion and runoff, and aggregation are strongly affected by processes such as soil wetting and drying, freezing and thawing, and weather-related behavior of soil organisms. Extrinsic variability from cultural influences may be expressed spatially and temporally. Drainage, tillage, vehicle traffic, overburden pressure, plant cover, and
4
CHAPTER 1
Table 1.2–2. Coefficients of variation and rank for spatial, temporal, and extrinsic variance component estimates for infiltrability (log values) and the slope of the water retention curve (from van Es et al., 1999). Infiltrability Source of variability
CV
Slope of retention curve
Rank
% Spatial Soil type Field scale Temporal Year to year Week to week Extrinsic Tillage Random (unexplained)
CV
Rank
%
5.8 0.0†
4 5
23.00 0.0
1 5
0.0 7.6
5 3
0.0 9.3
5 4
10.0 9.1
1 2
14.5 21.8
3 2
† The value 0.0 indicates that this variance component is very small or dominated by interactions with other variance components.
soil amendments may dramatically alter mean behavior and variability patterns. For example, tillage tends to homogenize soils spatially, but may cause greater temporal variability from loosening and subsequent settling and recompaction (Mapa et al., 1986; Starr, 1990). Van Es et al. (1999) studied the relative significance of various sources of variability (spatial, temporal, and management related) at different scales for soil infiltrability and water retention parameters. For infiltrability, tillage effects and temporal changes within growing seasons (weeks) were important sources of variability, while water retention was affected most by soil type (Table 1.2–2). For both properties, structured variability at the field scale, which is often the focus of field experiments, was insignificant. The relative significance of all sources of variability needs to be considered when designing soil studies because results may be seriously biased if observations are generalized over larger areas, longer time domains, or multiple management practices. 1.2.1.4 Structure of Variability The structure of soil variability has important implications for the gathering and analysis of soils data. Random sampling was introduced to ensure that estimates were unbiased and met the criterion of independent sampling under identical conditions. The basic model for this is (Fig. 1.2–1a): Yi = µ + ε i
[1.2–1]
where Yi is the realization of a soil attribute at location i, µ is the mean value for the spatial domain, and εi signifies a random error term. The attribute can be described through two statistical parameters, the first moment (mean) E[Yi] = µ and the second moment (variance)
[1.2–2]
SOIL SAMPLING AND STATISTICAL PROCEDURES
5
Fig. 1.2–1. One-dimensional trace representing different data structures. Lines connect sample data points and values are not continuous between points.
E[(Yi − µ)2] = σ2
[1.2–3]
These statistics are often assumed to be parameters of a normal (gaussian) probability distribution function, thereby allowing for a series of sophisticated statistical testing procedures. Subsequent research efforts demonstrated that this model is more the exception than the rule for soil properties and processes. The typical deviations from the above model are discussed in Sections 1.2.1.4.a through 1.2.1.4.f. 1.2.1.4.a Nonuniformity of the Mean (First-Order Nonstationarity) Within spatial and temporal domains, the soil property cannot be assumed to have the same expected value, but shows structural variation through a trend (Fig. 1.2–1b) or discontinuity (as assumed with soil maps, Fig. 1.2–1c). The generally accepted notions that soils can be mapped and that experimental designs are more efficient when blocking is used support the ubiquity of the nonuniform character of soils. Appropriate models are: Yi = µ + β(xi) + εi
[1.2–4]
6
CHAPTER 1
where β(xi) may indicate a constant or a function, both dependent on a spatial or temporal location xi. The presence and significance of a field trend can be identified (David, 1977; Davidoff et al., 1986). With first-order nonstationarity (i.e., Eq. [1.2–2] is invalid), sampling and experimental design can no longer be done with simple randomization, as the results may be biased by the imperfect allocation of sampling locations by the randomization process. Observations are best made through stratified sampling methods (see also Section 1.4). β(xi) may need to be estimated to allow for detrending of the data (Cressie, 1991; Burrough, 1983; Tamura et al., 1988). With few exceptions, soil properties and processes are highly nonstationary in the depth direction, leading to the general practice of analyzing and presenting soils data by depth interval. A special case of first-order nonstationarity is the presence of “hot spots”, requiring intensive sampling efforts to adequately describe the data (Gilbert, 1987). 1.2.1.4.b Periodicity Figure 1.2–1d relates to the occurrence of cyclical trends in spatial and temporal data and is a special case of nonstationarity of the mean. Observations taken in time are very commonly periodic as a result of diurnal and annual patterns of temperature, precipitation, and related processes. Periodicity is therefore often superimposed at different scales. Spatial periodicity is less common and tends to be associated with cultural practices such as ridge and furrow patterns or wheel traffic. The model for domains containing periodicity is: m
Yi = (a0/2) + Σ [ak cos (ωkxi) + bk sin (ωkxi)] k=1
[1.2–5]
where m is the number of superimposed periodic frequencies, k is the number subscript of a set of sine and cosine wave functions, ak and bk are the amplitude coefficients, and ωk are Fourier frequencies. Periodicity can be detected by spectral analysis, which decomposes the space or time data into superimposed sinusoidal components through Fourier transforms (e.g., McBratney & Webster, 1981; Section 1.6). In most cases, only one or two superimposed periodic frequencies dominate (m = 1, 2). Wavelet analysis is a similar but more sophisticated approach that can be applied to soils data where frequency patterns are not uniform in the domain (Lark & Webster, 1999, see also Section 1.8.3). Spectral methods have also been applied as diagnostic tools, for example, to estimate soil aggregate size and strength from micropenetrometer data (Grant et al., 1985). When periodicity is present or expected in the study domain, experimental and sampling schemes should guard against bias from systematic designs. 1.2.1.4.c Nonuniformity of the Variance (Second-Order Nonstationarity) Nonuniformity of the variance yields varying levels of uncertainty across a spatial or temporal domain (Fig. 1.2–1e), mostly resulting in inaccurate statistical tests. Data representation needs to account for this and sampling intensities may
SOIL SAMPLING AND STATISTICAL PROCEDURES
7
need to vary over the domain. Second-order nonstationarity generally is only present when the data are also first-order nonstationary, that is, the variance increases with the mean. This problem is often addressed by normalizing or standardizing the data (Cressie, 1991). 1.2.1.4.d Correlation in Space or Time Correlation in space or time (Fig. 1.2–1f) is very common for soils, and the assumption of independence among observations therefore rarely holds (Nielsen et al., 1973; Vieira et al., 1981; Russo & Bresler, 1981). In such cases, Yi is considered to be a regionalized variable, and the variance is expressed in terms of the relative spatial or temporal location (h):
or
E[(Yi − Yi+h)2] = 2γ(h)
[1.2–6]
E[(Yi − µ)(Yi+h − µ)] = C(h)
[1.2–7]
where γ(h) and C(h) are the semivariogram and autocovariance function, respectively. Geostatistics, state-space analysis, and time-series analysis (Sections 1.5 and 1.6) are statistical methods that can describe correlated data and allow for superior estimation techniques based on the fact that such data provide more information than random data. Correlated data also pose challenges for design of experiments, sampling schemes, and data analysis. Spatially balanced plot allocation improves efficiency (van Es et al., 1989; Lopez & Arrue, 1995) and protect against unequal precision in the analysis of experiments (Martin, 1986; van Es & van Es, 1993). Analysis by nearest neighbor methods (Papadakis, 1937; Gill & Sukla, 1985) and generalized least squares (Cressie, 1991) also allow for improved efficiency. Sampling designs can be altered to improve efficiency and account for redundant information of nearby observations (McBratney & Webster, 1983) and conscious decisions are required on design-based vs. model-based sampling efforts (Brus & de Gruijter, 1997; see also Section 1.4). 1.2.1.4.e Anisotropy Anisotropy exists when the magnitude of the variance is directional, which may be present with spatial data when observations are made in two or three dimensions. This may be expected in many soil studies as the spatial nature of soil forming factors is often directional along the land surface (e.g., erosion, sediment deposition). Evaluation of anisotropy is especially relevant when point data are used for spatial estimation, as is done with kriging (see Section 1.5). 1.2.1.4.f Nonnormal Frequency Distributions Nonnormal frequency distributions are sometimes observed with soil properties. Properties that describe mass transmission in soils, such as hydraulic and gas diffusivity, often show data ranges of orders of magnitude as a result of the incidental existence of macropores. This results in frequency distributions with skew-
8
CHAPTER 1
ness (the third moment) that is different from zero (Fig. 1.2–1g). The kurtosis (fourth moment) of the distribution is an indication of the central tendency of the observed data and may also indicate nongaussian distributions. If parametric tests are to be performed, the underlying assumption of normality (or in some cases other distributions) should be tested. Data may need to be transformed (e.g., to logarithmic values) or described by alternate probability density functions. The above data patterns often occur in combinations and may be superimposed. For example, field measurements of infiltrability often reveal the existence of a trend across a domain, a lognormal frequency distribution, spatial and temporal correlation, and an anisotropic variance structure. Universal, robust, and nonparametric (indicator) methods (Journel, 1983; Hawkins & Cressie, 1984; Goovaerts, 1997; see also Section 1.5) are designed to address correlated data structures that contain trends or are “poorly behaved”. 1.2.2 Variability and Scale 1.2.2.1 Scale of Research Domain Soil studies require choices on the study domain, the processes to be researched, the properties to be measured, and the measurement methods to be used, each having a scale component that affects the resulting data structure. The size and structure of variability are affected by the spatial and temporal domain. Larger domains tend to be more variable. Higher variability in turn may affect the mean response of a soil attribute when dealing with highly nonlinear systems (Kachanoski & Fairchild, 1994). In addition, the structure of variability is generally a function of the domain. Figure 1.2–2 shows a set of observations along a transect that appears to be stationary when considering the entire domain. If we were to consider only the part of the domain between Points A and B, the data would be considered to have a trend. If we considered only the section from C to D, the variance would be lower. The relative magnitude of variability at different scales will probably vary among properties and will also depend on the measurement method. Several studies showed that a large fraction (>80%) of the variability at the field scale (several hectares) may be measured within an area of about 1 m2 (McBratney & Webster, 1984; van Es et al., 1999). In the latter study, 66% of the variability in infiltration measurements among four soil types ranging from clay loam to loamy sand was measured within 1 m2. These examples illustrate that most of the variability may
Fig. 1.2–2. Data trace with scale-dependent properties (after Burrough, 1983).
SOIL SAMPLING AND STATISTICAL PROCEDURES
9
be encountered within very short distances and soil maps may only explain a small fraction of the variability of a soil attribute. Soil properties generally show correlated observations in space or time. The distinction between systematic (spatially or temporally structured) information and random variation is entirely scale dependent (Burrough,1983). Apparent random variation in a large domain will often reveal structure in a subregion when observations are made at smaller distances. This phenomenon may repeat itself over several scales. For example, a single unit on a soil map at the 1:100 000 scale will reveal multiple mapping units when a 1:10 000 scale is used. Vice versa, spatial patterns in a large domain are generally not visible when sampling a smaller domain within it (Yost et al., 1982). This implies that the variance, stationarity, and correlation structure of a soil attribute are not fixed properties, but depend in part on the scale of the research domain. Fractals have been applied to describe soil variation over a range of length scales. (see also Section 1.8.1; Burrough, 1983). 1.2.2.2 Scale of Observation The dependence of the measured response on the scale of observation (also called sample support) poses a serious challenge to all soils research. With increasing scale of the observational domain, that is, the size of the soil volume or area on which the measurement is made, the variance generally decreases. In some cases, the mean response may also be affected when the scale of observation is altered. This is to be expected with nonlinear processes (e.g., mass flux properties). As early as 1911, Mercer and Hall examined the variation in yields from small plots and demonstrated that the plot-to-plot variance decreased as the size of the plot increased, up to some limit (Mercer & Hall, 1911). Lauren et al. (1988) measured ponded saturated conductivity of in situ soil columns at five scales ranging from 884 to 120 000 cm3 and found a decrease in CV values from 186 to 81%, as well as a decrease in mean value from smaller to larger measurement domains. These relationships can be explained through the concept of the representative elementary volume (REV), as derived from continuum theory (Bear, 1972). At the microscopic level, soil properties and mass flow processes are highly variable and show a wide range (Fig. 1.2–3). For example, at the 0.01-mm measurement scale, a sand may have bulk densities ranging from
Fig. 1.2–3. Data ranges encountered when measuring bulk density of a sandy soil at different observational scales.
10
CHAPTER 1
0 to 2.65 Mg m3, depending on whether a pore, a particle, or fractions of both are sampled. With increasingly larger sample volumes, the bulk density becomes equivalently less variable as observations become more macroscopic and spatially integrated. The definition of an REV therefore is the smallest volume of soil that can represent the range of microscopic variations (Bear, 1972). For water flow processes, Bouma (1985) established guidelines for REVs on the basis of soil structure, suggesting that 20 elementary structural units (particles or peds) or a representative number of planar or cylindrical voids be contained in a REV. This is especially relevant with flux properties where incidental macropores can greatly influence flow processes. Although the REV itself is an elusive concept, especially in nonstationary domains, and specific values cannot be defined for any soil type and process, the concept in practice provides a researcher with guidance on identifying an acceptable measurement scale. Representative domains of observation similarly need to be defined in the time domain. Figure 1.2–4 shows 1-h measurements of field-saturated hydraulic conductivity for two growing seasons on a silt loam and a clay loam soil, which is influenced by initial moisture conditions (van Es et al., 1999). Single-time samplings will result in a biased estimate for overall infiltration behavior for any of the sites or years. Multiple measurement times improve the estimation of the mean, but may
Fig. 1.2–4. One-hour measurements of field-saturated hydraulic conductivity for two growing seasons on a silt loam and a clay loam soil.
SOIL SAMPLING AND STATISTICAL PROCEDURES
11
result in wrong inferences on the dynamic nature of the attribute, for example, by making an initial measurement in May 1992 and a second measurement in August 1993 and concluding that the conductivity has improved with time. Clearly, many measurement times are needed to adequately estimate the mean value of the attribute, or its temporally dynamic behavior. DeGloria (1993) used the concepts of minimum time step (MTS) and minimum time period (MTP) for quantification and simulation modeling of soil behavior. For example, processes related to soil genesis occur over long time intervals with MTP and MTS values of 105 and 104 d, respectively, while movement of soluble contaminants should be measured with values of 100 and 10−2 d, respectively, to adequately assess the dynamics of the process. Larson and Pierce (1994) suggested the use of control charts to assess the temporally dynamic nature of soil qualities. With increasing size of the observational domain, the frequency distribution logically transforms with changes in the variance and, in some cases, the mean. In addition, the shape of this frequency distribution may alter towards normality, as defined by the Central Limit Theorem. For example, measurements of mass flux behavior at the 10-cm scale (e.g., infiltration rings) may be lognormally distributed, while they are likely to be normally distributed at the 10-m scale (e.g., rotating boom rainfall simulators). Similarly, neutron emissions from a source in neutron thermalization measurements for soil water content (Section 3.1.3.10) are governed by a Poisson distribution, but the temporal integration of longer counts yields a normal distribution. In many cases, nongaussian behavior may be an indication of an inadequate scale of observation. Besides measuring at a scale that adequately represents the process, researchers need to balance the variable costs associated with measurements at different scales. Increasing the size of the measurement domain generally yields more precise (i.e., lower variance) results, but also tends to be more costly and time-consuming, or prevents a good distribution of samples in the domain. Measurements on smaller volumes may be more variable, but many more observations can be made and they can be better distributed over the research domain. In addition, logistical considerations such as greater sampling ease with small equipment may play a role in the selection of the measurement equipment. Generally, a scale of observation can be defined that reasonably balances the needs for representation and precision on one side with logistical ease on the other. If, for a given cost expenditure, the sample size (nj) and the variance (σj2) associated with different measurement techniques are known (e.g., Lauren et al., 1988), their ratio provides a measure of relative precision of a measurement approach. If (n1/σ12) > (n2/σ22)
[1.2–8]
then Measurement Approach 1 provides greater precision than Approach 2. Imaging techniques based on remote sensing (Sections 3.1.3.7 and 3.1.3.8) are a special case for dealing with variability and scale issues. Observations do not simply make up a sample that represents a larger population, but in fact the entire population is sampled through measurement of reflectance or emission. Assessing the true behavior is less affected by inadequate sampling, but mostly by the ability of the measurement technique itself to accurately and unbiasedly estimate a soil
12
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property. Scale issues may play a role in that estimates are based on the mean response within a single pixel, and increasing resolution increases the researcher’s ability to detect spatial structure. In summary, studies of the soil need to be conducted with a solid understanding of the variability structure of the measured attributes, both in terms of the design of sampling efforts and the methods of data description and parameterization. This requires some a priori knowledge on the stochastic characteristics of the variable, which is generally not available. When costly and extensive studies are conducted, an investment in a reconnaissance survey is generally warranted. At all times, it should be recognized that dealing with variability is inherent to soils research and that mean response, variance, frequency distribution, stationarity, and correlation depend on the measurement technique itself, the scale of the research domain, and the scale and precision of observation. 1.2.3 References Addiscott, T.M., and R.J. Wagenet. 1985. Concepts of solute leaching in soils: A review of modelling approaches. J. Soil Sci. 36:411–424. Bear, J. 1972. Dynamics of fluids in porous media. Elsevier, New York. Boesten, J.J.T.I. 1991. Sensitivity analysis of a mathematical model for pesticide leaching to groundwater. Pest. Sci. 31:375–388. Bouma, J. 1985. Soil variability and soil survey. p. 130–149. In J. Bouma and D.R. Nielsen (ed.) Soil spatial variability. Proc. Soil Spatial Variability Workshop, Las Vegas, NV. 1984. PUDOC Wageningen, the Netherlands. Bouma, J., and H.A.J. van Lanen. 1987. Transfer functions and threshold values: From soil characteristics to land qualities. p. 106–111. In K.J. Beek et al. (ed.) Proc. ISSS/SSSA workshops on quantified land evaluation procedures. Publ. no. 6. Int. Inst. for Aerospace Surv. and Earth Sci., Enschede, the Netherlands. Brus, D.J., and J.J. de Gruijter. 1997. Random sampling or geostatistical modeling? Choosing between design-based and model-based sampling strategies for soil. Geoderma 80:1–59. Burrough, P.A. 1983. Multiscale sources of variation in soil. I. The application of fractal concepts to nested levels of soil variation. J Soil Sci. 34:577–597. Cressie, N.A.C. 1991. Statistics for spatial data. John Wiley & Sons, New York, NY. David, M. 1977. Geostatistical ore reserve estimation. Elsevier, New York, NY. Davidoff, B., J.W. Lewis, and H.M. Selim. 1986. A method to verify the presence of a trend in studying spatial variability of soil temperature. Soil Sci. Soc. Am J. 50:1122–1127. DeGloria, S.D. 1993. Visualizing soil behavior. Geoderma 60:41–55. Gilbert, R.O. 1987. Statistical methods for environmental pollution monitoring. Van Nostrand Reinhold, New York, NY. Gill, P.S., and G.K Sukla. 1985. Efficiency of nearest neighbor balanced block designs for correlated observations. Biometrika 72:539–544. Goovaerts, P. 1997. Geostatistics for natural resources evaluation. Oxford Univ. Press, Oxford, UK. Grant, C.D., B.D. Kay, P.H. Groenevelt, G.E. Kidd, and G.W. Thurtell. 1985. Spectral analysis of micropenetrometer data to characterize soil structure. Can. J. Soil Sci. 65:789–804. Hawkins, D.M., and N. Cressie. 1984. Robust kriging—A proposal. Math. Geol. 16:3–18. Heuvelink, G.B.M,, and P.A. Burrough. 1993. Error propagation in cartographic modeling using Boolean logic and continuous classification. Int. J. Remote Sens. 14:3505–3505. Holden, N.M., A.J. Rook, and D. Scholefield. 1996. Testing the performance of a one-dimensional solute transport model (LEACHC) using response surface methodology. Geoderma 69:157–174. Journel, A.G. 1983. Nonparametric estimation of spatial distributions. Math. Geol. 15: 445–468. Kachanoski, R.G., and G.L. Fairchild. 1994. Field scale fertilizer recommendations and spatial variability of soil test values. Better Crops 78:20–21. Kickert, R.N. 1984. Sensitivity of agricultural ecological system models and implications for vulnerability to toxic chemicals. Environ. Toxicol. Chem. 3:309–324.
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Lark, R.M., and R. Webster. 1999. Analysis and elucidation of soil variation using wavelets. Eur. J. Soil Sci. 50:185–206. Larson, W.E., and F.J. Pierce. 1994. The dynamics of soil quality as a measure of sustainable development. p. 37–51. In J.W. Doran et al. (ed.) Defining soil quality for a sustainable environment. SSSA Spec. Publ. 35. SSSA and ASA, Madison, WI Lauren, J.G., R.J. Wagenet, J. Bouma, and J.H.M. Wösten. 1988. Variability of saturated hydraulic conductivity in a Glossaquic Hapludalf with macropores. Soil Sci. 145:20–28. Lopez, M.V., and J.L. Arrue. 1995. Efficiency of an incomplete block design based on geostatistics for tillage experiments. Soil Sci. Soc. Am. J. 59:1104–1111. Mapa, R.B., R.E. Green, and L. Santo. 1986. Temporal variability of soil hydraulic properties with wetting and drying subsequent to tillage. Soil Sci. Soc. Am. J. 50:1133–1138. Martin, R.J. 1986. On the design of experiments under spatial correlation. Biometrika 73:247–277. McBratney, A.B., and R. Webster. 1981. Detection of ridge and furrow patterns by spectral analysis. Int. Stat. Rev. 49:45–52. McBratney, A.B., and R. Webster. 1983. How many observations are needed for regional estimation of soil properties? Soil Sci. 135:177–183. Mercer, W.G., and A.D. Hall. 1911. The experimental error of field trials. J. Agric. Sci. 4:107–132. Nielsen, D.R., J.W. Biggar, and K.T. Erh. 1973. Spatial variability of field-measured soil–water properties. Hilgardia 42:215–259. Papadakis, J.S. 1937. Methode statistique pour des experiences sur champs. Bull. 23. Inst. Amel. Plantes a Salonique. Persicani, D. 1996. Pesticide leaching into field soils: Sensitivity analysis of four mathematical models. Ecol. Modell. 84:265–280. Russo, D., and E. Bresler. 1981. Effect of field variability in soil hydraulic properties on solutions of unsaturated water and salt flows. Soil Sci. Soc. Am. J. 45:675–681. Schaap, M.G., and F.J. Leij. 1998. Database-related accuracy and uncertainty of pedotransfer functions. Soil Sci. 163:765–779. Starr, J.L. 1990. Spatial and temporal variation of ponded infiltration. Soil Sci. Soc. Am. J. 54:629–636. Tamura, R.N., L.A. Nelson, and G.C. Naderman. 1988. An investigation on the validity and usefulness of trend analysis for field plot design. Agron. J. 80:712–718. van Es, H.M., C.B. Ogden, R.L. Hill, R.R. Schindelbeck, and T. Tsegaye. 1999. Integrated assessment of space, time, and management-related variability of soil hydraulic properties. Soil Sci. Soc. Am. J. 63:1599–1608. van Es, H.M., and C.L. van Es. 1993. Spatial nature of randomization and its effects on the outcome of field experiments. Agron. J. 85:420–428. van Es, H.M., C.L. van Es, and D.K. Cassel. 1989. Application of regionalized variable theory to largeplot field experiments. Soil Sci. Soc. Am J. 53:1178–1183. Vieira, S.R., D.R. Nielsen, and J.W. Biggar. 1981. Spatial variability of field-measured infiltration rate. Soil Sci. Soc. Am. J. 45:1040–1048. Wösten, J.H.M., C.H.J.E. Schuren, J. Bouma, and A. Stein. 1990. Functional sensitivity analysis of four methods to generate soil hydraulic functions. Soil Sci. Soc. Am. J. 54:832–836. Wösten, J.H.M., and M.Th. van Genuchten. 1988. Using texture and other soil properties to predict the unsaturated soil hydraulic functions. Soil Sci. Soc. Am. J. 52:1762–1770. Yost, R.S., G. Uehara, and R.L. Fox. 1982. Geostatistical analysis of soil chemical properties of large land areas. I. Variograms. Soil Sci. Soc. Am. J. 46:1028–1032.
Published 2002
1.3 Errors, Variability, and Precision R. R. ALLMARAS, USDA-ARS, University of Minnesota, St. Paul, Minnesota OSCAR KEMPTHORNE, Iowa State University, Ames, Iowa
1.3.1 Introduction The experimenter is confronted nearly every day with the examination of results from his or her own experiments as well as those of others. There is a need to know how the data were obtained and what confidence can be placed in the numerical results. Significant aspects of these matters involve the principles and methods of statistics, and the objective of this section is to describe the basic ideas of statistics that are relevant to errors of observations and numbers derived from these observations. A measurement is a quantification of an attribute of the material under investigation, directed to the answering of a specific question in an experiment. The quantification implies a sequence of operations or steps that yields the resultant measurement. Thus, the concept of measurement may be said to include not only the steps used to obtain the measurement, but also use of the measurement by the experimenter to draw conclusions. The measurement process provides a result that serves as part of the basis upon which an experimenter makes a judgment about the attribute under investigation. Some judgments may require a more reliable basis than others. The desired reliability in measurements will depend on the purpose for which measurements are to be used, but the degree of reliability may be limited by resources available to the experimenter. The experimenter may control reliability by choosing a measurement process making use of a number of relevant scientific principles, by controlling attributes of the environment in which measurement is made, and by repeating the measurements. Different combinations of these control alternatives may be suitable for a given measurement process; the suitability of a particular combination of control alternatives varies among measurement processes and depends on the reliability desired. The number, which the experimenter uses to judge about the substance under investigation, may not be a single measurement but may be a derived number, that is, some function of several measurements utilizing the same or different measurement processes. The best function will be dictated by the scientific nature of the investigation as well as by the theory of combination of observations and related statistical concepts. 15
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1.3.2 Classification of Measurement Errors From the standpoint of measurement errors, the simplest situation is that of obtaining an attribute of an object that is not affected by the measuring process. In that case, the measuring process can be applied again and again to the unchanged object, as, for example, determining the length of a bar of steel. Repeated application of the measuring process to this unchanged object, following the directions laid down in the specifications, will yield a sequence of numbers. Variation in the numbers arises because no sequence of operations or “state of nature” is perfectly reproducible. In other words, no human or machine can do exactly the same sequence of operations again and again, and identical circumstances and measurements cannot be achieved perfectly. If the measurement is coarse, the lack of reproducibility of the measuring process may have no effect, as for instance would be the case with measuring the length of a 14.6-cm rod to the nearest centimeter with a ruler graduated in centimeters. In general, however, the lack of reproducibility of operations will produce variability. For instance, the instruction to bring the pH of a solution to 6 by adding 1.0 M HCl can be performed only to a certain degree of perfection, depending on the indicator and the operator. Supposedly simple operations like weighing a precipitate will not lead to the same answer upon repetition with a sufficiently sensitive balance. Of course, much of the training in elementary analytical sciences is directed to the performance of operations in a manner to achieve negligible variability in results and conclusions, but this ideal can rarely be achieved. The variability among results of a measuring process applied to a constant object may be called measurement error or, for emphasis, pure measurement error. In contrast to the above situation, we may imagine that the object of measurement, which may be a batch of material, is heterogeneous and can be measured only in parts that are unlike, but that the measurement process is perfectly reproducible for each part. If we could apply the measuring process to all parts of the whole under these ideal circumstances, there would be no error in the final result, but the circumstances under which one can process the whole are very rare. To give an obvious example, suppose we wish to characterize the K status of plots of land in connection with a study of K-fertilizer uptake. We cannot process all the soil of the plot for obvious reasons. We, therefore, must determine the status of the land by drawing samples and applying a measurement process to these samples. The samples will not be equal with regard to the attributes under examination, and the variation among the sample results will lead to uncertainty in the final result. Variation of this type, in which it is supposed that the measuring process itself is exactly reproducible, is called pure sampling error. In practice, few measurement situations lie at either of the two extremes we have mentioned. In the majority of cases, the measurement process destroys the object being measured, so that repetition of the measuring process on the object of measurement cannot be performed. A common way out of this difficulty in soil analysis is to homogenize the sample by passing it through a fine sieve and mixing it thoroughly. The homogenized sample is then subdivided, and one performs repeated measurements on subsamples, which one has good reason to regard as identical. When analyses are made in this way, the total error of an observation on a subsample will include both measurement error and sampling error, the latter because
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the total sample from which the subsamples are derived is only a small part of the whole for which information is desired. In soil, of course, there is always a lower limit to the subsample size that can be taken without sampling error from even a sample that has been finely ground and mixed. This problem is of considerable importance in some kinds of soil analysis. The measurement process consists of a sequence of operations, and at each step in the sequence there will be a certain lack of perfect repeatability. The measurement error may, therefore, have a structure, in the sense that part arises from Step 1 of the sequence of operations, part from Step 2, and so on. Similarly, the total operation of sampling may in each particular situation be broken down into distinct steps, perhaps according to the sampling design, and each step will introduce a part of the sampling error. The total error is that arising from both measurement and sampling. In practice, we must attempt to control the total error, so that conclusions based on results will not differ in any important respect from conclusions that would be made in the absence of error, or so that the effect of the error on subsequent conclusions can be assessed. In general, error in the sense discussed above is not the result of incorrect procedure; also, it is to be distinguished from mistakes in following the directions, which will lead to large deviations of single measurements from other measurements made with the same measurement process.
1.3.3 Scientific Validity of Measurements In general, the measurement process is aimed at the characterization of an object of measurement in terms of scientific concepts. We might, for instance, wish to measure the total content of combined N in a soil that contains appreciable amounts of NO3−. We can imagine a highly reproducible Kjeldahl analysis wherein no additives were used to assure conversion of the NO3− to NH4+ in the digestion. The measurement is flawed because the total content of combined N is sought, while the measurement process does not measure all the N present as NO3−. The occurrence of such a defect in the measurement process can be found only on the basis of sound scientific principles or by special tests of validity. Underlying any measurement process applied to an object of measurement is what one may term the scientific true value, which can be visualized as being approached more and more closely by refinement of measurement operations and by appropriate refinements in the process. For example, in the instance just considered, the scientific true value would be approached more closely if the process were changed to include NO3−. Scientific relevance of the measurement process is not a matter of the theory of error, which involves the “wandering” of the results. Clearly, however, the theory of error enters into what we may call the validation of a measurement process, because such validation will require the comparison of results obtained in different ways, each way having its own peculiar error characteristics. It is because the scientific validation of a measurement process involves such comparisons that the experimenter must have some awareness of the concept of error and of procedures for drawing conclusions in the presence of error.
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It is, one supposes, obvious that examination of the results given by a single measurement process cannot per se lead to scientific validation of the measurement process, but it is equally obvious that validation involves consideration of the variability exhibited by the results of applying the measurement process. Therefore, we take up the problem of characterizing the data one will obtain by repeating the measurement process. 1.3.4 Characterization of Variability Suppose that we have at hand a large sample of soil that can be subdivided into a very large number of smaller samples, to each of which we can apply the measurement process. We suppose that a large number of the smaller samples have been analyzed. Under these circumstances, the variability among analytical values may arise from pure measurement error, sampling (of the large sample) error, or both. We can then construct a frequency distribution or histogram of the resulting numbers, which could have the appearance of Fig. 1.3–1, in which the area of the block between a and b is the relative frequency of numbers between a and b, and the units of area are chosen so that the total area of all the blocks is unity. This histogram is an empirical or observed distribution. We have supposed that we could have obtained a large number of results, and we can imagine making successive histograms in which the intervals become progressively narrower until we have essentially a continuous curve rather than a curve which proceeds by finite steps. The distribution may be called the true distribution underlying the whole measurement operation, and it may take a particular mathematical form, of which examples will be given below.
Fig. 1.3–1. Frequency distribution of arbitrary measurements.
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The mean of this essentially infinite population, which will be hypothetical in all cases (i.e., the arithmetic average of an infinitely large number of results), may be called the statistical true value or limiting mean associated with the measurement process and the object of measurement. It may also be called the operational true value, in the sense that it is a mean value associated with operations of the measurement process. This statistical true value may be different from the scientific true value that is being sought, for one or more of the following reasons: 1. The specifications of the measurement process may be inadequate. The deficiency may be one of selectivity, in which the process does not permit inclusion of all that is desired, or it may include more than is intended. Alternatively, the specifications may not provide for the problem encountered with some measurements, in that the quantitative expression of the property being measured is not entirely independent of the nature of the material on which the analysis is performed. Inadequate specification of a measurement process may be characterized as a scientific deficiency of the process, and the resulting deviation of the statistical true value from the scientific true value may be termed scientific bias. 2. The auxiliary apparatus or materials used in making the measurement may be faulty, so that the results tend to be too high or too low. Differences between the scientific and statistical true values arising in this way may be called measurement bias. 3. The process for obtaining the samples to be measured may result in selection of samples that are not representative of the whole. A difference between the scientific and statistical true values arising in this way may be called sampling bias. These are very generalized statements of what may go wrong with a measurement process. Another source of discrepancy arises from the human element. We can imagine two different operators analyzing similar samples of the same soil by the same method but obtaining different distributions because of differences in performance such as filling pipettes to consistently different levels. The purpose of training is, of course, to attempt to eliminate this type of personal effect as much as possible, but one cannot assume totally adequate performance. The spread or dispersion of the distribution of Fig. 1.3–1 must now be considered. One can imagine two measurement processes with Distributions 1 and 2 in Fig. 1.3–2, in which the means of the two distributions are the same, but Distribution 2 is clearly more spread out than Distribution 1. If the sampling procedure in the two processes is the same, so that for instance the two distributions were estimated by first drawing 2000 samples and then partitioning these into two sets of 1000 samples, one set for each measurement process, then Process 1 may be said to give more precise measurements than Process 2. Precision is inversely related to the variability among results obtained by applying a measurement process again and again to an object of measurement or to samples from a population that is the object of measurement. This variability can be represented by the construction of a histogram or frequency distribution of the results, and we can imagine a mathematically defined form for the distribution that
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CHAPTER 1
Fig. 1.3–2. Frequency distributions expected in two different arbitrary situations of reproducibility of the measurement process or sample homogeneity.
would result from an infinite number of repetitions. Usually one is satisfied with a mathematically defined distribution that is developed from a model of the measurement process. The most common distribution considered is the normal distribution, and most of our ideas on precision and modes of handling imprecision are based on the normal distribution. Let us suppose that the total error of an observation, say e, is made up additively of components, say, e1 arising from taking a prescribed aliquot imperfectly, e2 arising from not adding the prescribed amount of one reagent, and so on. Then e = e1 + e2 + ... + en. If these component errors have a mean of zero and are independent, the total error will follow a distribution close to the normal distribution. If f(x)dx denotes the relative frequency of e in the interval x to x + dx, then f (x) =
______ 1 x2 exp − _____ ‰ σ%2π && ‰ 2σ2
[1.3–1]
Classical writers often used the expression h in place of 1/σ to seek a positive relation to precision. This gives the familiar bell-shaped curve of error characterized by one parameter σ, which is called the standard deviation, or by h. We give in Fig. 1.3–3 the two curves for (i) σ = 1 or h = 1 and (ii) σ = 0.5 or h = 2. A numerical quantity for which the relative frequency of possible values is specified is called a random variable. It is a purely mathematical exercise to determine the frequency with which errors lie in particular ranges, and the most useful symmetrical ones are given in Table 1.3–1. To facilitate presentation, the situation is considered where σ = 1 and Eq. [1.3–1] is integrated between specified limits corresponding to par-
SOIL SAMPLING AND STATISTICAL PROCEDURES
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Fig. 1.3–3. Frequency distribution of random variable x for two values of h and σ.
ticular segments of the random variable. An empirical distribution that is virtually normal in appearance is presented in Fig. 1.3–4. In practice, one may not get a normal distribution because sets of observations have common errors, and for other reasons. The fact that the normal distribution of error is determined by one parameter, σ (or h), makes the definition of a measure of precision for such errors easy to specify. If then we have normally distributed errors, the matter of quantifying precision is simple. It is now more common to use σ, which is called the standard deviation, so that the lower the value of σ, the higher the precision. The concept of probable error was frequently used in the past. Its meaning is given in Table 1.3–1. Instead of considering one standard deviation, only 0.674 σ is considered. The frequency of occurrence of errors within ± 0.674 σ is equal to the frequency of occurrence of errors outside this range, so that the confined proportional frequency is 0.50.
Table 1.3–1. Frequencies of errors in particular segments for a normal distribution with zero mean and unit standard deviation. Segment of the random variable
Proportion of the total
−0.674 to 0.674 −1.0 to 1.0 −1.96 to 1.96 −2.58 to 2.58
0.50 0.67 0.95 0.99
Usual name of segment denoted Probable error Standard deviation 95% confidence region 99% confidence region
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CHAPTER 1
Fig. 1.3–4. Histogram and distribution of Kjeldahl determinations of N in 20 aliquots taken from a homogeneous (NH4)2SO4 solution (personal communication with D.R. Timmons, Ames, IA, 1976).
1.3.5 Skewed Frequency Distributions Frequency distributions may be skewed for numerous reasons. For example, a distribution may be a composite of multiple distributions, or there can be functional relations between variance and mean of the distribution. Frequently the experimenter may observe that a plot of the frequency vs. the magnitude of the random variable reveals a positive skew such as in the lognormal distribution of pore-water velocity presented in Fig. 1.3–5. This type of distribution can be detected by methods such as illustrated in Table 1.3–2 and Fig. 1.3–5. After the observations are listed in increasing order, their cumulative probability is plotted vs. the observation (V) or a function of the observation, g(V) = lnV, both as in Fig. 1.3–5B. A normal distribution of the observation or of a function of the observation produces a linear plot. The parameters computed in Fig. 1.3–5 are the same as those computed in the usual manner in Table 1.3–2; moreover, the observed goodness of linear fit in Fig. 1.3–5 gives confidence in use of the distribution parameters. The density of the lognormal frequency is calculated from 1 (ln x − µ)2 f (x) = ______ exp − _________ xσ%2π && — 2σ2 f (x) = 0
for x < 0
for x ≥ 0 [1.3–2]
where µ and σ are the mean and standard deviation of lnx as the variate. ∆x is the class interval selected, such as 20 cm d−1 in Fig. 1.3–5A.
SOIL SAMPLING AND STATISTICAL PROCEDURES
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Many soil properties approximate a lognormal distribution. Nielsen et al. (1973) found that saturated hydraulic conductivity, pore-water velocity, and a scaling coefficient that relates horizontal heterogeneity of soil water flow properties were
Fig. 1.3–5. (A) Histogram and distribution of pore-water velocity measured 20 times in a Panoche silt loam (fine-loamy, mixed, superactive, thermic Typic Haplocambids), and (B) graphical determination of transformation to achieve normality (data from Biggar & Nielsen, 1976).
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CHAPTER 1
Table 1.3–2. Twenty observations of pore-water velocity, V in cm day−1, and corresponding statistical parameters (Biggar & Nielsen, 1976). i
Vi†
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2.6 5.6 7.0 8.0 8.7 9.8 14.0 15.5 21.4 21.6 22.1 24.8 36.3 42.2 46.5 52.0 77.4 89.7 126.6 131.4
(lnVi)†
(i − 0.5)/n‡
ui
0.96 1.72 1.95 2.08 2.16 2.28 2.64 2.74 3.06 3.07 3.10 3.21 3.59 3.74 3.84 3.95 4.35 4.50 4.84 4.88
0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975
−1.96 −1.44 −1.15 −0.93 −0.76 −0.60 −0.45 −0.32 −0.19 −0.06 0.06 0.19 0.32 0.45 0.60 0.76 0.93 1.15 1.44 1.96
† When X = g(V ), then a linear fit of u vs. g(V ) can be used to determine what form of g(V ) is normally distributed. These constants may then be used to reconstruct the theoretical relation. ‡ (i − 0.5)/n approximates the area under the cumulative probability function P(u) and u = (x − µ)/σ is u the corresponding upper limit in P(u) = [1/(2π)]∫−∞ exp[−(x2/2)]dx.
all lognormally distributed. Meanwhile, soil water content and dry bulk density were normally distributed. The apparent diffusion coefficient for Cl− is also lognormally distributed (Van de Pol et al., 1977). The lognormal distribution is often observed for size (diameter) distributions of aggregates and primary particles (Gardner, 1956), dispersed clay size (Austin, 1939), exchangeable Ca and Sr (Menzel & Heald, 1959), and heights of constant length pins impinging on a newly tilled soil surface (Allmaras et al., 1966). All of these distributions, whether normal or lognormal, may be determined by using the methods illustrated in Table 1.3–2 and Fig. 1.3–5. Analogous to the situation with a normally distributed random variable, where the magnitude is the additive effect of a large number of small independent causes, the arithmetic magnitude of the lognormally distributed random variable is the multiplicative effect of a large number of independent causes. This analogy is discussed by Aitchison and Brown (1957) and Gaddum (1945). As an example, we may consider the diameter of an aggregate of soil particles in relation to two causes that may bring about a change in diameter. A proportionate-effect hypothesis predicts that the diameter change, resulting from the action of a cause, is some proportion of the initial diameter. Let the initial diameter be d1, the diameter after the first cause acts be d2, and the final diameter after the second cause acts be d3. The effect resulting from the first cause is (d2 − d1)/d1 = e1 and that after the second cause acts is (d3 − d2)/d2 = e2. Hence, d3 = d1(1 + e1)(1 + e2) or lnd3 = lnd1 + ln(1 + el) + ln(1 + e2) or lnd3 = lnd1 + f1 + f2, where f1 = ln(1 + e1) and f2 = ln(1 + e2). With
SOIL SAMPLING AND STATISTICAL PROCEDURES
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Table 1.3–3. Number of particle emissions per unit of time from polonium observed by scintillation counting, and the χ2 goodness of fit to the Poisson distribution (Rutherford & Geiger, 1910). Number of α particles observed per unit time x
Number of times observed in 2608 trials fo
fo x
Expected number of times based on Poisson distribution† fe
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
57 203 383 525 532 408 273 139 45 27 10 4 0 1 1
0 203 766 1575 2128 2040 1638 973 360 243 100 44 0 13 14
54 210 407 525 508 394 254 141 68 29 11 4 2 1, the variance of y–St can be estimated by: 1 v(y–St) = __ A2
L
Σ Ah2 v(y–h) h=1
[1.4–10]
where v(y–St) is the estimated variance of y–h: v(y–h) =
nh 1 _______ Σ (yhi − y–h)2 i=1
nh (nh − 1)
[1.4–11]
with nh = sample size in the hth stratum. The standard deviation is estimated by s(y–St) –St&. Confidence intervals are calculated in the same way as with SRS (see = %v&(y &&) Eq. [1.4–4]). The method of estimating means, fractions, or SCDFs (after 0/1 transformation) in a domain depends on whether the areas of the domain within the strata are known. If they are, then the mean of the jth domain, Y–j, is estimated by 1 Y–$j = __ Aj
Σh Ahj y–hj
[1.4–12]
with Ahj = area of domain j within stratum h, Aj = total area of domain j, and y–hj = sample mean of domain j within stratum h. The variance of Y–$j is estimated by
SOIL SAMPLING AND STATISTICAL PROCEDURES
1 v (Y–$j) = __ A2j where
59
Σh A2hj v (y–hj)
1 v(y–hj) = _________ nhj (nhj − 1)
[1.4–13]
nhj
Σ (y − y–hj)2 i=1 hij
[1.4–14]
with nhj = number of sample points falling in domain j within stratum h. If the areas of the domain within the strata are not known, they have to be estimated from the sample. Unbiased estimates to be substituted in Eq. [1.4–12] are: A$hj = Ah(nhj/nh)
[1.4–15]
A$j = Σ A$hj
[1.4–16]
and h
The variance is now larger because of the error in the estimated areas. It is estimated by 1 v(Y–$j) = __ A$2j
Ah2 n__ –$ 2 hj – 2 – Σh n________ (n − 1) — Σi (yhij − y hj) + nhj ‰ 1 − n (y hj − Y j) h
h
h
[1.4–17]
Sample Sizes. The sample sizes in the strata may be chosen to minimize the variance v(y–St) for a given maximum allowable cost, or to minimize the cost for a given maximum allowable variance. A simple linear cost function is: C = co + Σ chnh
[1.4–18]
with co = overhead cost, and ch = cost per sample point in stratum h. If we adopt this function, the optimal ratios of the sample sizes to the total sample size n are: nh AhSh/%c&h __ = _________ n Σ(AhSh/%c&)h
[1.4–19]
where the Sh are prior estimates of the standard deviations in the strata. This formula implies that a stratum gets a larger sample if it is larger or more variable or cheaper to sample. The total sample size affordable for a fixed cost C, given that optimal allocation to the strata is applied, is: (C − co) Σ (AhSh/%c&) h n = _________________ Σ(AhSh%c&) h
[1.4–20]
The total sample size needed to keep the variance below a maximum value Vm, again presuming that optimal allocation to the strata is applied, is: n = (1/Vm)(ΣWhSh%c&)( &) h ΣWhSh/%c h
[1.4–21]
where Wh = Ah/A. If the cost per point is equal for the strata, this reduces to: n = (1/Vm)(ΣWhSh)2
[1.4–22]
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If, instead of Vm, an absolute error d has been specified with an allowed probability of exceeding α, then Vm can be derived from d and α, according to Vm = d/u1−α/2, where u1−α/2 is the (1 − α/2) quantile of the standard normal distribution. When estimating areal fractions rather than means of quantitative variables, the above formulas for sample sizes can still be applied if Sh is replaced by %P &&1 − &&, Ph) where Ph is a prior estimate of the fraction in stratum h. h(&& Advantages. There are two possible reasons for stratification. The first is that the efficiency may be increased as compared with SRS, that is, smaller sampling variance at the same cost, or lower cost with the same variance. In this case, the stratification is chosen such that the expected gain in efficiency is maximized. In practice, this can be achieved by forming strata that are as homogeneous as possible. Also, if the cost per sample point varies strongly within the area, for instance with distance from roads, it is efficient to stratify accordingly and to sample the inexpensive strata more densely. Another reason for stratification may be that separate estimates for given subareas are needed. If the strata coincide with these subareas of interest then, as opposed to SRS, one has control over the accuracy of the estimates by allocating sufficient sample sizes to the strata. Disadvantage. With inappropriate stratification or suboptimal allocation of sample sizes, there could be a loss rather than gain in efficiency. This can occur if the stratum means differ little or if the sample sizes are strongly disproportional to the surface areas of the strata. If, for instance, one has strata with unequal areas and only a small sample in each, then these sample sizes are bound to be strongly disproportional because they must be integer numbers. 1.4.3.4 Two-Stage Sampling Restriction on Randomization. In Two-stage Sampling (TsS), as with StS, the area is divided into a number of subareas. Sampling is then restricted to a number of randomly selected subareas, in this case called primary units. Note the difference with StS where all subareas (strata) are sampled. In large-scale surveys this principle is often generalized to multistage sampling. Three-stage soil sampling, for instance, could use farms as primary units, fields as secondary units, and sample plots as tertiary units. Selection Technique. A version is described by which the primary units (PUs) are selected with replacement and with probabilities proportional to their area. An algorithm to make n such selections from all N PUs in the area is as follows: 1. Determine the areas of all PUs, A1... AN, and their cumulative sums, S1... SN, with k
Sk =
Σ Ai
i=1
[1.4–23]
2. Generate a (pseudo-)random number x from the uniform distribution on the interval (0, SN). 3. Select the PU of which the corresponding Sk is the first in the series that exceeds x.
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4. Repeat Steps 2 and 3 until n PUs are selected. An alternative, sometimes more efficient, algorithm works with a geographical representation of the area and its PUs: 1. Select a random point in the area as in SRS. 2. Determine with a point-in-polygon routine in which PU the point falls, and select this PU. 3. Repeat Steps 1 and 2 until n selections have been made. In the second stage, a predetermined number of sample points, mi, is selected within each of the PUs selected in the first stage. This is done in the same way as with SRS. If the geographical algorithm is applied, the random points used to select the PUs may also be used as sample points. If a PU has been selected more than once, an independent sample of points must be selected for each time the PU was selected. Example. Figure 1.4–2C shows four square PUs selected in the first stage and four points in each in the second stage. Notice the stronger spatial clustering compared with SRS in Fig. 1.4–2A. This is just a simple, notional example. It should be noted, however, that the PUs may be defined in any way that seems appropriate and that the number of sample points may vary among units. Statistical Inference. Means, areal fractions, and SCDFs (after 0/1 transformation) of the area are estimated by the simple estimator: y–Ts = 1n_
n
Σ y–i
[1.4–24]
i=1
with n = number of PU selections, and y–i = sample mean of the PU from selection i. The strategy (TsS, y–Ts) is p-unbiased. The variance is simply estimated by: 1 v(y–Ts) = ______ n(n − 1)
n
Σ (y–i − y–Ts)2 i=1
[1.4–25]
Notice that neither the areas of the PUs, Ai, nor the secondary sample sizes mi occur in these formulas. This simplicity is due to the fact that the PUs are selected with replacement and probabilities proportional to size. The effect of the secondary sample sizes on the variance is implicitly accounted for. To understand this, consider that the larger mi is, the less variable is y–i, and the smaller its contribution to the variance. –Ts). Confidence intervals The standard deviation is estimated by s(y–Ts) = %v(y &&& are calculated in the same way as with SRS (see Eq. [1.4–4]). The method of estimating means, areal fractions, and SCDFs in domains depends on whether the area of the domain, Aj, is known or not. If it is known, then the mean of the jth domain, Y–j, is estimated by: Y–$ = Y$j/Aj
[1.4–26]
where Y$j is an estimate of the total (spatial integral) of the variable y over the domain j. To estimate this total, we first define a new variable y′, which equals y every-
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where in the domain, but is zero elsewhere. The total of y over domain j equals the total of y′ over the area, and this is estimated as A times the estimated mean of y′, following Eq. [1.4–24]: A Y$j = A y–Ts ′ = __ n
n
Σ y–i′ i=1
[1.4–27] – where yi′is the sample mean of the transformed variable y′ from PU selection i. The variance of the domain mean is estimated by: A 2 ______ 1 v(Y–$j) = __ ‰ Aj n(n − 1)
n
Σ (y–i′ − y–Ts ′ )2
i=1
[1.4–28]
If the area of the domain is not known, it has to be estimated from the sample. An unbiased estimate to be substituted for Aj in Eq. [1.4–26] is: A A$j = _ n
n m __ij Σ i=1 m
[1.4–29]
i
with mij = number of points in PU selection i and domain j. Hence, the ratio estimator: n $j Σ y–i′ Y – i=1 __ ______ $ Y Rj = = n m ij A$j Σ ___ [1.4–30] m i=1
i
with estimated variance: A 1 v(Y–$Rj) = __ ______ $ ‰ Aj n(n − 1)
n m Σ y–i′ − Y–$Rj __ij i=1
‰
mi
2
[1.4–31]
Advantage. The spatial clustering of sample points created by TsS has the operational advantage of reducing the travel time between points in the field. Of course, the importance of this depends on the scale and the accessibility of the terrain. The advantage may be amplified by defining the PUs such that they reflect dominating accessibility features like roads and land ownerships. Disadvantage. The spatial clustering generally leads to lower precision, given the sample size. However, the rationale is that due to the operational advantage a larger sample size can be afforded for the same budget, so that the initial loss of precision is outweighed. 1.4.3.5 Cluster Sampling Restriction on Randomization. In Cluster Sampling (ClS), predefined sets of points are selected, instead of individual points as in SRS, StS, and TsS. These sets are referred to as clusters. Selection Technique. In principle, the number of clusters in the area is infinite, so it is impossible to create all clusters beforehand and to sample from this collection. However, only clusters that are selected need to be created, and selection of a cluster can take place via selection of one of its points. Hence the following algorithm:
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1. Select a random point in the area as in SRS; use this point as a “starting point”. 2. Find the other points of the cluster to which the starting point belongs by applying predetermined geometric rules corresponding with the chosen cluster definition. 3. Repeat Steps 1 and 2 until n clusters have been selected. A condition for this algorithm to be valid is that the geometric rules are such that the same cluster is always created regardless of which of its points is used as the starting point. A well-known technique satisfying this condition is random transect sampling with equidistant sample points on straight lines with a fixed direction. Given this direction, the random starting point determines the line of the transect. The other sample points are found by taking a prechosen distance in both directions from the starting point, until the line crosses the boundary of the area. Clusters thus formed will generally consist of a variable number of points, and the probability of selecting a cluster is proportional to the number of points in it. This is taken into account in the statistical inference. Example. Figure 1.4–2D shows four transects, each with four equidistant points. To limit the length of the transects, the area has first been dissected with internal boundaries perpendicular to the transects. Notice the spatial clustering and the regularity compared with SRS, StS, and TsS (Fig. 1.4–2A, 1.4–2B, and 1.4–2C). This is just a simple, notional example. It should be noted, however, that the clusters may be defined in any way that seems appropriate. Statistical Inference. For this type of design the same formulas are used as for TsS, clusters taking the role of primary sampling units. Advantages. As in TsS, the spatial clustering of sample points has the operational advantage of reducing the travel time between points in the field. In addition, the regularity may reduce the time needed to locate consecutive points in the cluster. The importance of these advantages depend on the scale, the accessibility of the terrain, and the navigation technique used. Disadvantages. As with TsS, the spatial clustering generally leads to lower precision, given the sample size. Again, the rationale is that because of the operational advantages a larger sample size can be afforded for the same budget, so that the initial loss of precision is outweighed. If the spatial variation has a dominant direction, the precision can be optimized by taking transects in the direction of the greatest change. Another disadvantage is that the sample size, that is, the total number of points in the clusters that happen to be selected, is generally random. This may be undesirable for budgetary or logistical reasons. The variation in sample size can be reduced by defining clusters of roughly equal size. 1.4.3.6 Systematic Sampling Restriction on Randomization. As with ClS, in Systematic Sampling (SyS), random selection is applied to predefined sets of points, instead of individual points as in SRS, StS, and TsS. The difference with ClS is that only one cluster is selected. In this sense, SyS is a special case of ClS. Note that the term cluster as used here
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does not refer to geographical compactness, but to the fact that if one point of a cluster is included in the sample, all other points are included too. Selection Technique The selection algorithm for ClS is used, with n = 1. Example. Figure 1.4–2E shows a random square grid. Notice the more even spatial spreading and the greater regularity compared with all other types of designs (Fig. 1.4–2A through 1.4–2D). Statistical Inference. Means, areal fractions, and SCDFs (after 0/1 transformation) of the area are simply estimated by the sample mean y–, as with SRS. The strategy (SyS, y–) is p-unbiased. This condition holds only if the grid is randomly selected, as is prescribed by the selection technique given above. With centered grid sampling, on the other hand, the grid is purposively placed around the center of the area, so that the boundary zones are avoided. This is a typical model-based strategy (see Section 1.4.4), which is p-biased. Unfortunately, no unbiased variance estimators exist for this type of design. Many variance estimators have been proposed in the literature; all are based on assumptions about the spatial variation. A well-known procedure is Yates’s method of balanced differences (Yates, 1981). An overview of variance estimation is given by Cochran (1977). A simple, often applied procedure is to calculate the variance as if the sample was obtained by SRS. If there is no pseudocyclic variation, this overestimates the variance, so in that case the accuracy assessment will be on the safe side. Means, areal fractions, and SCDFs (after 0/1 transformation) in a domain are simply estimated by the sample mean in this domain: 1 y–j = __ mj
mj
Σ yij i=1
[1.4–32]
where mj is the number of grid points falling in domain j. Advantages. Because only one cluster is selected, the clusters should be predefined such that each of them covers the area as well as possible. This is achieved with clusters in the form of regular grids that are square, triangular, or hexagonal. The statistical precision can thus be maximized through the definition of the grid. In addition, SyS has the same operational advantage as ClS; the regularity of the grid may reduce the time needed to locate consecutive points in the field. Again, the importance of this depends on the scale, the accessibility of the terrain, and the navigation technique used. Disadvantages. Because this type of design does not produce any random repetition, no unbiased estimate of the sampling variance is available. If the spatial variation in the area is pseudocyclic, the variance may be severely underestimated, thus making a false impression of accuracy. An operational disadvantage may be that the total travel distance between sample points is relatively long because of the even spreading of the points. Finally, SyS has the same disadvantage as ClS. The sample size (number of grid points that happen to fall inside the area) is generally random, which may be undesirable for budgetary or logistic reasons. The possible variation in sample size will often be larger than with ClS, and it will be more difficult to reduce this variation.
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1.4.3.7 Advanced Design-Based Strategies Apart from the basic strategies outlined in the above sections, a large number of more advanced strategies have been developed. This section outlines some of the major possibilities. 1.4.3.7.a Compound Strategies The basic strategies of the previous sections can be combined in many ways to form compound strategies. One example is given in Fig. 1.4–2F, where TsS has been applied, but with SyS in both stages instead of SRS. In this case, a square grid of 2 × 2 PUs was selected, and then a square grid of 2 × 2 points in each of the selected PUs. Notice that the total between-point distance is reduced as compared with SyS in Fig. 1.4–2E, that the risk of interference with possible cyclic variation has practically vanished, and that the operational advantage of regularity in the configuration still largely exists. Figure 1.4–2G shows another example of a compound strategy, stratified cluster sampling with four strata and two clusters in each stratum. The clusters are perpendicular transects, each with two points at a fixed distance. Notice that, due to the stratification, a more even spreading is obtained as compared with ClS in Fig. 1.4–2D, while the operational advantage of regularity still exists. See de Gruijter and Marsman (1985) for an account of perpendicular random transect sampling and an application in quality assessment of soil maps. The reason for combining two or more basic strategies is always an enhancement of advantages or mitigation of disadvantages of the basic strategies. As a final example, consider the situation in which the high precision and the operational advantage of regularity in SyS is wanted. However, it is desirable that the precision can be quantified from the data, without recourse to assumptions about the spatial variability. A possible solution is to adapt the two-stage–systematic compound strategy of Fig. 1.4–2F. In order to enable model-free variance estimation, the PUs could be selected at random instead of systematically, while maintaining grid sampling in the second stage. In that case, the variance can be estimated in the same way as with basic TsS. In devising a compound strategy, there are very often good reasons to stratify the area first, and then to decide which designs will be applied in the strata. It is not necessary to have the same type of design in each stratum. As long as the stratum means and their variances are estimated without bias, these estimates can be combined into unbiased overall mean and variance estimates using the formulas given in Section 1.4.3.3. If a variogram for the area is available, the variance of a compound strategy can be predicted, prior to sampling, using the Monte-Carlo simulation technique presented in de Gruijter (1999). In the case of stratification, this technique can be applied to each stratum separately, using different variograms if necessary. 1.4.3.7.b Spatial Systematic Strategies Most strategies discussed so far are spatial in the sense that primary sampling units and clusters are defined on the basis of geographical coordinates. Also, strata
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are usually defined that way. Given these definitions, however, the randomization restrictions do not refer to the coordinates of sample points. A category of more inherently spatial strategies exists of which the randomization restrictions explicitly make use of X and Y coordinates or distances in geographical space. Two examples are given. Figure 1.4–2H shows a systematic unaligned sample. This techniques was proposed by Quenouille (1949). The area is first divided into square strata, and one point is selected in each stratum, however, not independently. A random X coordinate is generated for each row of strata, and a random Y coordinate for each column. The sample point in a stratum is then found by combining the coordinates of its row and column. Notice in Fig. 1.4–2H the irregular, but still fairly even spread of the points. Figure 1.4–2I shows a Markov chain sample, a technique discussed by Breidt (1995). Again, notice the irregular but fairly even spread of the points. The underlying principle is that the differences between the coordinates of consecutive points are not fixed, as with systematic unaligned samples, but stochastic. These differences have a variance that is determined through a parameter φ, chosen by the user. Thus, Markov chain designs form a class in which one-per-stratum StR and systematic unaligned designs are special cases, with φ = 0 and φ = 1, respectively. The example in Fig. 1.4–2I was generated with φ = 0.75. The purpose of this type of strategy is to allow enough randomness to avoid the risk of interference with periodic variations and linear artifacts like roads, ditches, cables, and pipelines, while still maintaining as much as possible an even spread of the points over the area. 1.4.3.7.c Regression Estimators Suppose that an ancillary variable x is available that is roughly linearly related to the target variable y and known everywhere in the area, for instance, from remote sensing or a digital terrain model. Then this information can be exploited by using a regression estimator. For a simple random sample this is y–Lr = y– + b(X– − x–)
[1.4–33]
where y– is the sample mean of the target variable, x– is the sample mean of the ancillary variable, measured at the same points as y, X– is the areal mean of the ancillary variable, and b is the least squares estimate of the regression coefficient, that is, n Σ (yi − y–) (xi − x–) i=1 ______________ b= n Σ (xi − x–)2 [1.4–34] i=1
For large samples, say n > 50, the variance can be estimated by (Cochran, 1977): 1 v(y–Lr) = ______ n(n − 2)
n
Σ [(yi − y–) − b(xi − x–)]2 i=1
[1.4–35]
If the ancillary variable is not known everywhere in the area, but can be measured cheaply in a large sample, then the relationship can be used by measuring y only on a random subsample, and again applying a regression estimator. This tech-
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nique is known in the sampling literature as double sampling or two-phase sampling. Instead of the areal mean, X–, we now have the large sample mean –x ′, so that y–Lr = y– + b(x–′ − x–) with estimated variance (Cochran, 1977): 1 (x–′ − x–)2 sy2 − s2y,x _________ v(y–Lr) = s2y,x _ + _________ + —n Σ(xi − x–)2 n′
[1.4–36]
[1.4–37]
where n′ is the size of the large sample, and s2y,x is the estimated residual variance: 1 [Σ(y − y–)2 − b2Σ(x − x–)2] s2y,x = ____ i i n−2
[1.4–38]
The regression estimators given above have been generalized to stratified sampling and to the case with more than one ancillary variable. They have a great potential for natural resource inventory, but their application in practice seems underdeveloped. Brus (2000) discussed in detail the use of regression models in design-based estimation of spatial means of soil properties. 1.4.4 Model-Based Strategies In the model-based approach, the emphasis is on identifying suitable stochastic models of the spatial variation, which are then primarily used for prediction, given the sample data (see Section 1.5). The models can also be used to find efficient sampling designs, but the main focus is on model building and inference, not on sampling design. This is natural, because the approach was developed to cope with prediction problems in the mining industry, where the data had already been collected via convenience or purposive sampling (Section 1.4.1.5). Nevertheless, stochastic models of the spatial variation have been successfully used in optimizing spatial sampling configurations for model-based strategies. Three different forms can be distinguished. First, if no prior point data from the area are available, the model can be used to determine the optimal sampling grid for point kriging or block kriging, given an accuracy requirement. It has been shown (Matérn, 1986) that if the spatial variation is second-order stationary and isotropic, then equilateral triangular grids usually render the most accurate predictions, closely followed by square grids. In case of anisotropy, the grid should be stretched in the direction with the smallest variability. McBratney et al. (1981) presented a method to determine the optimal grid spacing for point kriging, given a variogram. This method uses the maximum kriging variance as quality criterion to be satisfied at lowest possible costs. In the case of a regular grid, the maximum kriging variance occurs at prediction points at the centers of grid cells. Figure 1.4–3 shows a graph of the maximum kriging variance as a function of grid spacing for square and equilateral triangular grid. A program (OSSFIM) and examples can be found in McBratney and Webster (1981). A similar method to determine the optimal grid spacing for block kriging is given by McBratney and Webster (1983). These methods are intended for large areas with a compact shape, so that boundary effects can be disregarded.
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Fig. 1.4–3. Maximum kriging variance of points as function of grid spacing for square and equilateral grids. (From McBratney & Webster, 1981.)
Second, if there is preexisting point data from the area, the model can be used to find good locations for additional sampling. To that end a contour map of the kriging variance is made; additional sampling is then projected preferably in regions with high variance, as this provides the largest reduction of uncertainty. This technique is practical and has found widespread application. It is only approximative, however, in the sense that it does not lead to an exact optimal configuration of sampling points. Third, if the area is small or irregularly shaped, then boundary effects cannot be disregarded and computationally more intensive methods are needed. Van Groenigen and Stein (1998) present such a method based on spatial simulated annealing. Using this method it is very easy to account for preexisting data points; at the start they are simply added to the new points and their locations are kept fixed during the optimization process. The method then renders an optimized configuration, as opposed to the approximative method described above. Another advantage of this method is its versatility. Different quality criteria for optimization can easily be built in. The algorithm is implemented in the SANOS software (freely available from the World Wide Web), which offers criteria for (i) optimizing variogram estimation, (ii) minimizing kriging variance, (iii) optimizing detection of
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Fig. 1.4–4. Optimization by simulated annealing in the presence of sampling constraints, boundaries and preliminary observations. (From van Groenigen et al., 2000.)
maximum contamination in multivariate studies, and (iv) minimizing the collocated ordinary cokriging variance. Figure 1.4–4 shows an example of a point configuration optimized by the van Groenigen and Stein method, taken from a multivariate soil contamination study in the Rotterdam harbor (van Groenigen et al., 2000). This was a complex study area, with many boundaries, sampling constraints (in this case buildings) and preliminary observations. The figure shows that the optimization method was able to find the empty spaces between the boundaries, buildings, and preliminary observations, and to spread new sample locations well within these spaces. The scope of model-based strategies is wider than that of design-based strategies. First, the data requirements are more relaxed. Data from convenience, purposive as well as probability sampling can be used for model-based inference, while design-based inference requires probability sampling. Second, model-based inference can be directed towards a wider class of target quantities, including local functions and functions defined by geographic neighborhood operations. An example of the latter is the total surface area of land patches consisting of a minimum number of adjacent pixels classified as suitable for a given land use. A local function which can only be predicted by a model-based strategy is, for instance, the spatial mean of a small domain (or block) with no sample points in it. The price paid by the model-based approach for its larger versatility is full dependency on a stochastic model of which the validity is more or less arguable. If the alternative of the design-based approach is not applicable, this dependency
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just has to be accepted. However, where the scope of the two approaches overlap (Section 1.4.3.1), one has the choice, as discussed in Section 1.4.2. 1.4.5 Composite Sampling Composite sampling is the technique of putting the material of individual samples together and to mix and analyze this. As only the composite samples are analyzed, the number of analyses is strongly reduced. The technique is often used in soil sampling because of its great advantage in saving laboratory costs. A vast amount of literature exists on this subject, both theoretical and applied, but a wellfounded and generally applicable methodology of composite sampling does not seem to be available. Therefore, some general guidelines are given here. The basic assumption in its most general form is that analyzing a composite gives the same result as analyzing the individual samples used to form the composite. Two special cases are mentioned here. The first case is where interest lies in the presence or absence of a qualitative variable, for instance, a species of soil microbe or a chemical substance. If the method used to determine presence or absence has a detection limit that is low enough, then a composite sample could be analyzed instead of individual samples separately. This case is often referred to as group screening or group testing. The second special case, more relevant to soil science, is where interest lies in the average value of a quantitative variable, for instance, phosphate content in the topsoil. Here the assumption is that analyzing a composite gives the same result as by averaging the values measured on individual samples. In other words, arithmetic averaging can be replaced by “physical averaging”. Of course, preassumptions are that averaging is meaningful, and that it is needed given the purpose of the project. We discuss these assumptions briefly. Averaging of Values Is Meaningful. This requires that the target variable is a quantitative property, precluding composite sampling when the target variable is measured on a nominal or an ordinal scale. Averaging of Values Is Needed. Taking a noncomposite sampling scheme as a point of departure, this presumption implies that, without compositing, the estimator of the target quantity would be a function of one or more arithmetic means of individual sample values. The simplest example of such an estimator is the unweighted sample mean, as used with simple random sampling (Section 1.4.3.2), systematic sampling (Section 1.4.3.6), Systematic unaligned sampling, or Markov chain sampling (Section 1.4.3.7). In these cases, all individual samples could in principle be put together into one composite. Other examples, involving multiple arithmetic means, are the estimators used with stratified sampling (Section 1.4.3.3), twostage sampling (Section 1.4.3.4), and cluster sampling (Section 1.4.3.5). In these cases, all individual samples belonging to the same stratum, primary sampling unit or cluster could in principle be put together. This requirement precludes compositing when the purpose is to estimate, for instance, a measure of dispersion (e.g., standard deviation or range), an extreme value, a quantile, or values at unsampled points via kriging. In these cases the estimators are not arithmetic means of sample values.
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Arithmetic Averaging Can Be Replaced by Physical Averaging. In order to make this basic assumption valid, three requirements must be met. 1. The target variable must be directly measured in the samples, or be defined as a linear transformation of one or more measured variables. Otherwise, if the target variable is a nonlinear transformation of one or more measured variables, the transformation of the mean value(s) from a composite sample is not equal to the mean of the transformed values from individual samples. Neglecting this fact can lead to an unacceptable systematic error. Examples of a target variable defined as a nonlinear transformation are: the indicator variable indicating whether or not the phosphate content in the topsoil exceeds a given threshold, available soil moisture content calculated with a nonlinear model from inputs measured at sample points, and pH as a logarithmic transformation of H activity. 2. After putting individual samples together and mixing, no physical, chemical, or biological interactions between the increments should take place that influence the value of the target variable. This precludes, for instance, compositing when the target variable depends on pH and some samples contain calcium carbonate while others don’t. Also, many soil physical measurements require soil samples to be undisturbed, which is often compromised by compositing. 3. Compositing reduces laboratory costs, but it introduces two interrelated sources of error: error by imperfect mixing of the composites and error by subsampling the mixed composite. Also, random measurement errors will cancel out less well in the case of composite sampling than with noncomposite sampling because fewer measured values are averaged. The additional error due to compositing should not enlarge the total error too much, and this puts a limit to the number of individual samples that can be bulked. The increase of the contribution of measurement error to the total error could be counteracted by taking multiple measurements from each composite while still preserving a cost advantage. Also, if mixing and subsampling are important error sources, one could make a number of smaller composites from random subsets of individual samples, instead of one large composite. Some influential theoretical publications on composite sampling are Duncan (1962), Brown and Fisher (1972), Rohde (1976), and Elder et al. (1980). Boswell et al. (1996) provide an annotated bibliography. Papers on composite soil sampling are Baker et al. (1981), Brus et al. (1999), Cameron et al. (1971), Carter and Lowe (1986), Courtin et al. (1983), Onate (1953), Ruark et al. (1982), Reed and Rigney (1947), Webster and Burgess (1984), and Williams et al. (1989).
1.4.6 Sampling in Dimensions Other than Two-Dimensional Space The above sections are focused on sampling in two-dimensional space, but in practice, very often other dimensions are involved: one-dimensional or three-dimensional space, time, or space–time. In general, the methodology presented for
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two-dimensional space can be easily transferred or adapted to these other dimensions, as is briefly outlined in the next sections. 1.4.6.1 Sampling in Three-Dimensional Space and at Depth One-dimensional spatial universes can have a horizontal or a vertical orientation. Horizontal one-dimensional universes are, for example, projected trajectories of roads or pipelines. The methodology presented for two dimensions is directly transferable to this situation. Sampling in vertical one-dimensional space (i.e., at depth) is practically always done at more than one location; hence, it is part of sampling in three-dimensional space. The universe of interest is very often embedded in three-dimensional space. Sample points would then have three coordinates (X, Y, and Z), and theoretically all three could be determined independently of each other, similarly to X and Y in two-dimensional sampling. That would typically lead to sampling at a single variable depth at each location. However, this is hardly ever done in practice. There are two main reasons to treat the Z coordinate differently, and to decompose the threedimensional sampling problem into a two-dimensional (horizontal) problem and a one-dimensional (vertical) problem. The first reason is when the target variable is defined as a function of soil properties at various depths, as is usually the case in the context of, for instance, plant growth and leaching. It is then logical to sample at these depths at each location. The second reason is when the target variable is defined at a point in three dimensions, for example, the concentration of a contaminant, but the target quantity is defined over three-dimensional space, for example, the three-dimensional spatial mean. In that case, although not a necessity, it is usually efficient to take samples at various depths at the same location. The sample is then designed and analyzed as a two-stage sample, with locations as primary units and soil profiles as secondary units (see Section 1.4.3.4). The methodology of sampling at depth is in principle the same as that for twodimensional space; however, cluster and two-stage sampling will usually be inefficient because their operational advantages in two-dimensional space do not hold for sampling at depth. The two dominant techniques in practice are purposive sampling at fixed depths and stratified random sampling, with soil horizons as strata and compositing of samples from the same horizon. 1.4.6.2 Sampling in Time Sampling in time is done, for example, to monitor the quality of groundwater at a single critical location. It is similar to sampling in 2D space in the sense that, although the practical aspects differ, the same principles, theory, and problems of choice play a role. In particular, the choice between the design-based and the model-based approach is again of paramount importance (see Section 1.4.2 for a general discussion). In the special case of sampling in time it should be added that cyclic variations (diurnal, annual) seem to be more common in time than in space. If this is true, then for sampling in time more caution is needed with model-based strategies involving systematic sampling, that is, constant time-intervals, because of a greater risk that the sampling interval interferes with some cyclic pattern of variation (see also Section 1.4.3.6). On the other hand, taking samples at constant in-
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tervals is often more convenient. Of course this advantage vanishes when a programmable automatic sampling device can be installed. For sampling in time, design-based strategies also have the advantage of greater simplicity and more robustness in the sense that the statistical inference from the sample data does not rely on the validity of a time series model. The scope of design-based strategies, however, is limited to estimation of parameters related to the universe or to subuniverses as a whole (e.g., means, totals, quantiles or distribution functions; see also Section 1.4.3.1). Random sampling is less usual in time than in space, nevertheless the same methodology as presented for two-dimensional space is applicable to sampling in time. Clearly, in two-dimensional spatial sampling, populations, domains, strata, and primary units are all areas, in temporal sampling they are periods of time. The advantages and disadvantages indicated for the various spatial sampling strategies in Section 1.4.3 hold, necessary changes being made, for sampling in time. Longterm monitoring projects often have no predetermined end, but budgets tend to be allocated annually. In that case, it is practical to take the budgetary years as strata, and to determine the sample size for each successive year from the available budget. There is one exception to the rule that the two-dimensional spatial sampling designs of Section 1.4.3 are applicable in time—the systematic unaligned type of design (Section 1.4.3.7.b) needs two dimensions. Instead, the Markov chain design (Section 1.4.3.7.b) is well suited to achieve a fairly even spread of sampling times, while still avoiding the risk of interference with (pseudo-)cyclic variations. One important purpose in temporal sampling is not covered by the designbased methods presented for two-dimensional spatial sampling: estimation or testing of a step trend. If interest lies in possible effects of a sudden natural change or certain human activities that start at a given point in time, then a relevant quantity to estimate may be the difference between the temporal means before and after the change: tc te 1 1 D = x–a − x–b = _____ xdt − ____ xdt te − tc mtc tc − tb mtb
[1.4–39]
where x–a and x–b are the temporal means after and before the change, respectively, tb and te are the beginning and the end time of the monitoring, and tc is the time of the change. This effect is simply estimated by: D$ = –x$a − x–$b
[1.4–40]
where –x$a and x–$b are estimators of the temporal means, depending on the applied type of sampling design (see Section 1.4.3). If the samples taken before and after the change are taken independently from each other, then the variance of D$ equals: V(D$) = V(x–$a) + V(x–$b)
[1.4–41]
that is, the sum of the true sampling variances of the estimated means. An estimate of V(D$) can simply be obtained by inserting the estimates of V(x–$a) and V((x–$b), as given in Section 1.4.3 for the various designs:
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v(D$) = v(x–$a) + v(x–$b)
[1.4–42]
A two-sided 100(1 − α)% confidence interval for D is given by: D$ ± t1−α/2 %&&& v(D$)
[1.4–43]
where t1−α/2 is the (1 − α/2) quantile of the Student distribution with degrees of freedom equal to the sum of the degrees of freedom on which the estimates v(x–$a) and v(x–$b) are based. The null-hypothesis of no effect (D = 0) can be tested against the alternative D ≠ 0 with the two-sided two-sample t-test. The null-hypothesis is rejected if the confidence interval of Eq. [1.4–43] does not contain zero. The simplest and most common model-based sampling strategy in time is to sample at equidistant points in time, and to use a time series model for statistical inference from the sample data. Time series analysis is a broad subject on which a vast literature exists; see Hipel and McLeod (1994) for a practical textbook.
1.4.6.3 Sampling in Space–Time 1.4.6.3.a Introduction The general remarks made in Section 1.4.6.2 about sampling in time also apply to sampling in space–time. In addition, two specific concepts need to be mentioned here: static and dynamic monitoring systems. With static systems, the samples are taken each time at the same locations. With dynamic systems, a new set of sampling locations is selected at each sampling time. The choice between a static and a dynamic system should be guided by operational as well as statistical considerations. Obviously, a static system has an operational advantage if the costs of repeated sampling at the same location are lower than for sampling at different locations with the same sample size. Common reasons for this are when finding sample points in the field is made easier by marking, or when sampling, measuring, or recording equipment is permanently installed at fixed points in the field. A possible statistical advantage of static systems is that estimation of temporal trends will often be more efficient than by dynamic systems. A statistical disadvantage is that while monitoring goes on, only the information on temporal variability increases, not that on spatial variability. The main advantage of dynamic systems is that they are much more flexible than static ones. This is because at each sampling time the system can be adapted to altered circumstances with respect to the spatial or temporal variabilities existing in the universe, the accumulating amount of information on these variabilities, the information needs, or the available budget. The importance of flexibility can hardly be overrated, especially for long-term monitoring. Rotational systems are a compromise between static and dynamic systems in that at each sampling time a fraction of the locations is rotated out of the sample and replaced by new ones. Advantages compared with static systems are greater flexibility and better spatial coverage. An advantage compared with dynamic systems is the potential for better efficiency in estimating temporal trends.
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1.4.6.3.b Design-Based Static Systems A static system can be viewed as a combination of a spatial sampling design and a temporal sampling design, such that at each sampling time all points of the spatial sample are sampled and vice versa. The inference will depend primarily on these two constituting designs. For both space and time a choice has to be made between the design-based (DB) and the model-based (MB) approach, so there are four possible combinations: DB in space and time, DB in space plus MB in time, MB in space plus DB in time, and MB in space and time. Only the first combination (i.e., purely DB systems) is dealt with in this section; the other combinations are discussed in the section on MB sampling in space–time (Section 1.4.6.3.e). The set of sampling locations for DB static systems can be selected by the designs described in Section 1.4.3 on DB sampling in space, while the set of sampling times can be selected by the methods discussed in Section 1.4.6.2 on sampling in time. Inference on (a parameter of) the Spatial Cumulative Distribution Function at any given sampling time can be done by applying the appropriate method from Section 1.4.3 to the data collected at that time. Inference on a space–time mean is done in two steps. First, for each sampling location the temporal mean is estimated from the data at that location, using the method associated with the temporal design. Then the space–time mean and its standard error is estimated with these temporal means as observations, using the method associated with the spatial design. This standard error accounts automatically for errors due to sampling in space and sampling in time, but not for possible spatial correlations between the estimated temporal means due to synchronized sampling at the locations. The same two-step procedure can be followed for estimating space–time totals. If the difference between the spatial means at two different times, D = Y–2 − – Y 1, must be estimated or tested, then a possible temporal correlation between the estimated means Y–$1 and Y–$2 should be taken into account. A simple, implicit way of doing this is similar to the two-step procedure discussed above and is based on the fact that the temporal difference between two spatial means is equal to the spatial mean of the temporal differences. So, first calculate the temporal difference at each location, then apply the appropriate method of inference from Section 1.4.3 to these differences. In the case of classical testing this procedure leads to the t-test for paired observations. The same procedure can be followed for inference on the spatial mean of any temporal trend parameter, such as the difference between the temporal means before and after some event (step trend), or the average change per unit of time (linear trend). This, again, neglects spatial correlations due to synchronized sampling. 1.4.6.3.c Design-Based Dynamic Systems With dynamic systems, at each sampling time one is free to choose a spatial sampling design from Section 1.4.3 that seems most appropriate given the circumstances at that time. Because the samples taken at different times are mutually independent, the estimated means Y–$1 and Y–$2 are likewise mutually independent. Hence, the difference D is estimated by
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D$ = Y–$2 − Y–$1
[1.4–44]
and its standard error is simply estimated from the variances: s(D$) = %&&& v(Y–$1) & +&&& v(Y–$2)
[1.4–45]
In the case of classical testing this procedure leads to the two-sample t-test. More generally, the two-step procedure for static systems (estimating first temporally, then spatially) is now reversed into estimating first spatially, then temporally. For instance, inference on a space–time mean proceeds by first estimating the spatial mean at each sampling time (using the method associated with the spatial design at that time), and then estimating the space–time mean with these means as observations (using the method associated with the temporal design). Inference on totals and trend parameters is similar. Dynamic systems can be considered as a special case of two-stage sampling in space–time, using spatial sections of the universe at given times as primary sampling units, and sampling locations as secondary units. Therefore, the methods of inference for TsS, given in Section 1.4.3.4, can be applied. In the extreme case of only one location selected at each time, this is equivalent to Simple Random Sampling in space–time. Of course, one may reverse the order of space and time in the two stages above by using time series at given locations as primary units and sampling times as secondary units. Now the set of sampling locations remains fixed through time, as with static systems, which brings the same operational advantages. The difference with static systems is that sampling at the various locations is not synchronized, so that correlation due to synchronized sampling is avoided, and that the temporal design may be adapted to local circumstances. This kind of dynamic system is attractive when considerable spatial variation between time series is known to exist, and the mentioned operational advantages are real. The inference is as for static systems. 1.4.6.3.d Design-Based Rotational Systems Rotational sampling or sampling with partial replacement represents a compromise between static and dynamic systems. The rationale is to avoid, on the one hand, the unbalance in static systems that accumulate more data only in time. On the other hand, the relative inefficiency of dynamic systems for estimating temporal trends is partially avoided because repeated measurements are made at the same locations. The principle of rotational sampling is to divide the locations of an initial spatial sample into different rotational groups, and each time to replace one group by a new set of locations. The spatial mean at any time t0, Y–0, is estimated by the composite estimator (Rao & Graham, 1964; Skalski, 1990): Y–$′ = Q(Y–$′ + Y–$ − Y–$ ) + (1 − Q)Y–$ [1.4–46] 0
−1
0,−1
−1,0
0
where Q is a weighting constant (0 < Q < 1), Y–$0 is the estimator for t0 based on the entire sample at t0, Y–$0,−1 is the estimator for t0 based on only those samples common to t0 and the previous time t−1, Y–$−1,0 is the estimator for t−1 based on only those
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samples common to t0 and t−1, and Y–$−1 ′ is the composite estimator for t−1. The difference D is estimated by D$ = Y–$0′ − Y–$−1 ′
[1.4–47]
Rao and Graham (1964), in an extensive study assuming a finite population, calculated optimum values for Q and gains in efficiency over the simple estimators of level (Y–$0) and change (D). It appeared that when the temporal correlation is high, moderate gains (15–55%) for level and large gains (100–800%) for change are achieved. Keeping the sample size constant, the gains for level decrease with increasing number of repeated measurements, while the gains for change increase. Many different strategies of rotational sampling have been developed, including improved estimation procedures. In some strategies a given set of locations is rotated back into the sample after having been rotated out for some time. See Binder and Hidiroglou (1988) for a review on rotational sampling. 1.4.6.3.e Model-Based Sampling in Space–Time Broadly speaking, four approaches can be followed for model-based sampling in space–time, using: 1. Geostatistical model of variation in space–time (an extension of spatial geostatistics with the time dimension) 2. Multivariate time-series model (a generalization of univariate time-series, the response being the time-dependent vector of observations at the sampling locations) 3. Regionalized time-series model (a spatial geostatistical model with the parameters of time-series models fitted at the sampling locations as regionalized variables) 4. Space–time Kalman filter (a data assimilation technique with optimality properties, especially useful for short time-series) All four approaches involve complicated and highly specialized statistical techniques. The reader is referred to Kyriakidis and Journel (1999) for the geostatistical approach, Hipel and McLeod (1994) for time-series analysis, and Anderson and Moore (1979) and Binder and Hidiroglou (1988) for the Kalman filter approach. The geostatistical approach is generally considered to be less promising than the regionalized time-series and the Kalman filter approach, because it is often difficult to construct a realistic model of the variation in space–time. The same applies to the multivariate time-series approach. Although much depends on what data are available, it seems that in many practical situations the regionalized time-series approach and the Kalman filter approach are the most suitable candidates for modelbased inference from sample data. 1.4.7 References Anderson, B.D.O., and J.B. Moore. 1979. Optimal filtering. Prentice-Hall, Englewood Cliffs, NJ. Baker, A.S., S. Kuo, and Y.M.Chae. 1981. Comparison of arithmatic soil pH values with the pH values of composite samples. Soil Sci. Soc. Am. J. 45:828–830.
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Binder, D.A., and M.A. Hidiroglou. 1988. Sampling in time. p. 187–211. In P.R. Krishnaiah and C.R. Rao (ed.) Handbook of statistics. Vol. 6. Elsevier Science Publishers, Amsterdam, the Netherlands. Boswell, M.T., S.D. Gore, G. Lovison, and G.P. Patil. 1996. Annotated bibliography of composite sampling. Part A. Environ. Ecol. Stat. 3:1–49. Breidt, F.J. 1995. Markov chain designs for one-per-stratum spatial sampling. Survey Methodol. 21:63–70. Brown, G.H., and N.I. Fisher. 1972. Subsampling a mixture of sampled material. Technometrics 14:663–668. Brus, D.J. 2000. Using regression models in design-based estimation of spatial means of soil properties. Eur. J. Soil Sci. 51:159–172. Brus, D.J., and J.J. de Gruijter. 1997. Random sampling or geostatistical modelling? Choosing between design-based and model-based sampling strategies for soil (with Discussion). Geoderma 80:1–59. Brus, D.J., L.E.E.M. Spätjens, and J.J. de Gruijter. 1999. A sampling scheme for estimating the mean extractable phosphorus concentration of fields for environmental regulation. Geoderma 89:129–148. Cameron, D.R., M. Nyborg, J.A. Toogood, and D.H. Laverty. 1971. Accuracy of field sampling for soil tests. Can. J. Soil Sci. 51:165–175. Carter, R.E., and L.E. Lowe. 1986. Lateral variability of forest floor properties under second-growth Douglas fir stands and the usefulness of composite sampling techniques. Can. J. For. Res. 16:1128–1132. Cochran, W.G. 1977. Sampling techniques. 3rd ed. Wiley, New York, NY. Courtin, P., M.C. Feller, and K. Klinka. 1983. Lateral variability in some properties of disturbed forest soils in southwestern British Columbia. Can. J. Soil Sci. 63:529–539. de Gruijter, J.J. 1999. Spatial sampling schemes for remote sensing. p. 211–242. In A. Stein and F.D. van der Meer (ed.) Spatial statistics for remote sensing. Kluwer Acad. Publ., Dordrecht, the Netherlands. de Gruijter, J.J., and B.A. Marsman. 1985. Transect sampling for reliable information on mapping units. p. 150–163. In D.R. Nielsen and J. Bouma (ed.) Soil spatial variability. Proc. Workshop ISSS and SSSA. Las Vegas, NV. 30 Nov.–1 Dec. 1984. Pudoc, Wageningen, the Netherlands. de Gruijter, J.J., and C.J.F. ter Braak. 1990. Model-free estimation from spatial samples: A reappraisal of classical sampling theory. Math. Geol. 22:407–415. Domburg, P., J.J. de Gruijter, and D.J. Brus. 1994. A structured approach to designing soil survey schemes with prediction of sampling error from variograms. Geoderma 62:151–164. Domburg, P., J.J. de Gruijter, and P. van Beek. 1997. Designing efficient soil survey schemes with a knowledge-based system using dynamic programming. Geoderma 75:183–201. Duncan, A.J. 1962. Bulk sampling. Problems and lines of attack. Technometrics 4:319–343. Elder, R.S., W.O. Thompson, and R.H. Myers. 1980. Properties of composite sampling procedures. Technometrics 22:179–186. Hipel, K.W., and A.I. McLeod. 1994. Time series modelling of water resources and environmental systems. Elsevier, New York, NY. Kyriakidis, P.C., and A.G. Journel. 1999. Geostatistical space–time models: A review. Math. Geol. 31:651–684. Krishnaiah, P.R., and C.R. Rao (ed.) 1988. Sampling. Handbook of statistics. Vol. 6. North-Holland, Amsterdam, the Netherlands. Matérn, B. 1986. Spatial variation. 2nd ed. Springer-Verlag, New York, NY. McBratney, A.B., and R. Webster. 1981. The design of optimal sampling schemes for local estimation and mapping of regionalized variables: II. Program and examples. Comput. Geosci. 7:335–365. McBratney, A.B., and R. Webster. 1983. How many observations are needed for regional estimation of soil properties? Soil Sci. 135:177–183. McBratney, A.B., R. Webster, and T.M. Burgess. 1981. The design of optimal sampling schemes for local estimation and mapping of regionalized variables—I. Theory and method. Comput. Geosci. 7:331–334. Onate, B.T. 1953. Some statistical aspects of the use of composites in soil sampling. Philippine Agricult. 241–257. Patil, G.P., and C.R. Rao (ed.) 1994. Environmetal statistics. Handbook of statistics. Vol. 12. North-Holland, Amsterdam, the Netherlands. Quenouille, M.H. 1949. Problems in plane sampling. Ann. Math. Stat. 20:355–375. Rao, J.N.K., and J.E. Graham. 1964. Rotation designs for sampling on repeated occasions. J. Am. Stat. Assoc. 59:492–509.
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Reed, J.F., and J.A. Rigney. 1947. Soil sampling from fields of uniform and nonuniform appearance and soil type. J. Am. Soc. Agron. 39:26–40. Rohde, C.A. 1976. Composite sampling. Biometrics 32:273–282. Ruark, G.A., D.L. Mader, and T.A. Tattar. 1982. A composite sampling technique to assess urban soils under roadside trees. J. Arboriculture 8:96–99. Särndal, C.E., B. Swensson, and J. Wretman. 1992. Model assisted survey sampling. Springer-Verlag, New York, NY. Skalski, J.R. 1990. A design for long-term status and trends monitoring. J. Environ. Manage. 30:139–144. van Groenigen, J.W., G. Pieters, and A. Stein. 2000. Optimizing spatial sampling for multivariate contamination in urban areas. Environmetrics 11:227–244. van Groenigen, J.W., and A. Stein. 1998. Constrained optimization of spatial sampling using continuous simulated annealing. J. Environ. Qual. 27:1078–1086. Webster, R., and T.M. Burgess. 1984. Sampling and bulking strategies for estimating soil properties in small regions. J. Soil Sci. 35:127–140. Webster, R., and M.A. Oliver. 1992. Sample adequately to estimate variograms of soil properties. J. Soil Sci. 43:177–192. Williams, L.R., R.W. Leggett, M.L. Espegren, and C.A. Little. 1989. Optimization of sampling for the determination of mean radium-226 concentration in surface soil. Environ. Monitor. Assess. 12:83–96. Yates, F. 1981. Sampling methods for censuses and surveys. 4th ed. Griffin, London, UK.
Published 2002
1.5 Geostatistics S. R. YATES, USDA-ARS, George E. Brown, Jr. Salinity Laboratory, Riverside, California A. W. WARRICK, University of Arizona, Tucson, Arizona
1.5.1 Introduction Scientists studying processes at the soil surface have been long aware that the spatial and temporal variations of surface features profoundly affect their study and suitability for various uses. The variability in surface features is clearly illustrated in soil and geologic maps. Although the range of variation in maps implies knowledge about the spatial distribution of an attribute, the true understanding of the spatial variability in a mapped region is very limited. For example, the boundary between soil mapping units is not generally as distinct as implied by the continuity of the boundaries and the sudden change in the feature across the boundary, nor is the placement of the boundary as exact as implied on most maps. In addition, a region within a mapping unit is not as uniform as implied by the uniformity shown on the map. For example, the U.S. Soil Survey guidelines allow up to 25% of a mapping unit to be comprised of dissimilar soils (USDA, 1983). A study of the relative variability of selected soil properties sampled from within a mapping unit (NRC, 1993) found that the bulk density and soil pH have the least variability within mapping units; water retention, total sand, total clay, and cation exchange capacity have moderate variability; and organic matter content, soil thickness, estimated hydraulic conductivity, and exchangeable Ca, K, and Mg have high variability within mapping units. Other reviews have also shown that soil properties are highly variable in space (Jury, 1985; Jury et al., 1987; Mulla & McBratney, 2000) with observed ranges of coefficients of variation of 50 to 300% for saturated hydraulic conductivity, 25 to 100% for infiltration rates, and 20 to 350% for parameters describing unsaturated hydraulic conductivity. Soil variation is often viewed as a nuisance. However, variable conditions make certain locations better suited for specific activities than other locations and can be important to an overall ecosystem. If the entire land surface had uniform properties and conditions, it would be unlikely that the current rich diversity in plants and animals would exist. Spatial and temporal variability should be viewed as a benefit to our environment. In addition, any activity that significantly reduces the natural variability has the potential to reduce diversity and should be avoided. Soil variation, and its inherent unpredictability, can be problematic, especially for those who must determine the suitability of a landscape for a particular purpose. When it is necessary to determine the behavior of an attribute across the land sur81
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face, its variability can make landscape management more complex and expensive. For this situation, variability and the underlying uncertainty are undesirable, especially when a large investment of time and resources are being considered. This has led scientists to study surface variations in a systematic manner (Journel & Huijbregts, 1978; Cressie, 1990; Webster, 1994). The original goal of the early efforts was to improve estimating the location of recoverable resources and the understanding of soil forming processes. Since computers have become readily available to soil scientists, there has been an effort to quantitatively study the spatial and temporal variation of many soil processes (Burrough et al., 1994). Geostatistical methods are also used in a variety of other disciplines, including mining, geology, and more recently the biological sciences. The widespread adoption of geographic information systems and other computerized tools in land management is generating a rapid increase and interest in the use of geostatistics. Numerous books have been published on geostatistical methods and provide a valuable resource to those considering using or learning geostatistics. These include the classic treatment by Journel and Huijbregts (1978) as well as several introductory texts (Clark, 1979; Isaaks & Srivastava, 1989; Journel, 1989; Rendu, 1981). The latest text on geostatistics (Goovaerts, 1997) includes a discussion of recent developments, and Cressie (1991) is a comprehensive and advanced textbook on spatial statistics that includes sections on geostatistics. In addition, many computer programs are available to implement geostatistical procedures. A short list of Windows-based programs is given in an Appendix, Section 1.5.3.1. 1.5.1.1 Geostatistical Investigations Geostatistics is commonly used to map and identify the spatial patterns of given attributes across a landscape. Many investigations have been conducted that show the spatial distribution of a soil property throughout a region (Burgess & Webster, 1980; Webster & Burgess, 1980; Vieira et al., 1981; Vauclin et al., 1983; Warrick et al., 1986; Yates et al., 1986; Yates & Warrick, 1987; ASCE, 1990a, b; Webster, 1991; Gallichand et al., 1992; McBratney & de Gruijter, 1992; Odeh et al., 1992a, b; Zhang et al., 1992, 1999; Juang & Lee, 2000). Geostatistics offers a variety of techniques suitable for mapping studies, including combinations of linear or nonlinear estimation with one or more spatial attributes using the kriging method, inverse distance weighting, pooling, nearest neighbors, and other methods. Geostatistics can also be used to improve the efficiency of sampling networks (Burgess et al., 1981; Russo, 1984a; Warrick & Myers, 1987). Since sampling is generally time-consuming and expensive, it is desirable to obtain the maximum information from the fewest number of samples. In addition, when other correlated spatial attributes are available, the sampling efficiency can be improved using the cokriging or pseudo-cokriging methods (Clark et al., 1989; Myers, 1991; Zhang et al., 1999). Soil variability can be problematic to landscape managers who must decide whether a landscape is suitable for a particular use, or who must decide what actions are necessary to make a landscape suitable for a particular use. Early research used kriging to obtain a map identifying the spatial distribution of an attribute which could be used to obtain the locations in need of remediation (Zirschky, 1985;
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Zirschky et al., 1985). For some situations, the uncertainty of the attribute’s spatial distribution should be considered and might be in the form of a statement of the probability that a particular event has occurred. This information can be obtained using indicator kriging (Bierkens & Burrough, 1993; Smith et al., 1993; Juang & Lee, 1998) or disjunctive kriging (Yates et al., 1986; Yates & Yates, 1988; Finke & Stein, 1994; von Steiger et al., 1996). As changes occur in the use of a landscape, it is important to be able to predict the future effects in the landscape. This can be accomplished using processoriented predictive models together with the simulation of the spatial variability of key soil attributes. In this case, a random field can be derived (Mejia & RodriguezIturbe, 1974; Freeze, 1980; Mantoglou & Wilson, 1982) that has the same statistical behavior as the attribute of interest. In addition, if samples are available, the simulation can be conditioned, meaning the simulated random field reproduces the known sample values at the appropriate locations. The random field can then be used in predictive mathematical models so that the effect of spatially correlated model parameters can be studied (Bierkens & Weerts, 1994; Wang et al., 1997).
1.5.2 Using Geostatistical Methods 1.5.2.1 Sampling One of the most important considerations involving geostatistics is sampling. For studies of spatial variability, the sampling methodology can have a profound effect on the outcome. Missing key features, small-scale variability, the effect of boundaries, numbers of observations, sample placement, and other factors can affect the results of a geostatistical investigation. The importance of this topic is reflected in that most texts on geostatistical methods include a section discussing sampling issues (Section 1.4). Often the data have been collected with minimal thought given to the analysis. Although many of the methods can be illustrated with such a data set, the analyses used will depend on the overall objective of the study and the appropriate data collected. Consider the following soil data set (Rhoades et al., 1988). It contains 898 to 901 samples of four spatially dependent random functions: the volumetric water content, θ, the bulk density, ρb, the sodium content, Na, and the boron content, B. Two additional functions were obtained by taking the natural logarithm of the soil Na and B contents. A first step in conducting a post priori analysis is to become familiar with the data set by obtaining various statistical information. Table 1.5–1 contains some of this information. It is clear from Table 1.5–1 that the Na and B contents have a high coefficient of variation, which is consistent with other reports (Jury et al., 1987). All of the measured soil properties fail the Kolmogorov normality test, but after transformation to natural logarithms, the Na and B contents appear to be close to normally distributed. This is also borne out from the values of the skewness and kurtosis. If a spatially correlated attribute is correlated (Table 1.5–2) with another attribute, it stands to reason that the second attribute will also be spatially correlated.
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Table 1.5–1. Statistical information for data set. θ
Data set statistic Number of values Minimum Maximum Mean Median First quartile Third quartile Standard deviation Coefficient of variation Skew Kurtosis Kolmogorov-Smirnov statistic†
898 0.03 0.44 0.22 0.23 0.18 0.27 0.07 0.31 −0.22 −0.08 0.06
ρb
Na
ln(Na)
B
ln(B)
901 899 899 900 900 1.19 0.36 −1.02 0.14 −1.97 1.73 1844.6 7.52 42.7 3.75 1.44 66.06 3.04 2.44 0.17 1.45 22.4 3.11 1.09 0.09 1.40 5.73 1.75 0.47 −0.76 1.49 71.31 4.27 2.87 1.05 0.07 129.5 1.60 3.89 1.18 0.05 1.96 0.53 1.59 6.93 −0.55 5.74 0.003 4.48 0.30 0.67 53.1 −0.743 28.09 −0.60 0.097 0.306 0.049 0.28 0.063
† Kolmogorov-Smirnov critical value is 0.045 (α = 0.05) and 0.054 (α = 0.01).
The ln(B) is highly correlated with ln(Na). Since the Na content can be determined quickly and inexpensively, its values can be used to improve the estimates of the B content at a potentially lower cost. This example will be further illustrated below. One objective of a geostatistical analysis is to make inferences about the random function Z(x) given the available data, Z(x1), Z(x2),... Z(xn). These data are considered a single realization of the random field, Z(x). Ideally, multiple realizations of Z(x) would be available for the analysis. Since this is not generally possible, additional assumptions or hypotheses about the statistical nature of the underlying random fields are required to implement the geostatistical methods. In addition, this often implies that certain types of information are required to utilize the method. The most important of these assumptions are the hypotheses related to the behavior of the underlying random field, called stationarity, which concerns the behavior of the unknown realizations. This will be discussed further below. 1.5.2.2 Spatial Autocorrelation Geostatistical analysis can be used to determine if a spatial correlation exists. This is accomplished by one of several functions that describe spatial correlation: the autocorrelation function, the covariance function, or the variogram (e.g., semivariogram). This information can be used to determine if further geostatistical analysis is justified or whether classical statistical methods (those requiring independent samples) can be utilized. An example of the absence and presence of spatial correlation is shown in Fig. 1.5–1, where a single realization of a random field is mapped. It is clear from Table 1.5–2. Correlation coefficient, r.
θ ρb Na ln(Na) B ln(B)
θ
ρb
Na
ln(Na)
B
ln(B)
1 1 1 1 1 1
−0.49 −0.14 0.67 0.57 0.76
0.06 −0.25 0.58 0.83
0.14 −0.02 0.53
0.09 −0.16
0.19
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Fig. 1.5–1A that the spatial distribution has little or no correlation. This figure was prepared by simulating a random field (see Section 1.5.3.2 in the Appendix) with a zero correlation length. In Fig. 1.5–1B, the random field shows some continuity and behaves in a more coherent manner; hence, it is more spatially correlated.
Fig. 1.5–1. Simulated random functions: (A) the correlation scale is 0; (B) the correlation scale is 20. For both figures the mean, µ, is 5 and the variance, σ2, is 56.3. (µln(z) = 1.020, σln(z) = 1.086).
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1.5.2.2.a Variogram The most common function used in geostatistical studies to characterize spatial correlation is the variogram. The variogram, γ(h), is defined as one-half the variance of the difference between the sample values for all points separated by the distance h, γ(h) = (1/2)var[Z(x) − Z(x + h)] = (1/2)E{[Z(x) − Z(x + h)]2}
[1.5–1]
where var[ ] and E{ } indicate the variance and expected value, respectively. This function has advantages over the autocorrelation function and covariance function in that it can be used even if the variance is undefined (e.g., γ increases with h indefinitely). Therefore, it can be used for most situations. The estimator for the variogram, γ*(h), is calculated from the data using γ*(h) = [1/2N(h)]
N(h)
Σ [Z(xi) − Z(xi + h)]2
i=1
[1.5–2]
where N(h) is the total number of pairs of observations that are separated by a distance h. The variogram can be strongly affected by outliers in the data, and more robust estimators for the variogram have been proposed (Cressie & Horton, 1987; Omre, 1984). The variogram model is a mathematical description of the relationship between the variance and the distance, h, that separates observations (Fig 1.5–2). There are many equations that can be used to model a sample variogram. Shown in Fig. 1.5–2 are four widely used equations. Note that γ increases with the separation distance, h. A small, non-zero, value may exist for γ at h ≈ 0. This limiting value for γ is called the nugget variance and results from various sources of unexplained error, such as measurement error or variability that occurs at scales too small to characterize given the available data. Many variogram models have a limiting value for the variance at large h. This limiting value is called the sill and, in theory, should be approximately equal to the variance of the data. The range parameter, a, is the value for h where the sill occurs. In Fig. 1.5–2, only the spherical model has a precisely defined range. For a variogram that approaches the sill asymptotically, the range parameter is often defined as the point where the variogram is 95% of the asymptotic sill value. For the exponential model, the range parameter, b, would be approximately 1/3 of the range of the spherical model, for the Gaussian model b would be 1/(31/2). The linear model does not have a sill or a range and, therefore, the variance is undefined. Although many equations produce a good fit to a sample variogram, only certain models meet the requirement to be conditionally positive definite. Many geostatistical techniques require the inversion of matrices, and this ensures that the estimation variance will be positive and avoids singular matrices. The variogram is constructed by (i) calculating the squared difference (Eq. [1.5–2]) for each pair of observations in the data set [Z(xj), Z(xk)], (ii) determining the distance between each pair of observations (xj − xk), and (iii) averaging the squared differences for those pairs of observations with the same separation distance. If the observations are evenly spaced on a transect, the separation distances,
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Fig. 1.5–2. Variogram models and defining parameters.
hi, are multiples of the smallest distance separating observations, ∆ (i.e., adjacent observations). This leads to the following separation distances: h1 = ∆, h2 = 2∆,..., hn = n∆. When locations of the observations are placed on an irregular pattern, it is possible to construct the variogram by assigning pairs of observations to the appropriate lag interval. This is accomplished through a binning procedure. Bins are created with interval centers at distances hi = i∆, where ∆ is the basic lag interval. Each pair of observations is assigned to the ith bin if i∆ − δ/2 ≤ (xj − xk) < i∆ + δ/2, where (xj − xk) is the separation distance between the pair Z(xj) and Z(xk), and δ is the width of the lag interval. Usually δ is equal to ∆ so that all pairs are assigned to a bin. However, it is possible to have overlapping bins (i.e., δ > ∆), or gaps between bins (i.e., δ < ∆). There are several important considerations when calculating the variogram. First, it is often observed that spurious results occur when the separation distance becomes too large. This is usually because fewer pairs of observations exist for large separations due to finite boundaries. Another consideration is the number and width of lag intervals. The width of the lag interval can affect the sample variogram due to the number of samples and variation in the separation distances that fall into a particular lag interval. This is demonstrated in Fig. 1.5–3 where a histogram of the distances separating the data pairs is given for two cases. The information shown
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in Fig. 1.5–3A uses data from Fig. 1.5–1A, which were uniformly spaced. The center of the lag window was 3 (e.g., between 1.5 and 4.5). The reported lag distance is really the average of the pairs of observations that fall into the interval, in this case the average lagged distance, h2, is 3.4 (σ = 0.81), and γ2 = 0.41 (N(h2) = 53,938). It is apparent that there is a wide range in actual separation distances contributing to the average values. This leads to some imprecision in the sample variance for this lag interval compared with a situation where every sample pair within a lag interval had the same separation distance, which can occur for data collected on a transect with fixed distances between observations. In Fig. 1.5–3B, the observations were randomly distributed throughout the region and the center of the lag window was 300 m (e.g., between 150 and 450 m). The calculated variogram values are: h2 = 325.3 (σ = 89.5), and γ2 = 15.62 [N(h2) = 6369]. For this example, a large number
Fig. 1.5–3. Histogram of distances separating sample pairs and their frequency: (A) derived from simulated data shown in Fig. 1.5–1A; (B) derived from the data from Rhoades et al. (1988). In both cases, the basic lag separation distance represents the same fraction of total field size.
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of observation pairs are used to calculate a variogram value. In many cases this is not possible, but it is generally accepted that 30 or more pairs are sufficient to produce a reasonable sample variogram. The width of the lag interval can affect the variance. This is shown in Fig. 1.5–4, where γ and h are plotted as a function of interval width. This figure shows the effect of increasing the width of lagged interval on the calculated values of γ and h. For this example, the variogram is very stable as the width of the lag interval increases. However, the value for h is affected by the lag width for sizes greater than approximately 250. Overall, the averaging has little effect on the reported variogram, for this data set. Shown in Fig. 1.5–5 is the calculated variogram for the correlated and uncorrelated simulated fields shown in Fig. 1.5–1. These variograms were determined using a lag window of 3. This figure shows that the underlying spatial structure used in creating the simulated random fields is reproduced when calculating the variogram; for example, an uncorrelated random field produces a nugget variance that is nearly equal to the theoretical variance. When a large number of data are available for calculating the sample variogram, the relationship between γ and h usually produces a recognizable curve. This is shown in Fig. 1.5–6 for the Na and B contents, after a natural logarithmic transformation has been applied to the data. In both cases, the sample variogram has a non-zero nugget and a sill. When modeling the sample variogram, a choice must be made whether to use a spherical, exponential model or some other model. For this data set, either one will produce a good representation of the sample variogram. For illustrative purposes, the exponential model was selected and the model parameters are shown in the figure. The variogram provides information about the spatial correlation of the soil attributes. First, as the minimum sampling distance decreases, there is a tendency
Fig. 1.5–4. Relationship between γ, h, and the width of the lag interval.
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Fig. 1.5–5. Variogram models for correlated and uncorrelated attributes (see Fig. 1.5–1).
for the nugget variance to decrease. For this data set, the minimum distance separating samples is 48.53 m (Samples: 466, 467), and the nugget variance for both attributes is approximately 0.74, or about 52% [ln(B)] and 30% [ln(Na)] of the total variance. This indicates that some information at the smaller scales has been lost.
Fig. 1.5–6. Global variograms for ln(Na) and ln(B). The model parameters are for an exponential variogram model.
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The sample variogram also provides information about the distances that the samples are correlated. For both attributes, the range of the variograms is approximately 900 m (i.e., a = 3b). 1.5.2.2.b Autocovariance and Autocorrelation The spatial correlation can also be expressed in terms of the autocovariance or autocorrelation function. Under appropriate stationarity conditions, the covariance, C(h), and its estimator, C*(h), can be used in geostatistical analyses and are defined as C(h) = E[Z(x)Z(x +h)] −µ2 N(h)
C(h) = [1/N(h)]
Σ [Z(xi)Z(xi +h)] −µ2
i=1
[1.5–3]
Relationships between the autocovariance, C(h), the autocorrelation, r(h), and variogram functions are γ(h) = C(0) − C(h) γ(h → ∞) = var[Z(x)] = C(h → 0) r(h) = [C(h)]/[C(0)] = 1 − {[γ(h)]/[C(0)]}
[1.5–4]
where 0 ≤ r(h) ≤ 1. 1.5.2.2.c Integral Scale For a correlated random field, a measure of the distance for which the attribute is spatially correlated is the integral scale R (Russo & Bresler, 1981), defined as ∞
R = [2m0 hr(h)dh]1/2
[1.5–5]
where r(h) is the autocorrelation function. Using the integral scale and transforming the variogram model shown in Fig. 1.5–5 to an autocorrelation function, yields an integral scale of 18.3 and is close to the theoretical value of 20. 1.5.2.2.d Directional Variograms When a variogram depends only on the separation distance, it is said to be isotropic (e.g., no directional dependence). Often there is a preferred orientation for a spatially dependent attribute with higher spatial correlation in a certain direction. When this occurs, the directional dependence can be included in the variogram model. For many situations, the anisotropic variogram can be transformed into an isotropic variogram by a linear transformation. For this case, the anisotropy is said to be geometric. For example, consider Fig. 1.5–7A where two variograms, γ1 and γ2, respectively, are shown for two angular directions, α1 and α2. If the range pa-
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Fig. 1.5–7. Schematic of geometric anisotropy: (A) two variograms with ranges a1 and a2; (B) an ellipse is formed using several range values.
rameters, a1 and a2 (or some other suitable choice), for several angular directions describe an ellipse when plotted in a two-dimensional diagram, the spatial correlation function exhibits geometric anisotropy, as shown in Fig. 1.5–7B. This type of anisotropy provides important information about the spatial attribute, and obtaining estimates is especially easy, as will be discussed below. The variogram in an arbitrary direction, α, is calculated in the same manner as a global variogram, but only the sample pairs within both the distance class and the angular class are used to obtain the sample variance. In this case, a value for γ(hi) is obtained from the samples that fall within the segment, dhi and dα (Fig. 1.5–8). 1.5.2.2.e Stationarity A stationary random function, Z(x), has the same joint probability distribution for all locations xi and xi + h, where h is an arbitrary separation vector. Since insufficient information is available to determine the joint probability distribution,
Fig. 1.5–8. Defining elements of a directional variogram.
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hypotheses concerning the stationarity of the random field are needed to augment the incomplete knowledge about its behavior. Various assumptions concerning the stationarity can be introduced to relax the requirement that the samples be collected from multiple realizations, which may be impossible if the sample location is altered while collecting the sample. In each case, it is assumed that the joint distributions do not depend on the location (i.e., xi and xj) of the samples and allows each separation to be viewed as a separate realization, enabling statistical inferences to be made (Journel & Huijbregts, 1978). Second-Order Stationarity. For this hypothesis, the expected value of Z(x) and the covariance function exists and both are independent of spatial location, therefore E[Z(xi)] = µ C(h) = E[Z(xi + h)Z(xi)] − µ2
[1.5–6]
for all i. A consequence of second-order stationarity is that the variogram must also exist. Since the variogram can be written in terms of the covariance function, the variance (i.e., the covariance at zero lag) must be finite. Intrinsic Hypothesis. The intrinsic hypothesis is the working hypothesis for the most common geostatistical methods. It is a weaker assumption than secondorder stationarity and requires that the expected value of Z(x) exists and be independent of position. Unlike second-order stationarity, the intrinsic hypothesis requires a finite variance that is independent of position for the increments [Z(x + h) − Z(x)] for all h (Journel & Huijbregts, 1978, p. 33). Therefore, the variogram may exist even if the variance is undefined. For this reason, the variogram is the preferred measure of spatial correlation. Drift. Some soil attributes are known to change in a systematic manner across a landscape. For these situations, the expected value may depend on location {i.e., E[Z(x)] = µz(x)}, and therefore, may not be stationary. The terms trend and drift are often used to describe this type of behavior. Example: Stationary Random Fields. Determining if a random function is nonstationary and accounting for the nonstationary behavior if present can produce many difficulties. The scale at which you observe a phenomenon may affect one’s interpretation of drift. For example, if samples of a random function are obtained and plotted as in Fig. 1.5–9A, it appears that there is a trend in the attribute’s response from one end of the sampling sequence to the other. However, if additional samples of the same random function are obtained (Fig. 1.5–9B), it becomes clear that the random function is stationary, in that the mean and variance are independent of position along the transect. This illustrates one of the difficulties in addressing nonstationary trends in data. The interpretation may depend on the amount of data collected, the scale of the observations and one’s knowledge of the characteristics of the data and underlying statistical distribution. Determining if a data set exhibits stationary or nonstationary behavior is problematic. A simple approach to check for nonstationary behavior is to calculate the
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Fig. 1.5–9. Simulated stationary random function: (A) a portion of the entire transect shown in (B).
variogram and determine if, at large separation distances, it increases more rapidly than a γ = |h|2 curve. This often indicates nonstationary conditions (Journel & Huijbregts, 1978, p. 39–40). Other approaches are available to address nonstationary trends in a data set (Webster & Burgess, 1980; Cressie, 1991). 1.5.2.3 Geostatistics and Estimation Kriging is a procedure that produces a best linear unbiased estimate of an attribute at an unmeasured site, requiring only a known spatial correlation function, for example, the variogram. Geostatistics is a powerful tool for investigating various soil, hydrologic, agricultural, and biological problems. One of the most common uses is for mapping the spatial distribution of an attribute. Using kriging, the spatial pattern of an attribute can be obtained throughout a region. Although the process produces numeric values, the attribute is usually presented as contours or surface diagrams that allow immediate visualization of the spatial patterns. In addition to the estimates, the kriging method provides an estimation variance, giving an indication of the quality of the estimate. There are several common methods for producing kriging equations. Of these, ordinary kriging is most common and has been adopted in several commercial and public-domain programs (see Burrough et al., 1994; Section 1.5.3.1). Since geostatistical methods are fairly general in nature, it is possible to use surrogate information in the estimation process. This can be in the form of qualitative information or quantitative information that can be transformed into nonparametric data. There is generally a benefit to the use of additional information provided that it is correlated to the attribute of primary interest. It can be especially advantageous if samples of the auxiliary attribute are easy and inexpensive to obtain. When two correlated variables are used in the estimation process, it is called multivariate or cokriging. In terms of accuracy, a map produced using cokriging is generally superior to that obtained from kriging because the estimates used to create the map incorporate spatial information from each attribute through their interattribute correlation. Kriging has been used by many researchers studying various spatially related phenomena. Vauclin et al. (1983) used kriging and cokriging to predict available water content, pF = 2.5 values, and sand content in soils. McBratney and Webster
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(1983) applied cokriging to study silt and sand content. Carr and Myers (1984) used cokriging to analyze satellite data. Yates and Warrick (1987) and Mulla (1988) estimated soil water content using kriging and cokriging that included the bare-soil surface temperature and the sand content as auxiliary attributes. Using soil map delineations, Stein et al. (1988) showed that cokriging resulted in an average increase in precision of about 10% in maps of 30-yr average moisture deficit. Gallichand et al. (1991, 1992) used kriging to make estimates of the hydraulic conductivity to improve the design of an irrigation drainage system in a 35 000-ha region of the Nile Delta. Nash et al. (1992) used cokriging to estimate the vegetative cover in an arid range land. Zhang et al. (1992) showed that estimates of soil texture could be improved using cokriging and associated spectral properties, and also used cokriging to study solute concentration in a field soil (Zhang et al., 1997). Vaughan et al. (1995) studied the effect of water content on estimates of soil salinity using the cokriging method. Wen and Kung (1993) and Li and Yeh (1999) used cokriging in the estimation of certain hydraulic properties for saturated and unsaturated conditions. Zhang et al. (1999) used cokriging to study the distribution of nitrate in soil and used data at shallow depths to improve the estimates of the soil nitrate in deeper soils layers. Clearly, kriging is a powerful tool and can be useful in a wide variety of investigations. 1.5.2.3.a Ordinary Kriging Ordinary kriging is commonly used because the mean value of a random function need not be known or assumed. Consider a situation where the Na content in a field soil is sampled at many locations. Generally, the areal extent of each sample (e.g., size) will be very small compared with the size of the field, and it is appropriate to consider each sample a point in two-dimensional space. The underlying distribution of Na in the field is the random function, Z(x), and the n samples of Na [e.g., Z(x1), Z(x2),..., Z(xn)] are random variables that are used in the kriging process. Kriging Estimator. The kriging estimator, Z*(x), provides an estimate of an attribute at a location x0 that was not sampled. The estimator is written as a linear combination of the observed values, Z(xi), that is, n
Z*(x0) =
Σ λiZ(xi)
i=1
[1.5–7]
where the observation at each location is weighted by λi. The value for λi depends on its proximity and orientation to x0 and to the other sample locations, xi. Since by definition, n
E[Z*(x0)] =
Σ λ E[Z(xi)] and E[Z(xi)] = µ i=1 i
[1.5–8]
the estimates will be unbiased {i.e., E[Z(x) − Z*(x0)] = 0}, provided that the weight factors satisfy n
Σ λi = 1
i=1
where µ is the mean value of Z(x).
[1.5–9]
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The kriging method produces a best-linear, unbiased estimator. This means that the variance of errors has been minimized. This can be accomplished using a Lagrangian multiplier, β (Journel & Huijbregts, 1978), to solve the kriging system of equations. From a practical viewpoint, this doesn’t introduce any difficulties, since the Lagrangian multiplier is obtained along with the weight factors by solving the following matrix equation n
Σ λ γ(xk − xi) + β = γ (xk − x0) i=1 i
[1.5–10]
where k = 1,2,3,..., n. If four sample values are used to estimate Z(x0), Eq. [1.5–10] could be written in the expanded notation γ11 γ21 U γ31 γ41 I1
γ12 γ22 γ32 γ42 1
γ13 γ23 γ23 γ23 1
γ14 γ24 γ24 γ24 1
1 1 1 1 0
[ O
λ1 λ2 U λ3 [ λ4 IβO
=
γ10 γ20 U γ30 [ γ40 I1O
[1.511]
where γ12 = γ(x1 − x2). Kriging Variance. Once the weight factors and Lagrangian multiplier are known, the estimation variance can be obtained from σk2 = β +
n
Σ λiγ(xi − x0) i=1
[1.5–12]
Block Kriging. Obtaining estimates at a point in space is useful for making maps of a spatially variable attribute. However, at times it is required to obtain an estimate of the average value of a spatial attribute over a region, which can be accomplished using block kriging. Block estimates can be obtained even if the samples represent points in space. To provide a block estimate, the average autocorrelation function is used on the right-hand side of the kriging equations shown in Eq. [1.5–10], and no modification is necessary to the left-hand side of Eq. [1.5–10]. Therefore, γi0, in Eq. [1.5–11] is replaced with an average spatial autocorrelation function appropriate for the area to be estimated, which is –γ (xk,v) = (1/v)∫vγ(xk − x→)dx→
[1.5–13]
where the block, v, may be either a transect, area, or volume. Estimating a spatially dependent attribute over a region causes a reduction in the estimation variance. This is due to the averaging of the small-scale fluctuations of the random function over the region. Incorporating Eq. [1.5–12] and [1.5–13] into the kriging equations produces the following block-averaged kriging variance σk2 = β +
n
Σ λi–γ (xi − v) − –γ (v,v)
i=1
[1.5–14]
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where –γ (v,v) is the variance within the block, v, and is defined as → → → → –γ (v,v) = (1/v2)∫v∫vγ(ξ − ξ ′)dξ dξ ′ →
[1.5–15]
→
where the vectors ξ and ξ ′ each independently describe the domain v. 1.5.2.3.a Validation The accuracy of kriging depends on the quality of the variogram. The shape of the variogram near the origin is very important (i.e., this is the low variance region). However, many times there are few data pairs in this part of the variogram, so fitting a model becomes problematic. One way to assess the variogram model is to use the jack-knifing technique (see Vauclin et al., 1983; Russo, 1984a) with cross-validation. In principle, an acceptable variogram model is one for which n
Σ [Z(xi) − Z*(xi)]/(σkn) ≈ 0 i=1
[1.5–16]
and n
Σ [Z(xi) − Z*(xi)]2/(σk2n) ≈ 1 i=1
[1.5–17]
where Z*(xi) is the estimated value at a sampling location, xi, with the known value [i.e., Z(xi)] at this location not used in the kriging process. 1.5.2.3.c Examples Isotropic Case, Kriging Matrix. The steps involved in kriging are straightforward. To obtain an estimate, a matrix equation must be formed and solved. The matrix equation is Γiiλ = γi0
[1.5–18]
where Γii is a square matrix with n + 1 rows and columns, n is the number of samples used to obtain the estimate, and λ and γi0 are column matrices with n + 1 rows. The matrix λ includes the unknown weight factors and Lagrangian multiplier, β, and are found by solving the matrix equation. For example, to obtain an estimate at the location, x0, (e.g., at x = 500, y = 500) using the four nearest sampling points (see Fig. 1.5–10), a matrix equation shown in Eq. [1.5–11] needs to be formed. There are three parts to forming Γii: the variance between sampling points, the Lagrangian multiplier, and the unbiasedness condition. The unbiasedness condition for kriging requires that the sum of the weight factors is unity (see Eq. [1.5– 9]). This is incorporated into the matrix equation by setting the first four columns of row 5 to 1; and the last row of γi0 to 1. The presence of the Lagrangian multiplier in Eq. [1.5–10] is incorporated into Γii by setting the first four rows of column 5 to 1; and the last row to 0. The next step is to account for the spatial correlation between sample pairs (i.e., Γii). This is accomplished by placing 0 in the main diagonal of the Γii matrix;
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Fig. 1.5–10. Distances separating sampling locations and the position of the estimation site.
since the variance for each sample with itself (e.g., [Zi − Zi]2 = 0, in Eq. [1.5–2], since h ≡ 0) is zero. The remaining off-diagonal terms in Γii are obtained by determining the separation distances between pairs of sampling points and evaluating the variogram model. Given the variogram model: γ(h) = 0.744 + 1.71[1 − exp(−h/288.2)] for h > 0 γ(h) = 0 for h ≡ 0
[1.5–19]
the distance between points 1 and 2 is 202 m, and the value for the sample variogram is 1.606. This value is placed in row 1, column 2 of Γii. Obtaining this information for the other pairs of samples fills out the n × n off-diagonal terms of the matrix. Next, the spatial correlation between each sample and the estimation site is incorporated into γi0. The distances from the estimation site (500,500) to the sampling locations (1, 2, 3 and 4), respectively, are 208, 6, 169, and 202 m. Using these h values, the γ(xi − x0) are calculated and placed in the column vector γi0 on the righthand side of Eq. [1.5–18]. For example, the distance between sample 1 and the estimation site is 208 m, and the variance is 1.623. This value is placed into the first row of γi0. Once the remaining variances have been calculated, the matrix equation is complete. The final step is to solve the matrix equation for the weight factors and Lagrangian multiplier λ = γi0 Γii−1
[1.5–20]
which allows the calculation of the kriging estimate and kriging variance. This
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Fig. 1.5–11. Directional variograms for ln(Na). An exponential model was used in (B) with variogram parameters: C0 =.745, C1 = 1.73, a = 570.4, Raniso = 2.36, α = 76°.
0 1.606 1.697 1.753 1 1.606 0 1.489 1.611 1 U 1.697 1.489 0 1.967 1 1.753 1.611 1.967 0 1 I 1 1 1 1 0
[ O
λ1 1.623 λ2 0.779 U λ3 [ = U 1.503 λ4 1.609 IµO I 1
[ O
yields the kriging estimate = 1.41, the kriging variance = 1.25, λ1 = 0.107, λ2 = 0.600, λ3 = 0.154, λ4 = 0.140, and µ = 0.154. It can be seen that the weight factors are not dependent solely on distance. For example, points 1 and 4 are approximately at the same distance from the estimation site, yet the weight for point 1 is 24% less than point 4. This is due, in part, to the placement of point 2, which is closer to the estimation site and located in the same direction as point 1. This configuration tends to deemphasize the importance of point 1 to the estimated value and results in a lower weight factor. Creating Maps Using Kriging. A map showing the spatial distribution of a given attribute, for example, ln(Na) content, and the estimation variance can be produced very easily and efficiently using kriging. A critical step in the process is determining the variogram. To describe the spatial distribution of ln(Na), an anisotropic variogram is needed with an anisotropy ratio of 2.4 with a principal direction at 76°. This is shown in Fig. 1.5–11 and includes two directional variograms oriented in 0° and 90° directions (Part A) and the directional ellipse (Part B). The ellipse provides evidence that the anisotropy can be adequately described using a geometric transformation. The length of each ray inside the ellipse is equal to the range of the directional variogram. The directional variograms were calculated in four directions using an angular window of ± 15° (see dα in Fig. 1.5–8). The basic lag interval was 150 m and the lag width (see dhi in Fig. 1.5–8) was 150 m. The dashed lines in Fig. 1.5–11B are the major and minor axes. The ratio of the major to minor axis length is the anisotropy ratio, Raniso. Having obtained the variogram, all the necessary information is available to produce a map.
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Fig. 1.5–12. Map of the ln(Na) using the ordinary kriging method and an anisotropic variogram. (A) Estimates of ln(Na); (B) the kriging variance.
Figure 1.5–12 shows the map of kriging estimates for the ln(Na) using the anisotropic variogram shown in Fig. 1.5–11. Also shown are the kriging error estimates (Fig. 1.5–12B). The anisotropic behavior is readily apparent in Fig. 1.5–12 by the elongated patterns in the α = 76° direction. Kriging estimates can also be obtained using a global (e.g., isotropic) variogram. This case is shown in Fig. 1.5–13. Comparison between Fig. 1.5–12 and Fig. 1.5–13 show the effects of anisotropy on the estimated values of ln(Na). 1.5.2.3.d Multivariate Kriging Many times a sample collection includes more than one spatially dependent property, for example, soil water content and bulk density, some of which may have spatial cross-correlation. When more than one cross-correlated property is available, the added information can be used to improve the estimates or as a method
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Fig. 1.5–13. Map of the ln(Na) using the ordinary kriging method and an isotropic variogram. (A) Estimates of ln(Na); (B) the kriging variance.
for increasing the sampling efficiency. There are several methods that allow for the use of multiple random functions, including cokriging (Journel & Huijbregts, 1978; Myers, 1982), pseudocokriging (Clark et al., 1989; Myers, 1991), and kriging with regression (Odeh et al., 1995), among others. Of these, cokriging is the most commonly used method and pseudocokriging offers the potential to dramatically increase sampling efficiency. To use the cokriging method, both the autocorrelation and the cross-correlation functions must be determined. Generally, this would involve determining the variogram for each random function together with the cross-variograms, for each pair of random variables. Using cokriging with two correlated attributes improves the estimate at an unsampled location. This is accomplished by the inclusion of the auxiliary variable’s spatial correlation and the intervariable correlation. In general, there are no re-
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strictions to the number of attributes used in the estimation process or number of observations of each attribute. The sampling efficiency can be improved, however, if one of the attributes has fewer observations, or if many of the sampling locations are noncoincident. The decision about the number of observations of each variables is an important consideration of the sampling design. Commonly there are fewer observations available for the attribute of primary interest, especially if sampling this attribute is costly or difficult. If an easy-to-sample, correlated auxiliary attribute can be identified, then the accuracy or the sampling efficiency can be improved by using cokriging. This situation is often called an undersampled problem. Another example occurs when two attributes are highly correlated and are of equal interest and equal sampling difficulty. An improvement in the overall estimation of both attributes is possible provided that most of the sample locations for each attribute are independent of each other. This has the effect of increasing the sampling density in the field. As the cross-correlation increases, sampling each attribute at different locations becomes nearly the same as sampling them at the same location. Cross Variogram. The spatial correlation between variables can be determined by using a modified variogram model (e.g., Eq. [1.5–1] and [1.5–2]). This modified function is called the cross-variogram, γ1,2(h), and describes the relationship between separation distance and the spatial correlation between the two data attributes. The cross-variogram is defined as γ1,2(h) = (1/2)var[Z1(x) − Z2(x + h)] = (1/2)E{[Z1(x) − Z1(x + h)][Z2(x) − Z2(x + h)]}
[1.5–21]
where Z1(x) and Z2(x) indicate attributes 1 and 2, respectively (e.g., the Na content and B concentration). The sample cross-variogram is calculated using γ*1,2(h) = {1/[2N(h)]}
n
Σ {[Z1(xi) − Z1(xi + h)][Z2(xi) − Z2(xi + h)]} i=1
[1.5–22]
Positive Definite Condition. It is imperative that the variance of a linear combination of random variables be positive. For cokriging, this requires a condition be placed on the cross-variogram function (Myers, 1984) in addition to the requirement that the variogram be conditionally positive definite. To ensure that the variance will remain positive, the cross-variogram function must satisfy the CauchySchwartz inequality, i.e., |γi,j| ≤ pγi,i(h)γj,j(h)
[1.5–23]
These conditions are sufficient to ensure that the variance will be positive. Cross-Variogram Example. Shown in Fig. 1.5–14 is a global (isotropic) cross-variogram for ln(Na) and ln(B) which was calculated using Eq. [1.5–22]. The solid line is an exponential model with nugget, sill, and range of 0.49, 1.06, and 215
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Fig. 1.5–14. Isotropic cross-variogram between ln(Na) and ln(B).
m, respectively. The cross-correlation between attributes becomes small (e.g., high variance) as the separation distance approaches 600 m. Utilizing the sample variograms for ln(Na) and ln(B) together with the cross-variogram, cokriging can be implemented. Pseudo-Cross-Variograms. One disadvantage to the standard approach for calculating the cross-variogram is that only the locations common to both attributes can be used. Estimating the cross-variograms requires a large number of locations where data is collected for both variables, which is often not the case in practice and may not be the most efficient method of sampling when a significant correlation between variables exists. For these situations, it may be better, in terms of sampling efficiency, to sample only one variable at a location since an estimate of the other variable is possible using various correlation relationships. Clark et al. (1989) and Myers (1991) presented one such approach to determine the cross-correlation structure from variables that are not sampled at the same location. This is a variation of cokriging that uses what is termed a pseudocross-variogram function, g(h). The advantage of using this approach is that it is not necessary to sample both attributes at the same sampling locations to obtain a pseudo-cross-variogram. However, some knowledge of the covariance is necessary, and may require that some samples be collected for all variables at the same spatial locations. This work is an important step in improving the efficiency of the cokriging method. The pseudo-cross-variogram, gj,k(h), is defined as gj,k(h) = gj,k(−h) = (1/2)var[Zj(x) − Zk(x + h)] = g*j,k(h) − (1/2)[µ2j − 2µjµk + µk2]
[1.5–24]
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Table 1.5–3. Pseudo-cross-variograms. No. of pairs
Distance
48 1584 2028 4300 7789 9652 10618 10631
106.9 316.6 210.0 680.2 1180.8 1922.4 3038.5 2542.2
g12*(h)
g21*(h)†
5.95 6.04 5.68 6.14 6.31 6.60 6.88 6.80
6.00 6.04 5.69 6.14 6.31 6.60 6.88 6.80
m
† Variable 1 and 2 are, respectively, ln(Na) and ln(B); µ1 = 3.04; µ2 = 0.172; σ11 = 2.57; σ12 = 1.58; σ22 = 1.39.
and the estimator is g*1,2(h) = (1/2)E[Z1(xi) − Z2(xi + h)]2 N(h)
= {1/[2N(h)]} i=1 Σ [Z1(xi) − Z2(xi + h)]2
[1.5–25]
for j = 1 and k = 2. In general, gj,k(h) is a variance that depends only on the separation distance, h, but is not a variogram function unless j = k (Myers, 1991). If Z(x) is written as Y(x) + µz, where Y(x) is second-order stationary and µz is the mean of Z(x), and if the pseudo-cross-variograms gj,k and gk,j are symmetric, gj,k(h) can be rewritten in terms of a variogram, variances, and a covariance (Myers, 1991) γj,k(h) = gj,k(h) − [(1/2)(σj − 2σj,k + σk)] gj,k(h) = g*j,k(h) − (1/2)[µ2j − 2µjµk + µk2]
[1.5–26]
where σj and σk are the variances, σj,k is the covariance between attributes j and k, and γj,k(h) is the cross-variogram. Equation [1.5–26] provides a relatively simple method to obtain the cross-variogram and does not suffer from the requirement that many of the samples of each attribute be at the same locations. In fact, to obtain the cross-variogram with this method, if σj,k is known, none of the attributes have to be sampled at the same location. If σj,k is unknown, some samples of both attributes will be needed. Comparing Cross- and Pseudo-Cross-Variograms. Table 1.5–3 shows the calculated pseudo-cross variances for the ln(Na) and ln(B) data sets. The means, variances, and cross-correlation were obtained from Tables 1.5–1 and 1.5–2. It is clear that the symmetry relationship holds—compare g*12(h) with g*21(h)—and that for small separation distances, the cross-variogram and pseudo-cross-variogram are similar (see Fig. 1.5–15). At larger separation distances, however, the pseudocross-variogram deviates from the sample cross-variogram. Kriging Estimator. The multivariate (cokriging) estimator is defined as a linear combination of the available spatial attributes. For two random functions, Z1(x) and Z2(x), the estimator for variable Z1(x), is
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Fig. 1.5–15. Comparison of a traditional cross-variogram with one obtained using the pseudo-cross-variogram. n
Z1*(x0) =
Σ i=1
λiZ1(xi) +
m
Σ k=1
ωkZ2(xk)
[1.5–27]
where n and m are the number of samples of Z1(x) and Z2(x), respectively, and λi and ωk are the associated weight factors for Z1(x) and Z2(x), respectively. The cokriging estimator is required to be unbiased and have minimum estimation variance. This produces the following constraints on the weight factors n
Σ λi = 1
i=1
m
and
Σ ωk = 0
k=1
[1.5–28]
The Lagrangian multipliers, β1 and β2, occur from the minimization process and are caused by the presence of the constraints on the weights. This cokriging system of equations (see Journel & Huijbregts, 1978, p. 325) is n
m
Σ λiγ1,1(xI − xi) + k=1 Σ ωkγ1,2(xI − xk) + β1 = γ1,1(xI − x0)
i=1 n
m
Σ λiγ2,1(xK − xi) + k=1 Σ ωkγ2,2(xK − xk) + β2 = γ2,1(xK − x0) i=1
[1.5–29]
where I = 1,2,..., n; K = 1,2,..., m; the γ1,1 and γ2,2 are the variogram functions for Z1(x) and Z2(x), respectively; and γ1,2 and γ2,1 are the cross-variogram between Z1(x) and Z2(x). The cokriging equations can also be written in terms of the pseudo-crossvariogram (Zhang et al., 1999): n
m
n
m
Σ λiγ1,1(xI − xi) + k=1 Σ ωkg1,2(xI − xk) + β1 = γ1,1(xI − x0) i=1 Σ λig2,1(xK − xi) + k=1 Σ ωkγ2,2(xK − xk) + β2 = g2,1(xK − x0) i=1
[1.5–30]
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Kriging Estimation Variance. A value for the estimation variance can be obtained from σ21,ck = β1 +
n
m
Σ λiγ1,1(xi − x0) + k=1 Σ ωkγ1,2(xk − x0) i=1
[1.5–31]
Example: Comparing Kriging and Cokriging. Shown in Fig. 1.5–16 are maps of the ln(B) from kriging (Part A) and cokriging (Part B). The 900 available points of ln(B) and ln(Na) were used to develop these maps. It is clear that very little additional spatial information is obtained from cokriging when all the attributes are sampled at coincident locations. When fewer ln(B) samples are available (Fig. 1.5–17), in this case 94, significant additional information is offered by cokriging by utilizing the large num-
Fig. 1.5–16. (A) Ordinary kriging with 900 ln(B) samples; (B) ordinary cokriging with all available ln(B) and ln(Na).
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ber of ln(Na) content samples (i.e., 899). It is evident that the spatial resolution of ln(B) from cokriging approaches the resolution achieved when all the ln(B) were used (see Fig. 1.5–16A). In this case, it is highly advantageous to use the ln(Na) as surrogate information, when few ln(B) samples are available. 1.5.2.4 Geostatistics and Uncertainty There are several geostatistical approaches that provide information that can be used to characterize spatial uncertainty. This type of information can be especially valuable for management decision making. These methods include: indicator, probability, and disjunctive kriging. Geostatistics has long been used as a tool for management decision making. The earliest applications include economic optimization of mining operations. Other examples include Russo (1984b), who used
Fig. 1.5–17. (A) Ordinary kriging with 94 ln(B) samples; (B) ordinary cokriging with 94 ln(B) and all available ln(Na) samples (undersampled case).
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geostatistics to aid in managing the salinity of a heterogeneous field; Zirschky (1985) and Zirschky et al. (1985), who used geostatistics for determining reclamation strategies for the cleanup of hazardous waste sites. This was accomplished using kriging to obtain estimates of the contaminant concentration in the affected areas and led to locations where reclamation was necessary. Yates et al. (1986) illustrated the use of disjunctive kriging as a tool for decision making with regard to management of salinity in agricultural fields. The method produced the probability that the electrical conductivity would exceed a prescribed threshold level. Yates and Yates (1988) used disjunctive kriging to provide probabilistic information on septic tank set-back distances to ensure that drinking water would remain free from virus contamination. This approach involved coupling geostatistics and a simple predictive model. Gallichand et al. (1991, 1992) used kriging to provide information to aid in the design of drains for agricultural fields. They estimated the hydraulic conductivity and produced contour maps that enabled the design of a drainage system that reduced the area where the height of the water table would exceed a prescribed limit, providing a more efficient drainage system. An advantage to the use of indicator kriging (i.e., nonparametric) or disjunctive kriging (i.e., nonlinear) geostatistical methods is that the conditional probability that a measured indicator variable is above some prescribed tolerance level can be easily obtained (Journel & Huijbregts, 1978; Yates et al., 1986; Yates 1986). Given a known threshold value where the attribute of concern becomes a problem (or nuisance) and the ability to estimate the conditional probability, geostatistics offers a means for providing valuable information that can be used when making management decisions. For example, soil B content can have a deleterious effect on plant growth. Maas (1986) gave 1 to 2 g m−3 as the concentration range where soil B begins to affect health of moderately sensitive plants. For tolerant plants, the threshold level may be as high as 15 g m−3. For illustrative purposes, a value of 1 g m−3 can be used as the threshold (i.e., cutoff) level, zc, in a geostatistical investigation of the management of B content, where the conditional probability is obtained. In addition to the threshold level, a criterion is needed to determine if management action is needed. Since it is undesirable for the plants to be negatively affected by soil B, a suitable criterion would be that the probability of exceeding 1 g m−3 is low (i.e., perhaps 30–40%). When the estimated conditional probability that the soil B content is greater than 50%, indicating an unacceptable risk, the site is in need of moderate corrective action. If the probability is high (i.e., perhaps > 75%), intensive corrective action is needed. 1.5.2.4.a Indicator and Probability Kriging Indicator kriging has found considerable use in the mining applications (Journel, 1984; Lemmer, 1984; Sullivan, 1984) and is finding utility in the soil and environmental sciences (Bierkens & Burrough, 1993; Bierkens & Weerts, 1994). Probability kriging (Sullivan, 1984; Juang & Lee, 2000) is an extension of the indicator kriging method where more of the experimental information is used in the estimation process. Probability kriging is a form of cokriging that uses both the indicator function and the cumulative distribution function. Probability kriging has
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been shown to provide a lower estimation variance, less conditional bias, and better resolution for cases where only a few data points surround an estimation site. A discussion comparing indicator, probability, and disjunctive kriging is given by Journel (1984). Indicator and probability kriging are based on nonparametric information, unlike ordinary kriging. Indicator Function. Indicator kriging makes use of an indicator function that depends on the magnitude of a sample compared with a specified cutoff (or threshold) level, zc, that is, i[Z(x); zc] =
1 90
if Z(x) ≥ zc otherwise
[1.5–32]
where i[ ] is the indicator function. The indicator kriging estimator is the expected value of the indicator function. Therefore, the kriging estimator is the conditional probability that the indicator function is greater than or equal to the specified threshold level. Once the data is transformed, unbiased estimates can be obtained using ordinary kriging. An interesting aspect to indicator and probability kriging is that the variogram function must be determined for each cutoff level since the function i[Z(x); zc] differs for each zc, and thus, the weight factors will also depend on the cutoff level, zc. Although the use of indicator or probability kriging is straightforward, difficulties may be encountered when attempting to estimate the cumulative probability distribution (Journel, 1984; Sullivan, 1984). The probability distribution is obtained by performing indicator kriging with several cutoff levels. Since a spatial estimate of the probability for a particular cutoff level is obtained independently of the other cutoff levels, the estimated cumulative distribution function may decrease for increasing zc, have negative values or values greater than unity. These difficulties are called order-relation problems and can be eliminated by obtaining the cumulative distribution function in a nonindependent manner (Sullivan, 1984) or by using variograms that have relational consistency (Journel & Posa, 1990). Example Using Indicator Kriging. As with all kriging methods, a variogram is needed to implement indicator kriging. The first step in the procedure is to transform the ln(B) data into indicator values using Eq. [1.5–32]. Once completed, every data value that is greater than 1 g m−3 is transformed into a 1 and all other values to 0. Then, the variogram is calculated using standard methods and is shown in Fig. 1.5–18. The sample variogram model is also shown in this figure. Using the indicator variogram and the ordinary kriging method, a map of the conditional probability that the B content is greater than 1 g m−3 can be obtained (Plate 1.5–1). For this example, all the data are presented as natural logarithms. The areas where the probabilities of exceeding ln(1 g m−3) are high can be readily distinguished from other areas in the field. In this map, all values of probability less than 0.5 are green and indicate areas suitable for growing crops. Those areas in yellow and red are in need of corrective action. If only a portion of the domain can receive corrective action, say due to expense, the procedure shows where the corrective measures are most needed.
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Fig. 1.5–18. Indicator variogram for ln(B) when Zc = 1 g m−3.
1.5.2.4.b Disjunctive Kriging Disjunctive kriging (DK) is a nonlinear geostatistical method that is generally more accurate that linear kriging methods. Disjunctive kriging is a generalized form of kriging and is identical to the linear or lognormal kriging estimators if a random variable is univariate and bivariate normally or lognormally distributed and has a known mean value (Rendu, 1980; Journel & Huijbregts, 1978). There are several requirements needed before using the disjunctive kriging method on a random function with an arbitrary distribution. First, a transform exists, ϕ(Y), that is unique, invertible, and produces a function with Gaussian distribution. Second, the random function produced by the transformation, Y(x), is univariate normal with mean zero and unit variance and pairs of samples form a bivariate normal distribution. Further, the random function must be second-order stationary, since the variance must exist. Kriging Estimator and Variance. The disjunctive kriging estimator is written as a series of nonlinear functions where each function depends on only one transformed sample value, Y(xi) (see Journel & Huijbregts, 1978; Yates et al., 1986) n
Z*DK(x0) =
n
∞
Σ fi[Y(xi)] = Σ Σ fikHk[Y(xi)] i=1 i=1 k=0
[1.5–33]
where n is the number of samples, and fi[Y(xi)] is an unspecified functional relationship that will be expressed as Hermitian polynomials Hk. This can be done since any distribution can be written as an infinite series of Hermite polynomials where, fik, are the Hermitian coefficients and depend both on the distribution of the data and the order of the Hermitian polynomial (i.e., i and k). The fik are analogous to the weight factors, λi, used in ordinary kriging. The transform function, ϕ[Y(x)], is introduced to determine the unknown fi values. This transform can also be written in terms of Hermitian polynomials ϕ[Y(x0)] = Z(x0) =
∞
Σ CkHk[Y(x0)]
k=0
[1.5–34]
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Values for Ck, which define the transform, are found using the properties of orthogonality and can be obtained using n
Ck = (1/%2π &&k!) Σ ωiϕ(vi)Hk(vi)exp(−vi2/2) i=1
[1.5–35]
where vi and ωi are abscissas and weight factors, respectively, for Hermitian integration (Abramowitz & Stegun, 1970). This results in the following unbiased estimator Z*DK(x0) =
K
n
Σ Ck i=1 Σ bikHk[Y(xi)] k=0
[1.5–36]
where, for practical reasons, the series of Hermitian polynomials is truncated to K terms. The bik are the kriging weights that depend on the number of samples used in the estimation process and the order of the Hermitian polynomial, k. The kriging weights, bik, are found by solving the linear kriging equation for each k n
Σ bik(rij)k = (r0j)k i=1
j = 1, 2, 3,...., n
[1.5–37]
The disjunctive kriging is similar to the linear kriging system with the exception that the spatial autocorrelation function is raised to the kth power, and since the mean value for the transformed data is known (i.e., by definition is zero), a Lagrangian multiplier is not necessary and a form of simple kriging is used. The principal difference between ordinary and disjunctive kriging is that the kriging system must be evaluated k times for each estimate. The disjunctive kriging estimation variance is σ2DK =
K
n
Σ k!Ck2[1 − i=1 Σ bik (r0j)k] k=1
[1.5–38]
1.5.2.4.c Conditional Probability The DK method provides a systematic method for determining the conditional probability of exceeding a threshold value. Given a threshold value zc, which is transformed to yc, the conditional probability is P*DK = 1 − G(yc) + g(yc)
K
Σ (1/k!)Hk−1(yc)HK* [Y(x0)]
k=1
[1.5–39]
where G(yc) and g(yc) are the standard normal cumulative and probability density functions, respectively. The estimated conditional probability density function, P*DK(x0), can be found by taking the derivative of Eq. [1.5–39] with respect to yc. Example: Managing Soil Boron Content. The first step in the DK method is to create the transform relationship, ϕ[Y(x)]. This can be accomplished by sorting the original data set from smallest to largest in magnitude, assigning a probability to each data point (e.g., probability ≈ (i − 1/2)/n, where i is the sample num-
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Fig. 1.5–19. Variogram of the transformed ln(B) content, e.g., ϕ[ln(B)].
ber and n is the total number of samples) and calculating the corresponding Y(xi) for each Z(xi) by inverting the probability function. After the Y(xi) has been determined, a method is needed to interpolate between the sample values, since it is unlikely that the abscissas for Hermitian integration will be numerically equivalent to any datum. This can be accomplished by fitting a polynomial to the Y(xi) vs. Z(xi) values, interpolation, or some other similar method. The coefficients, Ck, which define the transform relationship (Eq. [1.5–34]) are given in Eq. [1.5–35] and can be obtained by using orthogonality and Hermitian integration, where the functional values for u(xi) at the abscissa points are found using interpolation. Once the Ck values are obtained, they can be tested by comparing the mean and variance of the samples with those obtained using the coefficients Ck, and the transformed data and ϕ[Y(x)] curve can be compared (see Yates et al., 1986). To create a map showing the spatial distribution of the ln(B) using disjunctive kriging, the variogram of the transformed variable, Yz(h), is required and is shown in Fig. 1.5–19. When using DK, the variogram function must be transformed into the autocorrelation function, where it is assumed that the underlying random function is second-order stationary (i.e., the variance exists). This function can be defined using the variogram from r(h) = 1 − γ(h)/γ(∞). For this example, the sample variogram is modeled using an exponential variogram model with a nugget value of 0.42, a sill of 0.91, and a range parameter of 275 m. Once the variogram and the Hermitian coefficients, Ck, have been obtained, maps can be created using Eq. [1.5–36] and [1.5–37]. Although disjunctive kriging produces a nonlinear estimator that is generally more accurate than linear estimators (Journel & Huijbregts, 1978), maps created using either DK or ordinary kriging will appear very similar and differences are minor or indistinguishable. Numerically, however, it can be shown that DK improves the estimation of ln(B) compared with ordinary kriging. To show this, 100 samples were randomly removed from the data set and used for testing the accuracy of the
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Table 1.5–4. Accuracy of the disjunctive kriging and ordinary methods. Average disjunctive kriging estimate Average ordinary kriging estimate Average disjunctive kriging variance Average ordinary kriging variance Sum of squares error, disjunctive kriging Sum of squares error, ordinary kriging
2.67 2.54 12.39 12.75 1552.5 1615.5
estimation procedure. This leaves 800 samples for use in estimating the variogram and for obtaining estimates. A comparison is made between the measured ln(B) and the estimated values for the 100 samples removed from the data set. The known sample value is not used in the estimation procedure. This information allows the performance of each kriging method to be determined, namely the average estimated value, the average estimation variance, and the sum of squares error (Table 1.5–4). Table 1.5–4 shows that the average kriging variance is slightly less for disjunctive kriging and that there is a reduction in the sum of squares error of 3.9%. Even though this represents a slight improvement in the estimates, it would be difficult to see any difference between the two methods when comparing their contour maps. Plate 1.5–2 shows the spatial distribution (Part A) and disjunctive kriging variance (Part B) for the ln(B) in a portion of the field. It has long been known that the kriging variance at a sample location is zero. For this data set, none of the locations on the grid of estimates coincided exactly with a sampling location. However, the estimates located near a sampling site have lower estimation variance than other sites. These can be identified in Plate 1.5–2 by the small circular areas. If the site was coincident to a sampling location, the centers would have an estimation variance that is zero. Example: Obtaining the Conditional Probability. An advantage of the DK method is that it allows the conditional probability that the attribute is greater than a prescribed threshold level to be determined in a straight forward and robust manner. Since it is undesirable for some plants, such as broccoli (Brassica oleracea var. botrytis L.), carrot [Daucus carota subsp. sativus (Hoffm.) Arcang.], and potato (Solanum tuberosum L.), to be exposed to soil B at concentrations exceeding 1 g m−3 (Maas, 1986), this threshold value can be used to determine if the exposure in this part of the field will be significantly higher. That is, the conditional probability that the soil boron content will exceed 1 g m−3 is greater than 75% can be determined. At locations where this occurs, plant growth is likely to be influenced unless some corrective action is taken. Conducting the kriging analysis using a maximum of 10 nearest neighbors provides estimates of the conditional probability that the B concentration exceeded 1 g m−3 (Plate 1.5–3). This figure shows a similar spatial behavior as the ln(B) in Plate 1.5–2A, with the highest probability zones located at large x (i.e., x ≈ 3500, y ≈ 1000). Although contour maps enable visualization of the spatial patterns of the ln(B) or conditional probability, it is unlikely that every point in the field could be managed independently of adjacent points. It may be better to obtain a field-averaged
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conditional probability using block kriging and for areas of the fields that can be managed independently from other areas. This would provide the appropriate information to land-use managers and be of help in choosing an appropriate action. An example of this approach is given by Yates et al. (1986). Using this approach, the costs of reclamation can be minimized if locations with high B levels receive more rigorous treatment than other areas. This method has the potential to provide a more efficient approach to managing agricultural lands. The ln(B) samples that are the basis of the analysis are optimally weighted using the underlying spatial correlation to produce the conditional probability. If other approaches are used to obtain the conditional probability, only the samples that are collected in a particular field can be used. This could increase the sampling cost, if precise estimates are required or if a large number of fields are evaluated. 1.5.3 Appendices 1.5.3.1 Selected Windows-Based Geostatistical Software 1.5.3.1.a GS+ for Windows Version 5.1 Gamma Design Software, P.O. Box 201, Plainwell, MI 49060-0201. Features: General statistics, Moran’s I Analysis, anisotropic variograms, automatic or interactive fit, fractal analysis, cross validation, block and punctual kriging, inverse distance weighting, cokriging, 2-D and 3-D mapping, frequency plots. 1.5.3.1.b S+SpatialStats MathSoft Inc., 101 Main Street, Cambridge, MA 02142. Features: Anisotropic variograms and correlograms, block and punctual kriging, ordinary and universal kriging, parametric and nonparametric trend surfaces, simulation of spatial random processes, local intensity estimation, spatial regression models, 2-D and 3-D mapping. Includes an interface with ARC GIS Software. 1.5.3.1.c gstat 2.1.0 Department of Physical Geography, Utrecht University, P.O. Box 80.115, 3508 TC Utrecht, The Netherlands. Features: Variogram, cross-variogram, covariogram or cross-covariogram calculation, nested variogram models, simple, ordinary and indicator kriging, multivariable universal kriging with arbitrary base functions, stratified universal kriging, conditional and unconditional Gaussian simulation, indicator simulation, block predictions for rectangular blocks or arbitrarily shaped blocks. 1.5.3.1.d Geostokos Toolkit Geostokos Toolkit, Geostokos Ltd., 36 Baker St., London W1M 1DG, England. Features: Scattergrams, distribution fitting, calculation and modeling of variograms, inverse distance estimation and trend surfaces, ordinary, universal, indicator and lognormal kriging, point and block (e.g., panels and 2-D polygonal areas) kriging, point and block kriging in 3-D, cokriging and simulation routines, cross-validation techniques. 1.5.3.1e Ecosse Software Geostokos (Ecosse) Limited, Alloa Business Centre, Alloa, Scotland, FK10 3SA, Central Scotland; Ecosse North America, LLC, Attn: William V. Harper, 7550 Satterfield Road, Columbus, OH, 43235-1819, USA.
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Features: Suitable for highly skewed sample values and/or trends in values, basic statistical analysis; histograms, scattergrams, nearest neighbor analyses; calculation and interactive modeling of semivariograms; cross-validation of semivariogram model and kriging process; ordinary, universal, lognormal, rank, and indicator kriging; conditional simulation for normal and lognormal values; principal components analysis, sequential gaussian simulation. 1.5.3.1.f Surfer Golden Software Inc., P.O. Box 281, Golden, CO 80402. Features: Maps of proportional symbol and categorical data, anisotropic variograms, detrending using polynomial regression, inverse distance, point or block ordinary kriging, minimum curvature, polynomial regression, triangulation, shepard’s method, radial basis functions, grid math to perform mathematic operations between grid files, advanced Macro language, 2-D, 3-D, relief or image outputs, Contouring and 3D graphics. 1.5.3.1.g Isatis 3.1 Ecole des Mines de Paris, Centre de Geostatistique, 38 Av. Franklin Roosevelt, 77210 Avon, France. Features: General statistics, variogram, covariance, correlogram, madogram, rodogram, variogram clouds and surface analysis, interactive or automatic variogram modeling, automatic drift identification using intrinsic random functions of order k, polygons of influence, moving average, moving median and moving dip average, inverse distance, least squares polynomial fit, discrete splines, point and block kriging and cokriging, ordinary and universal kriging, drift estimation, indicator kriging, disjunctive kriging, factorial kriging, several conditional and nonconditional simulations non conditional simulations, image processing, 2-D and 3-D output. Marketed by Geovariances and authorized resellers. 1.5.3.1.h Vesper 1.0 Australian Centre for Precision Agriculture, McMillan Building A05, The University of Sydney, NSW 2006, Australia. Features: Local & global variogram calculations, manual or automatic parameter fit, point and block kriging, ordinary, simple and lognormal kriging, quadratic trend, boundaries can be defined for the output of the estimates. 1.5.3.1.i GSLib/WINGSLib GSLib, Department of Applied Earth Sciences, Stanford University, Stanford, CA 94305, USA; Statios LLC, 1345 Rhode Island Street, San Francisco, CA 94107, USA. Features: General statistics, declustering function, variogram and cross-variogram calculations in 2-D and 3-D, simple and ordinary kriging, kriging with trend and drift, cokriging, indicator kriging and cokriging, point or block kriging and cokriging, indicator principal components simulation, LU simulation, simulated annealing simulation, sequential gaussian simulation, sequential indicator simulation, turning bands simulation, cross validation. 1.5.3.2 Simulating Random Fields Figure 1.5–1 was generated using a geostatistical simulation algorithm that simulates a random field with know statistical properties. Using this approach values of a random function can be obtained at any (x,y) coordinate using N
f(x,y) = (2/N)1/2
Σ cos[Wm(x sinγm + y cosγm) + ϕm]
m=1
[1.5–40]
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where f(x,y) is a stochastic process (e.g., N[0, 1; λv]) and x and y denote the horizontal distances of the two-dimensional field, N is the number of terms and is generally greater than 50, γm and ϕm are chosen from a uniform distribution over the range 0 to 2π or U[0, 2π], and Wm is given by Wm = λv
2 1/2 1 ________ −1 9 — 1 − G(Wm) A
[1.5–41]
where the correlation length = 1/λv and G(x) is chosen from a uniform distribution, U[0,1]. The procedure is described in detail by Freeze (1980) and El-Kadi (1986). Other methods can be used to generate random fields such as the turning bands method (Mantoglou & Wilson, 1982).
1.5.4 References Abramowitz, M., and A. Stegun. 1970. Handbook of mathematical functions. Dover Publications, New York, NY. American Society of Civil Engineers. 1990a. Review of geostatistics in geohydrology 1. Basic concepts. ASCE Task Committee on Geostatistical Techniques in Geohydrology. J. Hydrol. Eng. 116:612–632. American Society of Civil Engineers. 1990b. Review of geostatistics in geohydrology 2. Applications. ASCE Task Committee on Geostatistical Techniques in Geohydrology. J. Hydrol. Eng. 116:633–658. Bierkens, M.F.P., and P.A. Burrough. 1993. The indicator approach to categorical soil data. 2. Application to mapping and land use suitability analysis. J. Soil Sci. 44:369–381. Bierkens, M.F.P., and H.J.T. Weerts. 1994. Application of indicator simulation to modelling the lithological properties of a complex confining layer. Geoderma 62:265–284. Burgess, T.M., and R. Webster. 1980. Optimal interpolation and isarithmic mapping of soil properties. II. Block kriging. J. Soil Sci. 31:333–341. Burgess, T.M., R. Webster, and A.B. McBratney. 1981. Optimal interpolation and isarithmic mapping of soil properties. IV. Sampling strategy. J. Soil Sci. 31:643–659. Burrough, P., J. Bouma, and S.R. Yates. 1994. State of the art in pedometrics. Geoderma 62:311–326. Carr, J.R., and D.E. Myers. 1984. Applications of the theory of regionalized random variables to the spatial analysis of landsat data. p. 55–61. In IEEE Computer Society Proc. Pecora 9 Spatial Techn. for Remote Sensing Today and Tomorrow. IEEE Computer Society Press, Silver Spring, MD. Clark, I. 1979. Practical geostatistics. Applied Science Publishers, London,UK. Clark, I., K.L. Basinger, and W.V. Harper. 1989. MUCK—A novel approach to cokriging. p. 473–494. In B.E. Buxton et al. (ed.) Proc. Conf. Geostat. Sensitivity and Uncertainty Methods for Groundwater Flow and Radionuclide Transport Modeling. Battelle Press, Columbus, OH. Cressie, N. 1990. The origins of kriging. Math. Geol. 22:239–252. Cressie, N. 1991. Statistics for spatial data. Wiley Interscience, New York, NY. Cressie, N., and R. Horton. 1987. A robust-resistant spatial analysis of soil water infiltration. Water Resour. Res. 23:911–917. El-Kadi, A.I. 1986. A computer program for generating two-dimensional fields of autocorrelated parameters. Ground Water 24:663–667. Finke, P.A., and A. Stein. 1994. Application of disjunctive cokriging to compare fertilizer scenarios on a field scale. Geoderma 62:247–263. Freeze, R.A. 1980. A stochastic-conceptual analysis of rainfall-runoff processes on a hillslope. Water Resour. Res. 16:391–408. Gallichand, J., D. Marcotte, and S.O. Prasher. 1992. Including uncertainty of hydraulic conductivity into drainage design. J. Irrig. Drain. Div. Am. Soc. Civil Eng. 118:744–755. Gallichand, J., S.O. Prasher, R.S. Broughton, and D. Marcotte. 1991. Kriging of hydraulic conductivity for subsurface drainage design. J. Irrig. Drain. Div. Am. Soc. Civil Eng. 117:667–681. Goovaerts, P. 1997. Geostatistics for natural resources evaluation. Oxford Univ. Press, New York, NY. Isaaks, E.H., and R.M. Srivastava. 1989. Applied geostatistics. Oxford University Press, New York, NY. Journel, A.G. 1984. The place of non-parametric geostatistics. p. 307–335. In G. Verly et al. (ed.) Geostatistics for natural resources characterization. Part 1. D. Reidel Publ. Co., Hingham, MA.
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Journel, A.G. 1989. Fundamentals of geostatistics in five lessons. Am. Geophys. Union, Washington, DC. Journel, A.G., and Ch.J. Huijbregts. 1978. Mining geostatistics. Academic Press, New York, NY. Journel, A.G., and D. Posa. 1990. Characteristic behavior and order relations for indicator variograms. Math. Geol. 22:1011–1025. Juang, K.W., and D.Y. Lee. 1998. Simple indicator kriging for estimating the probability of incorrectly delineating hazardous areas in a contaminated site. Environ. Sci. Technol. 32:2487–2493. Juang, K.W., and D.Y. Lee. 2000. Comparison of three nonparametric kriging methods for delineating heavy-metal contaminated soils. J. Environ. Qual. 29:197–205. Jury, W.A. 1985. Spatial variability of soil physical parameters in solute migration: A critical literature review. Electric Power Research Institute, Interim Report: EA-4228, Project: 2485-6, Sept. 1985. Jury, W.A., D. Russo, G. Sposito, and H. Elabd. 1987. The spatial variability of water and solute transport properties in unsaturated soil. Hilgardia 55:1–32. Lemmer, I.C. 1984. Estimating local recoverable reserves via indicator kriging. p. 349–364. In G. Verly et al. (ed.) Geostatistics for natural resources characterization. Part 1. D. Reidel Publ. Co., Hingham, MA. Li, B.L., and T.C.J. Yeh. 1999. Cokriging estimation of the conductivity field under variably saturated flow conditions. Water Resour. Res. 35:3663–3674. Maas, E.V. 1986. Salt tolerance of plants. Appl. Agric. Res. 1:12–26. Mantoglou, A., and J.L. Wilson. 1982. The turning bands method for simulation of random fields using line generation by spectral method. Water Resour. Res. 18:1379–1394. McBratney, A.B., and J.J. de Gruijter. 1992. A continuum approach to soil classification by modified fuzzy k-means with extra-grades. J. Soil Sci. 43:159–175. McBratney, A.B., and R. Webster. 1983. Optimal interpolation and isarithmic mapping of soil properties. V. Co-regionalization and multiple sampling strategy. J. Soil Sci. 34:137–162. Mejia, J.M., and I. Rodriguez-Iturbe. 1974. On the synthesis of random field sampling from the spectrum: An application to the generation of hydrologic spatial processes. Water Resour. Res. 10:705–711. Mulla, D.J. 1988. Estimating spatial patterns in water content, matric suction, and hydraulic conductivity. Soil Sci. Soc. Am. J. 52:1547–1553. Mulla, D.J., and A.B. McBratney. 2000. Soil spatial variability. p. A321–A352. In M.E. Sumner (ed.) Handbook of soil science. CRC Press, Boca Raton, FL. Myers, D.E. 1982. Matrix formation of co-kriging. Math. Geol. 14:249–257. Myers, D.E. 1984. Cokriging—New developments. In G. Verly et al. (ed.) Geostatistics for natural resources characterization. Part 1. D. Reidel Publ. Co., Hingham, MA. Myers, D.E. 1991. Pseudo-cross variograms, positive-definiteness, and co-kriging. Math Geol. 23:805–816. Nash, M.S., A. Toorman, P.J. Wierenga, and A. Gutjahr. 1992. Estimation of vegetative cover in an arid rangeland based on soil-moisture using cokriging. Soil Sci. 154:25–36. NRC. 1993. Ground water vulnerability assessment: Predicting relative contamination potential under conditions of uncertainty. National Academy Press, Washington, DC. Odeh, I.O.A., A.B. McBratney, and D.J. Chittleborough. 1992a. Soil pattern recognition with fuzzy-cmeans: Applications to classification and soil-landform interrelationships. Soil Sci. Soc. Am. J. 56:505–516. Odeh, I.O.A., A.B. McBratney, and D.J. Chittleborough. 1992b. Fuzzy-c-means and kriging for mapping soil as a continuous system. Soil Sci. Soc. Am. J. 56:1848–1854. Odeh, I.O.A., A.B. McBratney, and D.J. Chittleborough. 1995. Further results on prediction of soil properties from terrain attributes—Heterotopic cokriging and regression-kriging. Geoderma 67:215–226. Omre, H. 1984. The variogram and its estimation. p.107–125. In G. Verly et al. (ed.) Geostatistics for natural resources characterization. Part 1. D. Reidel Publ. Co., Hingham, MA. Rendu, J.M. 1980. Disjunctive kriging: A simplified theory. Math. Geol. 12:306–321. Rendu, J.M. 1981. An introduction to geostatistical methods of mineral evaluation. 2nd ed. South African Institute of Mining and Metallurgy, Johannesburg, South Africa. Rhoades, J.D, D.L. Corwin, and P.J. Shouse. 1988. Use of instrumental and computer assisted techniques to assess soil salinity. p. 50–103. In Symp. Proc.of Int’l Symp. in Solonetz Soils, Osijek, Yugoslavia. Russo, D. 1984a. Design of an optimal sampling network for estimating the variogram. Soil Sci. Soc. Am. J. 48:708–716.
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Russo, D. 1984b. A geostatistical approach to solute transport in heterogeneous fields and its applications to salinity management. Water Resour. Res. 20:1260–1270. Russo, D., and E. Bresler. 1981. Soil hydraulic properties as stochastic processes: I. An analysis of field spatial variability. Soil Sci. Soc. Am. J. 45:682–687. Smith J., J.J. Halvorson, and R.I. Papendick 1993. Using multiple-variable indicator kriging for evaluating soil quality. Soil Sci. Soc. Am. J. 57:743–749. Stein, A., M. Hoogerwerf, and J. Bouma. 1988. Use of soil-map delineations to improve(co)-kriging of point data on moisture deficits. Geoderma 43:163–177. Sullivan, J. 1984. Conditional recovery estimation through probability kriging—Theory and practice. p. 365–384. In G. Verly et al. (ed.) Geostatistics for natural resources characterization. Part 1. D. Reidel Publ. Co., Hingham, MA. USDA. 1983. National soils handbook. Soil Conservation Service, U.S. Gov. Print. Office, Washington, DC. Vauclin, M., S.R. Vieira, G. Vachaud, and D.R. Nielsen. 1983. The use of cokriging with limited field soil observations. Soil Sci. Soc. Am. J. 47:175–184. Vaughan, P.J., S.M. Lesch, D.L. Corwin, and D.G. Cone. 1995. Water content effect on soil salinity prediction—A geostatistical study using cokriging. Soil Sci. Soc. Am. J. 59:1146–1156. Vieira, S.R., D.R. Nielsen, and J.W. Biggar. 1981. Spatial Variability of field-measured infiltration rate, Soil Sci. Soc. Am. J. 45:1040–1048. von Steiger, B., R. Webster, R. Schulin, and R. Lehmann. 1996. Mapping heavy metals in polluted soil by disjunctive kriging. Environ. Pollut. 94:205–215. Wang, D., S.R.Yates, J. Simunek, and M.Th. van Genuchten. 1997. Solute transport in simulated conductivity fields under different irrigations. J. Irrig. Drain. Div. Am. Soc. Civil Eng. 123:336–343. Warrick, A.W., D.E. Myers. 1987. Optimization of sampling locations for variogram calculations. Water Resour. Res. 23:496–500. Warrick, A.W., D.E. Myers, and D.R. Nielsen. 1986. Geostatistical methods applied to soil science. p. 53–82. In A. Klute. (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron. Monogr. 9. ASA and SSSA, Madison, WI. Webster, R. 1991. Local disjunctive kriging of soil properties with change of support. J. Soil Sci. 42:301–318. Webster, R. 1994. The development of pedometrics. Geoderma 62:1–15. Webster, R., and T.M. Burgess. 1980. Optimal interpolation and isarithmic mapping of soil properties. III. Changing drift and universal kriging. J. Soil Sci. 31:505–524. Wen, X.H., and C.S. Kung. 1993. Stochastic simulation of solute transport in heterogeneous formations— A comparison of parametric and nonparametric geostatistical approaches. Ground Water 31:953–965. Yates, S.R. 1986. Disjunctive kriging. III. Cokriging. Water Resour. Res. 22:1371–1376. Yates, S.R., and A.W. Warrick. 1987. Estimating soil water content using cokriging. Soil Sci. Soc. Am. J. 51:23–30. Yates, S.R., A.W. Warrick, and D.E. Myers. 1986. Disjunctive kriging. II. Examples. Water Resour. Res. 22:623–630. Yates, S.R., and M.V. Yates. 1988. Disjunctive kriging as an approach to decision making. Soil Sci. Soc. Am. J. 52:1554–1558. Zhang, R., A.W. Warrick, and D.E. Myers. 1992. Improvement of the prediction of soil particle size fractions using spectral properties. Geoderma 52:223–243. Zhang, R., S. Shouse, and S.R. Yates. 1997. Use of pseudo-crossvariograms and cokriging to improve estimates of soil solute concentrations. Soil Sci. Soc. Am. J. 61:1342–1347. Zhang, R., P. Shouse, and S.R. Yates. 1999. Estimates of soil nitrate distributions using cokriging with pseudo-cross variograms. J. Environ. Qual. 28:424–428. Zirschky, J. 1985. Geostatistics for environmental monitoring and survey design. Environ. Int. 11:515–524. Zirschky, J., G.P. Keary, R.O. Gilbert, and E.J. Middlebrooks. 1985. Spatial estimation of hazardous waste site data.. J. Environ. Eng. 111:777–789.
Published 2002
1.6 Time and Space Series O. WENDROTH, ZALF, Institut für Bodenlandschaftsforschung, Müncheberg, Germany D. R. NIELSEN, University of California, Davis, California
1.6.1 General Information In Section 1.5, geostatistical approaches were introduced dealing with the identification of spatial dependence of observations, interpolation techniques based on spatial dependence, and sampling schemes and their associated sampling errors. Besides geostatistics, several other analytical tools from the field of time series analysis already applied in areas such as hydrology and economic sciences have been used in soil science during the last two decades. Although these tools are limited to onedimensional, or serial data, they allow for the decomposition of the variability of spatial or temporal data series, and to model processes of variables or vectors in space or time. This section focuses on spatial correlation, spectral analysis, and state–space modeling.
1.6.2 Auto- and Cross-Correlation In classical statistics, the correlation between two variables is derived from the linkage of their distributions, their covariance and their respective variances. For a particular series of spatial or temporal data, the spatial relation between observations separated by a certain distance, can be quantified accordingly by the so-called autocovariance function CZ(h), which is defined as: N−h
CZ(h) = (1/N)
Σ [Z(x + h) − µZ][Z(x) −µZ]
i=1
[1.6–1]
where N denotes the total number of observations of regularly sampled observations, and h is the separation or lag distance class over which observations Z(x) are separated. The mean of observations Z(x) is µZ. The autocorrelation coefficient rZ(h) for a respective lag class is then calculated by: rZ(h) = {[CZ(h)]/sZ2} with sZ2 being the variance of Z(x), which is defined as 119
[1.6–2]
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sZ2 = [1/(N − 1)]
N
Σ [Z(x) − µZ]2
i=1
[1.6.–3]
The definition given in Eq. [1.6–1] is the so-called sample autocovariance function (Shumway, 1988). Notice that the sum of the deviation products in Eq. [1.6–3] is not divided by the number of observations in the respective lag class N(h), but just by N. Salas et al. (1988) gave this definition to be valid for long series of data, where the number in the denominator will not strongly affect the result. Moreover, one may consider the above definition as a weighting of the autocorrelation coefficient, by which the decreasing number of observations with increasing lag distance results in a smaller rZ(h) coefficient. As an example data set, volumetric soil water content θ, Cl− distribution, and soil bulk density ρb were sampled every 10 cm at 10- to 15-cm soil depth after a Cl− tracer had been applied. The sampling volume was 100 cm3. The tracer had been applied approximately 8 wk after plowing, and soil samples were taken 1 d after tracer application perpendicular to the direction of plowing. Although samples were not taken along one row but from parallel rows, they were considered and analyzed as if they had been taken along one row, on the basis of the assumption that the plowing operation causes a regular repetitive pattern of the soil surface contour mainly pronounced perpendicular to the tillage direction. In Fig. 1.6–1, spatial processes of water content, Cl−, and bulk density are shown for the 100 sampling locations. Especially for the soil water content and Cl− concentrations, some cyclic behavior of the data through space is intimated. For bulk density data, such cyclic behavior is only observed in some parts of the series. Fractile diagrams (Warrick & Nielsen, 1980) show that water content and bulk density are normally distributed (Fig. 1.6–2a and 1.6–2c), while Cl− concentration appears to be lognormally distributed (Fig. 1.6–2b). Besides local fluctuations, Cl− data exhibit an increasing trend through the 100 observations. Autocorrelograms for the three variables are given in Fig. 1.6–3. With separation distances larger than zero, rZ(h) initially decreases at the first lags. After this decrease, rZ(h) increases again for water content and Cl− concentration (Fig. 1.6–3a and 1.6–3b). The autocorrelation function of these two variables exhibits peaks at regular lag intervals, indicating a periodic variation. While rZ(h) for the water content fluctuates around zero (Fig. 1.6–3a), it generally decreases while conserving the periodic fluctuations of Cl− content (Fig. 1.6–3b). This long decline of the autocorrelation function is due to the increasing trend that was observed in the original data (Fig. 1.6–1b). The autocorrelation function of bulk density (Fig. 1.6–3c) indicates no spatial correlation. The spatial association between two variables Zj and Zk can be quantified with the cross-covariance function Cjk(h) (Shumway, 1988), which is defined as: N−h
Cjk(h) = (1/N)
Σ [Zj(x + h) − µj][Zk(x) −µk]
i=1
[1.6–4]
The cross-correlation function rjk(h), that is, the normalized cross-covariance function, is derived from: rjk(h) = [Cjk (h)]/%&&& s2j sk2
[1.6–5]
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Fig. 1.6–1. (a) Soil water content, (b) Cl− concentration, and (c) soil bulk density across 100 locations.
The cross-correlogram for the series of soil moisture and Cl− is shown in Fig. 1.6–4a. At lag h equal to zero, the cross-correlation coefficient rjk(h) is the same as the classical correlation coefficient. In the right-hand side of the cross-correlation function, the leading relation between water content and the logarithm of Cl− concentration is manifested (Shumway & Stoffer, 2000); that is, in the relation between water content and Cl− concentration, water content is considered at h lag intervals ahead of Cl− concentration. Whereas in the left-hand side of the cross-correlogram, this relation is given for water content at h lag intervals behind Cl− concentration. In both directions, observations are not only correlated over several locations, but they are even spatially related with a periodically repeating pattern. Only a very weak periodic correlation is indicated in the cross-correlogram between Cl− concentra-
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Fig. 1.6–2. Fractile diagrams of (a) soil water content, (b) Cl− concentration, and (c) soil bulk density.
tion and bulk density (Fig. 1.6–4b) and water content and bulk density (not shown), but this is a common result when a variable that is spatially well structured is combined with another variable with a weak structure. It is obvious that prior to the interpretation of a cross-correlogram, the autocorrelograms of both underlying variables are essential. However, from both, auto- and cross-correlograms of this data set, cyclic variation components seem to underlie water content and Cl− concen-
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Fig. 1.6–3. Autocorrelation functions of (a) soil water content, (b) Cl− concentration, and (c) soil bulk density.
tration. Hence, analytical tools for better understanding and quantifying statistical relationships should be applied, as explained in the section below. 1.6.3 Spectral Analysis Spectral analysis consists of a set of analytical tools that allows for the decomposition of cyclic or periodically repeating variation of regularly sampled onedimensional data. These tools have been widely used in time series analysis for modeling climatic and weather events, but also for the spatial domain in soil science (e.g., Trangmar et al., 1985; Kachanoski et al., 1985a, b). Long-term time series of soil water state variables and daily temperature data may provide opportunities for applying these techniques in soil science. Moreover, since many agricultural systems
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Fig. 1.6–4. Cross-correlation functions of (a) soil water content and Cl− concentration and (b) Cl− concentration and soil bulk density.
are managed in a regular or spatially repeated design, for example, tillage (Nielsen et al., 1983), planting (Samra et al., 1993), furrow and ditch systems, sprinkler irrigation systems, and experimental plots that have to be laid out in a regular instead of a random design (Bazza et al., 1988), there is potential to learn about underlying processes by decomposing cyclic variability components using spectral analytical tools. Preferably this is done in a direction perpendicular to the main tillage or cultivation operation. Each spatial or temporal data series Ys can be considered as a combination of an infinite number of sinusoidal waves. Each wave can be characterized by its amplitude, its wavelength, and its phase angle φ. The amplitude A is the height of the maximum above the base line, hence, one-half of the range of data over which one sine wave occurs. The wavelength λ is the length of one sine wave, also called one period in the case of time domain applications (Davis, 1986). λ is the reciprocal of the frequency τ. The phase angle describes the shift by which a sine wave is advanced either relative to itself or to other waves. The combination of k different waves in the s-direction can be expressed as the Fourier transform Ys =
∞
Σ αkcos(2πτks) + βksin(2πτks)
k=1
[1.6–6]
with the coefficients ak and bk being the respective amplitudes (Shumway, 1988).
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Fig. 1.6–5. Power spectra of (a) soil water content, (b) Cl− concentration, and (c) soil bulk density.
For spectral analysis, the autocovariance function Cx(h) of the data series xs is represented as a Fourier transform resulting in the power spectrum ∞
fx(τ) = h=−∞ Σ Cx(h)exp(−2πiτh)
[1.6–7]
where i2 = −1 (Shumway, 1988). In spectral analysis, the covariance is evaluated for its sinusoidal shape. In the power spectrum, the wave shape behavior of the covariance for each considered wavelength is then manifested by a signal or peak of that respective wavelength.
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In Fig. 1.6–5, power spectra of water content, Cl− concentration, and bulk density series are shown. First, spectra obtained without smoothing (L = 1) are examined. Both, water content and Cl− concentration show three distinctive peaks at frequencies of 0.1, 0.2, and approximately 0.4 (Fig. 1.6–5a and 1.6–5b). The respective wavelengths are calculated by dividing the sampling distance (10 cm) by the frequency τ. Hence, the three identified frequencies refer to wavelengths of 100, 50, and 25 cm, respectively. Without any physical proof, the 100-cm wavelength could result from the plow width, and the 25-cm wavelength from the plow share. The 50-cm wavelength results from another tillage implement running behind the plow and overlapping one-half of the plow width. In the power spectrum of bulk density (Fig. 1.6–5c), many peaks appear, manifesting the strong fluctuation and low spatial autocorrelation of the data. Interestingly, at this scale of observation, the bulk density, a variable commonly used in models to determine storage and transport processes, does not reflect a meaningful periodic behavior that would be comparable to that of water content or Cl− concentration. In order to assess the meaning of the relative magnitudes of the spectral peaks, the intervals of confidence level α for a respective spectrum fx(τk) are determined via (Shumway, 1988) 2Lfx$ (τk) 2Lf$x (τk) ________ ≤ fx (τk) ≤ __________ 2 2 χ 2L (α/2) χ 2L (1 − α/2)
[1.6–8]
with 2L degrees of freedom and L denoting the smoothing constant. In the case that the measured series of length T is not a power of 2, the series has to be extended for spectral analysis to T′. The degrees of freedom are then derived from 2LT T′−1. Confidence intervals for the three most meaningful peaks in the spectra of soil water content and Cl− concentration are given in Table 1.6–1. These intervals were calculated for the spectra that were obtained without any smoothing (Fig. 1.6–5, L = 1). Another evaluation criterion is the F statistics, where the hypothesis is examined that two spectra, which were calculated for the same frequency but with different degrees of freedom because of a different smoothing procedure, are equal. The F value for comparison of two spectra is (Shumway, 1988) F2L1,2L2 = [f$1(τk)]/[f$2(τk)]
[1.6–9]
Table 1.6–1. Peak periods in periodograms (Fig. 1.6–5) for water content and log(Cl− Conc.) and their respective confidence intervals, determined from χ2 distribution. Frequency λ−1
Spectrum f$
90% confidence interval
Soil water content
0.101 0.195 0.398
0.0134 0.0023 0.0033
(0.0045, 0.2618) (0.0008, 0.0445) (0.0011, 0.0643)
Log(Cl− Conc.)
0.101 0.195 0.398
23.3 3.2 6.6
(7.8, 454.8) (1.1, 63.1) (2.2, 128.0)
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The spectra obtained for smoothing constants L = 3 and 7 are shown in Fig. 1.6–5 for the three variables considered. Notice that smoothing with L = 3 and 7 attenuates the spectra compared with those obtained with L = 1 for soil water content and Cl− concentration, indicating that the periodicity of the main variance components is conserved (Fig. 1.6–5a and 1.6–5b). However, with increasing smoothing, the spectra for bulk density are not attenuated as for the other two variables considered, but periodic lengths exhibiting peaks differ and indicate that cyclic variance behavior of this variable is not strongly manifested at characteristic wavelengths (Fig. 1.6–5c). The F statistics calculated according to Eq. [1.6–9] are given for the respective smoothing combinations and resulting degrees of freedom in Fig. 1.6–6, for L = 1 and 3 yielding the F(2,5), and for L = 1 and 7 yielding F(2,11). Only for the combination of L = 1 and 7 were significant signals obtained for water content and Cl− concentration (Fig. 1.6–6a and 1.6–6b). None of the bulk density spectra were indicated to be significant by the F statistics. However, had those of the three main peaks of the spectra for water content and Cl− that were smaller than the theoretical F value been considered meaningless, important information on variability sources would have been lost. Longer series and higher sampling densities are two options to extract cyclic variability sources. Similar to the way the spatial dependence of two variables over different separation distances h is quantified in the cross-covariance function, their cross-correlation can be manifested in a frequency-dependent way, that is, the cross-spectrum, which is defined as: fjk(τ) =
∞
Σ Cjk(h)exp(−2πiτh)
h=−∞
[1.6–10]
It consists of two components, the cospectrum and the quad-spectrum. The crossspectrum shows whether the cross-covariance function of two series fluctuates at regular or periodic intervals. Analogous to the coefficient of determination for regression analysis, the cyclic association of two series can be evaluated for each particular frequency with the squared coherence by: κjk (τ) = |fjk (τ)|2/[fj (τ) fk (τ)]
[1.6–11]
The squared coherence is close to one if both series have characteristic fluctuations at the respective frequency. Notice that for both the cross-spectrum and the squared coherence the phase angle is not relevant; that is, the periodic fluctuations of the original data series do not have to be on top of each other, but can be lagged by some distance. The cross-spectrum of water content and Cl− concentration is shown in Fig. 1.6–7a. As could be expected from the power spectra of each of the respective variables and from the cross-correlation function, both variables are strongly associated at frequencies of 0.1, 0.2, and 0.4 cm−1. The intensity of their associated signals, that is, the degree of linkage between the amplitudes at a respective frequency is manifested by the magnitude of the cross-spectrum. On the other hand, the squared coherence only exhibits the frequencies at which the data series coincide
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Fig. 1.6–6. F statistics for power spectra of (a) soil water content, (b) Cl− concentration, and (c) soil bulk density, shown in Fig. 1.6–5.
regardless of the intensity of the signal in the power and cross-spectra, as can be seen in Fig. 1.6–7b. For other long and short wavelengths, which are meaningless compared with those at λ = 0.1, 0.2, and 0.4 cm−1, common periodic fluctuations exist. When a periodic impact is expected, which then should be investigated, sampling distance must be shorter than one-half of the length of one period. 1.6.4 State–Space Analysis State–space analysis is a tool borrowed from time series analysis that allows for the description of a spatial or temporal process based on two components. One
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Fig. 1.6–7. (a) Cross-spectrum and (b) squared coherence of water content and Cl− concentration.
is the physical or auto- or cross-covariate linkage between subsequent states of a vector Xs. The other is the identification of signal and noise; that is, observations can be treated as an indirect reflection of the true state of a system. Both, the physical or empirical model and the observation include variability, uncertainty, and noise components that are quantified in the analysis. Two basic equations underlie state–space analysis. One is the state equation: Xs = G(X)s−1 + ωs
[1.6–12]
where s denotes the position in space or time, G is the transfer function described as any physical or empirical equation, and ωs is the error due to the model’s failure. ωs is also called the model uncertainty and is needed because a model can never fully describe the process. Its variance is Q with units of variance per time. Since the system’s state is related to its magnitude at each previous location, this technique does not require an assumption regarding stationarity (Shumway, 1985). The second equation is the observation equation: Ys = MsXs + vs
[1.6–13]
With this equation, the true state of the vector Xs is related to the observed vector Ys by a calibration function Ms. The random noise vs, with variance R associated with the observation, is also called the measurement uncertainty. This variability
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component is due to the lack of reproducibility of an observation, the calibration error of an instrument, and the changing range of influence over the measurement range of an instrument. Both variance components are accounted for when the state vector’s process is described. Moreover, the process of variance of the state vector is described so that confidence bands around the series’ process are provided at each location s. The Kalman filter (Kalman, 1960) was developed for the purpose of prediction with associated variance. A scheme of the state–space or Kalman filter prediction is given in Fig. 1.6–8. As input information to the Kalman filter algorithm, a model equation and initial estimates for model parameters are given. Starting from the system’s state at s = 1, the state Xs is predicted for s = 2 using Eq. [1.6–12]. At position s = 2, an observation Ys is available. In the Kalman filter, the prediction is compared with the observation. Both are weighted against each other with the socalled Kalman-gain matrix Kgs. If the model equation reflects the process with a relatively large certainty and the measurements are relatively uncertain, then a larger weight is given to the prediction Xsp than to the observation. The result is the updated prediction Xsu: Xsu = Xsp + Kgs(Ys − MsXsp)
[1.6–14]
In Eq. [1.6–14] the Kalman-gain matrix Kgs is: Kgs = PspMs′(MsPspMs′ + R)−1
[1.6–15]
Besides the state, the state variance is predicted by: Psp = Gs−1Ps−1(Gs−1)′ + Q
[1.6–16]
Then, Psp is updated by the Kalman-gain resulting in Psu as follows: Psu = Psp − KgsMsPsp
[1.6–17]
The updated step usually causes a reduction of the variance. At those positions where no observation is available, the prediction of the state and the variance is not updated. Inasmuch as the previously predicted variance is the initial condition for the new interval, this lack of update causes the variance to increase. At any position
Fig. 1.6–8. Scheme of the Kalman filter applied in state–space models.
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for which an observation exists, the predicted state and variance can be updated again, resulting in a reduction of variance. State–space modeling can be applied for predicting a series with only occasional updating if the model equation is known and reliable estimates of its parameters and variances are available. In this case, the scheme indicated in Fig. 1.6–8 is valid. When model parameters have to be estimated, the Kalman filter is repeated iteratively while parameters are optimized. The output (Fig. 1.6–8) of one step is the beginning of the next iteration. Like in other optimization problems, difficulties can arise from local minima and from nonlinearity of physical equations. For further details, see Gelb (1974). State–space analysis has been used for predicting economical time series, for remote control of missiles, for voice detection in electronic engineering, for predicting spatial processes of soil physical state variables such as soil water content and temperature (Morkoc et al., 1985), for describing crop yield variability (Wendroth et al., 1992), for determining the linkage between aerial photographs and crop and soil state variables (Nielsen et al., 1999; Wendroth et al., 1999a), for modeling lake water budget (Assouline, 1993), for field estimation of soil physical properties (Katul et al., 1993), and recently for space–time optimization of field soil hydraulic properties (Cahill et al., 1999). The purpose of many applications is not only to predict a process, but to inversely estimate model parameters of empirical or physical equations. Unlike other estimation techniques, the above-mentioned uncertainty components are accounted for in the estimation. In the following sections, state–space analysis is applied in one case to estimate the statistical association between spatially sampled data, and in another case where the process of field soil water content is described based on physical equations. 1.6.4.1 Autoregressive State–Space Model for Spatial Processes For a spatial data set, state–space analysis is applied to data series of grain yield, nitrate content and remotely sensed normalized vegetation index (NDVI) observations (Fig. 1.6–9). For numerical reasons in the estimation of state–space coefficients, original data Z(x) are normalized with respect to their mean µZ and standard deviation σZ by: Γ(x) = [Z(x) − (µZ − 2σZ)]/(4σZ)
[1.6–18]
By this normalization, the variables become dimensionless, have normalized means of 0.5, and their spatial linkage relative to each other becomes more easily visualized in the same graph or figure. From Fig. 1.6–9a, local association between grain yield and nitrate content becomes obvious. Locally, both data sets fluctuate similarly. However, their relative magnitudes are not consistent across the transect. For example, along positions 1–10 and 62–77 grain yields are everywhere greater than nitrate content, while along nearly all positions between 30 and 50, grain yields are smaller than nitrate content. A similar behavior is evident in Fig. 1.6–9b for grain yield and NDVI. Except for a few fluctuations of each of the three variables, the variation over short distances is smaller than that over long distances; that is, for short distances, autocorrelation coefficients would be high and would decrease over longer distances.
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An autoregressive equation that accounts for spatial relations between subsequent states used in the state–space analysis is: Xi = ΦXi−1 + ωi
[1.6–19]
This equation is a specified case of Eq. [1.6–12] where the matrix Φ includes the transfer or autoregression coefficients. These coefficients are optimized using the EM-algorithm (Shumway & Stoffer, 1982). As an example, state–space analysis is applied to three different scenarios. In Scenario 1, all observations are considered. Hence, the updating step is possible at each position. The resulting state estimate is given in Fig. 1.6–10a. Because the data have been normalized with Eq. [1.6–16], the coefficients reflect the relative impact on the different variables at location i − 1. As the model result, ± 2 standard error of estimate are given as the 95% confidence intervals. If the updating step is only possible at every other location in the second scenario, because of neglecting yield observations at every even-numbered position, the transfer coefficients change slightly, and the 95% confidence range of observations increases (Fig. 1.6–10b). It can be noticed that extreme data fluctuations are outside of the 95% confidence range. However, the general process of the series is well described. If only every fourth yield observation is considered in the third scenario, the relative impact of
Fig. 1.6–9. Spatial process of grain yield in association with (a) soil NO3 content and (b) normalized vegetation index (NDVI), respectively.
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a neighboring yield observation is still relatively high (Fig. 1.6–10c). However, the 95% confidence range hardly increases compared with that of the second scenario. 1.6.4.2 State–Space Analysis for Time Series So far, an equation of empirical origin has been employed to describe a spatial process. The spatial model was based on the state vector’s spatial covariance, while the transition coefficients were estimated. Next, the Kalman filter is combined with a physically based equation in order to describe a series’ process in time. Katul et al. (1993), Parlange et al. (1993), and Wendroth et al. (1993) derived parameters of physical soil properties using the Kalman filter in combination with a physical
Fig. 1.6–10. State–space description of grain yield (Yi), soil nitrate (Nit), and normalized vegetation index (NDVI) considering (a) each, (b) every other, and (c) every fourth observation for the estimation.
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equation and a numerical optimization routine (Press et al., 1989). Here, boundary conditions, physical properties, and measurement and model variance R and Q, respectively, are assumed to be known, and the process of soil water content at three soil depths shall be described in a manner similar to Wendroth et al. (1999b). The purpose is to compare predictions of water content in two scenarios for investigating the effect of intermittent updating of the prediction at times when an observation is available. In the first scenario, water content time series is described with an explicit equation: dθ/dt = −K(θ)(dH/dx) − qj
for θ ≤ θmac
[1.6–20]
In this equation, z denotes depth, K(θ) the hydraulic conductivity as a function of soil water content θ, H the hydraulic head, and qj the flux across the upper boundary of a soil compartment. This is a simple explicit form of the water transport equation. Time discretization is kept variable, and time intervals are especially short during rainfall events. In the case considered, the soil was macroporous, and rapid water flow was anticipated at times of high rainfall intensity, as no surface water in puddles could be observed. Rapid water flow was assumed to occur when the water content exceeded a threshold θmac. The macropore space, that is, the total porosity minus θmac, was considered to be approximately one volume percent. For further details, see Wendroth et al. (1999b). In the second scenario, the Kalman filter prediction, Eq. [1.6–20] is expanded by a model uncertainty component and then considered as the state equation, equivalent to Eq. [1.6–12]. Describing the K(θ) function with a simple exponential form K(θ) = 10a+bθ
[1.6–21]
yields the prediction of variance underlying the water content series’ process: dP(t)/dt = [−2P(t) (dH/dz) 10ab10bθ ln(10)] + Q
[1.6–22]
Daily precipitation and evaporation, measured, predicted, and Kalman filter predicted water content series are shown in Fig. 1.6–11 for depths of 10, 30, and 50 cm, respectively. Measurement uncertainty R and spectral density Q were assumed 0.0004 (cm3 cm−3)2 and 0.0004 (cm3 cm−3)2 d−1, respectively. These values were based on experiences from previous experiments. At the 10-cm depth, predictions of θ based on Eq. [1.6–20] underestimate the measured water contents for almost the entire period. Especially after rainfall periods, the decay in θ is predicted too rapidly. Some uncertainty sources are given here that could have caused this result: 1. The assumption that hydraulic properties determined in the laboratory for the three selected soil depth compartments are an appropriate reflection of the field physical properties 2. The estimation of the upper boundary condition 3. Equation [1.6–20]
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4. The numerical discretization of the water transport equation 5. The empirical description of K(θ) Although some of these error sources could be eliminated, one is usually confronted with these and other uncertainties in field water balance calculations. Water con-
Fig. 1.6–11. Daily precipitation, evaporation, and soil water content at three soil depths. Soil water content was predicted using Eq. [1.6–18], and in combination with the Kalman filter.
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tent at the 30-cm depth is overpredicted in the dry period between Day 50 and 100. However, it is predicted with sufficient accuracy at the 50-cm depth. In comparison, the Kalman filter prediction is associated with the same error sources discussed above. However, the error sources are considered quantitatively, and the estimation is updated, while observations are weighed against predictions. The standard estimation error band includes the measured water contents except for the period between Day 55 and 95 at the 30-cm depth (Fig. 1.6–11). It becomes obvious that at times of precipitation, the estimation error band increases. This example illustrates the opportunity that for a physical description of a field situation, existing uncertainty sources can be compensated for by a stochastic technique accounting quantitatively for this uncertainty, in combination with occasional state observations. 1.6.5 References Assouline, S. 1993. Estimation of lake hydrologic budget terms using simultaneous solution of water, heat, and salt balances and a Kalman filtering approach: Application to Lake Kinneret. Water Resour. Res. 29:3041–3048. Bazza, M., R.H. Shumway, and D.R. Nielsen. 1988. Two-dimensional spectral analyses of soil surface temperature. Hilgardia 56:1–28. Cahill, A.T., F. Ungaro, M.B. Parlange, M. Mata, and D.R. Nielsen. 1999. Combined spatial and Kalman filter estimation of optimal soil hydraulic properties. Water Resour. Res. 35:1079–1088. Davis, J.C. 1986. Statistics and data analysis in geology. 2nd ed. Wiley and Sons, New York, NY. Gelb, A. 1974. Applied optimal estimation. Massachusetts Institute of Technology Press, Cambridge, MA. Kachanoski, R.G., D.E. Rolston, and E. De Jong. 1985a. Spatial and spectral relationships of soil properties and microtopography. I. Density and thickness of A-horizon. Soil Sci. Soc. Am. J. 49:804–812. Kachanoski, R.G., E. De Jong, and D.E. Rolston. 1985b. Spatial and spectral relationships of soil properties and microtopography. II. Density and thickness of B-horizon. Soil Sci. Soc. Am. J. 49:812–816. Kalman, R.E. 1960. A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 8:35–45. Katul, G.G., O. Wendroth, M.B. Parlange, C.E. Puente, and D.R.Nielsen. 1993. Estimation of in situ hydraulic conductivity function from nonlinear filtering theory. Water Resour. Res. 29:1063–1070. Morkoc, F., J.W. Biggar, D.R. Nielsen, and D.E. Rolston. 1985. Analysis of soil water content and temperature using state–space approach. Soil Sci. Soc. Am. J. 49:798–803. Nielsen, D.R., P.M. Tillotson, and S.R. Vieira. 1983. Analyzing field-measured soil water properties. Agric. Water Manage. 6:93–109. Nielsen, D.R., O. Wendroth, and F.J. Pierce. 1999. Emerging concepts for solving the enigma of precision farming research. p. 303–318. In P.C. Robert et al. (ed.) Proc. 4th International Conference on Precision Agriculture, Minneapolis, MI. Parlange, M.B., G.G. Katul, M.V. Folegatti, and D.R. Nielsen. 1993. Evaporation and the field scale soil water diffusivity function. Water Resour. Res. 29:1279–1286. Press, W.H., B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. 1989. Numerical recipes—The art of scientific computing (FORTRAN version). Cambridge University Press, Cambridge, MA. Salas, J.D., J.W. Delleur, V. Yevjevich, and W.L. Lane. 1988. Applied modeling of hydrologic time series. Water Reources Publications, Littleton, CO. Samra, J.S., S.S. Grewal, and G. Singh. 1993. Modeling competition of paired columns of Eucalyptus on interplanted grass. Agrofor. Syst. 21:177–190. Shumway, R.H. 1985. Time series in soil science: Is there life after kriging. p. 35–60. In D.R. Nielsen and J. Bouma (ed.) Soil Spatial Variability. Proc. Workshop ISSS/SSSA, Las Vegas, NV. Shumway, R.H. 1988. Applied statistical time series analysis. Prentice Hall, Englewood Cliffs, NJ. Shumway, R.H., and D.S. Stoffer. 1982. An approach to time series smoothing and forecasting using the EM algorithm. J. Time Ser. Anal. 3:253–264.
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Shumway, R.H., and D.S. Stoffer. 2000. Time series analysis and its applications. Springer, New York, NY. Trangmar, B.B., R.S. Yost, and G. Uehara. 1985. Application of geostatistics to spatial studies of soil properties. Adv. Agron. 38:45–94. Warrick, A.W., and D.R. Nielsen. 1980. Spatial variability of soil physical properties in the field. p. 319–344. In D. Hillel. Applications of soil physics. Academic Press, New York, NY. Wendroth, O., A.M. Al-Omran, C. Kirda, K. Reichardt, and D.R. Nielsen. 1992. State–space approach to spatial variability of crop yield. Soil Sci. Soc. Am. J. 56:801–807. Wendroth, O., G.G. Katul, M.B. Parlange, C.E. Puente, and D.R. Nielsen. 1993. A nonlinear filtering approach for determining hydraulic conductivity functions in field soils. Soil Sci. 156:293–301. Wendroth, O., P. Jürschik, A. Giebel, and D.R. Nielsen. 1999a. Spatial statistical analysis of on-site crop yield and for site-specific management. p. 159–170. In P.C. Robert et al. (ed.) Proc. 4th International Conference on Precision Agriculture, Minneapolis, MI. Wendroth, O., H. Rogasik, S.Koszinski, C.J.Ritsema, L.W. Dekker, and D.R. Nielsen. 1999b. State–space prediction of field-scale soil water content time series in a sandy loam. Soil Till. Res. 50:85–93.
Published 2002
1.7 Parameter Optimization and Nonlinear Fitting JIÌÍ ŠIMæNEK, George E. Brown, Jr. Salinity Laboratory, USDA-ARS, Riverside, California JAN W. HOPMANS, University of California, Davis, California
1.7.1 Introduction Experimentalists often collect data that later need to be summarized to infer or investigate cause–effect relationships. The data sets and derived relationships can be either static or dynamic. Soil water retention data (relating soil water matric head with soil water content; Section 3.3) or hydraulic conductivity data (relating unsaturated hydraulic conductivity with soil water matric head or water content; Section 3.5) represent typical examples of such data sets. The data can be expressed in a graphical form by simply drawing an eye-balled curve, or in functional form by fitting a curve from a selected class of functions through the data. The fitting process is called curve fitting or model fitting, depending on whether an arbitrary or theoretically derived function was selected to describe the behavior of the physical and/or chemical system under observation. For example, fitting a soil water retention curve (e.g., van Genuchten’s [1980] equation) can be considered as curve fitting since the equation is almost completely empirical, whereas fitting a solute breakthrough curve can be viewed as a model fitting process if the fitted function is an analytical solution of the convection–dispersion equation (see Sections 6.3 and 6.5). Alternatively, fitting the van Genuchten–Mualem unsaturated hydraulic conductivity function (van Genuchten, 1980) is a combination of curve and model fitting since the conductivity function is derived theoretically from a pore-size distribution model, but uses an empirical retention function. Although, in principle, curve and model fitting are not much different, curve fitting is more arbitrary. One typically selects an arbitrary function and the best-fit criterion is often formulated independent of statistical considerations (Bard, 1974). For model fitting, the functional relationship is well defined and only the parameters are unknown. Thus, functions obtained from curve fitting summarize the available data without necessarily increasing insight into the nature of observed processes and are generally limited to the measurement range. The process of model fitting is closely related to parameter estimation. Theoretically derived models (i.e., relationships describing a particular physical process) include parameters that describe physical properties of the observed system (e.g., the saturated hydraulic conductivity, porosity, diffusion coefficient, cation exchange capacity, infiltration rate, or others). Hence, one would expect that fitting 139
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a correct model produces estimated parameters that describe these physical properties with a certain accuracy. Consequently, the results of model fitting can often be extrapolated to beyond the measurement range. By comparison, parameters obtained by curve fitting generally have little or no physical significance. A simple case of curve or model fitting arises when fitting a functional relationship between a dependent variable y and a set of n variables xi, i = 1,..., n. A more complex problem arises when the fitting model is represented by one or more differential equations, with the measured variables representing either primary or secondary variables (the latter being derived from the primary variables). A typical example of such a problem involves the transient variably saturated water flow equation. The model is represented by the Richards equation, while the measured variables are matric heads and/or water contents (primary variables), or water fluxes and the sample soil water storage (secondary variables). To match the model with experimental variables, selected parameters in the soil hydraulic properties are usually fitted to functional relationships. In this section we briefly describe the process of parameter estimation. We include a discussion on least-squares and maximum-likelihood estimators, minimization techniques, significance of optimized parameters and their confidence intervals, and goodness of fit. Readers interested in more details are encouraged to study the texts of Bard (1974) and Beck and Arnold (1977), among many others. Detailed descriptions of proven experimental techniques that utilize parameter estimation to estimate soil hydraulic and transport properties are presented in Sections 3.6.2 and 6.6, respectively. 1.7.2 Maximum-Likelihood and Weighted Least-Squares Estimator The general approach in model fitting is to select a merit or objective function that is a measure of the agreement between measured and modeled data, and which is directly or indirectly related to the adjustable parameters to be fitted. The best-fit parameters are obtained by minimizing (or maximizing, depending on how the function is defined) this objective function. If no model and measurement errors exist, this minimum value would be zero. However, even if the model is perfect, experimental errors will generally create a non-zero minimum value for the objective function. An excellent discussion on data modeling and optimization can be found in Press et al. (1992). When measurement errors follow a multivariate normal distribution with zero mean and covariance matrix V, the likelihood function can be written as (Bard, 1974) β) = (2π)−n/2(detV)−1/2exp{(−1/2)[q* − q(β β)]TV−1[q* − q(β β)]} L(β
[1.7–1]
β) is the likelihood function, β = {β1, β2,..., βm} is the vector of optimized where L(β parameters, m is the number of optimized parameters, q* = {q*1, q*2,..., q*n} is a vecβ) = {q1, q2,..., q*n} is the corresponding vector of model pretor of observations, q(β dictions as a function of the unknown parameters being optimized, and n is the number of observations. The differences between measured and computed quantities, β), are called residuals. The likelihood function L(β β) is defined as the joint q* − q(β probability density function of the observations and identifies the probability of the
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data, given the parameter vector β (Bard, 1974). The maximum-likelihood estimate is that value of the unknown parameter vector β that maximizes the value of the likelihood function. Since the logarithm is a monotonically increasing function of its β) also maximize argument, the values of β that maximize the likelihood function L(β β). This property of the logarithm is frequently used in parameter identificalnL(β tion studies since lnL is a simpler function than L itself. Hence, Eq. [1.7–1] is reformulated as the log-likelihood or support criterion (Carrera & Neuman, 1986a) β) = −(L*/2) = −(n/2) ln(2π) − (1/2) ln(detV) ln L(β β)]T V−1 [q* − q(β β)] − (1/2) [q* − q(β
[1.7–2]
The notation of L* is used in Section 1.7.5. The maximum of the likelihood function must satisfy the set of m β-likelihood equations β)]/∂βi = 0 [∂ln L(β
i = 1,...., m
[1.7–3]
If all elements of the covariance matrix V are known, then the value of the unknown parameter vector β which maximizes Eq. [1.7–3] must minimize Φ: β) = [q* − q(β β)]TV−1[q* − q(β β)] Φ(β
[1.7–4]
which constitutes the last term of Eq. [1.7–2]. If information about the distribution of the fitted parameters is known before the inversion, that information can be included in the parameter identification procedure by multiplying the likelihood function by the prior probability density funcβ), which summarizes this prior information. Estimates that make use of tion, p0(β prior information are known as Bayesian estimates, and lead to the maximizing of β), given by a posterior probability density function, p*(β β) = cL(β β)p0(β β) p*(β
[1.7–5]
in which c is a constant that insures that the integral of the posterior probability density function is equal to one. The posterior density function is proportional to the likelihood function when the prior distribution is uniform. Inclusion of prior information leads to the following expression to be minimized β) = [q* − q(β β)]TV−1[q* − q(β β)] + (β β* − β)TVβ−1(β β* − β) Φ(β
[1.7–6]
where β* is the parameter vector containing the prior information and Vβ is a covariance matrix for the parameter vector β. The first term in Eq. [1.7–6] penalizes for deviations of model predictions from measurements, while the second term penalizes for deviations of parameter estimates β from the prior estimate of the parameters β*; this prior estimate may be viewed as a reasonable first guess of β (McLaughlin & Townley, 1996). The second term of Eq. [1.7–6], also called the penalty function, ensures that the obtained parameter estimate is constrained to a physically meaningful range of values. The solution of the minimization problem,
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Eq. [1.7–6], is a compromise between the best-fit estimate associated with minimization of the first term, and the prior estimate associated with the second term (McLaughlin & Townley, 1996). Russo et al. (1991) showed that the use of a penalty function can significantly improve the uniqueness of the estimated parameters. The covariance matrices V and Vβ are also referred to as weighting matrices and provide information about measurement accuracy and correlation between measurement errors, V, and between parameters, Vβ (Kool et al., 1987). When the covariance matrix V has only non-zero diagonal elements, then the measurement errors are uncorrelated. No prior information about the optimized parameters exists if all elements of Vβ are equal to zero. In that case, the problem simplifies to a weighted least-squares problem β) = Φ(β
n
Σ wi[q*i − qi(ββ)]2 i=1
[1.7–7]
where wi is the weight of a particular measured point. The weighted least-squares estimator of Eq. [1.7–7] is a maximum-likelihood estimator as long as the weights, wi, contain the measurement error information such that wi = 1/σ2i = 1/variance of measurement error of q*i
[1.7–8]
However, in many applications, the measurement errors are either unknown or weights are not specified based on probabilistic assumptions. It is then difficult to interpret the resulting optimized parameters, their confidence intervals, correlations, and in general their relationship with the true parameter values (Bard, 1974). Weights significantly affect the shape and the absolute values of the objective function, and when selected arbitrarily, this arbitrariness is reflected in all subsequent statistical evaluations given in Sections 1.7.4 and 1.7.5. Improper selection of weights can influence not only the confidence regions of optimized parameters, but also the location of the minimum of the objective function (Hollenbeck & Jensen, 1998). The robustness of the least-squares criterion for the estimation of model parameters has recently been questioned by Finsterle and Najita (1998). They pointed out that the least-square criterion causes outliers to strongly influence the final values of optimized parameters. Hence, outliers (e.g., individual data points with large measurement errors, as is often the case with field measurements) can introduce a significant bias in the estimated model parameters. Finsterle and Najita (1998) studied several other more robust estimators with different error distributions that reduce the effect of outliers on the optimized parameters. For example, they suggest use of the least absolute deviates or L1 estimator if errors follow a double exponential distribution: β) = Φ(β
n
Σ (1/σi)|q*i − qi(ββ)| i=1
[1.7–9]
and the maximum-likelihood estimator for measurement errors that follow a Cauchy distribution:
SOIL SAMPLING AND STATISTICAL PROCEDURES
β) = Φ(β
143
n
Σ log{1 + (1/2)wi[q*i − qi(ββ)]2}
i=1
[1.7–10]
Still other estimators that do not correspond to any standard probability density function were suggested by Huber (1981) and Andrews et al. (1972). Robust estimators decrease the relative weights of outliers and thus make the estimated parameters less affected by the presence of random errors following a heavy tailed distribution (Finsterle & Najita, 1998). Uniqueness, identifiability, stability, and ill-posedness are terms often encountered in the parameter estimation literature. Since exact definitions and detailed discussions of these terms were given by Carrera and Neuman (1986b), we will only present a brief definition of each. The solution is said to be nonunique whenever the minimization criterion (i.e., the objective function) is nonconvex, that is, has multiple local minima or the global minimum occurs for a range of parameter values. The convexity of the objective function can be enhanced by inclusion of prior information, Eq. [1.7–6], in the analysis. The parameters are nonidentifiable when different combinations of parameters lead to a similar system response, thereby implying that a unique solution is impossible. Stability is achieved if the optimized parameters are insensitive to measurement errors, that is, small errors in the system response must not result in large changes in the optimized parameters. Finally, the inverse problem is ill-posed if the identified parameters are unstable and/or nonunique. Inverse problems used to estimate parameters of the unsaturated soil hydraulic functions are often ill-posed, but can become well-posed in case of welldesigned experiments for homogeneous soils with small measurement and model errors (Hopmans & Šimçnek, 1999). An ill-posed problem can be replaced by a wellposed problem by adding more or other type of measurements, and/or by constraining the value range of the set of adjustable parameters. 1.7.3 Methods of Solution Many techniques have been developed to solve the nonlinear minimization or maximization problem (Bard, 1974; Beck & Arnold, 1977; Yeh, 1986; Kool et al., 1987). Most of these methods are iterative by requiring an initial estimate β i of the unknown parameters to be optimized. The behavior of the objective function, β), in the neighborhood of this initial estimate is subsequently used to select a Φ(β direction vector v i, from which updated values of the unknown parameter vector are determined, that is, β i+1 = β i + ρ iv i = β i − ρ iRipi
[1.7–11]
The direction vector is computed so that the value of the objective function decreases, or Φi+1 < Φi
[1.7–12]
where Φi+1 and Φi are the objective functions at the previous and current iteration level, Ri is a positive definite matrix, pi is the gradient vector, and ρ i is the step size.
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Methods based on Eq. [1.7–11] are called gradient methods. The various gradient methods used in the literature (e.g., steepest descent, Newton’s method, directional discrimination, Marquardt’s method, the Gauss method, variable metric methods, and the interpolation–extrapolation method) differ in their choice of the step direction, vi, and/or the step size, ρ i (Bard, 1974). The steepest descent method uses ρ i = 1 and Ri = I, where I is an identity matrix. The Newton method uses ρi = 1 and β): Ri = H−1, where H is the Hessian matrix of Φ(β β) = (∂2Φ/∂βi∂βj) Hij (β
i, j = 1,...., m
[1.7–13]
The steepest descent method is often inefficient, requiring many iteration steps to reach the minimum, and consequently is not recommended. The Newton method is usually not recommended because it requires evaluation of second derivatives, thus making it computationally inefficient, especially for problems involving solution of partial differential equations. The Gauss–Newton method simply neglects the higher-order derivatives in the definition of the Hessian matrix and assumes that H can be approximated by a matrix N using only the first-order derivatives. For nonlinear weighted least squares this leads to H ≈N = JwT Jw
[1.7–14]
where Jw is the product of the Jacobian matrix J β)] β) ∂[qi* − qi(β ∂qi(β Jij = ____________ = − ________ ∂βj ∂βj
[1.7–15]
and the lower triangular matrix L of the Cholesky decomposition of V −1 (Jw = LJ). Marquardt (1963) proposed a very effective method, commonly called the Marquardt–Levenberg method, which has become a standard in nonlinear leastsquare fitting among soil scientists and hydrologists (van Genuchten, 1981; Kool et al., 1985, 1987). The method represents a compromise between the inverse-Hessian and steepest descent methods by using the steepest descent method when the objective function is far from its minimum, and switching to the inverse-Hessian method as the minimum is approached. This switch is accomplished by multiplying the diagonal in the Hessian matrix (or its approximation N), sometimes called the curvature matrix, with (1 + λ), where λ is a positive scalar, leading to H ≈ JwTJw + λDTD
[1.7–16]
where D is a diagonal scaling matrix whose elements coincide with the absolute values of the diagonal elements of the matrix N. When λ is large, the Hessian matrix is diagonally dominant, resulting in the steepest descent method. On the other hand, when λ is zero, Eq. [1.7–16] reduces to the inverse-Hessian method. A common strategy is to initially select a modest value of λ (e.g., 0.02) and then decrease its value as the solution approaches the minimum (e.g., multiply λ by 0.1 at each iteration step).
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The computationally most time-consuming part of both the Gauss–Newton and Marquardt–Levenberg methods is the evaluation of Jacobians. The sensitivity coefficients may be calculated using three different methods (Yeh, 1986): the influence coefficient method (finite differences), the sensitivity equation method, and the variational method. When using the influence method, one has to balance trunβ, with rounding errors that decrease with ∆β β. A cation errors that increase with ∆β common practice is to use a one-sided difference method, that is, to change the optimized parameters by 1% β + ∆β βej) − qi(β β) ∂qi qi(β ___ ≈ _______________ ∂βj ∆βj
[1.7–17]
β is the small increment of β (e.g., 0.01β β). With where ej is the jth unit vector, and ∆β a total number of m fitting parameters, the governing equation must be solved (m + 1) times during each iteration of the nonlinear minimization. A better estimate of the sensitivity coefficients can be obtained by using a central difference scheme β + ∆β βej) − qi(β β − ∆β βej) ∂qi qi(β ___ ≈ _____________________ ∂βj 2∆βj
[1.7–18]
However, since this scheme requires evaluation of the objective function for two additional parameter values for each optimized parameter, contrary to only a single evaluation for a one-sided difference scheme, the central difference approach is usually not recommended. Depending upon the problem being considered (e.g., measurement errors, number of optimized parameters, type of measurements), the objective function Φ may be topographically very complex without a well-defined global minimum and/or having several local minima in parameter space. The behavior of the objective function is often analyzed using contour plots of response surfaces. Specifically, the objective function is calculated for two selected perturbed parameters, while the other optimized parameters are kept constant, so that the contours can be graphically represented in a two-parameter plane. Contours of the objective function will reveal the presence of local minima, parameter sensitivity, and parameter correlation. Since a response surface analysis studies only cross-sections of the full parameter space, such an analysis can therefore only suggest how the objective function might behave in the full m-dimensional parameter space (Hopmans & Šimçnek, 1999). Minimization methods can be highly sensitive to the initial values of the optimized parameters. Depending upon the initial estimate, the final solution may not be the global minimum, but instead a local minimum. Consequently, when using gradient-type minimization techniques, it is generally recommended to repeat the minimization problem with different initial estimates of the optimized parameters and to select those parameter values that minimize the objective function. More robust minimization techniques have recently been used by Pan and Wu (1998), who used the annealing-simplex method; by Abbaspour et al. (1997), who used a sequential uncertainty domain parameter fitting (SUFI) method; and by Vrugt et al. (2001), who used a genetic algorithm.
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In recent years, the increased interest in robust minimization algorithms has led to global optimization techniques (e.g., Barhen et al., 1997) that uniquely search for the global minimum. These methods include fast simulated annealing, a stochastic approximation method, evolution algorithms, a multiple-level singlelinkage method, an interval arithmetic technique, taboo search schemes, subenergy tunneling, and non-Lopschitzian terminal repellers (Barhen et al., 1997), but have not yet, or only sparingly, been used for unsaturated zone flow and transport problems. 1.7.4 Correlation and Confidence Intervals Since parameter estimation involves a variety of possible errors, including measurement errors, model errors, and numerical errors, an uncertainty analysis of the optimized parameters constitutes an important part of parameter estimation. Parameter uncertainty analyses usually assume that the solution converges to a global minimum, and that the model error is zero, so that parameter uncertainty is limited to measurement errors only. A confidence region of the parameter estimates can then be defined exactly by using a maximum allowable objective function increment (from its minimum) β) − Φmin| ≤ ε |Φ(β
[1.7–19]
where Φmin is the best attainable value of the objective function Φ, and ε is the largest difference between risks that one is willing to consider as being insignificant (Bard, 1974). The set of values β that satisfies this equation is known as the ε−indifference region. If ε is sufficiently small so that Φ can be approximated by means of its Taylor series expansion, the indifference region has a typical m-dimensional ellipsoid-type shape. An exact confidence region for nonlinear problems can be obtained by contouring the objective function with reference to some fixed levels of confidence (ε). This procedure is rather computationally expensive, especially for problems involving numerical solution of partial differential equations, since it usually requires discretizing the parameter space and computing the objective function value for each grid point (Hollenbeck et al., 2000). The appropriate contour value ε for a desired level of confidence can be selected based on a chi-square or F distribution (Beck & Arnold, 1977; Press et al., 1992; Hollenbeck et al., 2000). The approximate estimate of the parameter standard error is based on the Cramer–Rao theorem (e.g., Press et al., 1992), defining an estimate of the lower bound of the parameter covariance matrix C ≥ H−1
[1.7–20]
Since this estimate of the standard error is derived from linear regression analysis, it holds only approximately for nonlinear problems. The inequality in Eq. [1.7–20] becomes an equality if the model is linear in the parameters β, and the confidence region becomes an ellipsoid. Under the stated assumptions, the parameter covariance matrix C (previously referred to as Vβ when used as a weighting matrix for prior information) can be estimated directly from the variance, se2, of the residuals e (= q* − q)
SOIL SAMPLING AND STATISTICAL PROCEDURES
se2 = (eTe)/(n − m)
147
[1.7–21]
and the Jacobian or derivative matrix J (Eq. [1.7–15]) (Carrera & Neuman, 1986a; Kool & Parker, 1988) C ≈se2(JwTJw)−1
[1.7–22]
The estimated standard deviation of the parameter βj can then be determined from the diagonal elements of C as follows sj = %C && jj
[1.7–23]
from which γ% (γ = 1 − α) confidence intervals can be estimated using the Student’s t distribution βj,min = βj − tv,1−α/2sj βj,max = βj + tv,1+α/2sj
[1.7–24]
where v denotes the number of degrees of freedom (n − m). The uncertainty in the parameter estimates is underestimated when the parameters are correlated. A conservative way to find the confidence interval for correlated parameters is to find the projections of the confidence ellipsoid on the parameter axes by multiplying the parameter standard error sj with the square root of ε (Press et al., 1992; Hollenbeck et al., 2000). Correlation between optimized parameters can be estimated using the diagonal and off-diagonal terms of C. The parameter correlation matrix R can be directly obtained from the covariance matrix as follows Rij = Cij/(%C &&%C ii &&) jj
[1.7–25]
The correlation matrix quantifies changes in model predictions caused by small changes in the final estimate of a particular parameter i, relative to similar changes as a result of changes of the other parameter j. The correlation matrix reflects the nonorthogonality between two parameter values. A value of −1 or +1 suggests a perfect linear correlation, whereas 0 indicates no correlation. The correlation matrix may be used to select the nonadjustable parameters because of their high correlation with other fitting parameters. Interdependence of optimized parameters can cause a slow convergence rate and nonuniqueness, and increase parameter uncertainty. Although restrictive and only approximately valid for nonlinear problems, an uncertainty analysis provides a means to compare confidence intervals between parameters, thereby indicating those parameters that should be measured or estimated independently (Hopmans & Šimçnek, 1999). Alternatively, on the basis of the sensitivity analysis, one can collect observations of dependent variables at locations and times that will reduce confidence intervals and allow independent estimation.
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1.7.5 Goodness of Fit The maximum-likelihood approach leads to optimized parameters for a selected model without questioning the adequacy of the given model. Different criteria may be used to characterize the goodness of fit. The most popular criteria are given below. Absolute Error (AE): n
AE =
Σ |qi* − qi(ββ)|
i=1
[1.7–26]
Root Mean Squared Error (RMSE): n
Σ wi [q*i − qi (ββ)]2 i=1 RMSE = r_______________ n−m
[1.7–27]
Akaike Information Criterion (AIC): AIC = L* + 2m
[1.7–28]
(Akaike, 1974; Russo et al., 1991), where L* is the negative log likelihood for the fitted model (see Eq. [1.7–2]) and m is the total number of independently optimized parameters. For a Gaussian process, AIC can be estimated from the residual sum of squares (RSS) of deviations from the fitted model: AIC = n{ln(2π) + ln[RSS/(n − m)] + 1} + m
[1.7–29]
Bayesian Information Criterion (BIC): BIC = (AIC − 2m) + mlnn
[1.7–30]
(Akaike, 1977). φ): Hannan Criterion (φ φ = L* + cmln(lnn) [1.7–31] (Hannan, 1980), where c is a constant larger or equal to 2 (Carrera & Neuman, 1986a). Kashyap Criterion (dM): dM = L* + mln(n/2π) + ln|FM|
[1.7–32]
(Kashyap, 1982), where FM is the Fisher information matrix. The Kashyap criterion minimizes the average probability of selecting the wrong model among a set of alternatives. The best model is the one that minimizes AIC, BIC, φ, or dM. The Akaike and Bayesian information criteria, or Hannan and Kashyap criteria, all penalize for
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149
adding fitting parameters; that is, everything else being equal, the model with the smallest number of parameters is preferred. r2 Value. An important measure of the goodness of fit is the r2 value for reβ), values: gression of observed, qi*, vs. fitted, qi(β {Σwiq*i qi − [(Σwiq*i Σwiqi)/Σwi]}2 r2 = ________________________________________ *2 {Σwiqi − [(Σwiq*i )2/Σwi]}{Σwiq2i − [(Σwiqi)2/Σwi]}
[1.7–33]
The r2 value is a measure of the relative magnitude of the total sum of squares associated with the fitted equation, with a value of 1 indicating a perfect correlation between the fitted and observed values. Of the above goodness of fit criteria, the AE and the r2 value only relate observed and calculated quantities, whereas the RMSE, AIC, BIC, φ, or dM also take the number of optimized parameters into consideration. 1.7.6 Examples and Optimization Programs Many curve fitting and/or parameter optimization codes have been developed in the past two decades. One of the most widely used codes is the RETC (RETention Curve) program (van Genuchten et al., 1991) for estimating parameters in the soil water retention curve and hydraulic conductivity functions of unsaturated soils. The RETC program uses the parametric models of Brooks–Corey (Brooks & Corey, 1966) and van Genuchten (1980) to represent the soil water retention curve, and the theoretical pore-size distribution models of Mualem (1976) and Burdine (1953) to predict the unsaturated hydraulic conductivity function from observed soil water retention data (see Sections 3.3.4 and 3.6.3). Figure 1.7–1 shows one application in which RETC was used to simultaneously fit six hydraulic parameters to observed retention and conductivity data of crushed Bandalier Tuff (Abeele, 1984; van Genuchten et al., 1991). The observed hydraulic data were obtained by means of an instantaneous profile type drainage experiment involving an initially saturated 6-m-deep and 3-m-diameter caisson (lysimeter), as well as from independent laboratory analyses at relatively low water contents (Table 1.7–1). The soil hydraulic properties were described using the van Genuchten– Mualem model (van Genuchten, 1980) (see also Section 3.3.4): θ(hm) − θr 1 Se(hm) = ________ = __________ θs − θr (1 + |αhm|n)m
[1.7–34]
K(θ) = KsSel[1 − (1 − Se1/m)m]2
[1.7–35]
where θ is the volumetric water content (L3 L−3), hm is the soil water matric head (L), K is the hydraulic conductivity (L T−1), Se is effective fluid saturation (-), Ks is the saturated hydraulic conductivity (L T−1), θr and θs denote the residual and saturated water contents (-), respectively; l is the pore-connectivity parameter (-), and α (L−1), n (-), and m (= 1 − 1/n) (-) are empirical shape parameters. The above hydraulic functions contain six unknown parameters: θr, θs, α, n, l, and Ks.
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Fig. 1.7–1. Observed and fitted unsaturated soil hydraulic functions for crushed Bandalier tuff. The (a) calculated retention and (b) hydraulic conductivity curves are based on the van Genuchten (1980) model.
The objective function was defined as a weighted least-squares problem as follows β) = Φ(β
n1
n2
Σ w [θ*(hm,i) − θ(hm,i,ββ)]2 + W i=1 Σ wi[lnK*(θi) − lnK(θi,ββ)]2 i=1 i
[1.7–36]
where n1 and n2 are numbers of retention and hydraulic conductivity data pairs, respectively, θ*(hm,i) is the measured water content at the matric head hm,i, K*(θi) is the measured hydraulic conductivity for the water content θi, and W is the weight that insures that proportional weight is given to the two different types of data; that is, it corrects for the difference in number of data points and for the effect of having different units for θ and K (van Genuchten et al., 1991): n2
W = [n2
n2
Σ wiθ*(hm,i)]/[n1 i=1 Σ wi|ln K*(θi)|] i=1
[1.7–37]
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Table 1.7–1. Observed retention and conductivity data of crushed Bandalier tuff (Abeele, 1984). Matric head
Water content
Water content
Hydraulic conductivity cm d−1
cm −293.9 −322.5 −409.6 −453.2 −596.6 −641.5 −801.5 −860.1 −949.7 −1192.0 −1298.0 −1445.0 −1594.0 −1760.0 −1980.0 −16.9 −25.1 −27.2 −43.7 −61.5 −64.0 −66.4 −79.8 −85.2 −91.6 −91.2 −98.6 −104.8 −118.2 −122.4 −136.2 −142.6 −142.0 −150.2 −160.7 −169.9 −180.9 −190.9 −201.8 −204.1 −228.1 −232.6 −234.0 −229.3 −263.2 −287.3 −307.7
Laboratory data
Laboratory data 0.165 0.162 0.147 0.139 0.127 0.125 0.116 0.113 0.109 0.103 0.101 0.0988 0.0963 0.0915 0.0875
Caisson data 0.319 0.313 0.294 0.289 0.294 0.268 0.257 0.264 0.257 0.257 0.248 0.239 0.241 0.237 0.236 0.219 0.226 0.222 0.222 0.215 0.200 0.212 0.208 0.197 0.183 0.191 0.184 0.182 0.177 0.177 0.170 0.165
0.0859 0.0912 0.0948 0.0982 0.102 0.108 0.114 0.125 0.140 0.161
0.000411 0.000457 0.000719 0.00087 0.00149 0.00314 0.00377 0.00635 0.018 0.0303 Caisson data
0.165 0.169 0.175 0.177 0.180 0.183 0.184 0.191 0.196 0.199 0.201 0.205 0.210 0.214 0.215 0.218 0.219 0.219 0.224 0.234 0.235 0.239 0.239 0.247 0.256 0.256 0.256 0.262 0.267 0.288 0.293 0.294 0.313 0.320 0.330
0.0262 0.0317 0.0548 0.0852 0.0623 0.0940 0.0691 0.114 0.132 0.194 0.133 0.140 0.160 0.267 0.161 0.418 0.273 0.185 0.339 0.550 0.400 0.878 0.680 0.769 0.804 1.219 1.394 1.573 2.335 3.286 1.650 3.143 6.275 9.715 12.83
Although this type of weighting, as well as the logarithmic transformation of conductivities, has no support in the maximum-likelihood theory (Hollenbeck et al., 2000), it proved to be useful for a simultaneous fitting of retention and hydraulic
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Fig. 1.7–2. Breakthrough curves for nonreactive tritium as analyzed with the two-region physical nonequilibrium model.
conductivity data. The optimized parameters were as follows: θr = 0.028, θs = 0.325, α = 0.0113 cm−1, n = 1.58, l = 0, and Ks = 19.5 cm d−1. Effluent curves derived from the laboratory miscible displacement column experiments often have been analyzed using the CFITIM (van Genuchten, 1981), CXTFIT (Parker & van Genuchten, 1984; Toride at al., 1995), and STANMOD (Šimçnek et al., 1999b) codes. These programs may be used to solve the inverse problem by fitting analytical solutions of theoretical transport models, based on the advection dispersion equation, to experimental results (see Sections 6.3 through 6.5). Figure 1.7–2 shows measured and fitted breakthrough curves of a nonreactive (3H2O) solute for transport through a Glendale clay loam soil. A tritiated water pulse of 3.102 saturated pore volumes was applied to a 30-cm-long column, with the breakthrough curve being determined from the effluent (Table 1.7–2). An analytical solution for a two-region (mobile–immobile) physical nonequilibrium model (van Genuchten, 1981) was used for the analysis (see Eq. [6.3–42] in Section 6.3). The objective function was defined as the least-squares problem β) = Φ(β
n
Σ wi[c*(ti) − c(ti,ββ)]2
i=1
[1.7–38]
with weights wi equal to one. With the pore water velocity known (v = 37.5 cm d−1) and assuming that the retardation factor R is equal to 1 for 3H2O, only three parameters were optimized against the breakthrough curve: the dispersion coefficient D (= 15.6 cm2 d−1), the dimensionless variable β (= 0.823) for partitioning mobile and immobile water in nonequilibrium transport models, and the dimensionless mass transfer coefficient ω (= 0.870). Unfortunately, the optimized parameters are highly correlated with a positive correlation for parameters D and β with the correlation coefficient RDβ = 0.96, and a negative correlation for parameters D and ω with RDω = −0.93, and β and ω with Rβω = −0.985. The high correlation between optimized parameters leads to a relatively large uncertainty of final parameter estimates, cal-
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Table 1.7–2. Measured breakthrough curve of a nonreactive solute (3H2O) for transport through a Glendale clay loam soil (fine-silty, mixed, superactive, calcareous, thermic Typic Torrifluvents). Pore volume
Concentration
Pore volume
Concentration
0.512 0.599 0.686 0.73 0.817 0.904 0.992 1.079 1.166 1.253 1.34 1.428 1.558 1.646 1.754 2.016 2.604 3.125
0.001 0.016 0.082 0.138 0.296 0.465 0.593 0.685 0.764 0.806 0.85 0.901 0.915 0.923 0.947 0.967 0.981 1
3.342 3.516 3.712 3.842 3.951 4.038 4.125 4.255 4.386 4.516 4.777 5.037 5.385 5.818 6.251 6.791 7.331 7.439
0.986 1.015 0.971 0.838 0.638 0.48 0.353 0.236 0.166 0.118 0.066 0.038 0.018 0.008 0.004 0.002 0.006 0.0003
culated using the Student’s t distribution with the 95% confidence intervals, as follows: D ∈ (7.13, 24.0 cm2 d−1), β ∈ (0.76, 0.89), and ω ∈ (0.33, 1.41). Figure 1.7–3 shows the contours of the objective function, Eq. [1.7–38], calculated at the cross-sections through the global minimum. Notice the long valleys in the horizontal direction in the D–β plane and in the vertical direction in the β–ω plane, which explains the large confidence intervals for the D and ω variables, respectively. Also notice that the contours around the minimum are only approximately elliptical. Soil hydraulic parameters are increasingly being estimated from transient variably saturated flow experiments (see Section 3.6.2). Several optimization codes have been developed that can be used only for specific applications, such as one-step (Kool et al., 1985b) or multistep outflow (van Dam et al., 1994; Chen et al., 1999) experiments. More versatile codes that can be applied to a wider range of problems with various initial and boundary conditions and several different soil layers have also been developed (Kool & Parker, 1987; Šimçnek et al., 1998, 1999a). The SFIT model (Kool & Parker, 1987) may be used to estimate soil hydraulic parameters, while the HYDRUS-1D and HYDRUS-2D models (Šimçnek et al., 1998, 1999a) can estimate simultaneously or independently both soil hydraulic and solute transport parameters from one- and two-dimensional experiments, respectively. The SFIT and HYDRUS models also consider hysteresis in the unsaturated soil hydraulic properties. In addition to codes designed specifically for estimation of soil hydraulic properties, general optimization codes, such as PEST (Doherty, 1994), LM-OPT (Clausnitzer & Hopmans, 1995), and UCODE (Poeter & Hill, 1998), can be coupled with any parameter estimation problem. Alternatively, one can also interface optimization algorithms as listed in Press et al. (1992) with specific flow and transport simulation codes. In addition, software such as MS EXCEL can now be used simply and conveniently to solve a variety of parameter estimation problems (e.g., Wraith
Fig. 1.7–3. Contours of the objective function (Eq. [1.7–38]) as calculated at the cross-sections through the global minimum.
154 CHAPTER 1
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& Or, 1998). Many examples illustrating the versatility of parameter optimization methods for estimating soil hydraulic and solute transport parameters are presented in Sections 3.6.2 and 6.6, respectively. 1.7.7 Discussion During the past two decades, curve and model fitting have become relatively standard techniques to analyze a variety of biophysicochemical data involving the unsaturated zone. Such curve fitting codes as RETC and CXTFIT are now widely used for analyzing experimental data, such as retention and conductivity data, breakthrough curves, as well as other data. A major recent promise in applying parameter estimation and nonlinear fitting techniques is the effective coupling of optimization and advanced numerical codes, and applying these codes to complex transient flow and transport experiments (see Sections 3.6.2 and 6.6). Many commonly used inversion methods (e.g., CXTFIT) are based on analytical solutions for transport or Wooding’s solution for tension infiltrometry that require relatively simple initial and boundary conditions. The resulting optimization approach then often requires experiments that repeatedly achieve steady-state or equilibrium conditions. The form of the hydraulic and transport properties is also often severely restricted using analytical methods. Application of inverse modeling techniques can alleviate some of these difficulties. Coupling of parameter estimation and nonlinear fitting techniques with numerical models provides greater flexibility by allowing different experimental boundary and initial conditions. This is important especially for field experiments where it is difficult and expensive to control the initial and boundary conditions on a large scale; parameter estimation methods permit conditions encountered in the field to be analyzed more easily. In addition, more general soil hydraulic and transport property models can be used to better represent field behavior. Analytical methods were generally preferred some 20 yr ago because of limitations in numerical methods and computer technology, thus compromising experimental procedures for the purpose of keeping the mathematics as simple as possible. Recent advances in computer software and hardware now make it possible to couple sophisticated parameter estimation algorithms with state-of-the-art, integrated numerical flow and transport codes that do not sacrifice experimental aspects for numerical expediency. The limitations of this type of parameter estimation are mostly related to factors that determine the well-posedness of the solution. For example, which variables to measure and which parameters to optimize is not known a priori for flow and transport experiments. Also, it is not obvious, in advance, whether or not a given type of experiment, or a given data set, will result in a well-posed inverse problem and how many parameters can be uniquely estimated. Since it is not always clear what is causing nonunique or unstable solutions, an inverse problem always requires in-depth analysis to determine whether it is well-posed and, if not, the cause of the ill-posedness (see also Section 3.6.2). Because of its generality (in terms of the definition of the objective function, the possible combination of boundary and initial conditions, options for considering multilayered systems, and flexibility in selecting optimized parameters), pa-
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rameter estimation by inverse modeling can be an extremely useful tool for analyzing a broad range of steady-state and transient, laboratory and field, flow and transport experiments. 1.7.8 References Abbaspour, K.C., M.Th. van Genuchten, R. Schulin, and E. Shläppi. 1997. A sequntial uncertainty domain inverse procedure for estimating subsurface flow and transport parameters. Water Resour. Res. 33:1879–1892. Abeele, W.V. 1984. Hydraulic testing of crushed Bandelier tuff. Report no. LA-10037-MS. Los Alamos National Laboratory, Los Alamos, NM. Akaike, H. 1974. A new look at statistical model identification. IEEE Trans. Automat. Contr., AC-19, 716–722. Akaike, H. 1977. On entropy maximization principle. p. 27–41 In P.R. Krishnaiah (ed.) Applications of statistics. North-Holland, Amsterdam, the Netherlands. Andrews, D.F., P.J. Bickel, F.R. Hampel, P.J. Huber, W.H. Rogers, and J.W. Tukey. 1972. Robust estimates of location: Survey and advances. Princeton Univ. Press, Princeton, NJ. Bard, Y. 1974. Nonlinear parameter estimation. Academic Press, New York, NY. Barhen, J., V. Protopopescu, and D. Reister. 1997. TRUST: A deterministic algorithm for global optimization. Science 276:1094–1097. Beck, J.V., and K.J. Arnold. 1977. Parameter estimation in engineering and science. John Wiley & Sons, New York, NY. Brooks, R.H., and A.T. Corey. 1966. Properties of porous media affecting fluid flow. J. Irrig. Drain. Div. Am. Soc. Civ. Eng. 92:61–88. Burdine, N.T. 1953. Relative permeability calculations from pore-size distribution data. Petrol. Trans. Am. Inst. Min. Eng. 198:71–77. Carrera, J., and S.P. Neuman. 1986a. Estimation of aquifer parameters under transient and steady state conditions. 1. Maximum likelihood method incorporating prior information. Water Resour. Res. 22:199–210. Carrera, J., and S.P. Neuman. 1986b. Estimation of aquifer parameters under transient and steady state conditions. 2. Uniqueness, stability, and solution algorithms. Water Resour. Res. 22:211–227. Chen, J., J.W. Hopmans, and M.E. Grismer. 1999. Parameter estimation of two-fluid capillary pressuresaturation and permeability functions. Adv. Water Resour. 22:479–493. Clausnitzer, V., and J. W. Hopmans. 1995. Non-linear parameter estimation: LM_OPT. General-purpose optimization code based on the Levenberg–Marquardt algorithm. Land, Air and Water Resources Paper No. 100032. University of California, Davis, CA. Doherty, J. 1994. PEST. Watermark computing. Corinda, Australia. Finsterle, S., and J. Najita. 1998. Robust estimation of hydrologic model parameters. Water Resour. Res. 34:2939–2947. Hannan, E.S. 1980. The estimation of the order of an ARMA process. Ann. Stat. 8:1971–1081. Hollenbeck, K., and K.H. Jensen. 1998. Maximum-likelihood estimation of unsaturated hydraulic parameters. J. Hydrol. 210:1992–205. Hollenbeck, K., J. Šimçnek, and M.Th. van Genuchten. 2000. RETCML: Incorporating maximum-likelihood estimation principles in the hydraulic parameter estimation code RETC. Comput. Geosci. 26:319–327. Hopmans, J.W., and J. Šimçnek. 1999. Review of inverse estimation of soil hydraulic properties. p. 643–659. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, CA. Huber, P.J. 1981. Robust statistics. John Wiley, New York, NY. Kashyap, R.L. 1982. Optimal choice of AR and MA parts in autoregressive moving average models. IEEE Trans. Pattern Anal. Mach. Intel. PAMI-4(2):99–104. Kool, J.B., and J.C. Parker. 1987. Estimating soil hydraulic properties from transient flow experiments: SFIT user’s guide. Electric Power Research Institute Report, Palo Alto, CA. Kool, J.B., and J.C. Parker. 1988. Analysis of the inverse problem for transient unsaturated flow. Water Resour. Res. 24:817–830. Kool, J.B., J.C. Parker, and M.Th. van Genuchten. 1985. ONESTEP: A nonlinear parameter estimation program for evaluating soil hydraulic properties from one-step outflow experiments. Bull. 853. Virginia Agric. Exp. Stn., Blacksburg, VA.
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Kool, J.B., J.C. Parker, and M.Th. van Genuchten. 1987. Parameter estimation for unsaturated flow and transport models—A review. J. Hydrol. 91:255–293. Marquardt, D.W. 1963. An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11:431–441. McLaughlin, D., and L.R. Townley. 1996. A reassessment of groundwater inverse problem. Water Resour. Res. 32:1131–1161. Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513–522. Pan, L., and L. Wu. 1998. A hybrid global optimization method for inverse estimation of hydraulic parameters: Annealing-simplex method. Water Resour. Res. 34:2261–2269. Parker, J.C., and M.Th. van Genuchten. 1984. Determining transport parameters from laboratory and field tracer experiments. Bull. 84-3. Virginia Agric. Exp. Stn., Blacksburg, VA. Poeter, E.P., and M.C. Hill. 1998. Documentation of UCODE, A computer code for universal inverse modeling. Water-Resources Investigations Report 98-4080. U.S. Geological Survey, Denver, CO. Press, W.H., B.P. Flannery, S.A. Teukolsky, W.T. Vetterling. 1992. Numerical recipes, The art of scientific computing. 2nd ed. Cambridge University Press, Cambridge, Great Britain. Russo, D., E. Bresler, U. Shani, and J. C. Parker. 1991. Analysis of infiltration events in relation to determining soil hydraulic properties by inverse problem methodology. Water Resour. Res. 27:1361–1373. Šimçnek, J., M. Šejna, and M.Th. van Genuchten. 1998. The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media. Version 2.0. IGWMC-TPS-70. International Ground Water Modeling Center, Colorado School of Mines, Golden, CO. Šimçnek, J., M. Šejna, and M.Th. van Genuchten. 1999a. The HYDRUS-2D software package for simulating the two-dimensional movement of water, heat, and multiple solutes in variably-saturated media. Version 2.0. IGWMC-TPS-56. International Ground Water Modeling Center, Colorado School of Mines, Golden, CO. Šimçnek, J., M.Th. van Genuchten, M. Šejna, N. Toride, and F. J. Leij. 1999b. The STANMOD computer software for evaluating solute transport in porous media using analytical solutions of convection-dispersion equation. Versions 1.0 and 2.0. IGWMC-TPS-71. International Ground Water Modeling Center, Colorado School of Mines, Golden, CO. Toride, N., F.J. Leij, and M.Th. van Genuchten. 1995. The CXTFIT code for estimating transport parameters from laboratory or field tracer experiments. Version 2.0. Research Report no. 137. U.S. Salinity Laboratory, USDA-ARS, Riverside, CA. van Dam, J.C., J.N. M. Stricker, and P. Droogers. 1994. Inverse method to determine soil hydraulic functions from multistep outflow experiment. Soil Sci. Soc. Am. Proc. 58:647–652. van Genuchten, M.Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892–898. van Genuchten, M.Th. 1981. Non-equilibrium transport parameters from miscible displacement experiments. Research Report no. 119. U.S. Salinity Laboratory, USDA-ARS, Riverside, CA. van Genuchten, M.Th., F.J. Leij, and S.R. Yates. 1991. The RETC code for quantifying the hydraulic functions of unsaturated soils. EPA/600/2-91-065. USEPA, Office of Research and Development, Washington, DC. Vrugt, J.A., J.W. Hopmans, and J. Šimçnek. 2001. Calibration of a two-dimensional root water uptake model for a sprinkler-irrigated almond tree. Soil Sci. Soc. Am. J. 65:1027–1037. Wraith, J.M., and D. Or. 1998. Nonlinear parameter estimation using spreadsheet software. J. Nat. Resour. Life Sci. Educ. 27:13–19. Yeh, W.W-G. 1986. Review of parameter identification procedures in groundwater hydrology: The inverse problem. Water Resour. Res. 22:95–108.
Published 2002
1.8 Newer Application Techniques ALEX. B. MCBRATNEY, The University of Sydney, Sydney, Australia ALISON N. ANDERSON, New South Wales Farmers’ Association, Sydney, Australia R. MURRAY LARK, Silsoe Research Institute, Bedford, United Kingdom INAKWU O. ODEH, Australian Cotton Cooperative Research Centre, The University of Sydney, Sydney, Australia
Three new mathematically based techniques that have emerged as soil science tools in the last decade are described here. They are fractals, fuzzy sets, and wavelets, and they are presented in decreasing order of development and application in soil science to date.
1.8.1 Fractal Dimensions ALISON N. ANDERSON, New South Wales Farmers’ Association, Sydney, Australia ALEX. B. MCBRATNEY, The University of Sydney, Sydney, Australia
Most shapes found in nature are irregular, being largely devoid of straight lines. Classical Euclidean geometry, with its straight lines, spheres, and tetrahedra, does not describe such shapes. Soil structure is a case in point. Fractal geometry, introduced by Benoit B. Mandelbrot in 1975 (Mandelbrot, 1975), can be used to quantitatively describe the shape and irregularity of natural objects by estimating their fractal dimension. The dimension of a fractal object is not an integer. For example (as given in Bartoli et al., 1998), the surface fractal dimension of a fractal crumpled plane is between the Euclidean dimension of a plane (2 for a completely smooth surface) and that of a volume (3 for an infinitely crumpled plane). Since its formal introduction by Mandelbrot, fractal geometry has been applied to a number of scientific disciplines. Fractal geometry has been used to describe the architecture of soil and has aided in the description and prediction of physical, chemical, and biological processes occurring in the soil. It is hoped that fractal models may lead to a more accurate description of soil in process models than methods of classic geometry (Crawford et al., 1999). Fractal geometry also has a role in the quantification of the spatial variability of soil attributes. Due to the application of fractal geometry to soil science being relatively recent, the interpretation of fractal dimensions, how they are measured, and their ap159
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plication are varied. A variety of techniques used to measure fractal dimensions, and possible applications, have been described by Anderson et al. (1998) and Crawford et al. (1999). In this subsection we cover fractal theory and methods often used in soil science. Some specific examples relating to soil structure description, modeling soil physical processes, and describing soil spatial variability are given. Detailed treatment of the theory of fractal geometry is given in Mandelbrot (1982), Feder (1988), and Kaye (1989), for example. For more information about the applications of fractal geometry to soil science the reader is referred to Giménez and Rawls (1996), Senesi (1996), Giménez et al. (1997), Perfect (1997), Anderson et al. (1998), Baveye et al. (1998b), Hallett et al. (1999), and Pachepsky et al. (2000a). Pachepsky et al. (2000b) have compiled a bibliography on applications of fractals in soil science.
1.8.1.1 Theory A fractal object appears morphologically the same, regardless of the scale of observation. Mandelbrot (1989) describes fractals as “shapes whose roughness and fragmentation neither tend to vanish, nor fluctuate up and down, but remain essentially unchanged as one zooms in continually and examination is refined”. This is known as scale invariance or scaling. Although natural objects are not fractals in the strict mathematical sense, they often have similar features over a range of scales. Fractals may provide a useful characterization of soil heterogeneity between lower and upper length scale limits. The Menger sponge (Fig. 1.8–1), a wellknown mathematical fractal, has at least some features that may be useful to describe the properties of soil aggregates (Baveye & Boast, 1998). There is more that one type of fractal dimension that can be used to describe a fractal object. Practitioners should be clear about which dimension is appropriate for a given application. Fractal dimensions that have been used to characterize soil structure are discussed below. 1.8.1.1.a Mass Fractal Dimension Mass fractals have a nonuniform, self-similar mass distribution. For mass fractals the mass (M) inside a characteristic radius (r) scales according to M ∝ rDm
[1.8–1]
where Dm is the mass fractal dimension. The mass fractal dimension is less than the Euclidean dimension in which the fractal is embedded (the embedding dimension, de). The embedding dimension is equal to 2 in two-dimensional space and equal to 3 in three-dimensional space. The embedding dimension for a soil aggregate is 3. The embedding dimension for an image of soil structure (such as that produced from a soil thin section) is 2. A characteristic of mass fractals is scale-variant bulk density (ρb), since their mass scales with size or length scale. An equation given in Orbach (1986) relates density to the mass fractal dimension: ρb(r) = M(r)/V(r) = BrDm/Crde ∝ r(Dm−de)
[1.8–2]
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Fig. 1.8–1. The Menger sponge (after Mandelbrot, 1982).
where M is mass, V is volume, B varies according to the lacunarity, and C is a constant. Lacunarity defines the relative abundance of gaps (pores) in a mass fractal. The reader is referred to Rieu and Perrier (1998) for more information about lacunar fractal objects. Because Dm ≤ de, the density of a mass fractal decreases with increasing length scale. Chepil (1950), Lin (1971), Young and Crawford (1991), and Eghball et al. (1993) have reported decreases in bulk density for soil aggregates with increasing aggregate size (Fig. 1.8–2). This suggests that a mass fractal model of soil may be appropriate over at least some ranges in length scale.
Fig. 1.8–2. Bulk density–aggregate size relationship for a fine sandy loam (from a nonlinear fit to data from Chepil, 1950).
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Fig. 1.8–3. Two soil structures, pore space in black, with similar values of Dmp (the mass fractal dimension of the pore space), but with different values of dp (the spectral dimension of the pore space) due to structure (a) having more macropores that are connected. Values of Dmp are 1.786 for (a) and 1.785 for (b). Values of dp are 1.568 for (a) and 1.468 for (b). The side length of each image is 50 mm.
1.8.1.1.b Spectral Dimension The spectral dimension (d) is a measure of the connectivity of a fractal pathway. A large value of d reflects a more continuous, less tortuous pathway. Mass fractals with similar values of Dm, may have very different values of d (Fig. 1.8–3). Intuitively, the spectral dimension has an important role to play in fractal models of physical processes such as gas diffusion. 1.8.1.1.c Surface Fractal Dimension The surface fractal dimension (Ds) describes the irregularity or ruggedness of a perimeter or surface. A straight line is nonfractal and has a Euclidean dimension of 1. An irregular line displaying fractal scaling will have a value of Ds that approaches 2 as it becomes more space-filling. Similarly, a flat surface has a Euclidean dimension of 2, while a fractal surface has a value of Ds that increases towards 3 as the surface becomes increasingly volume-filling. If we were to step along an irregular perimeter (e.g., a coastline) with constant stride length x, the polygon created would have a perimeter that is an estimate of the actual perimeter length. The perimeter of the polygon, P(x), can be calculated by multiplying the number of steps taken by x. If we step along the irregular perimeter again with smaller stride length, we would notice that the estimated perimeter length would increase. The estimated perimeter would continue to increase as we reduce x, because fine-scale irregularity is included in the estimated perimeter as smaller steps are taken. This phenomenon is described by: P(x) ∝ x1−Ds
[1.8–3]
where Ds is a constant. Although an equation of this form was first proposed by Richardson (1961) following his work on the lengths of river banks, coastlines and
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frontiers, it was Mandelbrot (1967) that proposed that Ds be interpreted as a dimension, despite not being an integer. If we were to carry out the same procedure on a smooth perimeter such as a circle, the estimated perimeter would quickly become a scale-independent (i.e., independent of stride length) value. That is, P(x) would remain unchanged, regardless of the length x. As an example, a rugged coastline such as that found in the southern part of Norway has a value of Ds equal to 1.52 (Feder, 1988), while a smooth coastline, such as South Africa’s, has a value of Ds equal to 1.02 (Mandelbrot, 1967).
1.8.1.1.d Fragmentation Dimension Fragmentation is a common occurrence in nature. In agricultural soil, tillage, and planting operations result in soil fragmentation (Perfect, 1997). The distribution of fragment sizes will be related to the distribution of joints and preexisting planes of weakness (Turcotte, 1989). A number-size distribution is fractal when the number of fragments or objects, N, with a characteristic size greater than r scales with the relation: N ~ r−Df
[1.8–4]
where Df is the fragmentation fractal dimension (Turcotte, 1986, 1989). When zones of weakness exist at all scales, the fragmentation mechanism is scale-invariant and fits the power-law distribution given in Eq. [1.8–4]. Perfect (1997) reviews fractal models for the fragmentation (caused by multiple fractures at different length scales) of heterogeneous materials. Models are available for fragmentation of classical aggregates, aggregates with fractal pore space, and aggregates with fractal surfaces. These models, as reviewed by Perfect (1997), use probabilities of failure to predict the fragment-size distribution from knowledge of the geometrical properties of the original material. Mostly, Df has been used as an indicator of fragmentation caused by different tillage operations or as a result of different cropping strategies (e.g., Perfect & Kay, 1991; Eghball et al., 1993; Perfect et al., 1993; Rasiah et al., 1995). Highly fragmented soil, dominated by small aggregates, has a large value of Df, compared with soil dominated by large aggregates.
1.8.1.2 Quantifying Soil Structure Using Fractal Geometry Crawford et al. (1999) discuss the two possible ways of measuring soil structure: 1. To measure structural units directly from images of soil structure (e.g., Bartoli et al., 1991; Crawford et al., 1993; Peyton et al., 1994; Anderson et al., 1996; Gomendy et al., 1999) 2. To quantify structure based on an indirect evaluation of structural parameters from data on some soil process which is dependent on the underlying structure (e.g., Bartoli et al., 1991; Gomendy et al., 1999; Perfect, 1999)
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1.8.1.2.a Direct Methods for Estimating Fractal Dimensions Image analysis techniques can be used to estimate the mass, spectral, surface, and fragmentation fractal dimensions of soil. A variety of techniques that have been used in soil science research are discussed below. Researchers have developed computer programs to carry out fractal analyses on digitized images of soil structure. Images of soil structure are captured (e.g., by scanning a slide picture of a soil thin section), segmented into soil matrix and soil pore, and the resulting black and white soil image made binary. Each pixel in the image will be designated solid or pore space. The resolution of the image to be analyzed can have a pronounced influence on the results of a computerized fractal analysis (Anderson, 1997; Baveye et al., 1998a). Images of the highest possible resolution should be used for fractal analysis. Estimating the Mass Fractal Dimension. The mass fractal dimension (Dm) of soil structure is often estimated using box-counting methods (Pfeifer & Obert, 1989). Box-counting methods have been employed by Hatano et al. (1992), Hatano and Booltink (1992), Booltink et al. (1993), Peyton et al. (1994), Anderson et al. (1996), Bartoli et al. (1998), and Gomendy et al. (1999). Although box-counting methods used to estimate Dm in soil studies may vary slightly, the basic principle is the same for all of them. Box-counting involves covering a binary soil image with a grid, consisting of square boxes of size x. Grids with progressively larger boxes are placed over the soil image. As described in Anderson et al. (1996), the number of boxes (N) containing the phase of interest (e.g., soil matrix) is counted for each box size (x). As x is increased, N(x) decreases. The mass fractal dimension is estimated from a plot of ln N(x) vs. ln x, where Dm is equal to the absolute value of the slope. Similar to box-counting, Dm can be estimated by counting the number of pixels (N) residing in the phase of interest within radius (r) of some origin in the network. Crawford et al. (1993) used this method, with Dm being estimated from a plot of log N(r) vs. log r. The box-counting method can be used to analyze either the solid phase or the pore space. However, the solid phase and pore space cannot be simultaneously fractal (Crawford & Matsui, 1996; Rieu & Perrier, 1998). Crawford and Matsui (1996) show that it is most likely the solid phase that scales as a fractal, but that both phases can be characterized by power-law scaling exponents over a limited range in scale, making it difficult to distinguish between a fractal and its complement. Values, obtained by image analysis, for the mass fractal dimension of the pore space and solid phase vary considerably in the literature. Values will vary with soil type, image resolution, and scale. Ranges of values for different studies are given in Anderson et al. (1998). For isotropic structures the three-dimensional value of Dm can be approximated as 1 + Dm (obtained from a digitized soil image). However, isotropy is likely to be the exception rather than the rule for soil structure. A method for estimating the mass fractal dimension in three-dimensional space for anisotropic structures is given in Hatano and Booltink (1992).
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Estimating the Spectral Dimension. For any fractal network, the spectral dimension may be estimated by constructing a random walk through the network. From Orbach (1986): S(t) ∝ [(td/Dm)1/2]Dm ≡ td/2
[1.8–5]
where S(t) is the number of distinct sites visited in time t and is determined solely by d. The spectral dimension (d) has been estimated for both the pore space and solid phase of soil images (Crawford et al., 1993; Anderson et al., 1996; Anderson, 1997). Assuming the pore space is the phase of interest, a random pore pixel in a binary soil image is chosen from which a random walk by a particle is started. The walk progresses as the particle moves (steps) to any of the pore pixels that surround it (an 8-connected walk). Each step takes one unit of time (t). As the walk progresses, the number of steps taken (n) is counted, as is the number of distinct pixels visited, S(n). A plot of S(n) vs. log(n) yields a line of d/2 (Crawford et al., 1993). The random walk is usually terminated when the particle reaches the boundary of an image. The walk may also be terminated when a specified number (e.g., 100) of null steps (steps taken into pixels that have already been visited) have been taken. This is because it is likely that the particle is in a discrete (unconnected) void. Values of d have been calculated by averaging the results from a number of individual random walks. At present, it has not been specified how many walks are needed to be averaged to obtain a reliable estimate of d. Crawford et al. (1993) based their values of d on the average of 103 to 104 individual random walks. Anderson et al. (1996) and Anderson (1997) based their values on the average of 5000 individual random walks, after finding that values of d did not vary considerably when 2500 or more individual walks were averaged. Most fractals have d < 2 (Havlin & Ben-Avraham, 1987) and values in the range 1 < d < 2 have been reported for soil pore networks (Crawford et al., 1993; Anderson et al., 1996). Estimating the Surface Fractal Dimension. A number of image analysis methods are available to estimate the surface fractal dimension (Ds) of surfaces such as the pore-solid interface in soil images. The stride method discussed in Section 1.8.1.1.c is one method available for estimating Ds. Other methods include the boxcounting technique of Feder (1988) and Pfeifer and Obert (1989), and Minkowski’s sausage logic technique as discussed in Kaye (1989). With box-counting, the number of boxes with side x required to cover a boundary is determined, whereas disks of radius r are drawn around each point of a curve for Minkowski’s sausage logic method. As for the stride method, the estimated perimeter will increase as x and r are made smaller. Equation [1.8–3] can again be used to describe this phenomenon. Dilation and erosion logic (Flook, 1978) can also be used to estimate Ds. Anderson et al. (1996) used the difference (in pixels) between dilated and eroded pore space to estimate Ds for pore–solid interfaces. The pore space is first dilated (at the pore–solid interface) and then eroded for s = 1 to 50 pixels. The difference in the number of pixels in the eroded and dilated pore spaces is counted and divided by 2s. The surface fractal dimension is estimated from a plot of ln(difference in pixel
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count/2s) vs. ln(s). The value of Ds is considered to be equal to 1 plus the absolute value of the slope generated. A perimeter-area (or slit-island) method (Mandelbrot et al., 1984) was used by Hatano et al. (1992), Kamplicher and Hauser (1993), and Pachepsky et al. (1996) to determine Ds. This method is limited to similar nonconnected objects such as macropores (Bartoli et al., 1998). As given by Bartoli et al. (1998), the value of Ds of these objects can be found using the following power-law relationship between area (A) and perimeter (p): A ∝ pDs/2
[1.8–6]
Values of Ds obtained from binary images of soil structure will be in the range 1 ≤ Ds ≤ 2. Values of Ds obtained in two dimensions can be converted to values of Ds in three dimensions by adding 1 if isotropy can be assumed. Different values of Ds, at different scales, have been found (e.g., Pachepsky et al., 1996; Anderson et al., 1996). Estimating the Fragmentation Fractal Dimension. The fragmentation fractal dimension (Df) has been estimated from soil fragmentation data (both aggregate- and particle-size distribution data) by several researchers (e.g., Rieu & Sposito, 1991; Perfect et al., 1992; Eghball et al., 1993). Number-size distributions of fragments can be determined directly or indirectly from the mass-size distribution obtained by sieving and or sedimentation (Perfect, 1997). Perfect (1997) prefers the direct methods because they involve fewer assumptions and they include static and dynamic light scattering, scanning electron microscopy, image analysis, and manual counting (which will be discussed here). When the manual counting method is being used to determine a number-size distribution (as described in Perfect, 1997), fragment numbers are determined for subsamples of known mass from each size fraction. These values can then be multiplied by the total mass of material in each size fraction obtained by sieving. Because fragment sizes are most likely to vary continuously, rather than belong to discrete fragment size distributions, it is suggested that data be expressed as the cumulative number of fragments of length greater than or equal to l/lmax, that is, N(l/lmax), where l is the individual fragment length (approximated by the aperture size of the sieve on which the fragments are collected) and lmax is the length of the largest fragment present (approximated by the aperture size of the sieve on which the largest fragments are collected). The fragmentation fractal dimension can be estimated from a log-log plot of the cumulative number-size distribution (see Eq. [1.8–4]). However, estimations of Df may be biased using the linear method of estimating Df. Perfect et al. (1994) and Rasiah et al. (1995) use nonlinear fitting procedures that give more accurate estimates of Df. 1.8.1.2.b Indirect Methods of Estimating Fractal Dimensions Mercury Intrusion Porosimetry and the Water Retention Curve. Intrusion porosimetry can be used to determine the pore-size distribution of a porous material. Mercury can be forced into a soil sample that has been evacuated of water and gas, with more pores being intruded by Hg as the applied pressure is increased.
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A possible problem with this method is pore deformation as the Hg is forced into the soil. The Washburn (1921) equation relates the applied pressure to pore radius. Equation [1.8–7] (from Friesen & Mikula, 1987) was used by Bartoli et al. (1991) to estimate the surface fractal dimension of soil samples: dVp/dr = r(2−Ds)
[1.8–7]
where Vp is pore volume, r is the average pore radius, and Ds is the surface fractal dimension and is estimated from a double logarithmic plot of dVp/dr vs. r. A deficiency of the method is that in a case where a larger pore is only accessible by smaller pores, the larger pore will be counted as a number of smaller pores rather than a single larger pore. Mercury intrusion porosimetry has been used by Bartoli et al. (1991), Bartoli et al. (1992), Pachepsky et al. (1995), Bartoli et al. (1999), and Gomendy et al. (1999), for example, to estimate Ds. Mercury suction and intrusion experiments have the advantage that they take very little time compared with image analysis techniques (Bartoli et al., 1998). Ahl and Niemeyer (1989) and Gomendy et al. (1999) used the water retention curve to calculate surface fractal dimensions of wetted soil structures. As given in Bartoli et al. (1998), Ahl and Niemeyer (1989) derived the following analytical relation: Vp ∝ P(Ds−3)
[1.8–8]
where Vp is the cumulative pore volume, P is the matric potential (kPa), and Ds is the surface fractal dimension. Perfect (1999) developed a prefractal model for estimating the mass fractal dimension from water retention curves. The most accurate estimates of the mass fractal dimension are obtained when water retention data covering the entire potential range from saturation to zero water content are available. The mass fractal dimension controls the degree of curvature of the curve produced when relative saturation is plotted against the log of potential. Molecular Probing. The surface fractal dimension (Ds) of irregular surfaces can be estimated using adsorption data (e.g., Avnir et al., 1985; Sokolowska, 1989; Okuda et al., 1995). Adsorbates are used to probe the surface of sorbents. A highly irregular surface will adsorb a greater number of small molecules than a more uniform one. It is important that the number of probe molecules of any given size adsorbed by a sorbent is a function of the physical size of the probe only and not of the chemical characteristics of the probe (Okuda et al., 1995). Examples of adsorbates used to probe the surfaces of kaolinite, silica-gel, and sand by Okuda et al. (1995) are n-pentane, benzene, water, and N. Equations that can be used to calculate Ds from molecular probing experiments are given in Avnir et al. (1984). The equation used by Okuda et al. (1995) is: Nm(σ) ~ σ−Ds/2
[1.8–9]
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where Nm(σ) is the number of probe molecules of size σ required to cover the surface of the given sorbent. The value of Ds can be estimated from the slope of the line produced from a plot of Nm(σ) vs. σ on a double logarithmic scale. Okuda et al. (1995) used both the classical Brunauer–Emmett–Teller (BET) equation and a fractal BET equation from Fripiat et al. (1986) to estimate Nm(σ). Anderson et al. (1998) estimated Ds from specific surface area (SSA) data reported in Quirk (1955). The SSA data had been obtained by probing a Miami silt loam (1.2% for OC (Daniells & Larsen, 1991). These vague qualifications are better modeled by fuzzy sets. A symmetrical (Gaussian) model (Eq. [1.8–18]) is fitted to the pH values as: µA(pH) =
exp[−(pH − 7.5)2/3.0] 90
if 4 ≤ pH ≤ 10 otherwise
[1.8–31]
For example, fitting Eq. [1.8–31] to a topsoil pH value of 6.5 will yield a membership grade of a fuzzy subset of optimal pH for crop growth as: µA(pH) = exp[−(6.5 − 7.5)2/3.0] = 0.716531 An asymmetrical version of Eq. [1.8–20] was also fitted to a fuzzy subset of optimal OC values for crop growth as: µA(OC) =
1 exp[−(OC − 1.12)2/0.864] 90
if OC ≥ 1.2 if OC < 1.2 otherwise
[1.8–31]
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Table 1.8–4. Sample pH and organic C (OC) data as soil quality indicators, indicator (fuzzy) membership grades, and their joint-membership grades. Site ID
pH
µA(pH)
OC
µA(OC)
t-norm: (µA(pH))(µA(OC))
t-conorm: (µA(pH) + µA(OC)) − (µA(pH))(µA(OC))
0.978 0.888 1.000 1.000 1.000 0.936 1.000 1.000 0.766 0.941 0.567 0.659
0.806 0.545 0.857 0.966 0.989 0.639 0.981 0.808 0.478 0.916 0.219 0.472
0.996 0.957 1.000 1.000 1.000 0.980 1.000 1.000 0.912 0.998 0.734 0.903
% ed01001 ed03301 ed05801 ed05901 ed08701 ed09801 ed13901 ed15801 ed16301 ed17301 ed20301 ed21001
8.26 6.29 6.82 7.18 7.32 8.57 7.74 6.70 8.69 7.22 5.81 6.50
0.825 0.614 0.857 0.966 0.989 0.683 0.981 0.808 0.624 0.974 0.386 0.717
1.06 0.88 1.48 1.29 2.37 0.96 1.53 2.41 0.72 0.97 0.50 0.60
Fitting this model to an OC value of 0.65% yields µA(0.65) = exp[−(0.6 − 1.12)2/0.864] = 0.659241 This type of analysis can be applied to an array of soil data used as soil quality indicators to produce a corresponding array of soil quality (fuzzy) membership grades that could be combined into a joint fuzzy membership grade for mapping of the overall soil quality in a specific area. Table 1.8–4 shows examples using another subset of pH and OC data from the Edgeroi area of New South Wales, Australia. The joint membership grade can be obtained by the so-called t-norm and tconorm, which are respective variants of maximum and minimum operators. The application of these operators here (Table 1.8–4) is similar to the joint fuzzy membership function used for land evaluation by Burrough et al. (1992). Other examples of fuzzy sets application to land evaluation can be found in Chang and Burrough (1987), Burrough (1989), and Burrough et al. (1992). More recent examples can be found in de Gruijter et al. (1997). In conclusion, it is obvious that there is great potential for applying fuzzy set theory to soil science. In addition to examples presented here, other soil concepts or systems may be modeled, simulated, and even replicated with the help of fuzzy systems, not the least of which is human reasoning itself.
1.8.3 Wavelet Analysis R. MURRAY LARK, Silsoe Research Institute, Bedford, United Kingdom ALEX. B. MCBRATNEY, The University of Sydney, Sydney, Australia
Wavelet analysis has been developed since the early 1980s into a formidable body of theory with a wide range of applications. Hydrologists and others have applied it to their problems, some of which have involved analysis of data on soils. However, for the more central problems of pedometrics, wavelet analysis is in its early
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days. McBratney (1998) identified it as a method which might be used for analyzing soil information across a range of scales and Lark and Webster (1999) have written an introduction to the subject aimed at soil scientists and illustrated with an analysis of a set of soil data. Here the basic principles of wavelet analysis are described and continuous and discrete wavelet transforms are illustrated with some data from a soil transect. Given the costs of field sampling and laboratory analysis, most methods in pedometrics are designed for analysis of relatively small data sets. The development in recent years of sensors coupled to navigational systems permit the collection of large and more or less regular multidimensional arrays of data on soil properties— physical and chemical (e.g., Viscarra Rossel & McBratney, 1997; Kitchen et al., 1999). This opens up the possibility of using mathematical methods of analysis developed for signal and image processing. Such methods have been used in the past on large data sets collected for research (Webster, 1977), but may increasingly now be used on data sets generated routinely by sensor systems. A common approach to the analysis of signals and images is to decompose them on some appropriate basis functions. In short, the soil variable is treated as f(x), a function of x, the location in a space. The analysis builds f(x) out of the basis functions, which are basic mathematical building blocks. For a proper treatment of the subject the reader is referred to works on functional analysis such as that of Kreyszig (1978). Note that the theory in this discussion is developed and illustrated for one spatial dimension, but can readily be extended to two dimensions. One of the best known methods in signal processing is the Fourier transform in which f(x) consists of sinusoidal components (sine and cosine functions) with different frequencies. The phase and amplitude of the component of frequency s is contained in the Fourier coefficient F(s), obtained by the Fourier transform ∞
F(s) = ∫−∞ f(x)exp(−2iπxs)dx
[1.8–32]
The reader is reminded that the term in the integral after f(x) contains sine and cosine functions determined by Euler’s formula for complex exponents, which may be found in standard mathematics textbooks. It is possible to reconstruct f(x) from the Fourier coefficients by an inverse transform ∞
f(x) = ∫−∞F(s)exp(2iπxs)ds
[1.8–33]
The Fourier transform (Eq. [1.8–32]) and its inverse (Eq. [1.8–33]) may be thought of as defining a relationship between alternative representations of a data set. The representation of the data as a function of location, f (x), (in the spatial domain) is an alternative to the representation of the data as a complex function of frequency, F(s), (in the frequency domain). Transforming data from the spatial to the frequency domain can be useful in various ways. It provides a natural method for building up smoothed approximations to the original data by removing components of different frequency. These filtering methods allow spurious or irrelevant components (usually high frequency) to be removed from the data. Furthermore, if the data are regarded as a sample of a spatial variable, then there are statistical analyses which may be conveniently performed in the frequency domain. Such methods have been
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long-established in the analysis of time series (Box & Jenkins, 1976), and are related to geostatistical methods. One useful method is spectral analysis, partitioning the variance of the data into components of different spatial frequency. Webster (1977) was able to use the spectrum to relate periodic variation in properties of gilgai soil to periodic variation in microrelief. Despite the success of Webster’s study, Fourier methods have not been widely used in pedometrics. This is in part because of the assumptions which underlie the methods. The most serious problem for analysis of soil properties is that the amplitude and phase of each frequency component is constant in space. If the Fourier analysis of a soil property is to give insight into its spatial variability, then the contributions to variation at a given frequency must be uniform from one end of the data set to the other. This assumption is formally one of second-order stationarity, and may often not be plausible. Soil data often contain components of variation that fluctuate in their variance from place to place. Failure of the assumption also causes problems for geostatistical analysis. Wavelet analysis has been developed specifically to deal with data with transient or nonstationary features at different scales. In wavelet analysis the basis functions oscillate locally, but damp rapidly to zero on either side of their central location. They are thus little waves, or wavelets (Daubechies, 1992). Figure 1.8–10a shows a Mexican hat wavelet function, Fig. 1.8–10b shows the Morlet wavelet (which, unlike the other two is complex with real and imaginary parts), and Fig. 1.8–10c shows the first in Daubechies’s (1988) series of wavelets. A basic (mother) wavelet function is denoted ψ(x). To operate as a basis function it must have mean zero, that is, ∞
∫−∞ψ(x)dx = 0
[1.8–34]
and a squared norm of 1, ∞
∫−∞|ψ(x)|2dx = 1
[1.8–35]
To operate as a wavelet, ψ(x) must damp rapidly to zero on either side of its central location. This means that, when used as a basis function in an integral transform analogous to Eq. [1.8–32], it will only respond to the data f(x) in its immediate vicinity. The coefficient generated by the integral transform of a sequence of data with a single wavelet only corresponds to the data within a narrow window. To provide a complete analysis, the wavelet must be translated across the data, generating a set of local coefficients. Furthermore, the basic wavelet can be shrunk or dilated so that it analyzes the data at a particular spatial scale (denoted by the parameter λ > 0). The general formula for a dilation (scale λ) and translation (location x) of a mother wavelet ψ(x), is given by ψλ,x(µ) = (1/%λ &)ψ[(u − x)/λ]
[1.8–36]
The term 1/%λ & ensures that the square norm (Eq. [1.8–35]) is 1 at all scales. For simplicity λ is referred to as the scale in the remainder of this discussion. This is the normal usage, but note that it has the opposite sense to scale in cartography. Di-
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lating a wavelet (increasing λ) is equivalent to providing a more generalized representation of the data, normally achieved in cartography by representing more ground on a given area of map (and so reducing the scale). Scale plays a role in wavelet analysis equivalent to frequency in Fourier analysis. For a given wavelet, a particular scale may be defined as a band of frequencies (see Kumar & FoufoulaGeorgiou, 1994). 1.8.3.1 The Continuous Wavelet Transform λ is
The continuous wavelet transform (CWT) of the data at location x and scale
Fig. 1.8–10. Some commonly used mother wavelet functions: (a) the Mexican hat, (b) the Morlet wavelet (ω0 = 6, real part is solid line, imaginary part is broken line), and (c) Daubechies’s wavelet with two vanishing moments.
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Wf(λ,x) = ∫−∞ f(x)(1/%λ &)ψ[(u − x)/λ]du
[1.8–37]
Wf(λ,x) is a wavelet coefficient. Coefficients for small scales are derived from a narrow window (within which the wavelet function defined in Eq. [1.8–37] takes nonzero values). Thus, short-range (high-frequency) components are analyzed over a narrow window, while long-range components are analyzed over a wide window. This seems natural, and is an advantage of wavelet analysis over attempts to do localized Fourier analysis by imposing a windowing function that has a fixed width (Kumar & Foufoula-Georgiou, 1994). Once wavelet coefficients have been computed, the scalogram may be obtained. The scalogram for any location and scale is simply the squared modulus of the wavelet coefficient, |Wf(λ,x)|2. This is a localized measure of the contribution to the variance of the data since it refers to a specific location x and a window about that location determined by the scale, λ. The scalogram is therefore related to the frequency spectrum (as computed for gilgai soils by Webster, 1977) and the spectrum can be estimated from the scalogram (Abry et al., 1995). The key difference is the localized nature of the scalogram. The spectrum is estimated by integrating the scalogram over all locations for a fixed scale. The scalogram therefore retains information on spatial changes in the variability of the analyzed variable. Since a real data set is never continuous on the interval [−∞,∞], an approximation to the CWT is computed. This is done by incrementing the scale over small steps and computing the convolution of the data f(x) with the wavelet function ψλ,x. Torrence and Compo (1998) give a detailed description of the process. Clearly the wavelet coefficient cannot be computed for any location where the dilated wavelet function overlaps the ends of the data. Thus, the larger the scale, the fewer coefficients can be computed for any data set. A data set on a second variable at the same locations as f(x) is denoted g(x). ——— —— A cross-scalogram may be defined as the product Wf(λ,x)W g (λ, x) , where the overbar denotes the complex conjugate. The cross-scalogram measures local, scalespecific contributions to the covariance of the two properties. An example of the CWT is now presented. The data are at 90-cm intervals on a transect and are measurements of soil electrical conductivity at depths of 0 to 30 and 30 to 90 cm. Figure 1.8–11 shows the basic data. There are localized features that could not sensibly be analyzed on the basis of assumptions of stationarity. The data were analyzed using Torrence and Compo’s (1998) CWT algorithm. The Morlet wavelet (shown in Fig. 1.8–10) was used with scales λi, λi = λ020125j
j = 0, 1, 2,...
[1.8–38]
with λ0 the finest scale of the analysis (two units here, i.e., 180 cm). Figure 1.8–12 shows the scalograms for top soil (0–30 cm, Fig. 1.8–12a) and subsoil (30–90 cm, Fig. 1.8–12b) conductivity. The shape of the figure illustrates the reduction in the number of wavelet coefficients with increasing scale. A number of features are apparent. It is clear that variation is not uniform across the transect. In the subsoil, variation at scales approximately 64 units is greatest from around position 300 on the transect, and variation at all scales up to about 64 units is small from about po-
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Fig. 1.8–11. Soil conductivity on a transect at 90-cm intervals (broken line 0–30 cm depth, solid line 30–90 cm depth).
sition 500 to 600. At positions up to 300, variation is more uniformly distributed between the scales. From position 600 to the end of the transect, variation at scales between 8 and 32 units is quite important. The cross-scalogram is complex because the wavelet function is complex. Figure 1.8–13 shows the real part, called the co-scalogram by analogy with the co-spectrum. It is clear that conductivity at the two depths is most strongly correlated at scales of over 32 units from position 300 onwards. The two depths are related over shorter scales around positions 350 and 450, but elsewhere the co-scalogram values at the shorter scales are generally small or negative. This scale dependence may reflect dependence of conductivity on a property that is correlated between the two depths at coarser scales (e.g., soil texture related to parent material), but which fluctuates at shorter intervals because of factors that do not apply at both depths. 1.8.3.2 The Discrete Wavelet Transform In the CWT scale and location are (quasi)continuous variables. The discrete wavelet transform (DWT) is based on a discretization of scale and location. This results in certain useful properties of the transform, discussed below. In the DWT, a basic scale dilation step, λ0, is fixed. This is conventionally two intervals on the data. The discrete scales are then set at λ = λ0m,
m = 0, 1, 2,....
[1.8–39]
The wavelet at scale λ is then translated over locations x = nx0λ, where x0 > 0.
n = 1, 2,....
[1.8–40]
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Fig. 1.8–12. (a) Top soil conductivity scalogram and (b) subsoil conductivity scalogram. The unit distance is 90 cm.
It is clear that dilating the wavelet (increasing m) increases the size of the step between successive locations in Eq. [1.8–40]. An analogy has been drawn with microscopy where, after increasing the magnification (reducing λ), we scan an image in smaller steps from one field of view to the next. The discretized scales and locations give rise to a discrete form of the translated and dilated wavelet function ψ(x) 1 x − nx0λ0m ψm,n(x) = ____ ψ ________ m %λ && ‰ λ0m 0 and of the DWT
[1.8–41]
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Fig. 1.8–13. Co-scalogram of topsoil and subsoil conductivity. The unit distance is 90 cm.
1 x − nx0λ0m ________ Dm,n = ____ ∫ f(x)ψ dx m %λ && ‰ λ0m 0
[1.8–42]
The DWT coefficients, Dm,n can be used to reconstruct f(x), i.e., ∞
f(x) = m=−∞ Σ
∞
Σ D ψ (x) n=−∞ m,n m,n
[1.8–43]
The coefficients for an appropriate wavelet function represent a complete orthonormal basis for any real data set of finite variance. This means that the set of coefficients for a given m (n = 1, 2,....) correspond to an additive component of the data, at the corresponding scale, which is independent of corresponding components at other scales. Furthermore, the wavelet is orthogonal with dilations of itself at any scale, so the coefficients at a scale are independent of each other except in so far as spatial correlation within the data imposes dependence. The DWT therefore provides a basis for decomposing a set of data (in one or two dimensions) into independent additive components with different spatial scales. This is a multiresolution analysis of the data. More details of the procedure are given by Lark and Webster (1999), Kumar and Foufoula-Georgiou (1994), and Mallat (1989). In short, N = 2P data are decomposed into up to P independent components. P−1 of these can be regarded as reconstructions of the data at scales λ = 2, 4, 8,...., 2(P − 1). The last, the Smooth component, can be regarded as a smoothed representation of f(x), which only contains variation at scales λ ≥ 2P. Because the components of the data obtained by the DWT for each scale are orthogonal, their variances sum to that of the original data, and may be computed from the squared DWT coefficients. These components of variance have been called wavelet variances by Percival (1995), who showed that they could be a useful alternative to variance estimators used in signal analysis. Once again, because
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each DWT coefficient describes a small segment of the data at a given scale, the DWT provides a basis for identifying locations at which the variance of the sampled property appears to change (Whitcher et al., 2000). Daubechies’ (1988) wavelet shown in Fig. 1.8–10c is commonly used in multiresolution analysis by the DWT. A practical problem in implementing the procedure is to adapt it to the finite interval of real data. Cohen et al. (1993) propose a solution, which is used here. This only allows components to be extracted for scales up to 2(P−2). The adapted DWT, using Daubechies’s (1988) first wavelet was applied to the last 512 of the subsoil conductivity data. Figure 1.8–14a shows the resulting components at different scales. The form of the underlying wavelet is apparent in the components. Figure 1.8–14b shows the accumulated components. Starting from the smooth representation of the data, the components for each scale are added in turn (from the largest to the smallest scale). The top graph represents the sum of all components and so is a perfect reconstruction (bar rounding errors) of the original data. The DWT is effectively based on a tiling of the data into N/λ tiles at each scale. This can lead to artifacts when the data contain discontinuities, depending on whether the discontinuity occurs at the boundary between two adjacent tiles or in the middle of one. These problems can be avoided by computing a series of DWT for different tilings of the data and then averaging. This is done by repeatedly moving the starting point of the data forward by one interval, conducting the DWT, then moving the starting point again. Several DWTs will then be evaluated at any one location, and the corresponding components may be averaged. This approach has another use when analyzing data where N is not an integer power of two. The averaging procedure is applied to N′ = 2P < N data in a moving window. Lark and Webster (1999) used this procedure on their data. The results of the procedure, using a moving window of 512 data, on the 696 data in the transect of conductivity data are shown in Fig. 1.8–15 and Fig. 1.8–16, the averaged components at each scale. The components are smoother than those in Fig. 1.8–14, although they are, of course, still additive components of the original data. Although the wavelet basis in this figure is different from that used to obtain the scalograms in Fig. 1.8–12, the same general picture is seen. All scales contribute to variation around positions 350 and 450, and from position 600 to the end of the transect. Up to position 300 variation is limited, and most is seen at small scales. The components illustrate well the localized nature of the wavelet analysis. At any scale, the contribution to variation can change markedly from place to place. 1.8.3.3 Prospects for Wavelet Analysis The discussion above aims to introduce and illustrate the basic ideas of wavelet analysis. As was stated in the introduction, the application of wavelets in pedometrics is still very new, so possible areas of application are proposed rather than illustrated. Three general areas are identified. 1.8.3.3.a Basic Signal Processing—Filtering and Data Compression Wavelet analysis is an effective way of denoising data, without requiring any assumption that the noise is homogeneous across the signal. The term noise in this
Fig. 1.8–14. (a) Additive components of the last 512 subsoil conductivity data obtained by the discrete wavelet transform and (b) accumulated components adding information at each scale in turn.
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Fig. 1.8–15. Additive components of top soil conductivity obtained by the shift-averaging procedure.
context might denote a spurious high-frequency component of the data, introduced from parts of the sensing system. It might also refer to small-scale soil variation (in the sense of λ) that is not relevant to a particular application. For example, in precision agriculture the farmer will only be interested in soil variation at scales where inputs can be varied spatially. The wavelet coefficients from a DWT may be thresholded to identify those which appear to be significant. A smoothed representation may then be obtained by reconstructing the data, having first set to zero those coefficients that do not reach the threshold (Donoho et al., 1995). This is also an efficient way of compressing large data sets, which may often be important as sensing systems are more widely used. 1.8.3.3.b Identification of Stationary Regions One valuable application of densely measured data is as a variable for cokriging (Section 1.5) more sparsely measured properties (such as those which have
Fig. 1.8–16. Additive components of subsoil conductivity obtained by the shift-averaging procedure.
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to be determined by manual sampling and laboratory analysis). This application rests on assumptions of second-order stationarity (the intrinsic hypothesis). The wavelet transform of the densely measured data (DWT coefficients or scalogram components from the CWT) could be analyzed to identify spatial regions within which stationarity seems to be a plausible assumption. The work of Whitcher et al. (2000) on detecting change points in the variance of a variable using DWT coefficients could offer one way of tackling this problem. Cokriging could then be conducted within apparently stationary subregions of the area. 1.8.3.3.c Identification of Localized Relationships among Variables Making use of soil information often depends on being able to interpret it, particularly in the light of other information such as maps of crop yield. Relationships among variables may be complex. They may be scale dependent, and shortrange variation in two variables which are uncorrelated might obscure underlying relationships. There are geostatistical methods for dealing with scale-dependent correlation (Goovaerts & Webster, 1994), but they assume stationarity of the variation. As was seen in the cross-scalogram of the soil conductivity at two depths (and shown by Lark and Webster (1999) for soil texture at two depths) wavelets can analyze scale-dependent covariation of variables where the variation is not stationary. Another complex feature of relationships among soil and crop variables is spatial variation in the underlying relationship itself, regardless of scale. If soil water is limiting on yield in some parts of a field, then potentially limiting effects of pH will not be expressed there, while pH and yield are related elsewhere where soil water is adequate. An overall analysis of the pH–yield data will obscure the relationship where it exists. However, the cross-scalogram of the properties will identify localized expressions of a pH limitation to yield, regions within which the two properties are related. The wavelet analysis may therefore contribute to the interpretation and management application of large spatial data sets arising from new sensing systems. To conclude, wavelet analysis is a powerful technique. As more and more rapid sensing methods for soil become available, it is likely that both CWT and DWT will contribute strongly to their analysis. Three possible broad areas of application have been identified. It is to be hoped that these will be developed, and is very likely that more will emerge. 1.8.4 References Abry, P., P. Goncalves, and P. Flandrin. 1995. Wavelets, spectrum analysis and {1/f} processes. p. 15–29. In A. Antoniadis and G. Oppenheim (ed.) Lecture notes in statistics. Number 103: Wavelets and statistics. Springer-Verlag, New York, NY. Ahl, C., and J. Niemeyer. 1989. The fractal dimension of the pore-volume inside soils. Z. Pflanzenernaeh. Bodenkd. 152:457–458. Anderson, A.N. 1997. Soil fractal geometry. Ph.D. thesis. The University of Sydney, Australia. Anderson, A.N., and A.B. McBratney. 1995. Soil aggregates as mass fractals. Austr. J. Soil Res. 33:757–772. Anderson, A.N., A.B. McBratney, and J.W. Crawford. 1998. Applications of fractals to soil studies. Adv. Agron. 63:1–76. Anderson, A.N., A.B. McBratney, and E.A. Fitzpatrick. 1996. Soil mass, surface, and spectral fractal dimensions estimated from thin section photographs. Soil Sci. Soc. Am. J. 60:962–969.
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Published 2002
Chapter 2 The Solid Phase 2.1 Bulk Density and Linear Extensibility R. B. GROSSMAN AND T. G. REINSCH, USDA-NRCS, National Soil Survey Center, Lincoln, Nebraska
2.1.1 Introduction Bulk density has application to nearly all soil studies and analyses. The current activity in soil quality, soil sufficiency, and sequestration of C has increased interest in bulk density, particularly of surface layers. Diversity among soil layers in strength, thickness, and depth necessitates several kinds of methods that may not result in the same values. The four methods for this presentation are core, clod, excavation, and radiation. For most agricultural soils and for organic soils, the core method is applicable. The clod method has the advantage that samples can be taken while the soil is dry and moistened to a standard state in the laboratory. It has the further advantage that both moist and dry bulk density may be obtained. From these densities, derivative quantities can be calculated, one of which is linear extensibility (Section 2.1.6). For soils high in larger rock fragments, for fragile horizons, and for thin horizons, excavation methods are desirable (Page-Dumroese et al., 1999). Radiation methods are nondestructive and can be done in situ. Extraction of cores from below the water table may be necessary, in particular for organic soils. For this, several methods have been developed (Sheppard et al., 1993). Organic soil materials are not treated in detail. Sheppard et al. (1993) and Parent and Caron (1993) treat the subject thoroughly. Method D 4531-86 (American Society for Testing and Materials [ASTM], 1999a) describes the use of paraffin to coat samples of organic soil materials. The approach is a variation of the clod method. Engineers place much emphasis on bulk density measurements. A number of standards of the ASTM are discussed. The review by Blake and Hartge (1986) has been closely followed for several of the methods given below. Culley (1993) provides an excellent succinct review. The emphasis in the method descriptions is on how to obtain the sample volume. The reader’s conversance with the mass determination part of the procedure is assumed. 2.1.1.1 Principles Soil bulk density is the mass per unit volume of soil. In agriculture, the reference mass is after oven-drying, and the volume is for the 2-mm portion to obtain the volume (Section 2.2). Andraski (1991) and Page-Dumroese et al. (1999) provide illustrations of the differences between bulk densities inclusive and exclusive of the rock fragments. In engineering, the mass may include either or both the >2-mm fraction and water. The volume in engineering nearly always is for very moist or wet conditions (Soil Survey Division Staff, 1993). The engineers’ wet density is inclusive of the mass of water, commonly for the satiated condition, and may be inclusive of the fraction >2 mm. The engineers’ dry density is the same as the agricultural bulk density, except that the fraction >2 mm may be included. When applying SI units, the unit kilograms per cubic meter gives large numbers, and the more common practice is to use the units tonnes per cubic meter, grams per cubic centimeter, or megagrams per cubic meter, which all give equivalent numerical values in the range of 1.0 to 1.7. Engineers regularly use pounds per cubic foot. Other terms for bulk density, which are now obsolete, have been used, such as, volume weight, bulk specific gravity, and apparent specific gravity. 2.1.1.2 Variability The spatial variability of bulk density is about 10% of the mean, according to the literature reviewed. Mason et al. (1957) studied about 1000 sites for which land use as well as the soil series were identified. The standard deviations for the sites were 0.13 g cm−3 for A horizons, 0.10 g cm−3 for B horizons, and 0.074 g cm− 3 for C horizons. O’Conner (1975) reported a coefficient of variation for clod, core, and sand cone methods, taken collectively, from 1 to 12%. Smeck and Wilding (1980), using the clod method, found a coefficient of variation also in the range of 1 to 3%. Warrick and Nielsen (1980) reported a coefficient of variation of 7% for three field studies. Excluding A horizons, Cassel and Bauer (1975) reported standard deviations of 0.07 to 0.14 g cm−3. Aljibury and Evans (1961) reported a standard deviation of 0.14 g cm−3 for the upper subsoil of fields within a map unit. Mausbach and Gamble (1984) used the range between the second and third quartile as a measure of variability within a class for clod bulk densities of duplicate pedons of soils of the Midwest. The range was 0.2 to 0.3 g cm−3 for loamy soils, mostly of glacial origin, and 0.1 to 0.2 g cm−3 for silty soils of loessial and alluvial deposits. For the compliant cavity-excavation procedure (Section 2.1.3), triplicate measurements within a linear distance of 1 m on the same tillage feature (e.g., shoulder, traffic, interrow) of uniform cultivated loess-derived soils fall within a range of 0.1 g cm−3.
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As mass measurements are generally accepted to have low variability, the volume measurements contribute most to variability of bulk density. Method D 293794 (ASTM, 1999c) stipulates that the volume of the core for determination of the compaction of fill must not be less than 850 cm−3, which is slightly larger than the volume of a 10 by 10 cm core. The usual sample size for the clod method is 400 to 700 cm−3. Procedures are described here in which the soil volume is measured from the change in distance after excavation from a reference using a ruler. An uncertainty in the distance of 0.1 to 0.2 cm is likely. The uncertainty in the volume depends on the thickness of the sample. For a thickness of 2 cm, the uncertainty in the volume is 5 to 10%. For 5 cm, it is 2 to 4%. In reference to the number of samples required, the literature is inconsistent. For example, Mader (1963) and Mollitor et al. (1980) reported coefficients of variation for forested soils of the northeastern United States of 10% or less. Culley (1993) stated that “…four samples should be sufficient to estimate the mean density to within 10% of the true value, 95% of the time for a uniform soil type.” On the other hand, Terry et al. (1981) reported that for two forested soils in North Carolina, 5 to 20 core samples, 7.5 cm in diameter, are needed for a precision of 0.10 g cm−3. The authors determined that the average standard deviation for duplicate clods of 5000 samples in the USDA Soil Survey Laboratory is 0.04 g cm−3. 2.1.1.3 Application Bulk density has two principal groups of uses: (i) conversion of data (percentage, cmol kg−1) to a volume basis and (ii) characterization of the soil fabric. Conversion from weight to volume for a soil layer involves bulk density of the < 2-mm fabric and the volume of rock fragments: Av = {AwρbL[1 − (V>2/100)]}/F
[2.1–1]
where Av is the amount of a quantity (e.g., cation exchange capacity or organic C) for a specified area and thickness at or near field capacity, Aw is the amount for a specified weight of the 2-mm fraction, and F is a factor that is dependent on the units of area and the weight to which the quantity pertains. For an area of 1 m2 and assuming weight percentage data, F is 10. The determination of V>2 is discussed in Section 2.1.1.4 below. In agriculture, characterization of the bulk density of the < 2-mm fabric has several purposes. The most common applications are ease of root penetration (Pierce et al., 1983), prediction of water transmission (Rawls et al., 1998), and as an indicator of soil quality (Lal et al., 1998; Larson & Pierce, 1994; Arshad et al., 1996; Doran & Parkin, 1994). In engineering, bulk density is used to predict strength, overburden weight, and Ksat. For overburden weight, the rock fragments and the water usually would be included. Bulk density usually has more applicability for predicting root growth or movement of free or low suction water if the layer is massive, platy or has weak structure, or the structural units are large (10 cm or more across horizontally). If the structural expression is strong (except platy) and the units are small, bulk den-
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sity is less predictive because the large pores between the structure units are controlling. An additional but less general application of bulk density concerns evaluation of change with time of the surficial layers of cropland, in particular soil organic C. For comparison over time, the mass of the material 2-mm component. Both mass and volume of the >2-mm fraction are subtracted from the total mass and volume. The National Soil Information System (NASIS) of the National Cooperative Soil Survey (USDA-NRCS, Soil Survey Div. and USDA Tech. Center, 1998) contains volume estimates of the >2-mm fraction. If the information in NASIS is unavailable or not applicable, mass percentages may be measured and converted to a volume basis. In the National Cooperative Soil Survey, the masses of the >250-mm, 75- to 250-mm, and the Pass 10 (2–75 mm as a percentage of the 2 = W>250 + W75–250 + {(100 − Pass 10)[(W>250 + W75–250)/100]}
[2.1–6]
where W>2 is mass of coarse fragments (>2-mm diam.), W>250 is the mass of fragments (>250-mm diam.), W75–250 is the mass of fragments (75–250 mm diam.), and 100 − Pass 10 is the mass 75- to 2-mm diam. for the 2 is converted to volume: V>2 = W>2(ρbx/ρ>2)
[2.1–7]
where W>2 and V>2 are the mass and volume of rock fragments, respectively, ρ>2 is the density of the rock fragments, and ρbx is the bulk density inclusive of the rock fragments up to size x calculated by: ρbx = 100/{(W>2/ρ>2) + [(100 − W>2)/ρb]}
[2.1–8]
where ρb is the bulk density of the 2 is obtained by Eq. [2.1–6] or an alternative approach.
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Commonly, it is necessary to combine the weight percentage of the 2- to 75mm fraction for the 75 mm. Equations [2.1–7] and [2.1–8] are used to convert the weight of the 2- to 75-mm fraction to a volume percentage for the 2 = V>75 + {V′2–75[1 − (V>75/100)]}
[2.1–9]
where V>75 is the volume of rock fragments >75 mm, and V′2-75 is the volume 2 to 75 mm for the 75 is equal to V75–250 + V>250. As indicated, V75–250 would be obtained commonly from the area of the soil exposure, and V>250 from other sources. V75–250 and V>250 are combined as follows: V>75 = V>250 + {V75–250[1 − (V>250/100)]}
[2.1–10]
Under conditions that the rock fragments touch and are not separated by 2mm material. In all instances, the volume of rock fragments is below 100%. As a general rule, the weight of the sample should be 50 to 100 times the upper limit of the size class of concern (Test D 2488-93, ASTM, 1999b). For the 75-mm limit, the minimum weight would be about 50 kg. For 250 mm, the minimum weight would be 1 to 2 Mg. For many situations, 75 mm is the upper limit that can be weighed practicably in the field. Areal estimates are made for the larger fractions. The area of the exposure must be 50 to 100 times the cross-sectional area of the upper size of rock fragment considered (Test D 2488-93, ASTM, 1999b). For the 75- to 250-mm fraction, soil sampling exposures are commonly large enough for an areal evaluation. Above 250 mm in diameter, the area of the sampling exposure is usually too small, and the information from the map-unit component description or large excavations for other purposes must be employed. The size fraction on which mass is obtained may be other than the 75 mm. As an example, for the frame method (Section 2.1.3.2.a), 20 mm is the upper limit. Equation [2.1–9] gives the volume percentage for the size weighed. This volume percentage is reduced for the volume of the >2-mm fraction larger than was weighed. This reduced volume percentage and the volume of larger rock fragments are summed to obtain the percentage overall. The proportion of a sample occupied by rock fragments changes during drying, especially in soils showing high shrinkage. It is possible to estimate the volume of rock fragments when soil is at a water content below field content, thusly: V>2 = V′>2(ρb33/ρbxi)[1 − (V′>2/100)]}
[2.1–11]
where V>2 is the volume of the >2-mm fraction at field capacity, V′>2 is the measured volume of rock fragments at water contents below field capacity, ρb33 is the bulk density of the 2-mm material and the water may be desired: V>2 V>2 (θm2/100)] ρbt = ___ ρ + 1 − ___ ρ + __________________ ‰ 100 >2 —‰ 100 b — 100
[2.1–12]
where ρbt is the total bulk density inclusive of rock fragments and water, V>2 is the volume of rock fragments at or near field capacity, ρ′>2 is the density of the fraction >2 mm, ρb is the bulk density of the 2/100). 2.1.2 Core Method 2.1.2.1 Introduction A cylinder is inserted into the soil, and the sample is obtained within the cylinder. The volume of the sample is that of the inserted cylinder. The soil must remain within the cylinder on withdrawal from the soil. In nearly all instances, the soil remains in the cylinder because of its coherence. An exception for soil materials with low coherence is a device designed for below the water table, which has a gate that closes and retains the sample (piston sampler in Sheppard et al., 1993). On the other hand, the soil must be weak enough that the cylinder can be inserted without causing appreciable disruption of either soil or sampler. This usually requires that the soil be moist. The sample commonly extends the length of the core and is trimmed flush at the bottom. Rock fragments >5 mm hamper insertion of the cylinder. The simplest design is a single cylinder sharpened on one end. Other designs are described below. A more complex sampling device consists of a cylinder that is surrounded by a rotary bit that removes surrounding soil as the cylinder is inserted. Highly specialized devices have been designed to sample organic soils (Sheppard et al., 1993) and frozen soils (Tarnocai, 1993). Insertion of cylinders with plastic liners has been described by Robertson et al. (1974) and by Mielke (1973). The cores can then be segmented after extraction and the bulk density determined for subdivisions. Among other firms, the Giddings Machine Co., Fort Collins, Colorado, sells both plastic and aluminum liners for tubes of several diameters.1 An important advantage of core methods is that samples of standard dimensions may be obtained. Other determinations, such as Ksat, can be made on a sample that has a standard size and shape. 2.1.2.2 Method We divide the various core methods arbitrarily into those ~ 5 to 15 cm long (short) and into longer cores ~>50 cm. 1 Trade and company names are included here and elsewhere for the benefit of the reader, and do not imply endorsement or preferential treatment of the product listed by USDA.
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2.1.2.2.a Short Cores Topp et al. (1993) reviewed the use of cores for water rentention measurements. The cores for bulk density should be at least 75 and preferably 100 mm in diameter, and the height preferably should not exceed the diameter. The cylinder wall should be 0.5 to 0.6 mm. See Fig. 2.1–1. Page-Dumroese et al. (1999) compare cores 50 and 100 mm in diameter. Bulk densities for the smaller diameter were larger. Cylinders are inserted with force or impact energy. A single cylinder may be used, usually with the bottom sharpened. A superior design consists of two concentric cylinders. Usually the outer cylinder has an integral beveled cutting head. The inside of the outer cylinder is machined to provide a lip that has the thickness of the inner cylinder. The inner cylinder rests on this lip and is flush with its outer edge. Force or impact energy is applied to the outer cylinder and through it to the integral cutting head. As the cutting head advances downward, soil enters the inner cylinder. Usually the inner cylinder is filled to the top. The inner cylinder is removed, and the bottom is trimmed flush. Impact energy is commonly supplied with a sliding hammer. One device consists of an outer cylinder to which a 40-cm-long handle is attached. A weight with a central hole is used as a sliding hammer on the handle. Rope may be used to raise and drop the weight. Alternatively, the weight is attached to a rod that travels within the handle and impacts on the top of the handle. For compact soils, the upper limit of the impact energy ranges from about 10 J for 50-mm-diameter cores to 50 J for 100-mm diameter. It is better to apply steady
Fig. 2.1–1. Typical double-cylinder, hammer-driven core sampler for obtaining soil samples for bulk density. From Blake and Hartge (1986).
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force than impact energy if the soil is subject to shattering. Rogers and Carter (1987) found that the Ksat was lower if impact energy was used instead of force; however, the bulk density did not differ appreciably. Chong et al. (1982) described a portable frame that is anchored to the soil and permits insertion of the cylinder with steady force. Sharma and De Datta (1985) described a core sampler for the bulk density of satiated, puddled soil as is found in rice (Oryza sativa L.) paddies. To calculate the bulk density for any of the short-core methods, it is necessary to remove the excess soil protruding from the bottom and top of the core. If the top and bottom of the sample is flush with the ends of the cylinder, then the volume of soil is equal to that of the core. If the sample does not fill the cylinder, the empty space above the sample is determined and subtracted from the volume of the cylinder to obtain the volume of the soil. The approach has the advantage that the thickness can be as small as 2 cm, which reduces compaction and increases specificity of depth. The volume of the space above the soil may be measured by pouring glass beads into the core and striking off flush (Culley, 1993). Beads of nominal diameter 260 µm are suggested. The mass of beads necessary to fill the core is measured and converted to a volume using a prior determination of the bulk density of the beads. The core may be inverted and the void space on the underside of the core measured in the same fashion. The sum of the void space for the top and the bottom of the cores is subtracted from the total volume of the core to obtain the volume of the soil sample. An alternative method to determine the soil volume for incompletely filled cores is to obtain the difference in distance from a common reference to the bottom of the empty core and to the top of the soil in the partially filled core. A piece of retractable ruler is used. A depth-measurement tool for the ruler is made from a compression coupler 7.5 cm long formulated from chlorinated polyvinyl chloride. Flat neoprene washers 1.9 cm in diameter are placed in each end. Sectors are cut from each washer large enough so that the ruler has free movement. The use of beads is probably more accurate than measuring with a ruler; however, the beads are difficult to use in wet or windy conditions. 2.1.2.2.b Long Cores Long core samplers are typically used by engineers and geologists to extract cores from deep borings. Test designations D 1587-94 (ASTM, 1999d), D 158698 (ASTM, 1999e), and D 2937-94 (ASTM, 1999c) describe various long driving cores. Kelley et al. (1947) describe a device that consists of two cylinders with cutting knives located between, which cuts away the soil as the inner core is pushed downward. The inside tube is split longitudinally. Cores 10 to 15 cm in diameter are obtainable. Samples may be obtained to 2 m. Holtzclaw et al. (1975) described a device similar to that described by Kelley et al. (1947) that can obtain samples to 30 m. Wit (1962) described a sampler that is suitable for soft sediments below the water table. The water within the core may be retained. Tarnocai (1993) described various devices for use in frozen soils. Sampling organic soils presents special problems. A number of imaginative samplers have been fabricated (Sheppard et al., 1993). Two of these are briefly discussed. The Piston Post sampler consists of two concentric cylinders with the out-
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side having a cutting edge. A piston is in the inner tube. The piston is pushed downward while the cutting core is advanced. A vacuum is formed that acts to retain the sample in the inner tube. The other is the Macauley sampler device. A half cylinder is forced into the soil and removed. Bulk density is determined on segments cut from the half core of soil. Cores 2 to 10 cm in diameter may be obtained. The sleeve method, D 4564-93 in ASTM (1999f), is designed for cohesionless soil materials high in fine gravel. The core length is intermediate between the long and short separation for this discussion. The cylinder is worked into the soil material by a rotary motion. 2.1.2.3 Comments Short cores are the more commonly used. The soil must be coherent enough that the sample remains in the core, and rock fragments >5 mm cannot be abundant. For the most part, the cores are completely filled. It is the practice with short core methods to sample to a standard depth irrespective of short-range change in soil morphology. The considerable height of the core (10 cm being the most common) may exceed the thickness of the soil layer of interest. An additional limitation is that insertion of the cores may compact the soil sample. The variable-height method reduces these limitations. The excavation procedures to be discussed (Section 2.1.3) are less subject to the problems of rock fragments, thinness of the layer, and compaction. During the late 1940s and early 1950s, there was much activity on short-core methods. The efforts may have been stimulated by the active program of the Soil Conservation Service on near-surface physical evaluation. Mason et al. (1957) used the results of these efforts to evaluate the variability of physical properties. Lutz (1947) described a method in which a 6-cm-diameter sampling can is placed in a drive cylinder so that the edge of the can protrudes and forms the cutting edge. Lutz also developed a double-core method. Lotspeich and Laase (1961) attached a remote hydraulic pump to the Lutz sampler for insertion horizontally in pit wells. The sampling can configuration of Lutz was modified by Jamison et al (1950). They attached a sliding hammer to deliver impact energy and increased the diameter of the can to 9 cm. Uhland (1949) described an apparatus with a double cylinder head and a sliding hammer. Pikul et al. (1979) designed a square tube with slots on three sides though which a knife is inserted to cut out increments of soil 10 mm or more in height. Heinonen (1960) described a cylinder with opposing slots through which a knife is inserted to segment the core. For organic soil materials, Parent and Caron (1993) placed a core on a block of soil material and cut away the excess while pushing the cylinder into the soil material. Compaction may occur due to insertion of the cylinder. Designation D 293794 (ATSM, 1999c) contains a relationship between the thickness of the core wall and the ratio of the calculated areas based on the outside and the inside diameter: Ar = [(De2 − Di2)/Di2]100
[2.1–13]
where Ar is the ratio of the area of the end of the cylinder to that of the soil diameter. De is the outside diameter of the core, and Di is the inside diameter. Ar for the cylinder should be 10 to 15%.
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2.1.3 Excavation Method 2.1.3.1 Introduction Excavation methods have applicability to layers that can be described as being one or more of the following: cohesionless, high in rock fragments >5 mm, or thin (20-mm fractions. Attached rocks and large roots are excluded from the field sample. 2.1.3.3 Comments Sand-Cone Method. Blake and Hartge (1986) stated that the chief source of error is determining that the sand in the excavation is level with the base plate. The method cannot be run in the rain, and large changes in relative humidity may affect the packing density of the sand. Flint and Childs (1984) report that compaction of the sand is a significant problem. An advantage is that the volume of the excavation usually is appreciable, over 1000 mL, which reduces error. Flint and Childs (1984) presented a method that is similar to the sand-funnel method, but use plastic spheres. The sample volume of the device of Flint and Childs is five times the sand-funnel method (Blake & Hartge, 1986). It is also considerably lighter. Flint and Childs developed a reservoir base that compensates for slope and removes the need to flatten the soil if the sample is taken on a steep slope. Rubber-Balloon Method. Blake and Hartge (1986) stated that an important source of error is position of the top of the water relative to the bottom of the base plate. The error is similar in principle to that in the sand-funnel method. The upper limit of the excavation volume is about 1000 mL, the same as for the sand-funnel method. Blake and Hartge (1986) pointed out that the volume of the balloon is only 2 mL and can be neglected. For fragile soil layers, the pressure of the distended bal-
THE SOLID PHASE
217
loon may enlarge the test hole (Designation D 2167-94, ASTM, 1999h; and personal observation by the senior author). The method was developed for fill material that usually is much stronger than many kinds of fabric encountered in agricultural soils. Compliant-Cavity Method. The method was designed for fragile cultivated near-surface layers and O horizons of forestland soils. The diameter of the excavation is small enough to avoid the sample originating from more than one tillage feature (ridge vs. traffic interrow). Zones as thin as 2 cm can be measured. The volume of the sample commonly would be less than that for the sand-funnel or the rubber-balloon method, in the range of 200 to 800 mL, but may be increased. The thickness of the sample should exceed 2 cm. The method has the important advantage that it is unnecessary to flatten the ground surface on steep slopes or remove small irregularities. Thus, the surficial zone is usually not altered. On steep slopes, the thickness of the foam annulus is increased, and the upslope side is compressed more. The objective is that the water covers the bottom before running out the downslope side. The method by Howard and Singer (1981) is very similar to the complaint cavity method, but requires making the ground surface level. The polyurethane method (Mueller & Hamilton, 1992) also requires that the ground surface is flattened. The Saran-coated hole method of Shipp and Matelski (1965) uses a straight edge to measure the height of the water. Flattening the soil surface is not required in their procedure, but it would seem necessary. Polyurethane Method. The coefficient of variation is similar to that for the sand-funnel methods. The method may be run on vertical faces, which is not possible for other excavation methods. The materials required are relatively light. Ring Method. The method is quite rapid and robust. The diameter can range down to 15 cm. It is not necessary to excavate from the whole area within the ring. A limit of 2 cm on the minimum thickness of the sample should be considered. Frame Method. The area of 0.1 m2 may be increased or decreased, depending on local variability. The method is particularly well suited for surficial, thin, locally variable O horizons that contain appreciable rock fragments, as is common in forestland. The O horizon and shallow mineral layers in woodland soils present particular problems. One reason is that they are commonly thin, as little as 1 to 2 cm thick. Another reason is that they are usually very fragile. A third reason is the high local variability, and the final reason is that the soils commonly are high in rock fragments. The use of various core methods and the clod method is limited by rock fragments and by fragility. This leaves the excavation methods. The sand-funnel and rubberballoon methods have restricted applicability. For the compliant-cavity method, in order to have a large area, the amount of water becomes too large for routine use. The ring method suffers from the limitation that the rock fragments may make insertion impossible. This leaves the frame method. The size of 0.1 m2 is sufficient to encompass considerable local variability. Engineers have procedures for determining the bulk density of relatively large volumes. Method D 4914-89 (ASTM 1999i) uses sand, and Method D 5030-89 (ASTM 1999j) uses water. For both, the test excavations are lined with plastic. The
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CHAPTER 2
sand procedure specifies a volume of 0.03 to 0.17 m3, and the water procedure 0.08 to 2.8 m3. These methods are used if the size of the rock fragments invalidates the sand-funnel or balloon methods, both of which have an excavation volume of about 0.003 m3. An enlarged frame device may be an alternative. 2.1.4 Clod Method 2.1.4.1 Introduction This discussion largely follows Soil Survey Laboratory Staff (1996). The bulk density is the mass of the oven-dry sample divided by the volume of the clod. The bulk density is reported on a whole soil or 80 kPa) 9. Alcohol (e.g., ethanol)
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219
10. Reinforced paper towels or cheese cloth 11. Silt loam soil or similar fine porous medium for making hydraulic contact 12. Pressure plate extractor system with porous ceramic plates (see Section 3.3.2.4) 13. Drying oven (105°C) 14. Hot plate 15. Liquid vapor trap, a metallic enclosure over hot plate with a chimney and duct 16. Fume hood 17. Liquid detergent 18. Sieve, No. 10 (2-mm openings) 2.1.4.3 Procedure 2.1.4.3.a Preparation of Plastic Lacquer Prepare plastic lacquer with resin to solvent ratios of 1:4 and 1:7 on a mass basis. Place 2700 ± 200 mL of solvent (MEK or acetone) into each of two metallic pails. Add 540 or 305 g of resin to make 1:4 or 1:7 plastic lacquer, respectively. For the initial field and laboratory coatings, use the 1:4 plastic lacquer. Use 1:7 plastic lacquer for the last two laboratory coats. The 1:7 plastic lacquer serves essentially to close holes in the primary coat, conserves resin, and so reduces cost. In the field, mix solvent with a wooden stick. In the laboratory, stir solvent with a nonsparking, high-speed stirrer while slowly adding resin. Stir plastic lacquer for 30 min at 25°C. Store plastic lacquer in covered plastic or steel containers. Acetone may be substituted for MEK. 2.1.4.3.b In the Field Collect natural clods or pieces of the soil fabric, approximately 100 to 200 mL (fist-sized), from the face of the excavation. Three clods per horizon are recommended. Remove a piece of soil larger than the clod from the face of sampling pit. From this piece, prepare a clod by gently cutting or breaking protruded peaks and compacted material from the clod. If roots are present, trim the roots with shears. No procedure for sampling clods is applicable to all soils. Adjust field-sampling techniques to meet field conditions at time of sampling. Make a clothesline by stretching the rope between two fixed points. Tie the clod with fine copper wire or place clod in a hairnet. If the clod is dry, moisten the surface with a fine mist of water. Quickly dip the entire clod into plastic lacquer. Suspend the clod from clothesline to dry. Dry the clod for 30 min or until the odor of solvent dissipates. If the field bulk density value is required, store the clods in waterproof plastic bags as soon as the coating dries to prevent water-vapor loss. Pack clods in rigid containers to protect them during transport. 2.1.4.3.c In the Laboratory Prepare a round stock tag with a sample identification number. Cut the copper wire of sufficient length to loop around the clod and leave length enough for
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hanging the clod. Record the mass (TAG) of the tag and the wire. Loop the wire around the clod, leaving two tails, one to which the tag is attached, the longer other tail for hanging the clod. Record the mass of the clod (CC1). Dip the clod into 1:4 plastic lacquer. Wait 7 min and then dip the clod into 1:7 plastic lacquer. Wait 12 min and then dip the clod into 1:7 plastic lacquer. Wait 55 min and then reweigh the clod. If the clod has adsorbed >3% plastic by clod weight or smells excessively of solvent, allow a longer drying time; then reweigh the clod and record the mass (CC2). With the diamond cut-off saw, cut a flat surface on the clod. Place the cut clod surface on a tension table, maintained at 5-cm tension. Periodically check the clod to determine if it has reached equilibrium by successive mass comparisons. When the clod has reached equilibrium, remove the clod and record the mass (WSC). If the cut clod is nonwettable and does not adsorb water, place the clod in a desiccator on a water-covered plate with a 0-cm tension. Submerge only the surface of the clod in the water. Add a few milliliters of alcohol. Use an in-house vacuum and apply suction until the clod has equilibrated at saturation. Remove the clod and record its mass (WSC). Place the clod into a pressure-plate extractor (for 33 kPa) or tension table (for 10 kPa). To provide good contact between the clod and the ceramic plate, cover the ceramic plate with a 5-mm layer of silt loam soil and saturate it with water. Place a sheet of reinforced paper towel or cheesecloth over the silt loam soil. Place the cut surface of the clod on the towel or cheesecloth. Close the container and secure the lid. Apply gauged air pressure of 33 kPa (1/3 bar). (The tension table procedure reported by Topp et al. (1993) uses saturated porous media as the tension medium, providing a simpler more rapid desorption.) When water ceases to discharge from the outflow tube, the clod is at equilibrium. Extraction usually takes 3 to 4 wk with the pressure plate or WSC, equilibrate the clod on the tension table and repeat the desorption process. Dip the clod into the 1:4 plastic lacquer. After 7 min, dip the clod into 1:7 plastic lacquer followed 12 min later by a second dipping into 1:7 plastic lacquer. Wait 12 min again and dip the clod into 1:7 plastic lacquer. After 55 min from the last dipping, reweigh the clod and record its mass (CC3). If the clod has adsorbed >3% plastic by weight or smells excessively of solvent, allow a longer drying time; then reweigh the clod. The clod should be waterproof and ready for volume measurement by water displacement. Suspend the clod below the balance, submerge it in water, and record its mass (WMCW). Dry the clod in an oven at 105°C until the mass is constant. Weigh the ovendry clod in air (WODC) and in water (WODCW) and record the masses. If the clod contains >5% rock fragments by weight, remove them from the clod. Place the clod in a beaker and place it on a hot plate. Cover the hot plate with a liquid vapor trap. Use a fume hood. Heat the clod on a hot plate in excess of 200°C for 3 to 4 h. The plastic coating disintegrates at temperatures above 200°C. After heating, the clod should appear black and charred. Remove the clod from the hot plate, lightly coat it with liquid detergent, and add hot water.
THE SOLID PHASE
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Wet sieve the cool soil through a 2-mm, square-hole sieve. Dry and record the mass of rock fragments (RF) that are retained on the sieve. If feasible, determine the rock-fragment density by weighing the fragments in air to obtain their mass and in water to obtain their volume. Alternatively, assign a rock fragment density. If the rock fragments are porous and have a density similar to the soil sample, do not correct the clod mass and the volume measurement for rock fragments. Correct bulk density for the mass and the volume of the plastic coating. The coating has an air-dry density of 1.3 g cm−3. The coating loses 10 to 20% of its airdry mass when dried in an oven at 105°C.
2.1.4.3.d Calculations The bulk density of < 2-mm fabric at 33 kPa, ρb33, is given by: ρb33 = (WODC − RF − ODPC − TAG)/ {[(CC3 − WMCW)/WD] − (RF/PD) − (MPC/1.3)}
[2.1–14]
where WODC is the oven-dry coated clod mass, RF is the mass of rock fragments, ODPC is the mass of oven-dry plastic coat = 0.85MPC, where MPC is the mass of plastic coat before oven-drying, TAG is the mass of tag and wire, CC3 is the mass of the equilibrated clod after four additional plastic coats, WMCW is the mass of coated clod equilibrated at 33-kPa tension, suspended in water, PD is the density of rock fragments, WD is water density. The mass of the plastic coat before oven-drying, MPC, is obtained from: MPC = (CC2 − CC1 + FCE)RV + (CC3 − WMC)
[2.1–15]
where CC2 is the mass of clod after three laboratory plastic coats, CC1 is the mass of clod before three laboratory plastic coats, RV is the percentage estimate of remaining clod volume after cutting to obtain flat surface (≈80%), WMC is the mass of coated clod equilibrated at 33-kPa tension, and FCE is the field-applied plastic coat mass estimate. FCE is estimated as: FCE = 0.5(CC2 − CC1)/3
[2.1–16]
The bulk density of 10 m (D 5195-91; ASTM, 1999l).
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2.1.5.2 Equipment and Procedures Instruments are available from several manufacturers, but none are given here. There is no common procedure for the several instruments on the market, and none will be given here. It is recommended that users carefully follow the manufacturer’s instructions, keeping in mind that instruments may malfunction and that there is an element of danger because of the radiation. 2.1.5.3 Comments It is important to follow a nuclear safety protocol to prevent radiation exposure. The regulations concerning use and transport of sealed source nuclear devices has contributed to a decline in the use of nuclear devices. Gamma ray transmission and back scatter are nondestructive and provide a way to monitor in situ changes in density over time. This type of application is dependent on an independent in situ measure of soil water content on a volumetric basis as provided by electromagnetic methods and neutron probes, as discussed in Section 3.1. The distance between the probes on the two-probe gamma gauge must be known. Calibration procedures for a two-probe gamma gauge are described by Bertuzzi et al. (1987). Dry bulk density results obtained with the probe are generally lower than those obtained with the core method. Absolute differences between the core and the two-probe gamma gauge are ± 0.01 to ± 0.08 g cm−3 (Bertuzzi et al., 1987; Rawitz et al., 1982). Rock fragments with a diameter >6 mm create artifacts visible in x-ray, computed tomography imagery (Petrovic et al., 1982). The size of the rock fragments were overestimated, and the 1). For the special case of H = d = 0, Eq. [3.4–11] collapses to the Wooding (1968) expression for steady infiltration from a shallow circular pond: qs/Kfs = Q/(πa2Kfs) = [1/(α*C3a)] + 1
[3.4–12]
where C3 = 0.25π. Note for this special case there are only capillarity and gravity components of flow (first and second terms on the right of Eq. [3.4–12], respectively). The impact of lateral flow divergence due to soil capillarity is accounted for by the C3a term. The soil macroscopic capillary length parameter (α*) in Eq. [3.4–11] and [3.4–12] represents the relative importance of the gravity and capillarity forces during infiltration (Raats, 1976). Large α* indicates a dominance of gravity over capillarity, which occurs primarily in coarse-textured and/or highly structured porous media. Small α*, on the other hand, indicates dominance of capillarity over gravity, which occurs in fine-textured and/or unstructured porous media. As a result, infiltration is often gravity-driven in coarse-textured and structured porous media (large α*), and capillarity-driven in fine-textured, structureless porous media (small
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Table 3.4–3. Impacts of water ponding depth (H), ring insertion depth (d), ring radius (a), and soil macroscopic capillary length (α*) on quasi-steady hydrostatic pressure flow, capillarity flow, gravity flow, and relative infiltration rate (qs/Kfs) out of a ring infiltrometer. Values calculated using Eq. [3.4–11]. H
d
a
α*†
Pressure flow
Capillarity flow
Gravity flow
qs/Kfs
cm−1
cm 5 5 5 5 5
5 5 5 5 5
5 10 20 40 60
0.12 0.12 0.12 0.12 0.12
0.637 0.465 0.303 0.178 0.126
1.061 0.776 0.504 0.297 0.21
1 1 1 1 1
2.698 2.241 1.807 1.475 1.336
5 5 5 5
3 5 10 20
30 30 30 30
0.12 0.12 0.12 0.12
0.246 0.224 0.183 0.134
0.41 0.374 0.306 0.224
1 1 1 1
1.656 1.598 1.489 1.358
10 20 40
5 5 5
30 30 30
0.12 0.12 0.12
0.448 0.897 1.793
0.374 0.374 0.374
1 1 1
1.822 2.27 3.167
5 5 5
5 5 5
30 30 30
0.36 0.04 0.01
0.224 0.224 0.224
0.125 1.121 4.483
1 1 1
1.349 2.345 5.707
†α* values selected from Table 3.4–4.
α*). The connection between the magnitude of α* and porous medium texture and structure makes it possible to estimate α* from soil texture and structure categories (Table 3.4–4). There are also ring infiltrometer methods that allow direct measurement of α* (pressure infiltrometer, twin-ring infiltrometer, multiple-ring infiltrometer). Note as well that α* has been identified as numerically equivalent to the reciprocal of the Green and Ampt (1911) effective wetting front pressure head (see Eq. [3.4–16] and associated discussion). Ring infiltrometer analyses based on quasi-steady infiltration for constant ponded head can all be derived from Eq. [3.4–11] or [3.4–12].
Table 3.4–4. Soil texture-structure categories for site-estimation of α* (adapted from Elrick et al., 1989). Soil texture and structure category
α* cm−1
Compacted, structureless, clayey or silty materials such as landfill caps and liners, lacustrine or marine sediments. Soils that are both fine textured (clayey or silty) and unstructured; may also include some fine sands. Most structured soils from clays through loams; also includes unstructured medium and fine sands. The category most frequently applicable for agricultural soils. Coarse and gravelly sands; may also include highly structured or aggregated soils, as well as soils with large and/or numerous cracks, macropores.
0.01 0.04 0.12 0.36
THE SOIL SOLUTION PHASE
821
3.4.3.2.a Single-Ring and Double- or Concentric-Ring Infiltrometers W. D. REYNOLDS, Agriculture and Agri-Fo od Canada, Harrow, Ontario, Canada D. E. ELRICK, University of Guelph, Guelph, Ontario, Canada E. G. YOUNGS, Cranfield University, Silsoe, Bedfordshire, England
Introduction. The single-ring and double- or concentric-ring infiltrometers (Fig. 3.4–7) are used primarily for measuring cumulative infiltration, I (L), infiltration rate, q = dI/dt (LT−1], and field-saturated hydraulic conductivity (or infiltration capacity), Kfs (L T−1). The traditional approach for determining Kfs is based on a simplified one-dimensional version of Eq. [3.4–11]. The single-ring infiltrometer is a single open-ended measuring cylinder, while the double- or concentric-ring infiltrometer consists of the measuring cylinder placed concentrically inside a second open-ended buffer cylinder (Fig. 3.4–7). Apparatus and Procedures. 1. The single-ring infiltrometer method typically uses a single measuring cylinder that is 10 to 50 cm in diameter and 10 to 20 cm long, although diameters as large as 100 cm are used occasionally. The measuring cylinder of the double- or
Fig. 3.4–7. Schematic of the single-ring and double- or concentric-ring infiltrometers.
822
CHAPTER 3
concentric-ring infiltrometer, on the other hand, is usually about 10 to 20 cm in diameter by 10 to 20 cm long, while the buffer cylinder is generally about 50 cm in diameter and the same length as selected for the measuring cylinder. For both infiltrometer designs, the cylinders should be sturdy (e.g., made of metal or high-density plastic), but thin-walled (e.g., 1–5 mm wall thickness) with a sharp outsidebeveled cutting edge at the base to minimize resistance and soil compaction or shattering during cylinder insertion (Fig. 3.4–7). Using an appropriate insertion technique (e.g., drop-hammer apparatus or hydraulic ram), insert the cylinder(s) into the soil to a depth of 3 to 10 cm (discussed further in comments section). Ensure that the cylinder(s) is (are) long enough to allow the desired depth of water ponding after the cylinder is inserted (i.e., if the desired ponding and insertion depths are 5 cm, then the cylinder should be at least 11 cm long). The cylinder(s) should be held as straight (vertical) as possible during the insertion process to encourage one-dimensional vertical flow in the soil. Scraping, leveling, or similar disturbance of the infiltration surface (i.e., soil surface inside the cylinder) is not recommended, as this may alter the soil’s hydraulic properties and thus produce unrepresentative results. 2. Prevent short circuit flow or leakage around the cylinder walls by lightly tamping the contact between the soil and the inside surface of the cylinders. Larger gaps between the soil and cylinder walls should be backfilled with powdered bentonite or fine clay. 3. Pond a constant head (depth) of water inside the measuring cylinder and measure the rate at which water infiltrates the soil. If the concentric-ring infiltrometer is being used, pond the same depth of water in the buffer cylinder as in the measuring cylinder (Fig. 3.4–7). It is not necessary to measure infiltration through the buffer cylinder, although this is done occasionally for comparing single-ring and concentric-ring infiltrometer results (i.e., the summed infiltration from both rings serves as a single-ring infiltrometer measurement). The depth of water ponding should be as small as circumstances allow, which is usually on the order of 5 to 20 cm. A simple and convenient method for simultaneously maintaining a constant ponding head and measuring the infiltration rate is to set up a Mariotte reservoir (Fig. 3.4–7). The height of the Mariotte bubble tube sets the depth of ponding, and the rate of fall of the water level in the Mariotte reservoir can be used to calculate the infiltration rate. Alternative approaches include the use of a float valve arrangement connected via flexible tubing to a gravity-feed reservoir (often useful for high infiltration rates), and simple manual addition of water (often useful for low infiltration rates). In the manual approach, some kind of pointer or “hook gauge” is positioned above the infiltration surface, and when the water surface in the cylinder drops to the pointer or hook gauge level, water is manually added to bring the water surface back up to a preset mark on the cylinder wall. Average infiltration rate in the manual approach is determined using the volume of water added and the time interval between additions. The depth of water ponding is estimated as the mid-way elevation between the mark on the cylinder wall and the height of the pointer or hook gauge. If the concentric-ring infiltrometer is used, separate flow and head controlling devices will be required for the measuring cylinder and the buffer cylinder in order to allow separate determination of infiltration through the measuring cylinder.
THE SOIL SOLUTION PHASE
823
4. Determine infiltration into the soil by monitoring the discharge through the measuring cylinder. Quasi-steady flow in the near-surface soil under the measuring cylinder is assumed when the discharge becomes effectively constant. The time required to reach quasi-steady flow (equilibration time) generally increases with finer soil texture, decreasing soil structure, increasing depth of water ponding (H), increasing depth of cylinder insertion (d), and increasing cylinder radius (a). Equilibration times can be as short as 10 to 60 min for relatively small cylinders (e.g., 5–10 cm diam.) and/or materials that are coarse textured or well structured (Scotter et al., 1982). Equilibration times as long as several hours to days may be required, however, for large cylinders (e.g., 30–60 cm diam. and larger) and/or materials that are moderate to fine textured and stuctureless (Scotter et al., 1982; Daniel, 1989). Analyses. One of the earliest and simplest single- and concentric-ring infiltrometer analyses for field-saturated hydraulic conductivity neglects the first two terms on the right of Eq. [3.4–11] and assumes, Kfs = qs
[3.4–13]
where qs (L T−1) is the quasi-steady infiltration rate and Kfs (L T−1) is field-saturated hydraulic conductivity. It is clear from Table 3.4–3, however, that Eq. [3.4–13] will overestimate Kfs by varying degrees (i.e., qs/Kfs > 1 in Table 3.4–3), depending on the magnitudes of H, d, a, and α*. Setting H and d to 5 cm (commonly used values) and α* to 0.12 cm−1 (value for many agricultural soils), Eq. [3.4–11] predicts that a measuring cylinder diameter of 5.2 m would be required in order for Eq. [3.4–13] to estimate Kfs within 5% (i.e., qs/Kfs = 1.05). A cylinder this large is often not practical, as it would be difficult to install, highly consumptive of water, and may require an excessively long equilibration time. Equation [3.4–11] can, of course, be applied directly for determination of Kfs; that is, qs Kfs = __________________________________ [H/(C1d + C2a)] + {1/[α*(C1d + C2a)]} + 1
[3.4–14]
Note, however, that the α* parameter must be either estimated from the soil texture and structure categories in Table 3.4–4, or measured using independent methodology. An additional analysis (Bouwer, 1966, 1986) applies the Green and Ampt (1911) equation to one-dimensional vertical flow within and below the measuring cylinder: qs/Kfs = (H/Lf) + [1/(α*Lf)] + 1
[3.4–15]
where Lf (L) is the depth from the infiltration surface to the wetting front. As with Eq. [3.4–14], the α* parameter must be selected from Table 3.4–4 or measured independently. Note that like Eq. [3.4–11], the first, second and third terms on the right of Eq. [3.4–15] represent the hydrostatic pressure, capillarity, and gravity components of infiltration, respectively. Note also that Eq. [3.4–15] predicts that qs decreases with increasing Lf (over time), and that qs → Kfs as Lf (and time) become large. Equation [3.4–15] does not, however, account for the lateral flow divergence
824
CHAPTER 3
in the pressure and capillarity terms which is implicit in the (C1d + C2a) term of Eq. [3.4–11]. Consequently, Eq. [3.4–15] still tends to overestimate Kfs under quasi-steady flow. An interesting application of Eq. [3.4–15] that attempts to address the flow divergence and α* estimation issues is provided by the air-entry permeameter method (Bouwer, 1966; Topp & Binns, 1976). Here, a single cylinder is inserted into the soil and a rapid-responding tensiometer is inserted through the infiltration surface to a predetermined depth, Lf, which is less than or equal to the cylinder depth, d (Lf is typically set at ~5 cm in clayey soils and 15 cm in sandy soils; Topp & Binns, 1976). Next, cumulative infiltration is measured with time during the initial transient phase of constant ponded head infiltration, and the tensiometer is monitored to determine the time required for the wetting front to reach Lf. When the tensiometer responds (i.e., the tensiometer reading increases suddenly to near zero when the wetting front reaches the porous tensiometer tip), the cylinder is sealed off from the atmosphere and the air-entry value (pressure head) of the wetted soil within the cylinder is measured using a Bourdon type vacuum gauge (Topp & Binns, 1976). For many soils (Bouwer, 1966; Elrick & Reynolds, 1992a, b), α* = −hf−1 ≈ −2(ha−1);
hf ≤ 0, ha ≤ 0
[3.4–16]
where hf (L) is the effective Green–Ampt wetting front pressure head (Green & Ampt, 1911) and ha (L) is the measured air-entry pressure head. Lateral flow divergence is thus avoided by setting Lf ≤ d so that flow is confined to one-dimensional by the cylinder, and greater accuracy in the α* determination is attempted by calculating it via the air-entry pressure head, rather than estimation from texture and structure categories (Table 3.4–4). Difficulties can arise, however, when a diffuse and/or irregular wetting front causes an ambiguous tensiometer response (e.g., no increase or a slow increase in tensiometer reading, rather than a rapid increase), and when the relationship between α* and ha is inconsistent or not well estimated by Eq. [3.4–16].
Table 3.4–5. Example single-ring and double- or concentric-ring infiltrometer data sheet and Kfs calculation. Measuring cylinder radius, a = 30 cm Cylinder insertion depth, d = 5 cm Depth of water ponding, H = 10 cm Macroscopic capillary length parameter, α* = 0.12 cm−1 Air-entry pressure head, ha = −17.6 cm Effective wetting front pressure head, hf = −α*−1 ≈ ha/2 = −8.3 cm Estimated depth to wetting front, Lf = 40 cm Measured quasi-steady infiltration rate, qs = Qs/πa2 = 1.82 × 10−3 cm s−1 Quasi-empirical constant, C1 = 0.316π = 0.9927 Quasi-empirical constant, C2 = 0.184π = 0.5781 Quasi-empirical constant, C3 = 0.25π = 0.7854 Eq. [3.4–12]:
Kfs = qs/{[1/(α*C3a)] + 1} = 1.3 × 10−3 cm s−1
Eq. [3.4–13]:
Kfs = qs = 1.8 × 10−3 cm s−1
Eq. [3.4–14]:
Kfs = qs/{[H/(C1d + C2a)] + [1/[α*(C1d + C2a)]] + 1} = 1.0 × 10−3 cm s−1
Eq. [3.4–15]:
Kfs = qs/[(H/Lf) + (1/α*Lf) + 1] = 1.2 × 10−3 cm s−1
THE SOIL SOLUTION PHASE
825
Example Calculations Using Eq. [3.4–12] through [3.4–15] (Table 3.4–5). Note that Eq. [3.4–12], [3.4–13], and [3.4–15] produce higher estimates of Kfs due to lack of full accounting for the hydrostatic pressure and soil capillarity components of flow out of the infiltrometer. (The Eq. [3.4–15] calculation is based on the steady flow analysis, not on the air-entry permeameter analysis.) As demonstrated in Table 3.4–3, the accuracies of Eq. [3.4–12], [3.4–13], and [3.4–15] depend strongly on ring radius and insertion depth, water ponding depth, and soil capillarity. Practitioners should therefore take these factors into account when deciding which analysis will be adequate for their purposes. Remember also that Eq. [3.4–12] and [3.4–14] require estimation or independent measurement of α*, while Eq. [3.4–15] requires estimation or measurement of both α* and Lf. Comments. 1. The intention with the double- or concentric-ring infiltrometer is to physically prevent flow divergence under the measuring cylinder by adding the outer buffer cylinder. It is assumed that infiltration through the annular space between the buffer and measuring cylinders “absorbs” the flow divergence, leaving only vertical flow under the measuring cylinder (vertical flow is well described by Eq. [3.4–13] and [3.4–15]). Unfortunately, laboratory, field, and numerical simulation tests show that the buffer cylinder is often not effective, with the quasi-steady infiltration rate from the measuring cylinder still being influenced substantially by flow divergence. Consequently, the accuracy of Eq. [3.4–13] and [3.4–15] is often not improved appreciably by the use of the concentric-ring infiltrometer. 2. As shown in Table 3.4–3, the accuracy of Eq. [3.4–13] for determining Kfs increases with increasing cylinder radius (a), decreasing depth of water ponding (H), increasing depth of ring insertion (d), and increasing macroscopic capillary length (α*). Consequently, the cylinder diameter should always be as large as possible and ponding depth as small as possible if Eq. [3.4–13] is to be used. Although large insertion depths (d) also improve the accuracy of Eq. [3.4–13], they should probably be avoided in most cases, as large d values tend to increase equilibration time and cause excessive soil disturbance. Insertion depth should, however, be great enough to prevent leakage under the cylinder wall, especially in cracked soils. When planning a study using ring infiltrometers, Eq. [3.4–11] and Table 3.4–3 can be used, along with practical considerations and estimates of α* and Kfs, to establish optimum values for H, d, and a. 3. Physical sources of measurement error in single- and concentric-ring infiltrometers include soil compaction during installation, short circuit flow along the cylinder walls, siltation of the infiltration surface, and gradual plugging of soil pores by deflocculated silt and clay particles. Soil compaction can be reduced by using small insertion depths and thin-walled cylinders with sharp cutting edges. Short circuit flow or leakage along the cylinder walls can be reduced by tamping the soil adjacent to the walls and/or by backfilling gaps between the soil and wall with powdered bentonite or fine clay. Siltation, which is the plugging of soil pores on the infiltration surface by resettlement of eroded silt and clay, can be minimized by using diffuser devices (such as a coarse sponge) to reduce the force of the incoming water. Soil pores on and under the infiltration surface can be gradually plugged by deflocculated silt and clay if the water used for the measurement has major cation (Na,
826
CHAPTER 3
Ca, Mg, K) concentrations that are greatly different from those of the resident soil water. This can be avoided by ensuring that the water used has major cation concentrations similar to those of the soil water. Fortunately, the major cation concentrations of local tap water are frequently sufficient to prevent deflocculation. When this is not the case, however, a laboratory facsimile of the soil water can be prepared with the appropriate cation concentrations. Alternatively, commercial flocculent (clarifier) can be added to the available water. Distilled or deionized water should never be used, as it strongly encourages deflocculation. Other sources of error in infiltrometer measurements include changes in soil permeability with depth (e.g., due to soil layering or horizonation), the presence of a soil crust, interference from a shallow water table (Youngs et al., 1996), and evaporation from the cylinder when infiltration is very slow. In addition, excessively long equilibration times can occur when measuring low permeability materials and/or when large cylinder diameters are used. 4. Given that infiltration and Kfs are highly variable parameters, replication of ring infiltrometer measurements is essential. Unfortunately, there is no way of predetermining the number of replicates required, as this depends on the variability encountered. In addition, if the cylinder diameter is not large enough to encompass a representative soil sample, very large and erratic variations may occur among replicate measurements (Youngs, 1987). For example, Youngs et al. (1996) found it necessary to use 20-m-diam. infiltration “ponds” (constructed using low, plastic-lined embankments to prevent lateral leakage) to sample adequately an extremely heterogeneous coastal area material. The best approach appears to be to continue taking randomly placed replicate measurements with appropriate diameter cylinders until a reasonably constant mean and standard deviation are obtained (see Sections 3.4.1 and 3.4.3.1 for further comment). For agronomic plot-scale work, replications of 10 to 20 or more per treatment are often required (Bouwer, 1986). 5. Further information on single- and double- or concentric-ring infiltrometers can be found in Bouwer (1966, 1986), Daniel (1989), and Youngs (1972, 1987, 1991a, b). Additional information on the air-entry permeameter can be found in Bouwer (1966) and Topp and Binns (1976). 3.4.3.2.b Pressure Infiltrometer W. D. REYNOLDS, Agriculture and Agri-Food Canada, Harrow, Ontario, Canada D. E. ELRICK, University of Guelph, Guelph, Ontario, Canada
Introduction. The pressure infiltrometer method (Fig. 3.4–8) is used primarily to measure field-saturated hydraulic conductivity, Kfs (L T−1), but can also be used to determine matric flux potential, φm (L2 T−1), sorptivity, S (L T−1/2), the macroscopic capillary length parameter, α* (L−1)], the effective Green–Ampt wetting front pressure head, hf (L), air-entry pressure head, ha (L), and water-entry pressure head, hw (L). The method is based on three-dimensional analyses of ponded infiltration through a single, 10- to 20-cm-diam. ring inserted 3 to 10 cm into unsaturated soil. The original and most widely used analyses for the pressure infiltrometer are based on the quasi-steady-state phase of constant head infiltration (Reynolds & Elrick,
THE SOIL SOLUTION PHASE
827
1990). More recently, transient constant head and falling head analyses have been developed (Elrick et al., 1995, 2002; Odell et al., 1998; Wu et al., 1999) for estimating Kfs, φm, and other parameters in low conductivity materials (i.e., Kfs ≤ 10− 7 cm s−1). This chapter describes the original quasi-steady analyses for constant head infiltration, with brief reference made to the various transient analyses. There are several similarities in apparatus, analyses, and procedures between the pressure infiltrometer method and the constant head well permeameter method described in Section 3.4.3.3. Apparatus and Procedure. 1. Using an appropriate insertion technique (e.g., block of wood and hammer, hydraulic ram, or drop-hammer) insert a 10- to 20-cm-diam. ring into the soil to a depth of 3 to 10 cm (Fig. 3.4–8). A 10-cm-diam. ring appears adequate for all but extensively cracked soils (Reynolds et al., 2000). The ring should be thin-walled and beveled to a sharp cutting edge at the base in order to minimize resistance and soil compaction or shattering during the insertion process (Fig. 3.4–8). It is advisable to construct the ring from stainless steel to prevent rusting. The ring should be
Fig. 3.4–8. Schematic of a driving apparatus, ring, and soil tamper for use in the pressure infiltrometer method (from Reynolds, 1993, p. 607).
828
CHAPTER 3
held as straight (vertical) as possible during the insertion process. If the ring tilts during insertion, no attempt should be made to straighten it. A slight tilt to the ring can be accommodated; however, if the tilt is significant the ring should be removed and reinserted at a new location. Scraping, leveling, or similar disturbance of the soil at the measurement site is not recommended, as this may alter the site’s soil hydraulic properties. Tall vegetation may have to be clipped to ≤1-cm height, however, to facilitate ring insertion and installation of the cap (Fig. 3.4–9). 2. After the ring has been inserted, the contact between the inside surface of the ring and the soil may have to be tamped lightly (using the blunt end of a pencil or similar implement) to minimize the possibility of short circuit flow along the inside wall of the ring (Fig. 3.4–8). 3. Attach the cap to the ring using wing bolts (Fig. 3.4–9) or some other arrangement to ensure a water-tight seal. Attach a constant head reservoir to the cap and set the reservoir to maintain an appropriate constant head of water, H (L), on the soil surface. The Mariotte reservoir system of the commercially available Guelph Permeameter is often convenient for this purpose (Soilmoisture Equipment Corporation, 1987). Alternative Mariotte reservoirs can be constructed from plastic cylinders of different diameters: small diameters (e.g., 2–3 cm) for increased accuracy when measuring slow flow rates into low-permeability soils and large diameters (e.g., 5–10 cm or more) for increased capacity when measuring fast flow rates into high-permeability soils. 4. The pressure head of water acting on the infiltration surface (H) is determined by measuring the height of water in the standpipe (Fig. 3.4–9). If a Mariotte
Fig. 3.4–9. Schematic of a Mariotte reservoir system for use in the pressure infiltrometer method (from Reynolds, 1993, p. 608).
THE SOIL SOLUTION PHASE
829
reservoir is used (such as that of the Guelph Permeameter), the infiltrometer is operating properly when air bubbles rise regularly up into the reservoir from the base of the air inlet tube. There should be no water leaks in the cap or at the seal between the cap and the ring (i.e., leakage past the O-ring). 5. The hydrostatic pressure of the ponded head of water may occasionally lift the ring out of the soil when the soil is wet, of low permeability, and the ponding head is large. This is usually remedied by simply placing a suitable counter weight on top of the infiltrometer cap. 6. The rate of water flow, Q (L3 T−1), into the soil is measured by monitoring the water discharge out of the reservoir. If a Mariotte reservoir system is used, discharge is conveniently measured by monitoring the rate of fall of the water level in the reservoir, R (L T−1), and multiplying by the reservoir cross-sectional area, A (L2) (i.e., Q = AR). When the discharge becomes constant (e.g., effectively constant R value for about 10 or more min), the flow rate out of the infiltrometer and into the soil is deemed to have reached quasi-steady state. For highly to moderately permeable soils (e.g., Kfs ≈ 10−2 to 10−5 cm s−1), the time required to achieve quasisteady-state flow (equilibration time) is usually short (e.g., 5–60 min), with equilibration time generally increasing as permeability decreases. For very low permeability soils (e.g., Kfs ≤10−7 cm s−1), equilibration times can be on the order of many hours to days, and only early-time transient flow procedures are practicable (discussed further in comments section). Analyses. The Kfs, φm, S, ha, and hw parameters can be determined using single-head (H1), two-head (H1, H2), or multiple-head (H1, H2, H3,...) analyses. The α* and hf parameters, on the other hand, can be obtained only from two-head or multiple-head analyses. The pressure infiltrometer analyses are similar to those used for the constant head well permeameter (Section 3.4.3.3), although different equations are used. The single-head pressure infiltrometer analysis is the most straight forward, as it requires only a single quasi-steady flow measurement at one ponded head of water (H1). It may, however, be less accurate than the two-head or multiple-head analyses for determination of Kfs, φm, and S and tends to be used when these parameters need to be known only within a factor of ≤2 (discussed further in comments section). The two-head and multiple-head approaches are used when greater accuracy in Kfs, φm, or S is desired, or when simultaneous measurements of the α* and hf parameters are required (Reynolds & Elrick, 1990). On a theoretical basis, the accuracy of the ha and hw parameters is the same regardless of which analysis procedure is used (see below). Single-Head Analysis. For the single-head approach, Kfs and φm are determined using (Reynolds & Elrick, 1990), Kfs = α*GAR1/[a(α*H1 + 1) + Gα*πa2]
[3.4–17]
φm = GAR1/[a(α*H1 + 1) + Gα*πa2]
[3.4–18]
where α* (L−1) is a soil texture–structure parameter (which must be selected from the soil texture and structure categories in Table 3.4–4 or determined by independent
830
CHAPTER 3
measurement), A (L2) is the cross-sectional area (cell constant) of the pressure infiltrometer reservoir, R1 (L T−1) is the quasi-steady rate of fall of the water level in the pressure infiltrometer reservoir, a (L) is the inside radius of the ring, H1 (L) is the steady pressure head of water on the infiltration surface (set by the height of the air tube and determined by measuring the height of water in the standpipe), and G is a dimensionless shape factor obtained using (Reynolds & Elrick, 1990), G = 0.316(d/a) + 0.184
[3.4–19]
where d (L) is the depth of ring insertion into the soil. Equations [3.4–17] to [3.4–19] are based on Eq. [3.4–11] (Reynolds & Elrick, 1990). Note that AR1 = Q1 (L3 T−1) is the quasi-steady flow rate out of the infiltrometer and into the soil. Equation [3.4–19] was developed for 5 cm ≤ a ≤ 10 cm, 3 cm ≤ d ≤ 5 cm, and 5 cm ≤ H1 ≤ 25 cm (Reynolds & Elrick, 1990); however, work by Youngs et al. (1995) indicates that these ranges can be extended substantially (e.g., 0 < d ≤ 10 cm; 5 cm ≤ H ≤ 100 cm or more) without significant reduction in accuracy. Sorptivity, S (L T−1/2), may be calculated using: S = [γ(θfs − θi)φm]1/2
[3.4–20]
where γ ≈ 1.818 is a dimensionless constant related to wetting front shape (White & Sully, 1987), θfs (L3 L−3) is the field-saturated volumetric soil water content, and θi (L3 L−3) is the initial or background, or antecedent, volumetric soil water content at the time of the measurement. The ha and hw parameters can be determined using the tension bottle attachment and procedures described in Fallow and Elrick (1996), and remembering that the standpipe (Fig. 3.4–9) must be sealed (using a rubber bung, clamp, valve, or some other method) to prevent it from allowing air into the end-cap. Two-Head and Multiple-Head Analyses. The two-head analysis determines Kfs and φm using two consecutively ponded heads (H1, H2) and the simultaneous equations (Reynolds & Elrick, 1990), Kfs = (G/a)[(Q2 − Q1)/(H2 − H1)]
[3.4–21]
€€φm = (G/a)[(H2Q1 − H1Q2)/(H2 − H1)] − (πG2)[(Q2 − Q1)/(H2 − H1)] [3.4–22] where H2 > H1, Q2 > Q1, and Qi = AiRi; i = 1, 2
[3.4–23]
with Ai (L2) representing reservoir cross-sectional area, and the other parameters are as defined above. The A parameter is subscripted here to indicate that pressure infiltrometers with two or more reservoir diameters (e.g., Soilmoisture Equipment Corp., 1987) may be advisable (or required) to maintain measurement precision if large differences occur between Q1 and Q2. The multiple-head approach determines Kfs and φm using two or more consecutively ponded heads and the regression equations (Reynolds & Elrick, 1990),
THE SOIL SOLUTION PHASE
831
Kfs = (G/a)(∆Q/∆H)
[3.4–24]
φm = (G/a)(Y − πa2Kfs)
[3.4–25]
where ∆Q/∆H and Y are the slope and intercept, respectively, of the linear least squares regression line through a plot of Q vs. H data, and the other parameters are as defined above. Equations [3.4–24] and [3.4–25] apply when two or more heads of water (H values) are set sequentially on the infiltration surface (i.e., H1 set first; H1 < H2 < H3...; ring not allowed to drain when changing from one head to the next higher head) to obtain the corresponding Qi = AiRi values (i.e., Q1, Q2, Q3,...). As with the two-head analysis, infiltrometers with two or more reservoir diameters (variable A) may be required to maintain measurement precision. Equations [3.4–24] and [3.4–25] are readily solved using the built-in regression function in most handheld calculators and computer spreadsheets. Equations [3.4–21], [3.4–22], [3.4–23], [3.4–24], and [3.4–25] give identical results for two consecutively ponded heads because linear regression is equivalent to simultaneous equations when only two Q vs. H data pairs are used. When Kfs and φm are determined using two or more successively ponded heads (H), it is also possible to obtain simultaneous measures of the macroscopic capillary length parameter, α* (L−1), and the Green–Ampt wetting front pressure head parameter, hf (L), using: α* = Kfs/φm = −hf−1
[3.4–26]
That is, the α* parameter does not have to be estimated from Table 3.4–4 or obtained from an independent measurement, as in the single-head approach. Sorptivity may still be calculated using Eq. [3.4–20], and the ha and hw parameters from the procedures in Fallow and Elrick (1996). Further discussion of the relative merits of the single-head and two-head or multiple-head analyses is given in the comments section. Example Calculations. An example pressure infiltrometer data sheet is given in Table 3.4–6 for a highly structured loam soil. Note that flow for the first ponded head (H1 = 10 cm) required about 20 min to reach a convincing quasi-steady state of Q1 = 0.5333 cm3 s−1. The second ponded head (H2 = 20 cm) required an additional 18 min to equilibrate to Q2 = 0.7467 cm3 s−1, and the third head (H3 = 40 cm) a further 16 min to equilibrate to Q3 = 1.1733 cm3 s−1. A total of 54 min and 2989 cm3 of water (reservoir cross-sectional area, A = 64 cm2) were required to complete the measurement of quasi-steady flow for the three sequentially ponded heads. For each head, the R value declined through an initial transient phase to a quasi-constant value. An effectively constant R over about 10 min is often used as the criterion for establishment of quasi-steady flow, although this may vary with soil type and conditions, ring diameter, and depth of ring insertion. A timing interval of 1 min for each head happened to be convenient for this test, but larger or smaller intervals may be more appropriate depending on the flow rates encountered. Note also that we consider the flow to be quasi-steady only because true steady flow will occur only in completely homogeneous soil, which probably never exists in natural environments. Hence, a reasonable estimate of steady flow, or “quasi-steady flow”, is considered acceptable (discussed further in the comments section).
832
CHAPTER 3
Table 3.4–6. Example pressure infiltrometer data sheet. These data are contrived to illustrate better the “typical” behavior. Actual measurements may differ in terms of, for example, flow rate, optimum timing interval, equilibration time, constancy of flow rate at quasi-steady state. Ring radius, a = 5 cm; ring insertion depth, d = 10 cm Mariotte reservoir cross-sectional area, A = 64 cm2 H1 = 10 cm Cumul. time, t
H2 = 20 cm
Reservoir scale Rate, R reading, L (∆L/∆t)
Cumul. time, t
H3 = 40 cm
Reservoir scale Rate, R reading, L (∆L/∆t)
Reservoir Cumul. scale Rate, R time, t reading, L (∆L/∆t)
min
cm
cm min−1
min
cm
cm min−1
min
cm
cm min−1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 1.1 2 2.8 3.5 4.3 5 5.6 6.3 6.9 7.5 8 8.4 9 9.5 9.9 10.5 11 11.5 12 12.5
-1.1 0.9 0.8 0.7 0.8 0.7 0.6 0.7 0.6 0.6 0.5 0.4 0.6 0.5 0.4 0.6 0.5 0.5 0.5 0.5
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
16 17.2 18.3 19.2 20 20.8 21.5 22.3 22.9 23.7 24.3 25.1 25.8 26.5 27.2 27.9 28.6
-1.2 1.1 0.9 0.8 0.8 0.7 0.8 0.6 0.8 0.6 0.8 0.7 0.7 0.7 0.7 0.7
40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
30.5 32 33.3 34.5 35.6 36.8 37.8 39 40 41.2 42.3 43.4 44.5 45.6 46.7
-1.5 1.3 1.2 1.1 1.2 1 1.2 1 1.2 1.1 1.1 1.1 1.1 1.1
R1 = 0.5 cm min−1 = 0.00833 cm s−1
R2 = 0.7 cm min−1 = 0.01167 cm s−1
R3 = 1.1 cm min−1 = 0.01833 cm s−1
Q1 = AR1 = 0.5333 cm3 s−1
Q2 = AR2 = 0.7467 cm3 s−1
Q3 = AR3 = 1.1733 cm3 s−1
Table 3.4–7 gives the Kfs, φm, S, α*, and hf results for the data in Table 3.4–6. The multiple-head and two-head approaches give essentially the same results, with Kfs = 3.5 × 10−3 cm s−1, φm = 7.6 × 10−3 cm2 s−1, S = 6.4 × 10−2 cm s−1/2, α* = 0.46 cm−1, and hf = −2.2 cm. Good correspondence between the two approaches (as in this example) can be expected when the soil inside and immediately surrounding the ring is relatively homogeneous and does not contain a strong antecedent water content gradient. However, if soil heterogeneity (e.g., layering, large cracks or macropores) and/or an antecedent water content gradient exist within the soil volume wetted by the measurement, then the multiple-head and two-head analyses may not correspond nearly as well as they do in this example. The multiple-head and two-head analyses may also yield unrealistic or invalid (i.e., negative) Kfs, φm, or α* values if soil heterogeneity or antecedent water content gradients are extreme within the measurement zone (discussed further in comments section). Visual inspection of the soil at the field site suggested that the α* = 0.36 cm−1 category in Table 3.4–4 was most appropriate. When this α* is used, the single-head analysis yields effectively the same Kfs as the multiple-head and two-head analy-
THE SOIL SOLUTION PHASE
833
Table 3.4–7. Example pressure infiltrometer calculations. Note that the use of three significant figures in this table is for illustration and comparison purposes only. Under normal circumstances, the natural variability of field-measured soil hydraulic parameters warrants no more than two significant figures. H1 = 10 cm; H2 = 20 cm; H3 = 40 cm; ∆θ = (θfs − θi) = 0.3 Head combination
Kfs cm
s−1
φm cm2 s−1
S cm
s−1/2
α*
hf
cm−1
cm
Multiple-heads, regression analysis (Eq. [3.4–24], [3.4–25], [3.4–20], [3.4–26]) H1, H2, H3
3.48 × 10−3
7.60 × 10−3
6.44 × 10−2
0.458
−2.18
Two-heads, simultaneous equations analysis (Eq. [3.4–21], [3.4–22], [3.4–20], [3.4–26]) H1, H2 H1, H3 H2, H3
3.48 × 10−3 3.48 × 10−3 3.48 × 10−3
7.57 × 10−3 7.59 × 10−3 7.62 × 10−3
6.43 × 10−2 6.43 × 10−2 6.45 × 10−2
0.46 0.459 0.457
−2.17 −2.18 −2.19
Single-head analysis (Eq. [3.4–17], [3.4–18], [3.4–19], [3.4–20]), α* = 0.36 cm−1 (Table 3.4–4) H1 H2 H3
3.40 × 10−3 3.42 × 10−3 3.44 × 10−3
9.45 × 10−3 9.51 × 10−3 9.57 × 10−3
7.18 × 10−2 7.20 × 10−2 7.22 × 10−2
n/a† n/a n/a
n/a† n/a n/a
Single-Head Analysis (Eq. [3.4–17], [3.4–18], [3.4–19], [3.4–20]), α* = 0.12 cm−1 (Table 3.4–4) H1 H2 H3
2.79 × 10−3 2.96 × 10−3 3.13 × 10−3
2.33 × 10−2 2.47 × 10−2 2.61 × 10−2
1.13 × 10−1 1.16 × 10−1 1.19 × 10−1
n/a n/a n/a
n/a n/a n/a
†Not applicable, as α* and hf are estimated via Table 3.4–4 and Eq. [3.4–26], respectively.
ses (i.e., Kfs = 3.4 × 10−3 cm s−1), and the φm and S estimates are higher by only about 25 and 12%, respectively (Table 3.4–7). When the next lower α* category is used (α* = 0.12 cm−1, Table 3.4–4), the single-head estimates of Kfs are lower than the multiple-head and two-head estimates, but only by 10 to 25% which is not significant. The corresponding single-head φm and S estimates, on the other hand, are substantially higher than the multiple-head and two-head estimates. These discrepancies occur because the assumed α* values are smaller than the “actual” value determined by the multiple-head and two-head analyses (i.e., α* = 0.46 cm−1). The discrepancies between the single-head and two-head or multiple-head estimates of Kfs and φm will increase as the difference increases between the “assumed” α* value (via Table 3.4–4) and the actual α* value (as determined by the two-head or multiple-head procedure or by independent measurement). As demonstrated in Table 3.4–7, sensitivity to α* is usually low for Kfs and high for φm and S, although the sensitivities are to some extent affected by soil type and/or condition, ring radius (a), and ponding head (H) (see comments for more detail). As with the two-head and multiple-head analyses, good correspondence can be expected among the single-head results when the soil is relatively homogeneous, but variable results might occur if the soil is heterogeneous. Comments. 1. In order to obtain values of Kfs, φm, S, α*, and hf that are representative of the soil, care must be taken to minimize smearing and siltation of the infiltration surface, compaction or shattering of the soil during ring insertion, and short circuit
834
CHAPTER 3
flow along the contact between the soil and the inside surface of the ring wall. Compaction and shattering, which occurs primarily in clay-rich soils, can usually be minimized by avoiding the soil when it is excessively wet (compaction prone) and when it is excessively dry (shattering prone). Prewetting hard and/or dry soils may allow easier ring insertion, but this may also change the antecedent Kfs, φm, S, α*, and hf values if the soil exhibits shrink–swell behavior. As mentioned above, short circuit flow can be minimized by lightly tamping the contact between the soil and the ring wall (Fig. 3.4–8), and by careful insertion of the ring. 2. Ponding should not occur around the outside of the ring during a measurement, as it usually indicates short-circuiting or some other flow disturbance that is not accounted for by the theory. The appearance of a wetting front on the soil surface around the outside of the ring (Fig. 3.4–9) is not a problem, however; and this is in fact a frequent occurrence when the depth of ring insertion is small (e.g., 3 cm) and/or soil permeability is low. 3. Slow flow from the pressure infiltrometer generally means that the soil is of low permeability, or that the infiltration surface may be smeared and/or compacted and/or silted up. Rapid flow from the pressure infiltrometer means that the soil is of high permeability, or that the soil may have been shattered during ring insertion and short circuit flow is occurring. A slow flowing pressure infiltrometer that uses a Mariotte reservoir can be susceptible to solar heating effects. Solar heating of the air in the Mariotte reservoir can prevent air tube bubbling and/or cause a changing head on the infiltration surface when flow is very slow. Shading of the Mariotte reservoir under these conditions usually prevents this problem. A Mariotte reservoir that is not bubbling may also have an air (vacuum) leak. 4. The time required for the pressure infiltrometer to reach quasi-steady flow (equilibration time) tends to increase as the permeability of the soil decreases, and as the ring radius and depth of insertion increases. Equilibration times on the order of 5 to 30 min often occur when using small ring diameters (≤10 cm) and small ring insertion depths (≤10 cm) in soils of high to moderate permeability (e.g., Kfs = 10−2 to 10−5 cm s−1). Equilibration times may increase to 60 to 120 min or more for larger ring diameters, larger ring insertion depths, and for low permeability soils. 5. The range of Kfs that can be measured using relatively small heads (H ≤ 50 cm) and the quasi-steady flow analyses is on the order of 5 × 10−2 to 1 × 10−6 cm s−1. Early-time transient flow analyses using much larger ponding heads (H >100 cm) and constant or falling head techniques are usually required for measuring lower Kfs values. Recent studies suggest that early-time transient flow analyses can reduce the minimum measurable Kfs to as low as 10−9 cm s−1 (Fallow et al., 1993; Elrick et al., 1995; Youngs et al., 1995; Gerard–Marchant et al., 1997; Odell et al., 1998). Early-time analyses may also be useful for estimating φm and S in low permeability materials (Elrick & Reynolds, 1992a; Elrick et al., 2002). Recently, Wu et al. (1999) developed an approximate transient analysis based on the cumulative infiltration curve. The method appears promising for moderate to low permeability materials, but has received little testing so far. It should also be cautioned that sometimes low permeability materials are sufficiently near saturation (e.g., wet landfill cap and liner materials) that the pressure infiltrometer analyses (Eq. [3.4–17] to [3.4–26]) cannot be applied because there is effectively no capillarity. Youngs
THE SOIL SOLUTION PHASE
835
et al. (1995) argued that in this situation certain modified piezometer analyses can be substituted, and suitable shape factors were developed for that purpose. 6. The primary advantages of the two-head and multiple-head analyses (Eq. [3.4–20], [3.4–21], [3.4–22], [3.4–26] or [3.4–20], [3.4–24], [3.4–25], [3.4–26]) are that simultaneous measures of Kfs, φm, S, α*, and hf are obtained, and an α* value does not have to be estimated from Table 3.4–4 or determined by independent measurements. An important limitation, however, is that unrealistic or invalid (i.e., negative) Kfs, φm, or α* values can occur when strong soil heterogeneity (e.g., extensive layering, cracks, worm holes, root channels), an extreme vertical antecedent water content gradient, or insufficient equilibration time, causes one or more of the quasi-steady flow rates to be inappropriate (e.g., decreasing steady flow rate with increasing pressure head). When the two-head or multiple-head analysis produces a negative Kfs and/or φm, or when the analysis produces an α* value that falls substantially outside the realistic range of 0.01 cm−1 ≤ α* ≤ 1 cm−1, then the singlehead analysis (Eq. [3.4–17], [3.4–18], [3.4–19], [3.4–20]) should be applied to each head (H value) and the resulting Kfs, φm, and S values averaged (Elrick & Reynolds, 1992b). Fortunately, the pressure infiltrometer method is not highly susceptible to soil heterogeneity or antecedent water content gradients, and the success rates (i.e., Kfs, φm, S, α*, and hf all valid and realistic) for the two-head and multiple-head analyses are frequently acceptable. The primary advantages of the single-head analysis for calculating Kfs, φm, and S (Eq. [3.4–17], [3.4–18], [3.4–19], [3.4–20]) are the increased speed of setting only one head (H1), and the avoidance of negative Kfs and φm values. A limitation, however, is the necessity for obtaining α* via site estimation (e.g., using Table 3.4–4) or independent determination using other methodology. This may result in Kfs, φm, and S values of reduced accuracy because of potential errors associated with estimating α* (i.e., selecting the wrong α* category in Table 3.4–4), or because of variability associated with soil heterogeneity and/or the use of alternative methodology to estimate α*. Fortunately, the single-head analysis yields Kfs values that are usually accurate to within a factor of ≤2 when α* is site-estimated and selected from the categories in Table 3.4–4 (see Table 3.4–7 and associated discussion). This level of accuracy is often sufficient for practical applications. 7. The H value(s) should be as large as practicable if one is mainly interested in Kfs, because Kfs is affected primarily by the pressure and gravity components of infiltration through the ring. On the other hand, if one is interested mainly in φm and/or S, then H should be as small as practicable because φm and S are affected primarily by the capillarity component of infiltration through the ring. These patterns are demonstrated in Table 3.4–7, where it is seen that the accuracy of the single-head Kfs estimates increase as H increases, while the accuracy of the corresponding single-head φm and S estimates decrease. It should also be kept in mind that φm, S, α*, and hf can be of low accuracy when obtained using ponded infiltration techniques (regardless of analysis procedure) because of the generally dominating influence of pressure and gravity flow relative to capillarity flow. 8. Once a pressure infiltrometer measurement has been completed, the infiltrometer ring and its contained soil provides a convenient intact soil core for laboratory measurement of saturated hydraulic conductivity (Section 3.4.2), the hydraulic
836
CHAPTER 3
conductivity–water content–pressure head relationships (Sections 3.3.2 and 3.6.1.1), various soil solid phase properties such as bulk density, porosity, particle-size distribution, organic C (Chapter 2), and many other soil physical and biological properties. 9. An important potential application of the pressure infiltrometer is for the siting, design, and monitoring of filter fields associated with on-site wastewater treatment facilities such as household septic systems and milking center washwater disposal facilities. The Percolation Test, or Perc Test, is currently used by most municipalities for this purpose. Unfortunately, the Perc Test is neither accurate nor scientifically sound because it lacks standardization, and it provides a “T-time” that depends not only on soil permeability, but also on the antecedent soil moisture conditions, dimensions of the Perc Test hole, and the depth of ponded water in the Perc Test hole (Elrick & Reynolds, 1986). This allows the Perc Test to be both misleading and easily manipulated to yield a desired result. The basic soil hydraulic properties for site selection, design, and performance assessment of filter fields should be Kfs and φm (Elrick & Reynolds, 1986). 10. A pressure infiltrometer is commercially available from Soilmoisture Equipment Corporation, Santa Barbara, CA. More detail on the construction, use, and performance of the pressure infiltrometer can be found in Elrick and Reynolds (1992a, b).
3.4.3.2.c Twin- or Dual-Ring and Multiple-Ring Infiltrometers W. D. REYNOLDS, Agriculture and Agri-Food Canada, Harrow, Ontario, Canada D. E. ELRICK, University of Guelph, Guelph, Ontario, Canada E. G. YOUNGS, Cranfield University, Silsoe, Bedfordshire, England
Introduction. The twin- or dual-ring and multiple-ring infiltrometers (Fig. 3.4–10) are used primarily for measuring field-saturated hydraulic conductivity, Kfs (L T−1), but can also be used to determine matric flux potential φm (L2 T−1), sorptivity S (L T−1/2), the macroscopic capillary length parameter α* (L−1), and the effective Green–Ampt wetting front pressure head hf (L). The methods are based on threedimensional analyses of steady ponded infiltration as represented by Eq. [3.4–11] or [3.4–12]. The methods employ adjacent measuring cylinders that are of different diameters—two cylinders in the case of the twin- or dual-ring infiltrometer, and three or more cylinders in the case of the multiple-ring infiltrometer. Apparatus and Procedures. The twin- or dual-ring and multiple-ring infiltrometer methods typically use cylinders ranging from about 5 to 50 cm in diameter and 5 to 20 cm long. Cylinder design, equilibration time, and procedures for cylinder installation, water ponding, and flow measuring are similar to those given for the single-ring and double- or concentric-ring infiltrometers (i.e., procedure Steps 1–4 in Section 3.4.3.2.a). The cylinders are installed individually (not concentrically as in the double-ring infiltrometer), and separated just enough to prevent the individual wetting fronts from merging before the flow measurements are completed
THE SOIL SOLUTION PHASE
837
Fig. 3.4–10. Schematic of the twin-, or dual-, ring and multiple-ring infiltrometers (plan view).
(Fig. 3.4–10). Multiple cylinders (three or more) should be clustered rather than placed in a line or circle (Fig. 3.4–10) to minimize the potentially confounding effects of soil variability (discussed further in comments section). To obtain good measurement sensitivity, it is often recommended that the cylinder diameters differ from each other by at least a factor of two (Scotter et al., 1982). The infiltration measurements may be conducted consecutively (i.e., on one cylinder at a time) using a single flow and head controlling device (e.g., Mariotte reservoir), or simultaneously (all cylinders at the same time) using multiple flow and head controlling devices. Analyses. With quasi-steady infiltration rates (qs) from two or more adjacent cylinders, Eq. [3.4–11] can be solved for both Kfs and α* using simultaneous equations or least squares regression approaches. These approaches can be applied in the analysis of both twin- or dual-ring infiltrometer data and multiple-ring infiltrometer data. Simultaneous Equations Approach. For constant depth of ponded water (H), the simultaneous equations solution of Eq. [3.4–11] for two different insertion depths and/or radii produces, Kfs = (T1q1 − T2q2)/(T1 − T2)
[3.4–27]
α* = (T2q2 − T1q1)/[q1(HT1 + T1T2) − q2(HT2 + T1T2)]
[3.4–28]
838
CHAPTER 3
where T1 = C1d1 + C2a1, T2 = C1d2 + C2a2, q1 is qs for Cylinder 1 (insertion depth, d1; radius, a1), and q2 is qs for Cylinder 2 (insertion depth, d2; radius, a2). For the special case of H = d = 0, Eq. [3.4–27] and [3.4–28] collapse to, Kfs = (a1q1 − a2q2)/(a1 − a2)
[3.4–29]
α* = (a2q2 − a1q1)/[C3a1a2(q1 − q2)]
[3.4–30]
which is the simultaneous equations solution for Eq. [3.4–12] (Wooding, 1968), where C3 = 0.25π = 0.7854 (Scotter et al., 1982). Due to the highly variable nature of qs, averages of replicated q1 and q2 values are frequently used in Eq. [3.4–27] through [3.4–30], rather than single measures of q1 and q2. An interesting feature of Eq. [3.4–27] and [3.4–28] is that they apply for any two-cylinder combination of insertion depth (d) and radius (a). In other words, it is possible to obtain Kfs and α* from these equations using two cylinders with different radius but the same insertion depth (i.e., a1 ≠ a2, d1 = d2), with same radius but different insertion depths (i.e., a1 = a2, d1 ≠ d2), or with different radius and different insertion depths (i.e., a1 ≠ a2, d1 ≠ d2). However, using different insertion depths only will generally not be as efficient or practical as using different radii. This is because the qs value is not as sensitive to ring insertion depth as it is to ring radius (Table 3.4–3), because changing ring insertion depth over an appreciable range increases installation effort, and because large ring insertion depth may result in excessive soil disturbance and an increase in equilibration time. Nevertheless, judicious changes to both ring radius and ring insertion depth may be useful for obtaining sufficient sensitivity for accurate solution of Eq. [3.4–27] and [3.4–28] (see example calculations). It is also possible to develop a simultaneous equations solution for Kfs and α* based on variable water ponding depth (H) and fixed values for cylinder radius and insertion depth: Kfs = [T(q2 − q1)]/(H2 − H1)
[3.4–31]
α* = (q2 − q1)/[q1(H2 + T) − q2(H1 + T)]
[3.4–32]
where T = C1d + C2a. Equations [3.4–31] and [3.4–32] are in fact equivalent to Eq. [3.4–21] and [3.4–22], respectively, for the pressure infiltrometer method. Equations [3.4–31] and [3.4–32] can therefore be applied to a single cylinder (i.e., two heads ponded successively in one cylinder), or to two cylinders (i.e., two adjacent cylinders of same radius but different ponded heads). Regression Procedures. For a constant depth of water ponding (H), Kfs and α* can be obtained by recasting Eq. [3.4–11] in the form qs = {[Kfs(α*H + 1)]/α*}(1/T) + Kfs;
T = C1d + C2a
[3.4–33]
which indicates that a plot of qs vs. 1/T produces a straight line with slope = Kfs(α*H + 1)/α* and qs-axis intercept = Kfs. For the special case of H = d = 0, Eq. [3.4–33] reduces to
THE SOIL SOLUTION PHASE
839
qs = (Kfs/C3α*)(1/a) + Kfs
[3.4–34]
where a plot of qs vs. 1/a gives slope = Kfs/(C3α*) and qs-axis intercept = Kfs. Smettem and Clothier (1989) used a variation of Eq. [3.4–34] in a multiple disc (tension) infiltrometer analysis. If variable H but fixed cylinder radius and insertion depth are used, then Eq. [3.4–11] becomes qs = (Kfs/T)H + Kfs[(1/(α*T) + 1]; T = C1d + C2a
[3.4–35]
where a plot of qs vs. H has slope = Kfs/T and qs-axis intercept = Kfs[(1/α*T) + 1]. Thus, Kfs and α* can be obtained using the slope and intercept relationships from Eq. [3.4–33], [3.4–34], or [3.4–35], plus two or more (qs, 1/T), (qs, 1/a), or (qs, H) data pairs, and a simple linear regression fitting routine that is available in most computer spreadsheets. Least squares regression relationships for Kfs and α* can also be derived that allow H, d, and a to vary simultaneously. They have the form: n
n
n
n
Σ [(Hi + Ti)/Ti]qi Σ (1/Ti2) − Σ [(Hi + Ti)/Ti2] Σ (q /T ) i=1 i=1 i=1 i=1 i i Kfs = __________________________________________ n n n Σ [(Hi + Ti)/Ti]2 Σ (1/Ti2) − {Σ [(Hi + Ti)/Ti2]}2 i=1 i=1 i=1 n
α* =
n
n
[3.4–36]
n
Σ [(Hi + Ti)/Ti]qi Σ (1/Ti2) − Σ [(Hi + Ti)/Ti2] Σ (q /T ) ________________________________________________ i=1 i=1 i=1 i=1 i i n
n
n
n
Σ [(Hi + Ti)/Ti]2 Σ (q /T ) − Σ [(Hi + Ti)/Ti2] Σ [(Hi + Ti)/Ti]qi i=1 i=1 i i i=1 i=1
[3.4–37]
where Ti = C1di + C2ai. The i subscript denotes measured H, d, a, and qs values for a particular cylinder, with i = 1, 2, 3,..., n where n is the number of cylinders (i.e., sets of H, d, a, and qs values) used to make the twin- or dual-ring (n = 2) or multiple-ring (n ≥2) infiltrometer measurement. Although Eq. [3.4–36] and [3.4–37] are rather more complicated than Eq. [3.4–27] through [3.4–35], they have the distinct advantage of applying for any combination of two or more sets of H, d, a, and qs values, as well as for the special case of H = d = 0 (Wooding, 1968) in which case Ti = C3ai where C3 = 0.25π. Given this general applicability, Eq. [3.4–36] and [3.4–37] may be preferred over the other equations. Also, Eq. [3.4–36] and [3.4–37] are readily solved (despite their relative complexity) using a computer spreadsheet or a simple computer program. Calculation of φm, S, and hf. Once Kfs and α* are calculated, the matric flux potential (φm), sorptivity (S), and the effective Green–Ampt wetting front pressure head (hf) can be determined using φm = Kfs/α*
[3.4–38]
S = [γ(θfs − θi)φm]1/2
[3.4–39]
hf = −α*−1
[3.4–40]
840
CHAPTER 3
Table 3.4–8a. Example ring infiltrometer data sets. The qs values were obtained by using Eq. [3.4–11] and assuming Kfs = 10−4 cm s−1 and α* = 0.12 cm−1. T = C1d + C2a, C1 = 0.9927, C2 = 0.5781. Data set number
H
d
a
T
qs cm s−1
cm 1 2 3 4 5 6
5 5 5 5 5 5
5 5 5 5 5 5
5 10 20 30 40 60
7.854 10.745 16.526 22.307 28.088 39.65
2.698 × 10−4 2.241 × 10−4 1.807 × 10−4 1.598 × 10−4 1.475 × 10−4 1.336 × 10−4
7 8 9 10
5 5 5 5
3 5 10 20
30 30 30 30
20.321 22.307 27.27 37.197
1.656 × 10−4 1.598 × 10−4 1.489 × 10−4 1.359 × 10−4
11 12 13 14
5 10 20 40
5 5 5 5
30 30 30 30
22.307 22.307 22.307 22.307
1.598 × 10−4 1.822 × 10−4 2.270 × 10−4 3.167 × 10−4
15 16 17 18
15 10 7 5
5 10 15 20
5 10 20 40
7.854 15.708 26.453 42.978
3.971 × 10−4 2.167 × 10−4 1.580 × 10−4 1.310 × 10−4
where γ ≈ 1.818 is a dimensionless constant related to wetting front shape (White & Sully, 1987), θfs (L3 L−3) is the field-saturated volumetric soil water content, and θi (L3 L−3) is the antecedent or background volumetric soil water content at the time of measurement. Example Calculations. The performance of the various simultaneous equations (Eq. [3.4–27] to [3.4–32]) and regression (Eq. [3.4–33] to [3.4–37]) solutions is illustrated in Table 3.4–8 using hypothetical data generated via Eq. [3.4–11], and assuming Kfs = 10−4 cm s−1 and α* = 0.12 cm−1. Exact results (i.e., Kfse/Kfs = α*e/α* = R2 = 1) indicate that all the conditions or assumptions regarding the values for H, d, and a are met, whereas inexact results indicate that one or more conditions or assumptions are not met. It is immediately clear in Table 3.4–8b,c that applying relationships that assume H = d = 0 (Eq. [3.4–29], [3.4–30], [3.4–34]) when H and d are in fact greater than zero can lead to substantial errors, even when R2 values are large (Table 3.4–8c). Substantial errors can also result from Eq. [3.4–27], [3.4–28], and [3.4–33] when H is not constant, and from Eq. [3.4–31], [3.4–32], and [3.4–35] when d and a are not constant (Table 3.4–8b,c). Consequently, both the magnitude and variation of H, d, and a cannot be neglected when analyzing twinring and multiple-ring infiltrometer data, and equations that take H, d, and a into account should always be used. Of the 11 equations presented, only Eq.[3.4–36] and [3.4–37] apply for two or more cylinders (i.e., n ≥2), and for all combinations of magnitude and variation in H, d, and a. Comments. 1. Comments 3 and 4 for the single-ring and concentric-ring infiltrometer methods (Section 3.4.3.2.a) concerning cylinder insertion depth, physical sources of error, and replication also apply for the twin-ring and multiple-ring infiltrometers.
THE SOIL SOLUTION PHASE
841
Table 3.4–8b. Example ring infiltrometer calculations—simultaneous equations plus Eq. [3.4–36] and [3.4–37]. The qs values were obtained by using Eq. [3.4–11] and assuming Kfs = 10−4 cm s−1 and α* = 0.12 cm−1. T = C1d + C2a, C1 = 0.9927, C2 = 0.5781. Data set (Table Eq. 3.4–8a) [3.4–27]
α*e/α*†
Kfse/Kfs† Eq. [3.4–29]
Eq. [3.4–31]
Eq. [3.4–36]
Eq. [3.4–28]
Eq. [3.4–30]
Eq. [3.4–32]
Eq. [3.4–37]
4.15 1.68 0.998 0.795 0.677
n/a n/a n/a n/a n/a
1 1 1 1 1
n/a n/a n/a
n/a n/a n/a
1 1 1
n/a n/a n/a
1 1 1
1 1 1
−0.528 −0.416 −0.278
1 1 1
H = d = 5 cm; a = 5, 10, 20, 30, 40, 60 cm 1, 2 2, 3 3, 4 4, 5 5, 6
1 1 1 1 1
1.78 1.37 1.18 1.11 1.06
7, 8 8, 9 9, 10
1 1 1
n/a n/a n/a
11, 12 12, 13 13, 14
n/a n/a n/a
n/a n/a n/a
15, 16 16, 17 17, 18
0.363 0.721 0.879
n/a‡ n/a n/a n/a n/a
1 1 1 1 1
1 1 1 1 1
H = 5 cm; a = 30 cm; d = 3, 5, 10, 20 cm n/a n/a n/a
1 1 1
1 1 1
d = 5 cm; a = 30 cm; H = 5, 10, 20, 40 cm 1 1 1
1 1 1
n/a n/a n/a
H = 15, 10, 7, 5 cm; d = 5, 10, 15, 20 cm; a = 5, 10, 20, 40 cm 0.363 0.992 1.04
4.25 4.13 4.68
1 1 1
0.127 0.362 0.552
0.214 0.896 1.02
† Ratio of calculated value (Kfse, α*e) to actual value (Kfs, α*) using the indicated data sets and values from Table 3.4–8a. ‡ Equation not applicable for that combination of H, d, and a values.
2. The primary advantage of the twin-ring and multiple-ring infiltrometer methods is simultaneous determination of the Kfs, α*, φm, S, and hf parameters. An important limitation, however, is that soil heterogeneity in the form of layering, horizonation, cracks, worm holes, or root channels can result in highly variable, unrealistic, and even invalid (e.g., negative) parameter values. This occurs primarily because the individual cylinders must occupy physically different locations, and thus the soil sampled by each cylinder frequently has different hydraulic properties. In addition, the coefficient matrices for the twin-ring and multiple-ring methods tend to be poorly conditioned, and hence small changes in coefficient values can produce large changes in the calculated parameter values. This “low precision” problem can often be ameliorated substantially, however, by using averages of replicated q1 and q2 measurements in Eq. [3.4–27] through [3.4–30]. One can also use replicated measurements in the regression-based analyses (Eq. [3.4–33]–[3.4–37]); that is, the random high–low variations in the replicate coefficient values are often largely cancelled when the replicates are grouped together in a regression. This approach is illustrated in Smettem and Clothier (1989) using the Wooding (1968) analysis (Eq. [3.4–34]). 3. Further information relating to the twin- or dual-ring and multiple-ring infiltrometer methods can be found in Scotter et al. (1982) and Smettem and Clothier (1989).
842
CHAPTER 3
Table 3.4–8c. Example ring infiltrometer calculations—regressions. The qs values were obtained by using Eq. [3.4–11] and assuming Kfs = 10−4 cm s−1 and α* = 0.12 cm−1. T = C1d + C2a, C1 = 0.9927, C2 = 0.5781. Parameter
Eq. [3.4–33]
Eq. [3.4–34]
Eq. [3.4–35]
Eq. [3.4–36]
Eq. [3.4–37]
Data Sets 1–6, Table 3.4–8a (H = d = 5 cm; a = 5, 10, 20, 30, 40, 60 cm) Kfse/Kfs† α*e/α*† R2
1 1 1
1.35 1.98 0.9492
n/a‡ n/a n/a
1 n/a n/a
n/a 1 n/a
Data Sets 7–10, Table 3.4–8a (H = 5 cm; a = 30 cm; d = 3, 5, 10, 20 cm) Kfse /Kfs α*e/α* R2
1 1 1
n/a n/a n/a
n/a n/a n/a
1 n/a n/a
n/a 1 n/a
Data Sets 11–14, Table 3.4–8a (d = 5 cm; a = 30 cm; H = 5, 10, 20, 40 cm) Kfse/Kfs α*e /α* R2
n/a n/a n/a
n/a n/a n/a
1 1 1
1 n/a n/a
n/a 1 n/a
Data Sets 15–18, Table 3.4–8a (H = 15, 10, 7, 5 cm; d = 5, 10, 15, 20 cm; a = 5, 10, 20, 40 cm) Kfse/Kfs α*e /α* R2
0.621 0.256¶ 0.9951
0.816 0.563 0.9876
6.28§ −0.345 0.9636
1 n/a n/a
n/a 1 n/a
† Ratio of calculated value (Kfse, α*e) to actual value (Kfs, α*) using the indicated data sets and values from Table 3.4–8a. ‡ Equation not applicable for that combination of H, d, and a values (Eq. [3.4–33] to [3.4–35]), or does not calculate the indicated value (Eq. [3.4–36], [3.4–37]). § Average T value used (23.2481 cm). ¶ Average H value used (9.25 cm).
3.4.3.2.d References Bouwer, H. 1966. Rapid field measurement of air entry value and hydraulic conductivity as significant parameters in flow system analysis. Water Resour. Res. 1:729–738. Bouwer, H. 1986. Intake rate: cylinder infiltrometer. p. 825–844. In A. Klute (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron. Monogr. 9. ASA and SSSA, Madison, WI. Daniel, D.E. 1989. In situ hydraulic conductivity tests for compacted clay. J. Geotech. Eng. (Am. Soc. Civ. Eng.) 115:1205–1226. Elrick, D.E., R. Angulo-Jaramillo, D.J. Fallow, W.D. Reynolds, and G.W. Parkin. 2002. Infiltration under constant head and falling head conditions. p. 47–53. In P.A.C. Raats, D.E. Smiles, and A.W. Warrick (ed.) Environmental mechanics: Water, mass and energy transfer in the biosphere. Geophys. Monogr. Ser. 129. AGU, Washington, DC. Elrick, D.E., G.W. Parkin, W.D. Reynolds, and D.J. Fallow. 1995. Analysis of early-time and steadystate single ring infiltration under falling head conditions. Water Resour. Res. 31:1883–1893. Elrick, D.E., and W.D. Reynolds. 1986. An analysis of the percolation test based on three-dimensional, saturated-unsaturated flow from a cylindrical test hole. Soil Sci. 142: 308–321. Elrick, D.E., and W.D. Reynolds. 1992a. Infiltration from constant-head well permeameters and infiltrometers. p. 1–24. In G.C. Topp et al. (ed.) Advances in measurement of soil physical properties: Bringing theory into practice. SSSA Spec. Publ. 30. SSSA, Madison, WI. Elrick, D.E., and W.D. Reynolds. 1992b. Methods for analyzing constant-head well permeameter data. Soil Sci. Soc. Am. J. 56:320–323. Elrick, D.E., W.D. Reynolds, and K.A. Tan. 1989. Hydraulic conductivity measurements in the unsaturated zone using improved well analyses. Ground Water Monit. Rev. 9:184–193. Fallow, D.J., and D.E. Elrick. 1996. Field measurement of air-entry and water-entry soil water pressure heads with a modified Guelph pressure infiltrometer. Soil Sci. Soc. Am. J. 60:1036–1039.
THE SOIL SOLUTION PHASE
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Fallow, D.J., D.E. Elrick, W.D. Reynolds, N. Baumgartner, and G.W. Parkin. 1993. Field measurement of hydraulic conductivity in slowly permeable materials using early-time infiltration measurements in unsaturated media. p. 375–389. In D.E. Daniel and S.J. Trautwein (ed.) Hydraulic conductivity and waste contaminant transport in soils. ASTM STP1142. American Society for Testing and Materials, Philadelphia, PA. Gerard-Marchant, P., R. Angulo-Jaramillo, R. Havercamp, M. Vauclin, P. Groenevelt, and D.E. Elrick. 1997. Estimating the hydraulic conductivity of slowly permeable and swelling materials from single-ring experiments. Water Resour. Res. 33:1375–1382. Green, W.H., and G.A. Ampt. 1911. Studies on soil physics. I. The flow of air and water through soils. J. Agric. Sci. 4:1–24. Lee, D.M., W.D. Reynolds, D.E. Elrick, and B.E. Clothier. 1985. A comparison of three field methods for measuring saturated hydraulic conductivity. Can. J. Soil Sci. 65:563–573. Leeds-Harrison, P.B., and E.G. Youngs. 1997. Estimating the hydraulic conductivity of soil aggregates conditioned by different tillage treatments from sorption measurements. Soil Tillage Res. 41:141–147. Nielsen, D.R., J.W. Biggar, and K.T. Erh. 1973. Spatial variability of field-measured soil-water properties. Hilgardia 42(7):215–260. Odell, B.P., P.H. Groenevelt, and D.E. Elrick. 1998. Rapid determination of hydraulic conductivity in clay liners by early-time analysis. Soil Sci. Soc. Am. J. 61:1563–1568. Raats, P.A.C. 1976. Analytical solutions of a simplified flow equation. Trans. ASAE 19:683–689. Reynolds, W.D. 1993. Saturated hydraulic conductivity: field measurement. p. 599–613. In M.R. Carter (ed.) Soil sampling and methods of analysis. Lewis Publ., Boca Raton, FL. Reynolds, W.D., and D.E. Elrick. 1990. Ponded infiltration from a single ring: I. Analysis of steady flow. Soil Sci. Soc. Am. J. 54:1233–1241. Reynolds, W.D., B.T. Bowman, R.R. Brunke, C.F. Drury, and C.S. Tan. 2000. Comparison of tension infiltrometer, pressure infiltrometer and soil core estimates of saturated hydraulic conductivity. Soil Sci. Soc. Am. J. 64:478–484. Rogowski, A.S. 1972. Watershed physics: Soil variability criteria. Water Resour. Res. 8:1015–1023. Scotter, D.R., B.E. Clothier, and E.R. Harper. 1982. Measuring saturated hydraulic conductivity using twin rings. Aust. J. Soil Res. 20:295–304. Smettem, K.R.J., and B.E. Clothier. 1989. Measuring unsaturated sorptivity and hydraulic conductivity using multiple disc permeameters. J. Soil Sci. 40:563–568. Smiles, D.E., and E.G. Youngs. 1965. Hydraulic conductivity determinations by several field methods in a sand tank. Soil Sci. 99:83–87. Soilmoisture Equipment Corporation. 1987. 2800K1 operating instructions. Soilmoisture Equipment Corp., Santa Barbara, CA. Topp, G.C., and M.R. Binns. 1976. Field measurements of hydraulic conductivity with a modified airentry permeameter. Can. J. Soil Sci. 56:139–147. Warrick, A.W., and D.R. Nielsen. 1980. Spatial variability of soil physical properties in the field. p. 319–344. In D. Hillel (ed.) Applications of soil physics. Academic Press, Toronto, Canada. White, I., and M.J. Sully. 1987. Macroscopic and microscopic capillary length and time scales from field infiltration. Water Resour. Res. 23:1514–1522. Wooding, R.A. 1968. Steady infiltration from a shallow circular pond. Water Resour. Res. 4:1259–1273. Wu, L., L. Pan, J. Mitchell, and B. Sanden. 1999. Measuring saturated hydraulic conductivity using a generalized solution for single-ring infiltrometers. Soil Sci. Soc. Am. J. 63:788–792. Youngs, E.G., 1972. Two- and three-dimensional infiltration: seepage from irrigation channels and infiltrometer rings. J. Hydrol. 15:301–315. Youngs, E.G., 1987. Estimating hydraulic conductivity values from ring infiltrometer measurements. J. Soil Sci. 38:623–632. Youngs, E.G., 1991a. Infiltration measurements—A review. Hydrol. Proc. 5:309–319. Youngs, E.G., 1991b. Hydraulic conductivity of saturated soils. p. 161–207. In K.A. Smith and C.E. Mullins (ed.) Soil analysis: Physical methods. Marcel Dekker, New York, NY. Youngs, E.G., D.E. Elrick, and W.D. Reynolds. 1993. Comparison of steady flows from infiltration rings in “Green and Ampt” soils and “Gardner” soils. Water Resour. Res. 29:1647–1650. Youngs, E.G., P.B. Leeds-Harrison, and D.E. Elrick. 1995. The hydraulic conductivity of low permeability wet soils used as landfill lining and capping material: analysis of pressure infiltrometer measurements. J. Soil Tech. 8:153–160. Youngs, E.G., G. Spoor, and G.R. Goodall. 1996. Infiltration from surface ponds into soils overlying a very permeable substratum. J. Hydrol. 186:327–334.
Published 2002
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3.4.3.3 Constant Head Well Permeameter (Vadose Zone) W. D. REYNOLDS, Agriculture and Agri-Food Canada, Harrow, Ontario, Canada D. E. ELRICK, University of Guelph, Guelph, Ontario, Canada
3.4.3.3.a Introduction The constant head well permeameter method (also known as the borehole permeameter method) is used primarily to measure field-saturated hydraulic conductivity, Kfs (L T−1), but can also be used to determine matric flux potential φm (L2 T−1), sorptivity S (L T−1/2), the macroscopic capillary length parameter α* (L−1), and the effective Green–Ampt wetting front pressure head hf (L). The method is based on a three-dimensional flow analysis for quasi-steady infiltration obtained by ponding one or more heads of water in a cylindrical borehole or “well” augered into the unsaturated (vadose) zone (Zangar, 1953; Reynolds et al., 1983, 1985; Philip, 1985; Reynolds & Elrick, 1986; Elrick et al., 1989). The Kfs, φm, and S parameters can be obtained using a single ponded head, two successively ponded heads, or multiple (three or more) successively ponded heads. The α* and hf parameters require successive ponding of two or more heads in the well; that is, they cannot be obtained from a single head analysis. Although there are several constant head well permeameter designs (see Item 9 in Section 3.4.3.3.e), only the “in-hole Mariotte bottle” system will be illustrated here. Note, however, that the procedures, equations, and analyses discussed in this section apply regardless of permeameter design. 3.4.3.3.b Apparatus and Procedures 1. Using a screw-type or bucket auger, excavate a 2- to 10-cm-diam. well to the desired depth (Fig. 3.4–11). The well should be cylindrical and have a reasonably flat bottom, as this shape is assumed in the well permeameter analysis (a flashlight works well for checking this). Computer simulations suggest that the bottom of the well should be at least 20 to 50 cm above the water table (or above the capillary fringe) to avoid possible interference caused by “mounding” of the water table up into the well (Reynolds, 1993). Auger-induced smearing and compaction of the well surfaces in fine-textured materials should be minimized within the measurement zone (Fig. 3.4–12), as this can result in unrepresentative Kfs, φm, S, α*, or hf values. Smearing and compaction can be minimized by not augering when the material is very wet (e.g., clayey soils should not be augered when they are wet enough to be “sticky”), by using a very sharp auger, by applying very little downward pressure on the auger, and by taking only small bites with the auger before emptying it out. The “two-finger, two-turn” rule for augering within the measurement zone seems to work reasonably well: Once the top of the measurement zone is reached, use only two fingers on each hand to apply downward pressure on the auger (i.e., the weight of the auger provides most of the downward pressure), and make only two complete turns of the auger before emptying it out. If inspection of the well reveals smearing or compaction within the measurement zone (a smeared and/or compacted surface generally appears smooth and polished under the light
THE SOIL SOLUTION PHASE
845
of a flashlight), steps should be taken to remove it. A small, spiked roller mounted on a handle (Fig. 3.4–11) seems to work reasonably well for this purpose (Reynolds & Elrick, 1986; Reynolds & Zebchuk, 1996). When the roller is run up and down the well wall several times the smeared or compacted surface is usually broken up and plucked off by the sharpened, paddle-shaped spikes. Other implements that have been used for removing smeared and compacted surfaces include a stiff cylindrical brush (available from many commercial auger and well or borehole permeameter suppliers), a pick-like plucking instrument (Campbell & Fritton, 1994; Bagarello, 1997; Bagarello et al., 1999), and soil peels made from quick-setting resin (Koppi & Geering, 1986) (see Section 3.4.3.3.e for further discussion). If the removal of
Fig. 3.4–11. Schematic of the well and a roller-type desmearing–decompaction apparatus for the constant head well permeameter method (adapted from Reynolds, 1993, p. 601).
846
CHAPTER 3
smeared or compacted surfaces results in a measureable increase in well radius, this new radius should be measured and used in the permeameter calculations (see Section 3.4.3.3.c). 2. Stand the well permeameter in the well and attach it to some kind of stabilizing apparatus (here we refer specifically to vertical permeameters based on inhole Mariotte bottle systems such as illustrated in Fig. 3.4–12; other systems may not require stabilizing). The stabilizing apparatus should hold the permeameter upright, give the permeameter good stability against wind, and carry the weight of the permeameter (when full of water) so that the water outlet tip does not sink into the base of the well during the measurement (Fig. 3.4–12). A simple tripod (similar to a surveyor’s transit or camera tripod) that clamps solidly to the permeameter reservoir works well for this purpose (Fig. 3.4–12). An alternative apparatus consists of a vertical steel rod driven into the soil (or threaded into a weighted base plate) and attached to the permeameter reservoir via a solid clamp (e.g., Vauclin et al., 1994). To prevent possible collapse of the well when measuring unstable materials, it is advisable to install either a well screen or backfill around the permeameter to the top of the measurement zone using pea gravel or coarse sand (Fig. 3.4–12). Backfill material (which must have a much greater permeability than the material being tested to avoid interference effects) also helps to reduce well siltation (discussed further in Section 3.4.3.3.e), as well as produce faster more uniform bubbling of the permeameter air tube when measuring low permeability materials. Depending
Fig. 3.4–12. Schematic of a Mariotte-type permeameter for use in the constant head well permeameter method (adapted from Reynolds, 1993, p. 602).
THE SOIL SOLUTION PHASE
847
on the design of the permeameter used (see Section 3.4.3.3.e), it may or may not be more convenient to prefill the permeameter reservoir with water before inserting the water outlet tube into the well. In this respect, permeameters of the in-hole Mariotte bottle type (Fig. 3.4–12) are more easily handled when empty. 3. Close the water outlet of the permeameter (in-hole Mariotte bottle design) by pushing the air tube down into the outlet tip, and then fill the permeameter with water (Fig. 3.4–12). Use water at ambient temperature to minimize the accumulation within the reservoir of bubbles of degassed air, which can obscure the permeameter scale. Do not use distilled or deionized water, as this may encourage clay and silt dispersion and subsequent siltation of the well surface during the measurement. In many cases, local tap water can be used, as its major ion concentrations are often sufficiently close to those of the soil water to prevent clay and silt dispersion. In porous media that are particularly susceptible to siltation (primarily materials with high silt content), it may be necessary to use native soil water, or water with flocculent added. Fill the permeameter reservoir to the top, leaving no air space. This minimizes overfilling of the well when flow is started (discussed further in Section 3.4.3.3.e). 4. Lift the air tube out of the outlet tip to establish and maintain the desired depth of water in the well (H) (Fig. 3.4–12). The air tube should be raised slowly to prevent a sudden rush of water against the well surface. This can erode the well (especially if backfill material has not been used), promote well siltation by stirring silt and clay into suspension, and cause excessive air entrapment within the porous medium. As mentioned above, perforated well liners or screens have been used in place of backfill material to protect the well surface and prevent collapse (e.g., Bagarello, 1997). The desired depth of ponding in the well (H) is obtained by setting the base of the air tube at the appropriate level, which is usually accomplished using a calibrated height marker and scale (Fig. 3.4–12). The permeameter is operating properly when air bubbles rise regularly up through the permeameter and into the reservoir (discussed further in Section 3.4.3.3.e). 5. For Mariotte-based permeameters, the rate of water flow or discharge, Q, out of the permeameter and into the porous medium (soil) is usually measured by monitoring the rate of fall, R, of the water level in the permeameter reservoir, and then multiplying by the reservoir cross-sectional area. Monitoring R can be accomplished using a scale attached to the reservoir (Fig. 3.4–12) and a stopwatch, or an automated pressure transducer–data logger system similar to that described by Ankeny (1992). Normally, R decreases with increasing time and approaches a constant value (Rs) as the flow rate becomes quasi-steady (Qs). Quasi-steady flow is usually assumed when effectively the same R value (Rs) is obtained over four or five consecutive R measurements (see Section 3.4.3.3.d). Once Qs is obtained, the Kfs, φm, S, α*, and hf parameters can be calculated as shown below. 3.4.3.3.c Analyses As mentioned above, the Kfs, φm, and S parameters can be determined using single-head (H1), two-head (H1, H2), or multiple-head (H1, H2, H3,....) analyses. The α* and hf parameters can be obtained only from two-head or multiple-head analyses, however. The well permeameter analyses are similar to those for the pressure infiltrometer (Section 3.4.3.2.b), although different equations are used.
848
CHAPTER 3
Single-Head Analyses. Traditional Single-Head Approach. One of the first constant head well permeameter analyses is that developed by Glover (in Zangar, 1953): Kfs = (CQs)/(2πH2)
[3.4–41]
where Qs (L3 T−1) is the quasi-steady flow rate out of the permeameter and into the soil, H (L) is the steady depth of water in the well (set by the height of the air tube), and C is a dimensionless shape factor given by: C = sinh−1(H/a) − [(a/H)2 + 1]1/2 + (a/H)
[3.4–42]
where a (L) is the radius of the well. For Mariotte-type permeameters (such as the one illustrated in Fig. 3.4–12), Qs is conveniently obtained by measuring the quasisteady rate of fall of the water level in the permeameter reservoir, R1 (L T−1), and then multiplying by the reservoir cross-sectional area, A (L2) (i.e., Qs = AR1). Although this early analysis (i.e., Eq. [3.4–41] and [3.4–42]) is simple and easy to use, it has some deficiencies: only Kfs is calculated, and the gravity and capillarity components of flow out of the well are not taken into account. As a result, φm, S, α*, and hf cannot be determined with this analysis, and the Kfs calculation can be of reduced accuracy for certain well geometries (H/a ratios) and soil conditions (discussed further in Sections 3.4.3.3.d and 3.4.3.3.e). Some practitioners recommend that H/a ratios ≥5 (i.e., H at least five times larger than a) should be used for the Glover analysis (Amoozegar, 1992), as this often reduces the error in Kfs resulting from lack of account for gravity and capillarity (discussed further in Sections 3.4.3.3.d and 3.4.3.3.e). Updated Single-Head Approach. Reynolds et al. (1983, 1985) and Elrick et al. (1989) extended the original Glover analysis to obtain more accurate estimates of the shape factor (C), and to account for the gravity and capillarity components of flow out of the well. In this updated analysis, Kfs and φm are determined using: Kfs = CQs/[2πH2 + Cπa2 + (2πH/α*)]
[3.4–43]
φm = CQs/[(2πH2 + Cπa2)α* + 2πH]
[3.4–44]
where α* (L−1) is a soil texture–structure parameter, Qs = AR1 (A and R1 as defined above), and the dimensionless shape factor, C, is read directly from the appropriate curve in Fig. 3.4–13, or calculated from empirical expressions such as (Zhang et al., 1998): C = [H/(2.074a + 0.093H)]0.754
for α* ≥ 0.09 cm−1
[3.4–45a]
C = [H/(1.992a + 0.091H)]0.683
for α* = 0.04 cm−1
[3.4–45b]
C = [H/(2.102a + 0.118H)]0.655
for α* = 0.01 cm−1
[3.4–45c]
THE SOIL SOLUTION PHASE
849
Fig. 3.4–13. Shape factor curves (C vs. H/a) for use in the constant head well permeameter method. H is depth of water in the well, and a is well radius (adapted from Reynolds & Elrick, 1987, p. 292). These curves apply for 1 cm ≤ a ≤ 5 cm, 0.5 cm ≤ H ≤ 20 cm, and 0.25 ≤ H/a ≤ 20.
which have been least squares fitted to the curves in Fig. 3.4–13. Figure 3.4–13 and Eq. [3.4–45] apply for 1 cm ≤ a ≤ 5 cm, 0.5 cm ≤ H ≤ 20 cm, 0.25 ≤ H/a ≤ 20, and are based on discrete data points derived from numerical simulations (Reynold & Elrick, 1987) (discussed further in Section 3.4.3.3.e). Alternative empirical C value expressions are given in Bosch and West (1998), but they seem to be less accurate than those in Zhang et al. (1998). For this updated single-head analysis, α* is selected from the soil texture and structure categories in Table 3.4–4, or determined from independent measurements (discussed further in Section 3.4.3.3.e). Note that Eq. [3.4–45a] applies for all α* ≥ 0.09 cm−1 (Reynolds & Elrick, 1987), and is thus the preferred C vs. H/a relationship for most natural soil textures and structures (i.e., the α* = 0.12 and 0.36 cm−1 categories in Table 3.4–4). Because the updated analysis accounts for gravity and capillarity, φm can be calculated (via Eq. [3.4–44]), and the Kfs calculation (Eq. [3.4–43]) may be more accurate than the Kfs provided by the Glover analysis (Eq. [3.4–41]) (discussed further in Sections 3.4.3.3.d and 3.4.3.3.e). Sorptivity, S (L T−1/2), may be calculated from the updated analysis using: S = [γ(θfs − θi) φm]1/2
[3.4–46]
where γ ≈ 1.818 is a dimensionless constant related to wetting front shape (White & Sully, 1987), θfs (L3 L−3) is the field-saturated volumetric soil water content, and θi (L3 L−3) is the initial or background or antecedent volumetric soil water content at the time of the measurement. Two-Head and Multiple-Head Analyses. The extensions of the constant head well permeameter analysis made by Reynolds et al. (1985) allow Kfs and φm to be calculated using two or more successively ponded heads (H) in the well. Advantages gained by this are that the α* parameter does not have to be estimated (via
850
CHAPTER 3
Table 3.4–4) or determined independently, and an estimate of the Green–Ampt wetting front pressure head parameter, hf, can be obtained. The use of two successively ponded heads allows the two-head analysis, while three or more successively ponded heads allows the multiple-head analysis. In the two-head analysis, Kfs and φm are calculated using simultaneous equations of the form (Reynolds & Elrick, 1986; Reynolds et al., 1985): Kfs = A(E2R2 − E1R1);
R2 > R1
[3.4–47a]
where E1 = H2C1/{π[2H1H2(H2 − H1) + a2(H1C2 − H2C1)]}; H2 > H1;
C2 > C1
[3.4–47b]
E2 = H1C2/{π[2H1H2(H2 − H1) + a2(H1C2 − H2C1)]}; H2 > H1;
C2 > C1
[3.4–47c]
and φm = A(F1R1 − F2R2);
R2 > R1
[3.4–48a]
where F1 = (2H22 + a2C2)C1/{2π[2H1H2(H2 − H1) + a2(H1C2 − H2C1)]}; H2 > H1;
C2 > C1
[3.4–48b]
F2 = (2H12 + a2C1)C2/{2π[2H1H2(H2 − H1) + a2(H1C2 − H2C1)]}; H2 > H1;
C2 > C1
[3.4–48c]
The subscripts on E, F, R, and C in Eq. [3.4–47] and [3.4–48] refer to their values at the two ponded heads, H1 and H2, where H1 is ponded first and the water level is not allowed to fall when changing from H1 to H2. The multiple-head analysis (Reynolds & Elrick, 1986) involves least squares regression fitting of: CiQi = P1Hi2 + P2Hi + P3;
i = 1, 2, 3,...., n;
n ≥2
[3.4–49a]
to CQ vs. H data, where Qi = ARi is the quasi-steady flow rate corresponding to the steady ponding depth, Hi, Ci is the C value (shape factor) corresponding to Hi/a, P1 = 2πKfs;
P2 = 2πφm;
P3 = Ciπa2Kfs
[3.4–49b]
and the rest of the parameters are as defined above. As with the two-head method, H1 is ponded first with H1 < H2 < H3 0.2H, s > H, and Σ∆y ≤ 0.25 y 0. Maasland and Haskew (1957) used a model similar to Eq. [3.4–51] in which ∆y/∆t is in feet per second and Ks is
THE SOIL SOLUTION PHASE
861
in feet per day and presented a set of graphs to obtain a shape factor equal to 100C for various hole geometries with and without an impermeable layer at the bottom of the hole. Later, Boast and Kirkham (1971) further extended the theory of water flow into a cylindrical auger hole, and presented C values for various hole geometries, and for various distances to an impermeable or infinitely permeable layer below the hole (Table 3.4–11). The ratio of C values calculated using Eq. [3.4–52] and the corresponding ones from Table 3.4–11 for s/H = 0 are between 0.73 and 0.85 for H/r ≤ 2 and between 0.87 and 1.04 for H/r ≥ 5. The ratios of C values obtained from Eq. [3.4–53] and Table 3.4–11 for s/H ≥ 0.5 are 20–30 cm min−1), fast-acting automated systems may be required. A fast-acting pressure transducer placed at the bottom of the hole and connected to a data logger or computer, or a reel-type recorder equipped with a float (van Beers, 1970; van Bavel & Kirkham, 1948) can serve this purpose. Downey et al. (1994) described a computerized probe system for rapid recording of water level rise in a well or auger hole.
1 0.75 0.5
1 0.75 0.5
1 0.75 0.5
1 0.75 0.5
1 0.75 0.5
1 0.75 0.5
1 0.75 0.5
1
2
5
10
20
50
100
0.37 0.40 0.49
1.25 1.33 1.64
5.91 6.27 7.67
18.1 19.1 23.3
51.9 54.8 66.1
186 196 234
447 469 555
0
0.35 0.38 0.47
1.18 1.27 1.57
5.53 5.94 7.34
16.9 18.1 22.3
48.6 52.0 63.4
176 187 225
423 450 537
0.05
0.34 0.37 0.46
1.14 1.23 1.54
5.30 5.73 7.12
16.1 17.4 21.5
46.2 49.9 61.3
167 180 218
404 434 522
0.1
0.34 0.36 0.45
1.11 1.20 1.50
5.06 5.50 6.88
15.1 16.5 20.6
42.8 46.8 58.1
154 168 207
375 408 497
0.2
0.33 0.35 0.44
1.07 1.16 1.46
4.81 5.25 6.60
14.1 15.5 19.5
38.7 42.8 53.9
134 149 188
323 360 449
0.5
0.32 0.35 0.44
1.05 1.14 1.44
4.70 5.15 6.48
13.6 15.0 19.0
36.9 41.0 51.9
123 138 175
286 324 411
1
4.66 5.10 6.43
13.4 14.8 18.8
36.1 40.2 51.0
118 133 169
264 303 386
2
116 131 167
255 292 380
5
0.32 0.35 0.44
1.04 1.13 1.43
4.64 5.08 6.41
13.4 14.8 18.7
35.8 40.0 50.7
115 131 167
254 291 379
s/H ∞
115 130 166
252 289 377
5
4.62 5.07 6.39
13.3 14.7 18.6
35.5 39.6 50.3
113 128 164
241 278 359
2
0.32 0.34 0.43
1.03 1.12 1.42
4.58 5.02 6.34
13.1 14.5 18.4
34.6 38.6 49.2
106 121 156
213 248 324
1
0.31 0.34 0.43
1.02 1.11 1.39
4.46 4.89 6.19
12.6 14.0 17.8
32.4 36.3 46.6
91 106 139
166 198 264
0.5
s/H for infinitely permeable layer
† r = radius of the auger hole, H = depth of water in the hole at equilibrium, y = average distance from the water level in the hole to the water table at equilibrium for two consecutive measurements, and s = distance between the bottom of the hole and an impermeable or infinitely permeable layer below the hole.
y/H
H/r
s/H for impermeable layer
Table 3.4–11. Values of C coefficient in Eq. [3.4–51] for an auger hole underlain by an impermeable or an infinitely permeable layer (after Boast & Kirkham, 1971).†
862 CHAPTER 3
THE SOIL SOLUTION PHASE
863
4. Timer. A stopwatch, wrist watch, or clock capable of reading in seconds is required if water level rise is to be monitored manually. 5. Data sheet. A one-page data sheet, similar to the one presented in Fig. 3.4–15 will speed up the process of data collection, and allows accurate calcula-
Fig. 3.4–15. An example of a completed data sheet for the auger-hole method.
864
CHAPTER 3
tion of Ks for each measurement. The form of Fig. 3.4–15 can be modified to suit individual needs. 3.4.3.4.d Procedure 1. To prepare the auger hole, clean the soil surface (i.e., remove plants and debris) where the Ks measurement is desired. Using a hand or motorized auger of the desired diameter, bore a hole with minimum disturbance to at least 30 cm below the water table. At a minimum, describe the soil texture and other pertinent morphological characteristics of the soil and note any soil layering within the depth range where Ks will be measured (an alternative procedure, described later, must be used for layered soils). The saturated zone below the water table must be unconfined. A sudden rise in the water level in the hole during its construction may indicate a confined aquifer condition below an overlying aquitard (i.e., impermeable or slowly permeable layer). After digging the hole to the desired depth, place the planer auger (same diameter as the auger) at the bottom of the hole and turn it slowly with a small downward force to clean and shape the base of the hole into a flat-bottomed cylindrical cavity. Note that the planer auger should not be used for boring the hole or extending its depth. If you suspect that the auger hole may collapse during the measurement (as may be the case for coarse-textured soils), insert a section of thin-wall perforated pipe (e.g., a section of well screen) into the hole immediately after preparing it. The outside diameter of this pipe should be the same as the auger hole diameter to avoid serious errors in the calculated Ks values. 2. For initial data collection, allow adequate time for the water level in the auger hole to equilibrate (i.e., rise or fall) to the same level as that of the water table, a condition known as static water level. Record on the data sheet (Fig. 3.4–15) the radius of the bottom section of the hole (r). Set a reference level at the soil surface to measure the depth of the hole (D) and the depth to the static water level or water table (E). (For symbols see Fig. 3.4–14.) Estimate the depth to any effectively impermeable (or highly permeable) layers below the hole (S) from existing information about the site or by boring holes at other locations around the measurement area. Alternatively, the depth S can be determined after completion of measurements by extending the hole and assessing the materials below its original depth. Record the data on the data sheet (Fig. 3.4–15) and determine the initial depth of water in the hole (H = D − E) and the distance between the bottom of the hole and the effectively impermeable (or highly permeable) layer below the hole (s = S − D). Next, calculate and record the H/r and s/H values for the hole. (Note that the error in Ks resulting from estimating S is 20), the A/r values do not change significantly for all s/r and hc/r values and can therefore be extrapolated. Extrapolation to H/r < 4, however, should be exercised with caution. The diameter of the cavity at the bottom of the piezometer tube should be as close as possible to the inside diameter of the piezometer tube. Otherwise, serious errors may result in the calculated Ks value. The piezometer method is often not practical for rocky and gravelly soils, as advancing a piezometer tube or establishing a seal between the tube and the soil is usually problematic. To speed the process of installing deep piezometers, bore an oversized hole (i.e., a borehole larger than the outside diameter of the piezometer tube) to just above the water table, and lower the piezometer tube into it. Then continue the installation process through the bottom of the oversized borehole using the normal installation procedure (i.e., using an appropriate sized auger to obtain a tight fit between piezometer tube and borehole). The gap above the water table between the piezometer tube and the oversized borehole is then backfilled with soil and/or bentonite. For measurements at great depths below the water table, an oversized borehole can be bored to within a distance larger than the desired hc above the depth interval under consideration (see King & Franzmeier, 1981). The piezometer tube can then be installed in the undisturbed section of the aquifer through the bottom of the oversized borehole. The space between the piezometer tube and the soil below the
THE SOIL SOLUTION PHASE
877
water table and to some distance above the water table should be sealed with soil (and/or bentonite) to eliminate the possibility of water flowing up the outside of the piezometer tube. The cavity at the bottom of the piezometer tube should be bored after the tube is installed and sealed. Due to the size of the cavity, the rate of water level rise in the tube is generally slow enough that the water level can be easily measured using manual or continuous water level recorder procedures. Bailers and slugs are often not suitable for lowering the water level in the piezometer tube because they initially raise the water level, which forces water from the cavity back into the soil. Topp and Sattlecker (1983) developed a hand-inflatable packing device that allows the horizontal and vertical Ks of a shallow aquifer to be measured in a single borehole by both the auger-hole and piezometer methods. Goss and Youngs (1983) determined shape factors for a horizontal cavity at the end of a piezometer tube inserted horizontally into the aquifer. Their procedure requires the construction of a large pit below the water table and sealing the pit walls to prevent water entry into the pit. As a result, this procedure is time consuming and difficult to perform, and appears to have only specialized applications. A laboratory sand tank comparison of the piezometer, auger hole, and multiple well methods is given in Smiles and Youngs (1965). 3.4.3.5.h References Frevert, R.K., and D. Kirkham. 1948. A field method for measuring the permeability of soil below a water table. Highw. Res. Board Proc. 28:433–442. Goss, M.J., and E.G., Youngs. 1983. The use of horizontal piezmeteres for in situ measurements of hydraulic conductivity below the water table. J. Soil Sci. 34:659–664. King, J.J., and D.P. Franzmeier. 1981. Estimation of saturated hydraulic conductivity from soil morphological and genetic information. Soil Sci. Soc. Am. J. 45:1153–1156. Kirkham, D. 1945. Proposed method for field measurement of permeability of soil below the water table. Soil Sci. Soc. Am. Proc. 10:58–68. Luthin, J.N., and D. Kirkham. 1949. A piezometer method for measuring permeability of soil in situ below a water table. Soil Sci. 68:349–358. Smiles, D.E., and E.G. Youngs. 1965. Hydraulic conductivity determinations by several field methods in a sand tank. Soil Sci. 99:83–87. Topp, G.C., and S. Sattlecker. 1983. A rapid measurement of horizontal and vertical components of saturated hydraulic conductivity. Can. J. Agric. Eng. 25:193–197. Youngs, E.G., 1968. Shape factors for Kirkham’s piezometer method for determining the hydraulic conductivity of soil in situ for soils overlying an impermeable floor or infinitely permeable stratum. Soil Sci. 106:235–237.
3.4.3.6 Other Saturated Zone Methods 3.4.3.6.a Introduction Although the auger-hole method (Section 3.4.3.4) and piezometer method (Section 3.4.3.5) are by far the most commonly used techniques for measuring saturated hydraulic conductivity (Ks) below the water table, several other methods are also available that can be preferable under certain circumstances. The most important of these include the two-well method (Childs, 1952), the four-well method (Snell & van Schilfgaard, 1964), the multiple-well method (Smiles & Youngs, 1963), the pit bailing test (Lomen et al., 1987; Boast & Langbartel, 1984; Bouwer & Rice, 1983), and the slug test (Bouwer & Rice, 1976). Readers interested in these tech-
Published 2002
THE SOIL SOLUTION PHASE
877
water table and to some distance above the water table should be sealed with soil (and/or bentonite) to eliminate the possibility of water flowing up the outside of the piezometer tube. The cavity at the bottom of the piezometer tube should be bored after the tube is installed and sealed. Due to the size of the cavity, the rate of water level rise in the tube is generally slow enough that the water level can be easily measured using manual or continuous water level recorder procedures. Bailers and slugs are often not suitable for lowering the water level in the piezometer tube because they initially raise the water level, which forces water from the cavity back into the soil. Topp and Sattlecker (1983) developed a hand-inflatable packing device that allows the horizontal and vertical Ks of a shallow aquifer to be measured in a single borehole by both the auger-hole and piezometer methods. Goss and Youngs (1983) determined shape factors for a horizontal cavity at the end of a piezometer tube inserted horizontally into the aquifer. Their procedure requires the construction of a large pit below the water table and sealing the pit walls to prevent water entry into the pit. As a result, this procedure is time consuming and difficult to perform, and appears to have only specialized applications. A laboratory sand tank comparison of the piezometer, auger hole, and multiple well methods is given in Smiles and Youngs (1965). 3.4.3.5.h References Frevert, R.K., and D. Kirkham. 1948. A field method for measuring the permeability of soil below a water table. Highw. Res. Board Proc. 28:433–442. Goss, M.J., and E.G., Youngs. 1983. The use of horizontal piezmeteres for in situ measurements of hydraulic conductivity below the water table. J. Soil Sci. 34:659–664. King, J.J., and D.P. Franzmeier. 1981. Estimation of saturated hydraulic conductivity from soil morphological and genetic information. Soil Sci. Soc. Am. J. 45:1153–1156. Kirkham, D. 1945. Proposed method for field measurement of permeability of soil below the water table. Soil Sci. Soc. Am. Proc. 10:58–68. Luthin, J.N., and D. Kirkham. 1949. A piezometer method for measuring permeability of soil in situ below a water table. Soil Sci. 68:349–358. Smiles, D.E., and E.G. Youngs. 1965. Hydraulic conductivity determinations by several field methods in a sand tank. Soil Sci. 99:83–87. Topp, G.C., and S. Sattlecker. 1983. A rapid measurement of horizontal and vertical components of saturated hydraulic conductivity. Can. J. Agric. Eng. 25:193–197. Youngs, E.G., 1968. Shape factors for Kirkham’s piezometer method for determining the hydraulic conductivity of soil in situ for soils overlying an impermeable floor or infinitely permeable stratum. Soil Sci. 106:235–237.
3.4.3.6 Other Saturated Zone Methods 3.4.3.6.a Introduction Although the auger-hole method (Section 3.4.3.4) and piezometer method (Section 3.4.3.5) are by far the most commonly used techniques for measuring saturated hydraulic conductivity (Ks) below the water table, several other methods are also available that can be preferable under certain circumstances. The most important of these include the two-well method (Childs, 1952), the four-well method (Snell & van Schilfgaard, 1964), the multiple-well method (Smiles & Youngs, 1963), the pit bailing test (Lomen et al., 1987; Boast & Langbartel, 1984; Bouwer & Rice, 1983), and the slug test (Bouwer & Rice, 1976). Readers interested in these tech-
878
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niques are referred to the original articles cited above, as well as to Bouwer and Jackson (1974), Amoozegar and Warrick (1986), and Amoozegar and Wilson (1999). 3.4.3.6.b References Amoozegar, A., and A.W. Warrick. 1986. Hydraulic conductivity of saturated soils: Field methods. p. 735–770. In A. Klute (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron. Monogr. 9. ASA and SSSA, Madison, WI. Amoozegar, A., and G. V. Wilson. 1999. Methods for measuring hydraulic conductivity and drainable porosity. p. 1149–1205. In R. W. Skaggs and J. van Schilfgaarde (ed.) Agricultural drainage. Agron. Monogr. 38. ASA, CSSA, and SSSA, Madison, WI. Boast, C.W., and R.G. Langebartel. 1984. Shape factors for seepage into pits. Soil Sci. Soc. Am. J. 48:10–15. Bouwer, H., and R.D. Jackson. 1974. Determining soil properties. p. 611–672. In J. van Schilfgaarde (ed.) Drainage for agriculture. Agron. Monogr. 17. ASA, Madison, WI. Bouwer, H., and R.C. Rice. 1976. A slug test for determining hydraulic conductivity of unconfined aquifers with completely or partially penetrating wells. Water Resour. Res. 12:423–428. Bouwer, H., and R.C. Rice. 1983. The pit bailing method for hydraulic conductivity measurement of isotropic or anisotropic soil. Trans. ASAE 26:1435–1439. Childs, E.C. 1952. The measurement of the hydraulic permeability of saturated soil in situ. I. Principles of a proposed method. Proc. R. Soc. London A. 215:525–535. Lomen, D.O., A.W. Warrick, and R. Zhang. 1987. Determination of hydraulic conductivity from auger holes and pits—An approximation. J. Hydrol. 90:219–229. Smiles, D.E., and E.G. Youngs. 1963. A multiple-well method for determining the hydraulic conductivity of a stratified soil in situ. J. Hydrol. 1:279–287. Snell, A.W., and J. van Schilfgaarde. 1964. Four-well method of measuring hydraulic conductivity in saturated soils. Trans. ASAE 7:83–87, 91.
Published 2002
3.5 Unsaturated Water Transmission Parameters Obtained from Infiltration BRENT CLOTHIER, HortResearch, Palmerston North, New Zealand DAVID SCOTTER, Massey University, Palmerston North, New Zealand We dedicate this to John Philip, who died tragically during the writing of this section. His insight formed the basis upon which our understanding of unsaturated water flow is predicated.
3.5.1 Basic Theory To describe quantitatively soil water movement, we usually use Richards’ equation. It says that the time rate of change of water content at any point in the soil results from the sum of two processes. One mechanism involves the driving force due to pressure gradients in the soil water; the other involves gravity. Both processes also depend on the soil permeability and the spatial gradient in the local water concentration. Richards’ equation can be written with either the pressure potential in head units (h), or the volumetric water content (θ), as the dependent variable (Philip, 1969), giving either (dθ/dh)(∂h/∂t) = ∇[K(h)∇h] + [dK(h)/dh](∂h/∂z)
[3.5–1]
∂θ/∂t = ∇[D(θ)∇θ] + [dK(θ)/dθ](∂θ/∂z)
[3.5–2]
or
where t is time, z is depth, K is the hydraulic conductivity, and D is the soil water diffusivity defined as K(θ)dh/dθ. There are a number of assumptions implicit in Richards’ equation. By assuming that unique and definable K(h) and θ(h) functions exist, any effects due to hysteresis, entrapped air, water repellency, and nonuniform wetting due to preferential flow are ignored. Further, the soil structure and pore-size distribution are assumed to be temporally stable. Equation [3.5–2] has more restricted application than Eq. [3.5–1]. Equation [3.5–2] cannot describe flow in layered soils, as the varying hydraulic properties across an interface mean θ(z) might not be continuous. It also fails if there is a saturated zone where h is positive, as there D becomes infinite. Despite the above limitations, a variety of analytical and numerical solutions to Eq. [3.5–1] and [3.5–2] is available. However, to use either of these equations, the hydraulic characteristics or unsaturated transmission parameters of the soil are needed. The three functional properties necessary to describe the hydraulic character of a soil are the water retentivity function θ(h), the hydraulic conductivity function K(θ) or K(h), and the soil water diffusivity function D(θ). Two are sufficient. 879
880
CHAPTER 3
Since D is defined in terms of K, h, and θ, if any two of these functions are known, the third is readily found. It is possible, albeit difficult, to measure in detail these functional dependencies in the laboratory, and even in the field. However, a simpler approach is also possible. As the rate of infiltration into unsaturated soil depends on both the conductive and absorptive soil properties, information about the transmission properties of a soil can be inferred from an inverse analysis of the temporal pattern in the rate of infiltration. The goal of this section is to describe how this can be done. As with all simplifications, this approach has its limitations, some of which we outline in the concluding section (Section 3.5.5). Infiltration is the movement of water into soil. In our discussion of the various infiltration measurements, we will encounter two soil properties that integrate the behavior of unsaturated water transmission. The first is the sorptivity, and the second is the macroscopic capillary length. John Philip coined the term sorptivity in 1957. Much earlier, in 1911, Green and Ampt had realized that the total amount of freely supplied water infiltrating into an initially unsaturated horizontal soil column is proportional to the square root of time. This gravity-free, capillary-induced absorption is conditional on the soil being uniform and initially possessing a constant water content. Philip called the proportionality constant of the cumulative infiltration divided by the square root of time, the sorptivity (S). Sorptivity represents the ability of a soil to absorb water by capillary processes (m s−1/2). Sorptivity depends on the soil’s pore-size distribution, on how dry the soil is initially (θn), and on the pressure potential imposed at the surface (hs). If hs is negative, it determines the surface water content (θs); if it is positive, then θs is the water content at satiation. A second integral property is called the macroscopic capillary length (λc). John Philip also introduced this concept. He first wrote about it in 1983, and again in 1985 (Philip, 1983, 1985). It was later referred to as the sorptive length. White and Sully (1987) discussed it in detail, and slightly modified Philip’s definition. It is thus defined as hs
θs
λc = (1/Ks)∫hn K(h)dh = (1/Ks)∫θn D(θ)dθ
[3.5–3]
where θn is some relatively low initial water content at which the hydraulic conductivity is negligibly small and hn the associated pressure potential head, hs is the pressure head at the imbibition surface and θs the associated water content at the surface, and Ks the hydraulic conductivity at that pressure head. If hs is positive, then Ks is the saturated hydraulic conductivity. While λc can be found for any hs, its definition in terms of the soil water diffusivity can only be used when hs ≤ 0 for the reason given above. The macroscopic capillary length can be thought of as a conductivity-weighted mean pressure head. It also represents the relative importance of capillarity and gravity during water movement (Philip, 1969). Large values indicate capillarity is more important than gravity. Typical λc values for field soils, when hs is 0, fall between 20 and 200 mm (White & Sully, 1987). The macroscopic capillary length is particularly useful if the unsaturated hydraulic conductivity can be described by the exponential relationship K = Ksexp[a(h − hs)]
[3.5–4]
THE SOIL SOLUTION PHASE
881
over the h range of interest. Then a is a constant with inverse length units, and λc = a−1
[3.5–5]
Note that Eq. [3.5–4] can only apply if hs is less than, or equal to zero. Also if hs is less than zero, and a value for Ks is obtained by fitting Eq. [3.5–4] to measured K(h) values, then Ks is not necessarily the saturated hydraulic conductivity. The sorptivity and macroscopic capillary length are related (White & Sully, 1987). It follows that λc = bS2/[Ks(θs − θn)].
[3.5–6]
For most soils the constant b has a value close to 0.55. From an inverse analysis of the pattern of infiltration we will show how S and Ks can be inferred, and the hydraulic properties of the soil estimated. 3.5.2 One-Dimensional Infiltration Equations and Their Use The cumulative, one-dimensional infiltration (I) into soil can often be easily measured as a function of time (t). This may be done in either the field or laboratory, with either saturated or unsaturated conditions at the surface. Water may be ponded on the soil surface using a double-ring infiltrometer (see Section 3.4.3.2) or using a single-ring infiltrometer driven far enough into the soil to ensure only vertical flow occurs (Talsma, 1969). Alternatively, a disk permeameter (see Section 3.5.4 below) may be used to impose a constant negative pressure potential at the surface during infiltration. One-dimensional flow can be ensured by carving out a soil pedestal, or by driving a thin-walled tube into the soil (Clothier & White, 1981). Infiltration equations relating I and t may then be fitted to the data, so that values can be obtained for S and Ks. For representative values to be obtained, White and Sully (1987) suggest that the minimum diameter for infiltration rings, or disk permeameters, be at least λc. For most soils this means diameters greater than about 100 mm. Diameters even larger than this are desirable for ponded infiltration because of the confounding influence of macropores. Simple descriptions of I(t) are possible for both the short- and long-time behavior during one-dimensional infiltration at a certain surface pressure head (often zero) into uniform soil that is initially uniformly dry. They were first made explicit by Philip (1957). At short times, the absorptive forces of capillarity dominate over gravity, so that the cumulative infiltration (I) can be given by I = St1/2
[3.5–7]
This equation also holds at all times for absorption into an effectively semi-infinite horizontal soil column, as discussed below for the Brute and Klute experiment. At long times, gravity dominates, and the infiltration rate (i) asymptotically approaches i = dI/dt → Ks
[3.5.8]
882
CHAPTER 3
These descriptions are implicit in the work of Green and Ampt (1911). When the equations below are used to infer both S and Ks from infiltration data, a wide range of observation times are required. Short-time data, when capillarity dominates, are needed to find S, but in some soils these are hard to obtain, as the initial infiltration is so rapid. Also long-time data are needed for the influence of gravity and Ks to become apparent. This happens when t > (S/Ks)2 (Philip, 1969). However, S and Ks are not independent variables; Ks is roughly proportional to S2 (Talsma, 1969) Equations [3.5–7] and [3.5–8] can sometimes be used directly to find S and Ks. Talsma (1969) demonstrated this by measuring the infiltration rate of water ponded in infiltration rings at short and long times. Smith (1999) found S even more simply from Eq. [3.5–7] by measuring the time it took for a 10-mm depth of water poured into an infiltration ring to soak into the soil. Often, however, the times during which Eq. [3.5–7] and [3.5–8] apply are either too short, or too long, to be of practical use. Thus, other equations have been developed to describe what happens in between. We consider only those simpler, physically based equations that have been proven sound (Haverkamp et al., 1988; Clausnitzer et al., 1998). Thus the socalled Kostiakov–Lewis and the Mezencev and Horton equations are neither considered, nor recommended. The simplest equation fitting these criteria is the two-term algebraic equation of Philip (1957): I = St1/2 + Apt
[3.5–9]
Here we treat Ap as an empirical constant. Theory suggests that 1/3Ks < Ap < 2/3Ks. Note that unless Ap = Ks, Eq. [3.5–9] does not apply at long times, as it is inconsistent with Eq. [3.5–8]. Swartzendruber (1987) derived the following three-parameter equation that overcomes this problem, I = [1 − exp(−Ast1/2)]S/As + Kst
[3.5–10]
where, again, As is an empirical dimensionless constant. Arguably, the oldest infiltration equation is that of Green and Ampt (1911). Philip (1957) showed this could be written in implicit form as t = I/Ks − [S2ln(1 + 2IKs/S2)]/2Ks2.
[3.5–11]
The theory behind this equation implies a rectangular wet front moving down through the soil. Another implicit equation, similar in form, but implying a different-shaped wet front, was given by Parlange et al. (1982) as t = I/Ks − S2[1 − exp(−2IKs/S2)]/2Ks2.
[3.5–12]
They argued that Eq. [3.5–11] and [3.5–12] bracket expected soil behavior and often give similar results. Parlange et al. (1982) and Parlange et al. (1985) give more complex three-parameter equations, but in practice uncertainties due to other factors will often make their extra complexity unwarranted.
THE SOIL SOLUTION PHASE
883
Table 3.5–1. Finding S and Ks from cumulative infiltration data. t
I
h
mm
h
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
26.6 40.9 51.9 64.1 73.5 84.0 94.1 104.3
0.04 0.08 0.12 0.17 0.21 0.26 0.30 0.35
S′ = Ks′ =
(t − t′)2
t′
0.0001 0.0003 0.0007 0.0008 0.0015 0.0019 0.0023 0.0025 Sum =
119 mm h−1/2 204 mm h−1
0.0102
When free water is applied to the soil surface during infiltration, the pressure head at the surface, that is the ponding depth (hs), affects the transient rate of infiltration, and in particular the value of S found from the above equations. In contrast, Eq. [3.5–8] applies for all ponding depths, and the value of Ks should be unaffected. The sorptivity at zero ponding depth, S0, can be estimated from the sorptivity (Ss) measured at some hs > 0 using the equation (White & Sully, 1987) S02 = Ss2 − 2(θs − θn)hsKs.
[3.5–13]
Given I(t) data relating cumulative infiltration with time, and a decision to describe it using one or more of the above equations, the remaining problem in the inverse procedure is how to extract the transmission properties of Ks and S. As Wraith and Or (1998) pointed out, this can be done readily using a spreadsheet such as Excel (Microsoft Corp., Seattle, WA), Quattro Pro (Corel Corp., Ottawa, ON, Canada), or Lotus 1-2-3 (IBM Software Group, Cambridge, MA). All can employ the embedded nonlinear optimization package. To demonstrate this, we use Excel to fit, using least squares, Eq. [3.5–12] to some experimental data for cumulative infiltration (I) as a function of time (t). The data relate to a sandy soil with a 22.5-mm depth of water ponded on top. The results are from Haverkamp et al. (1988). Table 3.5–1 gives these data, along with t′, the time fitted by assuming a given shape of I(t). The procedural objective is to minimize (t − t′)2. The procedure is as follows. The I(t) values are entered into two columns on a worksheet. Somewhere else on the worksheet are placed initial, guessed values for the sorptivity (S′) and saturated hydraulic conductivity (Ks′). From the first reading and a rearranged Eq. [3.5–7] a sensible initial guess for S′ is 26.6/0.051/2, say 119 mm/h1/2. From the last two readings and the finite difference form of Eq. [3.5–8], a sensible starting value for Ks′ is (104.3–94.1)/(0.40–0.35), say 204 mm/h. In a third column are placed estimates of time (t′ in Table 3.5–1) calculated from the I values, the parameter values, and Eq. [3.5–12]. In a fourth column (t − t′)2 values are calculated, and this column is summed. The objective is to minimize Σ(t − t′)2. To do this, Solver is called up from the Excel Tools Menu. The cell containing the sum of squares is chosen as the Target Cell, and Solver is set to minimize this by selecting improved guesses of S and Ks. When Solve is selected, Solver replaces the initial
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Table 3.5–2. Optimized values of the parameters S and Ks found by inverse analysis of the I(t) data in Table 3.5–1. Equation
h−1/2
S, mm Ks, mm h−1
[3.5–9]
[3.5–10]
[3.5–11]
[3.5–12]
Measured
91 --
108 183
94 148
103 193
102 153
guesses with the optimized values. A similar procedure can be used for Eq. [3.5–9], [3.5–10], and [3.5–11]. Except, of course, that for the explicit Eq. [3.5–9] and [3.5–10] it is the error in the estimated values of I, rather than t, that is minimized. The optimized values for S and Ks thus obtained are shown in Table 3.5–2. Also shown are the measured values reported by Haverkamp and Vauclin (1981). For this sandy soil, Eq. [3.5–11] seems to provide the best estimates, but this need not always be the case. In practical situations, any of the equations is probably suitable. Conversely, if the assumptions implicit in them are seriously contravened they might all be unsuitable. Physically rational interpretation of the results should always accompany inverse statistical procedures. To estimate what S would have been with zero ponding depth, the optimized S and Ks values from Eq. [3.5–12], for example, can be put into Eq. [3.5–13], along with the reported values of θs = 0.312, θn = 0.082, and hs = 22.5 mm, to give an estimate of S0 = 93 mm/h1/2 for hs = 0. The capillary length scale, λc, calculated from Eq. [3.5–6], is 107 mm. Figure 3.5–1 shows the measured K(h) relationship along with that implied by Eq. [3.5–4] with a as 1/107 mm−1, and Ks as the value from Eq. [3.5–12] in Table 3.5–2. It would be unwise to see such agreement between measured and estimated values as typical. However, in an integral sense, it may in some cases be more important to have S correct, than to have the correct functional form of K(h). This will be especially so for infiltration prediction, but not for prediction of unsaturated redistribution or root uptake, where K(θ) for lower values of θ needs to be known accurately. Furthermore, in the field, the assumptions implicit in Eq. [3.5–9] to [3.5–12] will rarely, if ever, be fully satisfied. Macropore flow may negate the assumption that the soil can be treated as a continuum. Soil structure and texture changes usually mean the hydraulic properties vary with depth. The initial water content will also rarely be uniform with depth. Some degree of subcritical water repellency of the surface soil is also likely to be present and to change with time. Hysteresis effects can be significant. On top of this, there is the well-known large and non-Gaussian spatial and temporal variability in transmission properties, which plagues prediction of the spatially integrated transport of soil water. Thus, the above analysis needs to be treated with caution when applied to field data. Nonetheless, insight into field processes is best gained through field study. 3.5.3 Horizontal Absorption—The Bruce and Klute Experiment A more precise description of the unsaturated hydraulic conductivity can be obtained if the water content distribution, or even just the sorptivity and wetting front
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Fig. 3.5–1. The measured hydraulic conductivity–soil water suction, K(h), relationship for Grenoble sand (Haverkamp & Vauclin, 1981) (solid line) and the relationship found from analysis of the infiltration data in Table 3.5–1 (dotted line).
distance, in a horizontal soil column is measured after a period of imbibition at a constant pressure head. Often this is most easily done with free water imbibition. In a classical experiment, Bruce and Klute (1956) first described how to do this. Kirkham and Powers (1972, p. 256–264.) and Klute and Dirksen (1986) provided more recent descriptions. The method is usually only applied to repacked soil. A later experimental innovation was the construction of the tube holding the soil from offset blocks, the so-called Perroux tube (Elrick et al., 1979) as shown in Fig. 3.5–2, along with the typical shape of the water content distribution found. This allows almost instantaneous cessation of water movement. A hammer is used to knock the blocks into alignment, and the holes in them out of alignment. This “freezes” the water in each sampled segment, so no further water movement can occur while the soil is removed from each segment for weighing, oven-drying, and reweighing. Bruce and Klute (1956) used the following theory. For horizontal water absorption, Eq. [3.5–2] becomes
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Fig. 3.5–2. The Perroux tube apparatus used for a Bruce and Klute experiment, and the typical water content profile shape found.
∂θ/∂t = ∂/∂x[D(θ)(∂θ/∂x)]
[3.5–14]
We define η as the Boltzmann variable so η = xt−1/2
[3.5–15]
where x is the distance from the imbibing surface over which a given θ has traveled during time t. Substituting Eq. [3.5–15] into Eq. [3.5–14] and integrating yields for the soil water diffusivity at some water content θ′ θ′
D(θ′) = −1/2 (dη/dθ)θ=θ′∫θn ηdθ.
[3.5–16]
Here θn is the initial water content in the soil column. The trouble with using Eq. [3.5–16] to find the diffusivity is that it involves differentiating experimental data, an error-prone exercise, as Clothier et al. (1983) showed. Shao and Horton (1998) have suggested an alternative approach. They showed how to obtain the parameters in van Genuchten’s (1980) commonly used functions for the hydraulic properties from a Bruce and Klute type experiment. These functions are θ = θr + (θs − θr)[1 + (α|h|)n]−m
[3.5–17]
and Ks{1 − (α|h|)n−1[1 + (α|h|)n]−m}2 K = __________________________ [1 + (α|h|)n]m/2
[3.5–18]
THE SOIL SOLUTION PHASE
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Fig. 3.5–3. Dimensionless water content profiles during a Bruce and Klute experiment, as a function of dimensionless distance, η/ηf, for various values of the shape parameter n.
where θr is the residual water content (often taken as the water content at a pressure head of − 150 m), n is a dimensionless shape parameter, m = 1 − 1/n, and α is a scaling parameter with inverse length units. Shao and Horton (1998) show that the water content distribution between the absorbing surface and the wet front, plotted as a function of the Boltzmann variable, can usually be well described by the expression θ = θs − (θs − θn)(η/ηf)n
[3.5–19]
where θs is the water content at the absorbing surface, ηf is the value of η at the wet front, and n is the shape parameter defined above. Note that, given free water is applied, θs is the water content at satiation. Figure 3.5–3 shows values of the dimensionless water content, (θ − θi)/(θs − θi), as a function of η/ηf for a range of n values. If the sorptivity had not been measured independently from the rate of water absorption, then optimized values of ηf and n can be obtained from the θ (η) data, using Excel and Solver as described above. However, S is usually known, as it is easily found from the cumulative inflow rate using Eq. [3.5–7], provided that the wet front has not reached the end of the column. In this case n can be found by optimizing its value in θ = θs − {nη/[(1 + n)S]}n(θs − θn)n+1.
[3.5–20]
Even more simply, if only S and the transformed wet front distance, ηf, are measured, then n can be calculated directly from n = S/[ηf(θs − θn) − S]
[3.5–21]
Observations of the soil darkening with wetting, visible through the acrylic blocks, usually allow ηf to be measured as a function of time during an experiment. Lastly, given that Ks has been measured independently, α is then found as
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α = 2Ks/Sηf{1/m[(θs − θn)/(θs − θr)]}1/n.
[3.5–22]
A numerical example of the use of the above equations may help the reader. Assume for a fine sand S is 1.47 mm s−1/2, ηf is 6.05 mm s−1/2, θs is 0.40, θn is 0.08, θr is 0, and Ks is 0.0112 mm s−1. From Eq. [3.5–21] we find that n is 3.15, and then from Eq. [3.5–22] we find that α is 2.65 m−1. Usually free water is applied, and the θs and Ks values in Eq. [3.5–19] to [3.5–22] will be the values at satiation, but if a negative pressure head is applied at the absorbing surface, they will be the water content and hydraulic conductivity at that pressure head. Note that the retentivity and K(h) data inferred from a Bruce and Klute type experiment are for wetting, in contrast with the more conventional draining data obtained from the pressure plate apparatus. 3.5.4 Three-Dimensional Infiltration Using Disk Permeameters The disk permeameter or tension infiltrometer (Perroux & White, 1988), illustrated in Fig 3.5–4, allows measurement of infiltration with a constant negative pressure head at the soil surface. A porous plate or cloth on the base of the permeameter allows a suction to be maintained inside. Commercial suppliers include Soilmoisture Equipment Corporation, 801 S Kellog Ave, Goleta, CA 93117, USA (www.soilmoisture.com); Soil Measurement Systems, 7090 N. Oracle Rd Suite 178, Tucson, AZ 85704-4383 USA (http://home.earthlink.net/-soilmeasure/); and OBJECTIF K, 11 rue des Granges Galand, BP 121, 37552 Saint-Avertin, France.
Fig. 3.5–4. Three-dimensional wetting with axial symmetry under a disk permeameter.
THE SOIL SOLUTION PHASE
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If a disk permeameter or infiltration ring is placed on the soil, and the flow underneath is not confined, three-dimensional flow with axial symmetry occurs. This is shown diagrammatically in Fig. 3.5–4. Compared with one-dimensional infiltration, in this three-dimensional axisymmetric flow, the importance of absorption relative to gravity is greater, depending on source geometry, the time required to reach a steady infiltration rate is reduced, and the steady state infiltration rate is greater. We describe three-dimensional disk permeameter methods under two headings, transient methods using early time observations, and steady-state methods using the infiltration rate at longer times. 3.5.4.1 Early-Time Observations J. -P. VANDERVAERE, Université Joseph Fourier, Grenoble, France Disk permeameters have been extensively used over the last decade to measure in situ the soil hydrodynamic properties close to natural saturation. Although the methods of analysis generally preferred are based on steady flow (see Section 3.5.4.b), several researchers have worked on finding an analytical solution for transient flow from a disk infiltrometer. Turner and Parlange (1974), Warrick and Lomen (1976), Warrick (1992), Smettem et al. (1994), Haverkamp et al. (1994), and Zhang (1997) proposed different approaches to account for the supplementary term due to the circular geometry of the source as compared with one-dimensional infiltration. Uncertainties about the time at which a steady infiltration regime is attained, together with the fact that much useful information is lost by ignoring the transient stage, have strengthened the need for a transient three-dimensional infiltration equation for disk infiltrometers. Analysis of transient flow means shorter experiments and smaller sampled volumes of soil, which makes more likely compliance with the necessary assumptions of homogeneity and uniform initial water content. The most recent expressions for transient infiltration (Warrick, 1992; Haverkamp et al., 1994; Zhang, 1997) have in common the following two-term cumulative infiltration equation: I = C1t1/2 + C2t
[3.5–23]
but they differ in the expressions used for the coefficients C1 and C2. Warrick (1992) found C1 and C2 expressions for small times by assuming a constant diffusivity. Zhang (1997) gives empirical values for C1 and C2. Haverkamp et al. (1994), using previous work by Turner and Parlange (1974) and Smettem et al. (1994), proposed the following physically based expressions valid for times not approaching steady state: C1 = S [3.5–24] C2 = Kn + {[(2 − β)/3](K − Kn)} + {γ/[r(θ − θn)]}S2
[3.5–25]
where the subscript “n” refers to the initial conditions, r is the disk radius, β is a parameter depending on the hydraulic diffusivity function and lying in the interval [0;1], and γ is a constant approximately equal to 0.75.
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Except on rare occasions where the soil is initially at a high moisture content, the condition Kn < K is fulfilled. Then, if an average value of 0.6 is taken for β, Eq. [3.5–23] to [3.5–25] reduce to: I = St + 7/15 Kt + {0.75/[r(θ − θn)]}S2t.
[3.5–26]
The first term of the right-hand side in Eq. [3.5–26] represents vertical capillary flow and dominates the infiltration during the initial stage. The second term represents gravity-driven vertical flow, and the third term represents lateral capillary flow. One important feature of Eq. [3.5–26] is that water moving laterally by capillary flow adds a term linear in time. Hence, Eq. [3.5–26] can be considered a three-dimensional extension of the Philip’s one-dimensional infiltration Eq. [3.5–9]. Other transient flow equations can be found in the literature, but in practice their utilization mostly involves solving a set of equations to find the two variables involved in the process, K and S. White and Perroux (1989) proposed a markedly different approach, in which hydraulic conductivity is estimated from paired sorptivity measurements. The relative weights of the three terms in Eq. [3.5–26] play a decisive role in quantifying the accuracy with which S and K can be estimated. For example, if the second term on the right-hand side is negligible compared with the third term, this means that the infiltration process is controlled by capillarity and a precise estimation of K is unlikely. Note that this kind of limitation is not unique to transient flow analysis, but also limits the accuracy of the methods on the basis of steady flow analysis described below. Different methods used to determine S and K with C1 and C2 values are described in Vandervaere et al. (2000b), depending on whether use is made of one or several disk radii and one or several applied pressure heads hS. A popular method to determine hydraulic conductivity, presented in White and Sully (1987), makes use of both transient and steady flow, the latter being described by Eq. [3.5–31] (see Section 3.5.4.2 below). As the parameter λc is calculated by Eq. [3.5–6], the method relies on obtaining an estimation of sorptivity. This is achieved by assuming that both gravity and lateral diffusion effects can be neglected at the beginning of the axisymmetric infiltration process. Cumulative infiltration is thus approximated by Eq. [3.5–7], which is equivalent to Eq. [3.5–26] with the two terms linear in time truncated. Then, S can be determined as the slope of I vs. t1/2 during a time interval TS, as small as possible, over which Eq. [3.5–7] is considered valid: S = [dI/d(t1/2)]Ts.
[3.5–27]
However, Eq. [3.5–27] must be used with care, as two difficulties arise in practice: 1. In the case of soils with important gravity flow (low λC), the term St1/2 can quickly become dominated by the K term in Eq. [3.5–19]; on the other hand, in the case of soils exhibiting important capillary effects (large λC), the term St1/2 can become quickly dominated by the S2 term in Eq. [3.5–19]. Thus, it cannot be guaranteed that infiltration data can be obtained to which Eq. [3.5–27] can be applied with reasonable confidence and precision. The chosen time interval TS is likely to influence strongly the calculated S value (Bonnell & Williams, 1986). It can
THE SOIL SOLUTION PHASE
891
be shown (Vandervaere et al., 2000b) that the chances of obtaining an acceptable sorptivity determination are maximized when S is close to an optimal value, Sopt, given by: Sopt = {r(θ − θn)K[(2 − β)/3γ]}1/2.
[3.5–28]
2. Disk permeameters are usually placed on a layer of sand to ensure hydraulic contact between the permeameter and the soil. The sand is normally chosen for its high conductivity, so that no impeding effect modifies the steady state infiltration flux. However, the effects of this layer on the first stages of infiltration can be important enough to mask the portion of the infiltration curve desired for analysis (Vandervaere et al., 1997, 2000a). An alternative to the use of Eq. [3.5–27] is (White et al., 1992): S = lim [dI/d(t1/2)]. t→0
[3.5–29]
However, if contact sand is used, it affects this limit. A better approach is to use the following transformation of Eq. [3.5–23] (Vandervaere et al., 2000a): dI/d(t1/2) = C1 + 2C2t1/2.
[3.5–30]
Plotting dI/d(t1/2) vs. t1/2 gives C1 (i.e., the sorptivity) and C2 by simple linear regression. The linearized form (Eq. [3.5–30]) also serves as a tool to examine the validity of the two-term algebraic Eq. [3.5–23] and, to a certain extent, to detect and eliminate the influence of the contact material. Examples are shown in Fig. 3.5–5 for the case of numerically simulated tests with a clay soil and in Fig. 3.5–6 for a laboratory test with a sand. The dangers of using inadequate parameter-fitting techniques are discussed in Vandervaere et al. (2000a). In practice, S can be determined from early-time observations in the manner described below. Using a spreadsheet such as Excel, t and I are entered into two columns in the form of successive readings, ti and Ii. In two other columns we enter 1/2 − t1/2 ). ti1/2 and dI/d(t1/2) calculated with the finite difference form (Ii+1 − Ii−1)/(ti+1 i−1 Table 3.5–3 gives an example with data relating to a 125-mm-radius disk infiltrometer test carried out on a loamy soil. Plotting dI/d(t1/2) vs. t1/2 (Fig. 3.5–7) allows identification of the contact material influence, which corresponds to the initial sharply decreasing portion of the graph (four points in this example) not considered for calculations. Performing a classical linear regression on the rest of the data set gives the value of the intercept C1 theoretically equal to sorptivity (0.138 mm s−1/2 in this example). Note that, in many situations, the instant corresponding to the end of the contact material influence may be easy to estimate, but a best-guess estimation is much better than simply ignoring this effect. Due to the uncertainty on the value of the parameter β, methods based on transient flow analysis appear less precise than those based on using the steady regime. However, the following must be kept in mind:
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Fig. 3.5–5. Simulated axisymmetric infiltration tests for Yolo Light Clay linearized with Eq. [3.5–30], for disk radius values of 125 mm (diamonds) and 24.25 mm (triangles).
Fig. 3.5–6. Laboratory infiltration tests performed with “S31” sand on a soil column (1D, triangles) and with an axisymmetric geometry (3D, diamonds). Data are linearized with Eq. [3.5–30].
THE SOIL SOLUTION PHASE
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Table 3.5–3. Finding S from early-time disk infiltrometer data. ti
Ii
ti1/2
dI/d(t1/2)
s
mm
s1/2
mm s−1/2
0 5 10 20 30 40 50 60 75 90 105 120 135 150 165 180
0 1.23 1.71 2.06 2.27 2.41 2.53 2.66 2.82 2.97 3.11 3.24 3.36 3.48 3.59 3.70
0 2.24 3.16 4.47 5.48 6.32 7.07 7.75 8.66 9.49 10.25 10.95 11.62 12.25 12.85 13.42
/ 0.539 0.371 0.244 0.188 0.165 0.172 0.178 0.179 0.184 0.186 0.185 0.185 0.184 0.190 0.200
1. The inaccuracy resulting from the possible range of β values ([0, 1]) is often minor compared with other sources of error, which are well known to people familiar with field experimentation. The natural variability of soil hydrodynamic properties, especially when roots, worms, or agricultural practices affect the topsoil structure, can lead to coefficients of variation of more than 100% for K and/or S. Thus, it is only the order of magnitude of these hydraulic variables that can reasonably be determined from field investigations.
Fig. 3.5–7. Field infiltration test performed with a disk infiltrometer. Data are linearized with Eq. [3.5–30] (triangles) and linear regression (plain line) is performed on the increasing portion of the data set.
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2. The assumptions of vertical homogeneity and uniform initial water content are better approximated during shorter transient experiments because of the smaller volume of soil involved.
3.5.4.2 Steady-State Observations BRENT CLOTHIER, HortResearch, Palmerston North, New Zealand DAVID SCOTTER, Massey University, Palmerston North, New Zealand
Wooding (1968) showed that, provided Eq. [3.5–4] and [3.5–5] apply, the steady infiltration rate (i) from a shallow surface pond of free water of radius r can be approximated as i = Ks[1 + 4λc/(πr)].
[3.5–31]
Note that in Eq. [3.5–31] the first term represents the effect of gravity, and the second term represents the effect of geometry plus capillarity. If we define Q as the volume of water soaking into the soil per unit time, then i is found as Q/(πr2). Most methods for inferring water transmission parameters from steady-state, three-dimensional disk permeameter and infiltration ring data depend on Eq. [3.5–31], as it can be applied for both zero and negative constant pressure head at the surface. Weir (1986, 1987) and White et al. (1992) showed that while Eq. [3.5–31] is an accurate approximation for large ponds or disks, it can be up to about 10% in error when r is small relative to λc. If we define a dimensionless variable τ by τ = r/2λc
[3.5–32]
then for values of τ < 0.5 the following equation is more accurate than Eq. [3.5–31] (Weir, 1987) τ = 2Kssin2(τ)/[πτ2sin(τ)cos(τ) + 2τ2sin2(τ)ln(τ) − 1.073τ4].
[3.5–33]
Table 3.5–4 compares the i/Ks values from Eq. [3.5–31] and [3.5–33] with values found by numerical solution (Pullan & Collins, 1987). This shows the accuracy of the two equations. Note that Eq. [52] in Philip (1986), also given as Eq. [3] and [6] in Smettem and Clothier (1989), is not accurate, and its use is not recommended. The dimensionless number i/Ks indicates the magnitude of the geometry plus capillarity term relative to the gravity term in Wooding’s equation. Table 3.5–4 shows that the two terms are of similar magnitude when τ is 0.6, or, for example, when r is 120 mm and λc is 100 mm. We describe two methods for finding the soil transmission parameters from steady-state disk permeameter data. This involves using the above equations, and separating the effects of gravity (Ks) and capillarity (λc). The first method of effecting this separation uses two or more disks of different diameters, with the same pressure head imposed at the surface (Smettem & Clothier, 1989). The second method uses a single disk diameter, but with two or more negative pressure heads imposed
THE SOIL SOLUTION PHASE
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Table 3.5–4. Values of i/Ks as a function of τ from Wooding’s equation, Weir’s equation, and a numerical solution. Dimensionless flux i/Ks τ = r/2λc
Eq. [3.5–31]
Eq. [3.5–33]
Numerical solution
0.001 0.005 0.01 0.05 0.1 0.4 0.6 1 4 10
637.6 128.3 64.7 13.7 7.37 2.59 2.06 1.64 1.16 1.06
639.6 129.7 65.8 14.4 7.80 2.80 2.36 -
640.0 129.8 65.8 14.4 7.81 2.75 2.16 1.69 1.17 1.07
sequentially (Ankeny et al., 1991). In a manner similar to that described above, the raw data can be easily analyzed using a spreadsheet with an embedded optimization package such as Excel with Solver. If data for disks of different sizes are obtained, the imposed values of r are placed in a spreadsheet column next to the measured values of i. Another column gives the fitted values of i from Eq. [3.5–31] or [3.5–33], calculated using the initial guesses of parameters Ks and λc, which are placed in other cells on the spreadsheet. A simple option is to take 100 mm as an initial guess for λc, and to take i for the largest ring or highest (closest to zero) surface pressure head as an initial guess for Ks. In another column, the squares of the differences between the measured and estimated values of i are calculated. This column is summed, and the cell containing the sum becomes the target cell for Solver. Solver is then used to optimize Ks and λc by minimizing the sum of squares in the target cell. For paired i values, this sum will be zero. To illustrate the above procedure, we take the topsoil data from Fig. 3 of Smettem and Clothier (1989), where i values of 14.2, 8.5, and 5.8 mm h−1 were found for disk radii of 22.5, 44.5, and 91 mm respectively. Our initial guesses are 100 mm for λc, and 5.8 mm h−1 for Ks. Using Eq. [3.5–31] gives values of 2.9 mm h−1 and 70 mm for Ks and λc, respectively. Using Eq. [3.5–32] and [3.5–33] gives values of 2.4 mm h−1 and 79 mm for Ks and λc respectively, and as τ is 1 m in radius, so that the centrifugal field in the model may closely approximate the uniformity of a gravitational field. If the sample is not considered a scale model but simply a medium subjected to a great force, it is possible to measure dynamic properties with a smaller, readily obtainable centrifuge. Methods based on this idea are the main subject of this section. Two types of implementations have been successfully used, one in which the means of controlling flow is internal to the centrifuge bucket, referred to here as internal flow control (IFC) (Nimmo et al., 1987), and the other in which the flow is controlled outside the centrifuge, called unsaturated flow apparatus (UFA) (Conca & Wright, 1998). In some experiments, transient instead of steady flow (e.g., Alemi et al., 1976; Nimmo, 1990), and even unstable flow (Culligan et al., 1997; Griffioen et al., 1997), have been investigated. So far these have not led to generally used techniques for soil property measurements, though they provide significant insights into the nature of unsaturated flow. The objectives of this section are to describe the steady-state centrifuge (SSC) apparatus and procedures, and to explain particular features of centrifugal force in the measurement of saturated and unsaturated hydraulic conductivity. The level of detail is not intended to suffice as a stand-alone instruction manual, but to allow the reader to understand each method and its corresponding apparatus, as well as the pros and cons.
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Basic Centrifugation. A centrifuge bucket is constrained to move in a circular path so that it accelerates toward the center. An object within the bucket therefore feels driven toward the bottom of the bucket. Because this force on the object within the bucket is directed away from the center of rotation, it is called a centrifugal (meaning “away from center”) force. As basic physics textbooks (e.g., Resnick et al., 1992) explain, centrifugal force is a pseudoforce, meaning that it exists only within the bucket, which is not a legitimate inertial reference frame because it is accelerated. But this only means that certain laws of mechanics do not apply; it does not mean that the object in the bucket feels no force or that the force is not centrifugal. The centrifugal force (Fc; N) increases linearly with distance from the center, and with the square of the speed of rotation. It can be expressed as Fc = mrω2
[3.6.1.1–2]
where m is the mass of the object in the bucket, r (m) is the distance from the center of rotation, and ω = 2π/tc is the angular frequency (rad s−1) with tc (s) being the time to complete one rotation. The factor rω2 is the centrifugal acceleration, analogous to gravitational acceleration (g). Equation [3.6.1.1–2] and others derived from it, which retain the explicit r dependence, permit the use of a sample whose height in the bucket is significant with respect to r, even though the force varies within the sample. When the object in the bucket is water in a porous medium, it is convenient to consider the centrifugal force per unit volume; that is, Fcv = ρwrω2
[3.6.1.1–3]
where ρw (kg m−3) is the density of water. This formula is the basis for derivations in this section, in which all pressure, centrifugal, and gravitational potentials are on a volumetric basis. Saturated Flow and Ksat Measurement in a Centrifugal Field. Especially for tight media, a technique using centrifugal force may be better than other techniques for measuring the saturated hydraulic conductivity (Ksat). Knowledge of Ksat is not only important in itself, but also as an indicator of the maximum measurable unsaturated K value. It allows operating speeds and flow rates for the unsaturated K measurements to be chosen more appropriately. The techniques available for measuring Ksat in a centrifugal field are adaptations of familiar gravity-driven, constant- and falling-head methods (Section 3.4.2). Figure 3.6.1.1–1b and 3.6.1.1–1c illustrate two ways of implementing such a Ksat measurement, with an internal (within the centrifuge bucket) or external (outside the whole centrifuge) constant or falling head (Nimmo & Mello, 1991). The internal or external option and the constant- or falling-head option permit four combinations, each with a somewhat different apparatus and formula. For the two internal head options (Fig. 3.6.1.1–1b), Nimmo and Mello (1991) derived the formula Ksat = (2qL)/[ρwω2(ro2 − r 2wa)] for the constant-head case, and the formula
[3.6.1.1–4]
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aL (ro + rwa)(ro − ri) Ksat = ___________ log ______________ Aroρwω2(t − ti) — (ro − rwa)(ro + ri)
[3.6.1.1–5]
for the falling-head case, where q is the flux density (m s−1), L is the sample length (m), ro (m) is the position at which the pressure potential ψ = 0 (Pa), as would be
Fig. 3.6.1.1–1. Diagrammatic configurations for various means of hydraulic property measurements. (a) No-inflow condition with controlled outlet head to establish a static equilibrium distribution of water content and matric potential for water retention measurement. Note that rb is the distance from the center of rotation to the bottom of the porous medium, regardless of the bulk water level. (b) Internally applied constant or falling head condition as used to measure Ksat in a standard-rotor centrifuge. Note that rwa is defined to supplement the dimensional definitions in (a). (c) Externally applied constant or falling head condition to measure Ksat in a UFA-rotor centrifuge.
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established by a free water surface, rwa (m) is the position of the inflow reservoir surface, a (m2) is the cross-sectional area of the reservoir, A is the cross-sectional area of the sample, t is time (s), and ri (m) is the position of the reservoir surface at time ti (s). For the external-head options (Fig. 3.6.1.1–1c), combining these equations with the appropriate representations of gravity-driven flow gives Ksat = (qL)/(ρwgz + 0.5ρwω2rb2)
[3.6.1.1–6]
for the constant-head case, and aL (gzi + 0.5ω2rb2) Ksat = _________ ln _________ Aρwg(tf − ti) — (gz + 0.5ω2r b2)
[3.6.1.1–7]
for the falling-head case, where z (m) is the height of water above the plane in which the sample rotates, and rb (m) is the position of the bottom of the sample. Unsaturated Flow in a Centrifugal Field. Steady-state conditions require a constant flow rate and a constant centrifugal force for a long enough time that both the water content and the water flux within the sample are constant. When these conditions are satisfied, Darcy’s Law relates K to the volumetric water content (θ) and the matric potential (ψ; Pa) for the established conditions. The air pressure is assumed to be atmospheric (zero) so the pressure potential and the matric potential can be equated (Koorevaar et al., 1983, p. 79). With centrifugal instead of gravitational force, Darcy’s Law takes the form q = −K[(dψ/dr) − ρwω2r]
[3.6.1.1–8]
Rearranging this equation yields q/K = ρwω2r − (dψ/dr)
[3.6.1.1–9]
In the case of hydrostatic equilibrium (i.e., q = 0), Eq. [3.6.1.1–9] leads, upon integration, to a description of the pressure potential profile by ψ(r) = [(ρwω2)/2](r2 − ro2)
[3.6.1.1–10]
Figure 3.6.1.1–1a gives a diagrammatical illustration of the method. At hydrostatic equilibrium, the matric potential gradient is equal and opposite to the centrifugal driving force. In the case of steady flow, if the driving force is applied with a centrifuge rotation speed large enough to ensure that dψ/dr n ρwω2r, any matric potential gradient that develops in the sample during centrifugation is insignificant; the flow is essentially driven by the centrifugal force alone. The flow equation then simplifies to q ≈ K(ψ)ρwω2r
[3.6.1.1–11]
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The ω threshold value for which the dψ/dr gradient becomes negligible depends on the soil hydraulic properties. Nimmo et al. (1987) discussed various possibilities for this gradient and presented model calculations showing that it becomes negligible at relatively low speeds for a sandy medium and at higher speeds for a finetextured medium. This centrifugally dominated situation is favorable for at least two reasons. First, once it has been verified, there is no need to determine the ψ gradient. Second, it normally results in a fairly uniform water content throughout the sample, permitting the association of the sample average θ and ψ values, determined from the sample weight and tensiometer readings, with the measured K. Repeat measurements with different q values (and perhaps, e.g., different speed) give additional points needed to define the K(ψ), K(θ), and θ(ψ) characteristics. It is important to know the pressure boundary condition at the outflow face. This pressure can be controlled by applying a known head using a thick outflow membrane in a vessel that ensures the formation of a water table at a known position, as was done by Russell and Richards (1938), by Nimmo et al. (1987) during steady flow applications of the IFC apparatus, and by Nimmo (1990) during a transient flow experiment. A schematic diagram of the apparatus to be used is depicted in Fig. 3.6.1.1–2. A shallow-lipped metal plate in the outflow reservoir is grooved on its upper face to prevent sealing to the porous ceramic plate with which it is in contact. The relevant water table in Fig. 3.6.1.1–2 is not shown in detail, but would exist in the lowest millimeters of the ceramic plate, at the height of the outer lip on the metal plate. This method requires that the ceramic plate has a high enough K value that (i) flow occurs easily through it, and water does not accumulate in the sample and (ii) the pressure difference between the water table and the top of the ceramic plate is negligibly affected by flow through the membrane. A similar apparatus can be used to apply a positive ψ in case a saturated K measurement needs to be made (Nimmo & Mello, 1991). A more widely used alternative to the above setup is to allow water to exit the sample freely (except for the unavoidable impedance of the screen or filter that supports the bottom of the soil sample). The common assumption is that ψ = 0 at the outflow face, though this has not been experimentally demonstrated. Although this assumption is not true in an exact sense, it may be a reasonable approximation. Theoretically, water will accumulate at a slightly positive pressure until eventually water drops form and exit at the outflow face. Analogously to the concept of air-entry pressure, the system would have a slightly positive “water exit pressure”. On the other hand, the matric potential may be somewhat negative if instabilities enhance the drainage of some outlet-face pores, an effect that may be enhanced by the centrifugal force. There are numerous effects that cause the net force field within the sample to deviate from uniformity, including: 1. The centrifugal force diverges radially from the axis of rotation, so that within the cylindrical sample it mostly is not parallel to the axis of the cylinder. 2. The centrifugal force varies substantially with r (i.e., along the sample axis), although the explicit r dependence in the equations accommodates for most aspects of this effect.
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Fig. 3.6.1.1–2. Cross section, in the plane of rotation, of apparatus fitting in a 1-L centrifuge bucket for establishing and measuring steady-state flow through the soil (Nimmo et al., 1992, 1994). A twin apparatus is used in the bucket opposite the first one.
3. Strictly, gravity should be vectorially added to the centrifugal force. In the IFC system, the direction of the net force is not a problem as long as the bucket swings properly to align with the net force, in which case the magnitude only needs correction at low speeds. In the UFA, both the direction and magnitude of the net force may create a problem at low speeds. 4. Coriolis forces are usually negligible for conditions of unsaturated property measurement. 5. Forces arising from angular acceleration and deceleration may be significant at the start and end of a centrifuge run, but are likely to be negligible as long as the constant-speed portion of the run is much longer than the duration of acceleration and deceleration.
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Fig. 3.6.1.1–3. Cross section, perpendicular to the plane of rotation, of the UFA rotor including two sample assemblies. The rotating seal and inflow tubes are not shown to scale to enhance the distinction of details. The filter paper and perforated plate at the outflow face of each sample are also omitted.
Apparatus and Supplies. Steady-State Centrifuge-Internal Flow Control. The basic SSC method with IFC uses the apparatus shown in Fig 3.6.1.1–2, consisting of water storage and flow control devices surrounding a soil column in a 1-L centrifuge bucket. This apparatus is subjected to centrifugal accelerations of up to 2000 g to generate measurable flow even at low K values. The reservoirs and porous plates control the water flow rate at the desired value, chosen to maintain a suitable state of unsaturated water content. Measurements of the change in mass of the various reservoirs indicate the volume flux (Q; m3 s−1). Dividing by the cross-sectional area of the sample gives q. When q becomes steady, Darcy’s Law applies, and K can be calculated from Eq. [3.6.1.1–8] or [3.6.1.1–11]. A wide variety of centrifuge types and rotor types can be used to obtain these measurements by the SSC-IFC approach. Steady-State Centrifuge–Unsaturated Flow Apparatus. The SSC–UFA method requires the use of a specific centrifuge and rotor, of which at least two models are commercially available. Figure 3.6.1.1–3 illustrates a UFA rotor designed for SSC measurements with certain Beckman centrifuges (Beckman Instruments, Fullerton, CA).1 In addition to the specific equipment requirements, the dimensions of the rotor and sample retainers of the UFA system differ from the IFC system, and it also employs a different way of delivering water to the sample. The rotor pictured here uses sample retainers 4.9 cm long and 3.3 cm in diameter, resulting in a cross-sectional area of 8.55 cm2 and a sample volume of 41.89 cm3. Somewhat larger sample sizes are possible with an alternative larger version of the UFA rotor. Rather than delivering water to each sample through use of an internal reservoir and ceramic plate applicator (internal flow control), water is supplied to each 1
The mention of brand names does not imply endorsement by the USGS.
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911
sample through external pumps attached (via tubing) to the UFA rotor and the use of a rotating seal within the UFA rotor. External pumps control the rate at which water is delivered to each sample during centrifugation. Many types of pumps are available for use with the SSC-UFA rotor; for example, intravenous pumps from the medical industry are commonly used. Water pumped into the rotor from an external water pump is dispersed over the sample through use of a specifically designed spreader. Testing of the evenness of dispersion has been performed by Conca and Wright (1998). Infused water, after passing through the sample, passes through a filter paper and perforated plate (both placed to prevent soil loss from the sample retainer) into an effluent collection chamber attached to the bottom of the sample retainer (Fig 3.6.1.1–3). The effluent chamber has calibration markings for measurement of outflow volume, useful for comparison to inflow when establishing steady-state flow. A centrifuge equipped with a strobe light can be used to observe the amount of effluent within the effluent collection chamber during centrifugation. Accessories and Supplies. Additional equipment used with either the IFC or UFA version includes a balance accurate to 0.01 g or better, for weighing samples and effluent after centrifugation, and a timer for recording the elapsed time of centrifugation. A tensiometer (or other suitable device, for conditions outside the tensiometer range) is needed for measuring the matric potential associated with a given centrifuge run. This should be a touch type (nonintrusive) tensiometer (e.g., with a disk membrane that can be brought into temporary contact with the surface of the sample). Supplies commonly needed include de-aired water, antibacterial agents, filter papers of various sizes, O-rings, lubricants, and miscellaneous hand tools. Other sample preparation and handling equipment that may be necessary can be found in other sections of this book, and other publications addressing centrifuge techniques. For use with the UFA apparatus, freshly de-aired water should be used for each centrifuge run, especially at high centrifuge speeds and low flow rates. Under these conditions, the infused water can be under high negative pressures. This can lead to dissolution of dissolved air and development of bubbles within the water delivery system, within the pumps, the tubing connecting the pumps to the rotor, or the rotor itself. Significant air bubbles can lead to inconsistent water delivery rates, pump failure and shutdown, and possible rotor face-seal damage. Antibacterial agents previously used include calcium selenate or chlorinated water solutions. Calcium selenate solutions (0.005 M CaSO4 and 0.005 M CaSeO4) work well and work for extended periods. In particular, calcium selenate prevents bacterial clogging of ceramic plates used with the SSC-IFC apparatus. However, because calcium selenate tends to precipitate easily, this solution could damage the rotating face seal when used in the SSC-UFA rotor. Chlorinated water (sodium hypochlorite, 0.3 mL L−1 water) is an adequate antibacterial agent for many applications, but is not as effective or as long lasting as calcium selenate. However, chlorinated water solutions are generally adequate for the more rapid measurements obtained with the SSC-UFA method. Procedure. Steady-state centrifuge methods rely on a sequence of runs whereby at each step, steady state is achieved and values of K, θ, and ψ (if desired)
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are determined. Information is obtained during drying if the sequence starts at saturation, or during wetting if it starts with a dry soil. Sample Preparation. Samples may be cored in the field directly into the appropriate retainer for centrifugation, or oversized minimally disturbed samples may be taken in the field and subsequently recored. For some applications, especially where reproducibility for experimental purposes is more important than correspondence of measured properties with those of in situ soils, artificial repacking is necessary. This is best done with a machine designed to pack samples to the desired bulk density with negligible segregation of particles by size or other characteristics (Ripple et al., 1973). For applications where less uniformity and quality is acceptable, bulk samples may be hand packed into the appropriate retainer by a fill and tamp (American Society for Testing and Materials, 1999) or slurry method. Planning the Sequence of Runs. K can be determined over a range of desired water content values by choosing appropriate flow rates and centrifuge speeds. The choice of parameters will fix the range of K values to be examined. An initial Ksat determination will give an indication of the maximum flux to apply during a sequence of unsaturated runs. If Ksat is relatively low, the driving force at the wettest run in the sequence must be sufficiently high and the flow rate must not exceed the measured Ksat inflow. Starting with a saturated sample, a stepwise decrease in the inflow rate using a precision metered pump or ceramic plate with known flow properties, along with a chosen rotational speed, allows measurement of the primary drying curve. Finer samples may need to start at relatively low flow rates, not exceeding Ksat, and be run at high speed, while coarser samples may start at higher inflow rates and be run at lower speed. In most cases of unsaturated SSC measurements, the K value that will be obtained in a given experimental run is essentially fixed by the choice of operating parameter values. Thus the purpose of each run is first to ascertain that the steadyflow conditions (and the nearly uniform profile conditions if desired) can be achieved, and second to obtain the θ and ψ measurements that correspond to the established value of K. Equation [3.6.1.1–8] or [3.6.1.1.–11] can be used to estimate values of K that will be obtained in a run. The choice of centrifuge speed depends on the characteristics of the sample and the desired range of K and water content to be explored. The two primary considerations are the driving force acting on the water and the compressive force acting on the solid medium. The most basic consideration is that the centrifuge speed must be great enough so that the centrifugal driving force is the major component of the net driving force. This is in contrast to the condition of q n K, in which the left side of Eq. [3.6.1.1–9] would be essentially zero. In that case, the forces would nearly balance and the profile would be essentially that of the hydrostatic case with the parabolic shape described by Eq. [3.6.1.1–10]. This condition would make it impossible to determine K accurately from a SSC measurement because the total driving force is a difference of two terms nearly equal in magnitude, one of which (dψ/dr) is poorly known. To prevent this problem, both the speed and the input q should be set higher. This will increase the centrifugal force relative to dψ/dr. In other words, the centrifuge speed must be great enough so that q is not negligible relative to K. The speed required to accomplish this is generally greater for finer-
THE SOIL SOLUTION PHASE
913
textured soils (Nimmo et al., 1994). Normally it is desirable to substantially exceed the basic speed minimum to assure that Eq. [3.6.1.1–11] applies. The criterion for this is that the centrifuge speed must be great enough so that q has a much larger magnitude than K. If speeds this high are not possible, lower speeds can be used as long as they exceed the basic minimum, though Eq. [3.6.1.1–8] will apply rather than Eq. [3.6.1.1–11], as will be explained in the section on calculations. If the centrifuge compresses the samples compared with their in situ condition, it will change the hydraulic properties at least slightly. The effect of centrifugal compaction on the hydraulic property measurements depends on the medium and the water content range of interest. This effect may be negligible, it may impose an upper limit to the running speed, it may increase the range of uncertainty associated with the results, or it may require correction when results are calculated (Nimmo & Akstin, 1988; Nimmo et al., 1994). Coarse-textured soils and low water contents exhibit the smallest effects. In practice, compaction effects may be negligible for sands and for finer-textured soils if water content values are not very high. For finer-textured soils at saturation, compaction is more likely to affect the results, for example, the factor-of-four correction worked out by Nimmo et al. (1994) for a silty medium. Achieving and Testing for Steady State. Each run must be carried out sufficiently long to achieve steady-state flow throughout the sample. One criterion is for the sample mass, and therefore the water content, to remain constant over time. Another criterion is that the in- and outflow rates should be equal, as determined from the prescribed pump rate (UFA) or the change in mass of the water in the inflow reservoir weight (IFC) and the change in the mass of the water in the outflow reservoir. If the centrifuge has a strobe light assembly and viewing port, the water level in the outflow reservoir can be monitored as the rotor is spinning to determine if the in- and outflow rates are equal. The time to achieve steady conditions may range from 1 h for high K values, to one or more days for low K values. For measurements on a series of samples that do not differ greatly in their hydraulic characteristics, it may be possible to develop a set program of run lengths coordinated with a sequence of K values. This program can then serve as a guideline for run lengths on the remaining samples. The sample is weighed at the end of each run and oven-dried after the last run to compute the water content corresponding to each steady-state measurement. The tensiometer (or alternative method if ψ is outside the tensiometer range) may be used after each run to obtain a matric potential value for each established steady-state water content. Calculations and Software. For the case of negligible ψ variation in the sample, K can be calculated directly from Eq. [3.6.1.1–11]. The sample midpoint (r = 8.7 cm for UFA and ~18 cm for IFC) is generally used in this calculation, being approximately the point of average ψ, θ, and centrifugal driving force. For cases where it is not yet known whether the ψ gradient is significant, a numerical model of unsaturated flow in a centrifugal field can furnish useful information about matric potential profiles, ψ(r), and driving forces to help in resolving this issue. The code used by Nimmo et al. (1994) uses a fifth-order Runge–Kutta algorithm with adaptive step size control (Press et al., 1989) to solve Eq. [3.6.1.1–8] for ψ(r), given
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an estimated K(ψ) function (represented by an empirical formula for convenience), the value of q for the run in question, and the centrifuge speed ω. The gradient dψ/dr within the sample can then be computed and compared with the magnitude of the centrifugal force. This comparison allows an evaluation of the run in terms of three categories: (i) the net driving force dominated by the centrifugal force, (ii) the net driving force significantly affected by both the matric potential gradient and the centrifugal force, and (iii) the net driving force insignificant because the centrifugal force and the matric potential gradient essentially cancel each other out. If the centrifugal force dominates, either Eq. [3.6.1.1–8] or [3.6.1.1–11] can be used to calculate the value of K for the run; if both types of forces are significant, Eq. [3.6.1.1–8] must be used; and if the forces cancel each other out, the run is not useful for K measurement and must be discarded. Experience with measurements on a variety of media indicates that the conditions of Category (ii) or (iii) are most likely to occur for dry conditions and low K values, although they sometimes also occur for the wettest and highest K runs when low centrifuge speeds are used. To optimize the determination of ψ(r) and K(θ,ψ), especially for those cases that fall into Category (ii) the calculations can be repeated with a revised estimate of K(ψ) that takes advantage of the newly computed ψ(r). This revision of K(ψ) is best done using results from all valid centrifuge runs for the sample in question. Unless all runs are dominated by centrifugal force, the next K(ψ) (after dψ/dr is recomputed) will differ somewhat from the previous one. The updated K(ψ) can in turn be used to again revise dψ/dr and recompute K(ψ). Iteration, using a Newton–Raphson scheme, can quickly produce an optimal K(ψ) and a best-estimate ψ(r) for computing the best K values to pair with the θ and ψ measured after each run. Regardless of which forces are significant, the computations require certain conversions of flow rate and driving force. Volume flux in milliliters per hour or cubic centimeters per hour must be converted to flux density (q) in centimeters per second by dividing by the cross-sectional area of the sample (8.55 cm2 for the samples frequently used with the UFA rotor; 19.63 cm2 for the 5-cm-diam. samples frequently used with the IFC apparatus). Centrifuge speed is usually known in revolutions per minute (rpm), which must be multiplied by a factor of π/30 to convert to radians per second to obtain angular velocity (ω). Because the equations in this section are derived and presented with potential on a volumetric basis, they take ψ in units of Pascals, and product K in units of m/s per Pa/m. For practical use, K can be converted to m/s by multiplying by ρwg (= 9807 Pascals per meter of water). Alternatively, if head units are used for ψ, centrifugal terms in the formulas must be divided by ρwg. Discussion and Conclusions. Comparison of Steady-State Centrifuge Techniques and Apparatus. The IFC and UFA systems have different features and capabilities. The UFA apparatus has the great advantage of being simpler to operate, typically requiring less than half as much time per measurement as the IFC apparatus. This advantage stems from the use of an easily adjustable pump to control the applied water flux. The UFA apparatus is commercially available, but costs several times as much as the IFC apparatus. For the latter, part is commercially available and part must be custom fabricated. The dimensions of the rotor and sample are significantly different. The UFA
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is limited to sample cylinders either 3 or 4 cm in diameter (depending on the rotor model), whereas the IFC normally accepts 5-cm-diameter samples and could be modified to handle much larger ones. Another significant IFC advantage is its greater radius of rotation of the sample (about 18 rather than 8 cm). The top of the UFA rotor sample retainer is 6.2 cm from the center of rotation, while the bottom is 11.2 cm from the center of rotation. This makes the maximum force achievable with the IFC more than twice as great as with the UFA. Additionally, the force field within the sample is much more uniform with the IFC than the UFA (varying by a factor of 1.8 from the inlet to the outlet face in the UFA and by about 1.2 in the IFC apparatus). Other points of comparison, such as the possible range of measurements, reliability of the controlled flux, and measurement uncertainties, generally require little attention. Future developments should include a much larger centrifuge, resulting in highly uniform force fields, and possibly internal metering pumps for flow control. Effects of Centrifugation on Measured Properties. In most cases, the biggest concern with the use of centrifugal force is that compaction can alter the structure of the material so as to influence K and water retention. For consolidated materials and media such as densely packed sands (Nimmo & Akstin, 1988), compaction may be negligible. For other cases, such as fine-textured or lightly compacted media, compaction and its effect on K and water retention may be significant. However, it may be possible to correct for compaction effects. In going from 1 to 200 g, Ksat for a compacted silt-loam-textured sample declined by a factor of four in the experiment of Nimmo et al. (1994). This might be a reasonable upper limit for the effect of centrifugation on unsaturated K for comparable media and conditions. For some highly structured soils, however, centrifugal methods may be unsuitable altogether. On the other hand, for some applications, centrifugal compaction is advantageous in permitting K to be measured under conditions of a simulated overburden pressure approximating that of the original condition of the sample. At accelerations of a few thousand g, a small mass placed on top of the sample can exert a mechanical pressure equivalent to that of hundreds of meters of overlying material. To simulate a given overburden, for greater uniformity of stress from the top to the bottom of the sample, it is better to use a large mass at low speed than a small mass at high speed. However, the speed must still exceed the minimum for a well-defined driving force as explained above. Nimmo et al. (1994) gave formulas and other considerations for generating effective overburden pressures. Differences in temperature between the laboratory and subsurface field conditions can also be minimized using a centrifuge with a controlled-temperature chamber. A fundamental concern is whether the centrifugal force distorts air–water interfaces in ways that affect the hydraulic properties. The most directly applicable evidence is from the test of the unsaturated version of Darcy’s Law by Nimmo et al. (1987), in which K at a fixed value of θ did not vary significantly as the centrifugal force varied from 200 to 1650 g. In this test, then, any effect of the variable force on air–water interfaces was insignificant for K determination. Hysteresis. Soil water retention is subject to hysteresis as exhibited by different water content values, corresponding to the same matric potential, during drying or wetting situations. The sequence of measurements can either start from near
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saturation (drying curve) or from a low water content (wetting curve), as desired. Preliminary centrifugation with low or zero applied water flux can bring the samples to a desired water content state from which to begin a wetting curve. Usually, the time required to approach steady-state conditions for a run on a wetting curve is greater by a factor of two or so, compared with the time needed for a comparable point on a drying curve. An issue related to hysteresis is its possible interference with the properties measured. During runs with significant ψ gradients, the water content varies within the sample. When the centrifuge stops for sample weighing or other reasons, water within the sample will redistribute from wetter regions to drier regions, so that the different parts of the sample are on different hysteretic curves. In many cases, for example, at low θ, where the retention curve has little sensitivity of θ to ψ, this effect should be negligible. In other cases, it should be kept in mind that the measured results are not perfectly representative of either a drying or a wetting curve. Conclusions. The measurement techniques described in this section combine the advantages of simplicity and accuracy with the speed and versatility afforded by centrifugal force. In most techniques, the time required to measure an unsaturated K value increases with decreasing water content. The use of centrifugal force permits a decrease in measurement time and/or an increase in the range of K values to be determined. Like some gravity-driven methods (e.g., Section 3.6.1.1.a), it can operate in a way that yields nearly uniform water content distributions. Steady-state centrifuge methods are useful in situations where compaction is not a serious problem or where it is desired to measure hydraulic properties with overburden pressures that approximate those under field conditions. They allow the body force to be set at a desired value, both for establishing a conveniently measured water flow rate and for simulating a desired overburden pressure on the sample. Steady-state centrifuge methods can measure K(θ) of samples with various textures across a wide range of K values that are important in applications such as the determination of deep percolation or aquifer recharge rates. Targeting of a specific water content for a certain application is aided by the ability of the SSC method to generate measurements quickly, to adjust the K to be measured, and to know the approximate values of both θ and ψ while a series of measurements is in progress. 3.6.1.1.c Wind and Hot-Air Methods LALIT M. ARYA, Soil Consultant, Oceanside, California
Wind Method. Principles. This method, first introduced by Wind (1968), and also known as the evaporation method, is especially popular in European laboratories. An initially saturated soil core, closed at the bottom, is allowed to evaporate under constant environmental conditions, while water loss is monitored as a function of time by weighing. Simultaneous measurements of the matric head (hm) are made with tensiometers inserted at two (e.g., Becher, 1971; Schindler, 1980) or more depths (e.g., Halbertsma & Veerman, 1994) in the soil core. The sample is oven-dried at the end of the ex– periment and average transient water contents for the bulk sample θ during the ex-
THE SOIL SOLUTION PHASE
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periment are back-calculated by adding incremental evaporative losses between successive observations to the final water content. The information is utilized to simultaneously obtain the soil water characteristic (SWC) and hydraulic conductivity function, K(θ) or K(hm), for the same sample. Because the evaporative flux occurs across the top surface of the sample, and hydraulic head gradients (dH/dz) can be calculated only between successive tensiometer depths within the body of the soil core, it is necessary to obtain a distribution of water contents and fluxes, appropriate for each tensiometer depth, from the experimental data. This procedure involves iterative calculations, as described below. Several researchers have contributed to the development, modifications, and refinement of the original procedure (e.g., Becher, 1971,1975; Tamari et al., 1993; Wendroth et al., 1993). Researchers at the Winand Staring Centre, The Netherlands, have developed an automated setup, including a computer program, to facilitate measurements and the necessary calculations (Halbertsma & Veerman, 1994). Equipment, Software, and Supplies. Figure 3.6.1.1–4 shows the setup of a laboratory evaporation experiment as used at the Winand Staring Centre for Integrated Land, Soil, and Water Research (Halbertsma & Veerman, 1994). The setup shows all the equipment that one would need. Included are a weighing scale, soil core, and tensiometer assembly, including polyvinyl tubing and a pressure transducer system. The weighing scale should have a resolution of 0.1 g or better with an accuracy of ± 0.1 g or better. The core itself can be of metal or polyvinyl chloride (PVC). The dimensions used in the Netherlands are 110 mm i.d. by 80 mm height; however, other dimensions should work just as well. Tensiometer cups are generally 6 mm o.d. by 55 mm length. Hence, the core size should be large enough to leave room around each tensiometer cup. A mercury-manometer system (e.g., Cassel & Klute, 1986) may be used when pressure transducers are not available. The system may be multiplied or expanded depending upon need and budget. Specific software for analyzing the data is not described in this section. It is expected that users will de-
Fig. 3.6.1.1– 4. Experimental setup for Wind’s evaporation method (after Halbertsma & Veerman, 1994).
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velop their own calculation procedures. Those interested in a software package may contact: Agricultural Research Department, Winand Staring Centre for Integrated Land, Soil, and Water Research (SC-DLO), P.O. Box 125, NL-6700 AC Wageningen, The Netherlands. Procedure for the Setup Developed at the Winand Staring Centre. 1. Soil cores should have predrilled holes at depths where tensiometer cups are to be inserted. Recommended depths are 10, 30, 50, and 70 mm from the bottom of the sample. 2. Obtain undisturbed core samples from the desired depths in the field, using a hydraulic jack or other means. Trim both ends of the core with a sharp knife. 3. Saturate the cores in the laboratory on a ceramic plate or blotting paper. Provide a few centimeters of standing water around the core. Saturation may take from a few days to 2 wk depending upon the texture and bulk density of the sample. After saturation, blot out the imbibed free water from the bottom and provide suitably designed lids at the top and bottom of the sample to prevent evaporation. 4. A tensiometer–pressure transducer assembly should be readied prior to starting the experiment. Ceramic-cup tensiometers, filled with de-aerated water and connected to pressure transducers via watertight and airtight polyvinyl tubing should be in place and pretested. Alternatively, a pressure transducer may be used in combination with stopcocks for switching from one tensiometer to another. 5. Insert the tensiometer cups into each of the four holes. It is necessary to make a passage into the soil prior to inserting the cups. The junction at the ceramic cup and the polyvinyl tubing should be provided with a rubber stopper of a size that completely seals the hole. This should be accomplished while connecting the ceramic cups to the tubing. 6. Place the sample on the weighing scale and allow it to equilibrate while keeping the top and bottom closed. Recommended time for equilibration is approximately 12 h. 7. Record tensiometer readings at all depths, as well as the initial mass of the sample. Open the top lid and begin evaporation. Continue taking mass and tensiometer readings at regular time intervals. Initial readings should be taken frequently. A time interval of 10 to 15 min is suggested for the first hour and 20 to 30 min for the second hour. Thereafter, readings may be taken at progressively increasing time intervals. The objective is to obtain detailed hm vs. t curves. 8. At the end of the experiment, oven-dry the sample and determine the water content and bulk density values. The experiment should be terminated when tensiometer data can no longer be relied upon, usually around hm ≈ −800 cm of water pressure. However, a reading of −650 cm may be a practical limit for commonly available tensiometers. – Calculations of hm(θ): Calculate the average water content values, θ of the bulk sample for each time the mass of the sample was measured and determine the
THE SOIL SOLUTION PHASE
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corresponding hm values from the tensiometers readings (one reading for each – depth). The values for θ are determined by adding successive evaporative losses to the final water content measured at the end of the experiment. For the correspon– ding times, calculate the corresponding depth-averaged matric head values, h m. Con– – struct an initial average soil water characteristic curve (SWC) by pairing h m and θ – – values for the core. This h m(θ) curve can be described by fitting a polynomial through the data points. Halbertsma and Veerman (1994) used a fifth-order polynomial for an experimental setup with four tensiometer depths. They assumed that the soil sample was homogeneous and that each tensiometer was at the midpoint of a compartment. In the second step, insert the measured values of hm for each of the four tensiometers into the polynomial, and obtain estimates of the corresponding water content values for each compartment, θ$–. From the θ$ values, calculate the estimated $ mean water content – values of the core, θ, for each time a set of measurements was taken. Compare θ$ with the corresponding measured average water content values, – – –$ θ, and compute the ratio (θ /θ ). Correct the θ$ values for a given time by multiplying – –$ with (θ/θ) value corresponding to the same time. Applying these steps to all observations would result in a new set of hm and updated θ$ values. Fit a new curve through the updated data points, using a polynomial that accurately describes the hm(θ$) relationship for the sample. Using the new polynomial, repeat the entire process to obtain the next updated hm(θ$) curve. A least squares method (e.g., Marquardt, 1963) may be used to obtain the best fit. Usually three to five iterations are needed to arrive at the final solution (Halbertsma & Veerman, 1994). Calculations of K(θ): The calculated hm(θ$) curve for the sample can be used to specify θ$ corresponding to each measured hm at each tensiometer position, and K(θ) can be calculated from z
m K(θ$)zm = [∫z0 (∂θ$/∂t)dz]/(∂H/∂z)zm
[3.6.1.1–12]
where t is the time, H is the hydraulic head (given by H = h + z), and z is the depth taken positive upward. Depth z0 represents the bottom boundary of the core (where the flux is zero), and depth zm is a depth midway between successive tensiometers. It should be noted that the hydraulic head gradient (∂H/∂z) can be calculated only at points between the tensiometers. Therefore, no K(θ$) values can be calculated for depths below the bottom tensiometer and above the top tensiometer. Comments. As indicated above, the evaporation method is popular in European laboratories. The method appears to compare well with other methods (e.g., Stolte et al., 1994). The hm(θ) and K(θ) functions are both obtained on the same sample. The equipment and setup are simple. However, the requirement that samples must be of homogeneous hydraulic properties cannot be assured when working with natural, undisturbed samples. In addition, the assumption that any measured hm and θ pairs apply uniformly to a tensiometer compartment may not be strictly true when a compartment’s thickness is large, and especially when steep gradients develop during the evaporation process. The main limitation of the method is that it provides data only in the range of tensiometric measurements, and only the desorption characteristic can be obtained. Large errors generally occur at or near saturation because of uncertainties in estimating small hydraulic gradients (Wen-
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droth et al., 1993). According to Wendroth et al. (1993), precision in defining hydraulic conductivities near saturation may be improved by increasing the initial evaporation rate using a fan. Mohrath et al. (1997) and Bertuzzi et al. (1999) list a number of error sources, which include errors due to: (i) uncertainties in the positions of the tensiometers in the sample, (ii) effects of temperature, (iii) calibration of the pressure-transducer system, and (iv) layering in the soil column. In addition, the method may overestimate the hydraulic conductivity of fine-textured soils (e.g., Schindler, 1992). The effects of these factors appear to be less on hm(θ) but more pronounced on K(θ) (Bertuzzi et al., 1999). Data obtained with the evaporation method may also be analyzed using parameter inversion methods (e.g., Simunek et al., 1998; Wendroth & Simunek, 1999). An advantage of parameter inversion is that data can be augmented with any independently measured water content (e.g., θ15bar or θs), giving a functional description for the entire SWC and K(hm). Also, fewer tensiometers may be needed, thus simplifying the experiment. Hot-Air Method. Principles. The hot-air method is different from the evaporation method in that drying of the soil occurs under forced hot air. Developed and tested by Arya et al. (1975), the method permits determination of the soil water diffusivity as a function of water content, D(θ). The diffusivity function can be converted to K(θ) if the soil water characteristic (SWC) for the sample is known. An initially saturated or near-saturated soil core, sealed at the bottom, is exposed to a stream of hot air, and is dried under the following initial and boundary conditions θ = θi θ = θo
x>0 x=0
t=0 t>0
where θi is the initial water content, θo the air-dry water content (i.e., surface water content at the end of the drying process), x the distance, and t the time. The analysis requires that the water content be a function of a Boltzmann type variable dependent on distance and the square root of time (e.g., Bruce & Klute, 1956; Gardner, 1959). Two conditions are necessary for the analysis to be valid: 1. The cumulative evaporation should be proportional to t1/2. 2. The water content at the bottom end of the core should remain unchanged at the end of the drying process; that is, the column must behave as though it were semi-infinite. The water content distribution in the soil core obtained under these restrictions can be analyzed to obtain D(θ), using the following equation θ
D(θ) = (1/2t)(dx/dθ)x∫θxi xdθ
[3.6.1.1–13]
Diffusivities can be converted to K(θ) using the relationship K(θ) = D(θ)(dθ/dh)
[3.6.1.1–14]
THE SOIL SOLUTION PHASE
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Available field and laboratory methods for determining K(θ) or D(θ) are generally limited to relatively high saturations, except perhaps centrifuge methods (Section 3.6.1.1.b), which can provide data at low water contents. The hot-air method, on the other hand, covers a wide range of saturations. It extends the K(θ) and D(θ) functions well into the dry range. Arya et al. (1975) applied the method to determine K(θ) functions for a loam soil and compared the results with field-measured data obtained using the instantaneous profile (Nielsen et al., 1964) and zero-flux (Arya et al., 1975) methods. The field data were limited to the wet range, but connected well with data obtained with the hot-air method. The method has attracted interest, especially in Europe (Ehlers, 1976; Bouma, 1977, 1982; van Grinsven et al., 1985; Wosten et al., 1986; van den Berg & Louters, 1986, 1988; Teiwes, 1988). Several investigators evaluated the method by comparing results with other methods (van Grinsven et al., 1985; Gieske & de Vries, 1990; Stolte et al., 1994). Results of these comparisons are mixed, but the main concern has been regarding the effects of temperature gradients on the determination of D(θ). Equipment, Software, and Supplies. 1. 2. 3. 4.
Weighing scale with a readability of 0.05 g, and accuracy of ± 0.05 g Electric hair dryer with variable speed Laboratory stand Metal soil cores of high thermal conductivity (e.g., brass, stainless steel, copper). The recommended dimensions are 36.5 mm i.d. by 76.5 mm length with 1-mm wall thickness. The metal sleeves should be machine cut and sharpened on one end. Two smaller pieces (~2 cm in length) should be cut from the same metal pipe and closed at one end by welding a round metal plate of the same diameter as the outer diameter of the core. These pieces will be placed on both ends of the soil core and taped with an air- and watertight tape to prevent evaporation during equilibration. A schematic of the core assembly is shown in Fig. 3.6.1.1–5. 5. A solid metal piston that fits snugly inside the core and slides with ease (Fig. 3.6.1.1–5) 6. Drying oven 7. Moisture cans and a wide-blade sharp knife Procedure. The procedure described below is only approximate. As will be clear from the comments and discussion, variations are possible, and users are advised to use trial and error to establish satisfactory experimental conditions for different soil materials and the intended range of water contents. 1. Obtain undisturbed soil cores by gently pushing the metal core sleeves vertically downward into the soil layer to be sampled. A metal ring of 1 to 2 cm long should be added to the top of the core sleeve to ensure that soil within the main core does not include a disturbed surface. Excavate the core from the soil and trim off the additional length from both ends. Obtain 8 to 10 cores from each depth. 2. Saturate the cores by placing them on a saturated ceramic plate or blotting paper. To start with, add 2 to 3 cm of free water around the cores to speed up saturation. The water level should be raised in increments every
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Fig. 3.6.1.1–5. Components of soil core assembly and suggested dimensions for the hot-air method of measuring soil water diffusivity.
3 to 4 h to bring the final level to within 2 to 3 cm from the top of the core. Cores of medium-textured soils should saturate in 3 to 4 d. Longer times may be necessary in some cases. The top of a near-saturated core should show a glistening appearance. However, 100% saturation may not be achieved due to entrapped air. 3. Remove the free water from around the core and let the excess water drain from the core for 24 h while preventing evaporation from the surface. A cover should be provided on top of the core while the drainage from the bottom end continues. Remove the cores from the saturation box and gently dab off excess free water from the bottom end using an absorbent tissue paper. Seal both ends of the core by placing closed ends of the sealing caps, and taping the seam with an air- and watertight tape (Fig. 3.6.1.1–5). 4. Place the cores on a flat surface in horizontal position, and allow them to equilibrate for 3 to 5 d. Turn the cores several times during this period to prevent buildup of moisture gradient across the core cross section.
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Fig. 3.6.1.1–6. Example of data for the hot-air method of measuring soil water diffusivity: (a) water content distribution at the end of the drying process and (b) the relationship of cumulative evaporation to the square root of time (data from Arya et al., 1975).
5. Select three cores randomly and determine their bulk densities and water contents. The average volumetric water content, θi, of the equilibrated cores is needed in Eq. [3.6.1.1–13] for calculating D(θ). 6. Evaporation can now be allowed to proceed from the remaining cores. A few trial runs are advised to establish drying conditions leading to a linear relationship between cumulative evaporation and t1/2. While this condition is nearly impossible to achieve at or near zero time, it should be achieved within the first 2 to 4 min of drying. An example is presented in Fig. 3.6.1.1–6. 7. Fasten an electric hair dryer on a laboratory stand with the air outlet pointing vertically downward. Select a core and place it in a vertical position under the hair dryer, with the top of the core facing upward. The distance between the top of the core and the air outlet of the dryer should be about 8 cm. Turn on the hair dryer at the cool air setting, and make adjustments to ensure that the air stream is received exactly on top of the core. Mark the position of the core so that it can be placed back exactly on the same spot after each removal for weighing. Remove the core and change the dryer setting to hot. Open the top seal of the core and measure the initial mass. Place the core under the hot air stream with the exposed soil surface pointing upward. Weigh the sample again after 1 min and quickly return it to its position under the hair dryer. Continue weighing and dry-
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ing at frequent intervals. Keep plotting the data to see if a linear relationship between cumulative evaporation and t1/2 exists. Adjustments of drying conditions may be necessary if there is significant deviation from linearity. A hotter setting on the dryer, a higher fan speed, and/or a closer distance between the soil surface and the air outlet of the dryer may provide an acceptable evaporative environment. Alternatively, samples may need to be prepared at a lower initial water content. 8. Continue drying and weighing for about 15 to 20 min, but no more than 20 min. Obtain the water content distribution with depth at the end of the drying process. Sampling should be as rapid as possible, since the water content distribution is assumed to be the same as when drying ceased. Open the bottom seal and push the sample upward in increments of about 2 to 3 mm. A suitably designed piston that fits the inner dimension of the core can be used for this purpose (Fig. 3.6.1.1–5). Slice the soil layers and place them in preweighed moisture cans. The sample thickness should be increased gradually towards the bottom end of the core. The sampling process should not exceed 4 to 6 min. Determine the gravimetric water content of the samples by drying them in the oven. Use the total ovendry mass of the sample and the inner volume of the core to calculate the sample bulk density. Convert gravimetric water contents to volumetric values by multiplying them with the sample bulk density and dividing by the density of water. A plot of water content data with depth should produce a distribution similar to the one illustrated in Fig. 3.6.1.1–6. Ideally, the water content at the bottom of the sample should be the same as θi obtained in Step 5. A lower value will result if the sample has been dried too long. However, because of errors and uncertainties (e.g., within-sample heterogeneity, between-sample variability, and temperature effects), accept the data if the bottom water content does not deviate from θi by more than 5%. 9. Samples can be routinely subjected to evaporation and the resulting data analyzed once Steps 6, 7, and 8 have been worked out satisfactorily. 10. Calculate D(θ) using Eq. [3.6.1.1–13]. Convert D(θ) to K(θ) using Eq. [3.6.1.1–14]. Please note that independent soil water characteristic (SWC) data for the sample must be available to calculate K(θ) Comments. The main advantages of the hot-air method are that it is rapid, requires no specialized equipment, and provides D(θ) or K(θ) data over a wide range of water content values, especially the dry range. By comparison, K(θ) data from other laboratory methods are generally limited to the higher water content range. The method, however, is only approximate because experimental conditions are not strictly consistent with theoretical assumptions. First, isothermal conditions do not exit. Second, the water content distribution measured at the end of the experiment is assumed to be the same as the distribution at the end of the drying process, whereas in reality some redistribution of water will occur during sampling for the gravimetric water content determinations. Third, pushing the sample from the wet end of the core may cause some compaction, and possibly some squeezing of water. Pushing the sample alternately from the wet end and then the dry end may offer some ad-
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vantage. However, this may still prove to be difficult if the soil is very sticky. In obtaining natural-structure field samples, one would generally use large diameter cores to ensure representativeness of the samples. However, pushing a large diameter core with a piston is more difficult unless a mechanical device is used. The core diameter recommended in the procedure described above may be too small, but convenient for postevaporation sampling for water content. Analysis on multiple samples from the same depth may partly overcome the loss of sample representativeness resulting from the small diameter core. The most serious concern, however, appears to be due to temperature effects on water flow. In some experiments, soil temperatures in excess of 90°C have been recorded, while air temperatures of up to 240°C have been required for sandy soils (van Grinsven et al., 1985; Dirksen, 1991). Because the samples are dried using hot air, a rise in soil temperature is expected; this increase will violate the assumption of isothermal conditions. The above 240°C air and 90°C soil temperatures, however, appear extreme. When metal cores of high thermal conductivity are used, excessive temperature gradients do not occur. It is reasonable to assume that heat dissipation occurring primarily by convection and radiation and secondarily by evaporation should keep the mean soil temperature well below the temperature of the forced-air stream, and prevent the development of excessive temperature gradients. On the other hand, soil cores encased in materials of low thermal conductivity (e.g., PVC pipes) are likely to heat and lead to large temperature gradients. The effect may be magnified if the air does not flow freely over the body of the core. Likewise, if the soil core diameter is large, conduction of heat from the soil to the metal surface, where it is dissipated, can be relatively slow. Hence, smaller diameter cores are likely to develop smaller temperature gradients. Arya et al. (1975) monitored temperature changes in a 3.65 cm i.d. by 7.65 cm long core of a sandy loam soil encased in a brass sleeve and exposed to an air stream at 90 to 100°C. Temperatures rose from the initial 20°C to about 37°C within 20 min of drying, but no more. An example of temperature profiles is shown in Fig. 3.6.1.1–7. Except in the immediate vicinity of the evaporating surface, temperature gradients remained 0, z = 0
[3.6.2–6]
where hm,i is the initial matric head, q denotes the flux density (L T−1), z = 0 is the bottom of the porous membrane or plate, z = L is the top of the soil core, h(0,t) is the water pressure head at the bottom of the porous membrane, and ha is either the pneumatic gas pressure applied to the top of the soil core (z = L) or the suction applied beneath the porous membrane (z = 0). For example, if h(0,t) = 1 cm (height of water above the bottom of the porous membrane), and ha = 80 cm (applied to the top of the soil core), then hm = −79 cm. Similarly, for the same water level in the burette, if an 80-cm suction head is applied to the water in the burette, hm = −79 cm. The objective function, Eq. [3.6.2–5], includes the matric head measurements inside the soil core (j = 1) and cumulative drainage volume (j = 2) vs. time. It is rec-
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ommended to select a relatively small time interval between measurements (input to data logger), and to smooth and eliminate data later, rather than selecting the number of required measurements (nj) a priori. Additional comments that provide guidance towards successful application of the multistep outflow method are: 1. Although the limitations with respect to the experiment or flow modeling are few, the inverse approach relies on the availability of a universally applicable nonlinear optimization algorithm. Problems with the parameter optimization technique generally are associated with the difficulty of defining an objective function that will yield unique and convergent solutions. Since ill-posedness of the inverse problem can be caused by correlation between the parameters to be optimized, uniqueness of the optimized parameters is generally increased by reducing the number of free parameters. For example, if parameter values can be measured independently, their values should be fixed or be included in the objective function as Bayesian estimates (Section 1.7). To further reduce the number of parameters to be optimized, we often make use of the relationship between m and n (m = 1 − 1/n) and couple the retention curve model with the conductivity curve model (van Genuchten, 1980). Furthermore, the tortuosity parameter of the conductivity function, l, is frequently set to a fixed value (l = 0.5 according to Mualem, 1976; l = −1 according to Schaap & Leij, 2000). 2. Because the Richards equation includes the soil water retention function, θ(hm), only by way of its derivative, ∂θ/∂t = C(hm)∂hm/∂t, where C(hm) = dθ/dhm, the residual and saturated water contents (θr and θs) are perfectly correlated. Consequently, only one of these two parameters may be optimized. In practice, θs, is independently measured from oven-drying at the conclusion of the outflow experiment. The parameters θr and θs can be estimated simultaneously only if some additional water content-related information is included in the optimization (e.g., the initial condition is given in terms of water content; the initial or final water volume in the sample is included into the objective function). 3. It must be stressed that uniqueness of the coupled retention and conductivity solution will depend on the selection of the hydraulic models and their ability to accurately represent the true soil hydraulic properties. For example, Zurmühl and Durner (1998) and Durner et al. (1999a) demonstrated that optimization with eight or more fitting parameters was stable and unique if bimodal hydraulic models were used for soils with a bimodal pore-size distributions. Furthermore, a nonsuitable retention function can lead to a meaningless estimate of the conductivity parameter, l (Durner et al., 1999b). 4. Extrapolation beyond the range of measurement is associated with a high level of uncertainty. To increase the range of validity of the optimized hydraulic functions, it is recommended to include independently measured soil water retention and/or conductivity data in the objective function. For example, with the experimental range limited by the largest applied pressure, soil water retention points at lower matric head values, as measured with the pressure plate extractor (Section 3.3.2.4) or the evaporation method (Section 3.6.2.4), may be included to augment the water content range of the optimized hydraulic functions. 5. Outflow methods provide few dynamic data at near saturation. This implies that the sensitivity of the method to the shape of the conductivity function near
THE SOIL SOLUTION PHASE
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saturation, and in particular the sensitivity for estimating the saturated conductivity parameter, Ks, is low. Nevertheless, it is frequently recommended not to fix Ks at the value of an independently measured saturated conductivity (Section 3.4) but to treat it as an empirical fitting parameter because this often improves the description of the overall conductivity function, particularly for undisturbed soils. However, disagreement between fitted and measured Ks may indicate that the shape of the assumed parametric form of the conductivity function near saturation might not be correct (Durner, 1994). 6. As was demonstrated by Eching and Hopmans (1993a), the multistep outflow experiment can be extended to optimize soil hydraulic parameters for soil wetting by reversing the time sequence of the pressure increments starting with an initially dry soil. Hence, combined analysis of a drying experiment followed by a wetting experiment allows simultaneous optimization of hysteresis (Schultze et al., 1996; Zurmühl, 1998). However, as reported by Durner et al. (1999b), poor agreements between optimization results and independent measurements might occur if an incorrect hysteresis model is used. 7. Recently, the choice of appropriate weighting factors in the objective function, Eq. [3.6.2–5], has been put into question, especially with regard to the calculations of the uncertainty of the optimized parameters. A poor choice of the measurement error can result in either too narrow or extremely large confidence intervals (Hollenbeck & Jensen, 1998a), thereby leading to incorrect interpretations of the optimization results. Using a parameter sensitivity analysis, Vrugt et al. (2001b) showed that uniqueness of a multistep outflow experiment without matric head measurements can be significantly improved by including in the objective function only the outflow measurements immediately following the applied pressure increments, when flow rates are highest, in addition to the total outflow at the end of each pressure step. 8. The ability of estimating soil hydraulic functions using inverse procedures has raised the question whether hydraulic properties might be influenced by the flow rate or boundary conditions. The effect of the flow dynamics can be caused by water entrapment or by discontinuity of water-filled pores for large matric head gradients (see Section 3.3.1). Research is ongoing (Wildenschild et al., 2001; Mortensen et al., 1998) to investigate these dynamic effects. Additional complications were reported by Hollenbeck and Jensen (1998b), who addressed the difficulty of experimental reproducibility, and Schultze et al. (1999), who cautioned that outflow experiments might include air-phase effects that should be incorporated by two-phase flow modeling instead of describing water flow by the traditional Richards equation (Eq. [3.6.2–1]) only. 9. Some have expressed concerns about the step-wise changes of the boundary condition, which may cause a flow behavior that does not occur in nature (van Dam et al., 1994). As an alternative to the multistep method, a continuous outflow method was tested using a gradual change of the pressure boundary condition. Durner et al. (1999b) discussed the comparison of the multistep outflow method with this continuous method. They concluded that the two methodologies are equally suitable for identifying the water retention parameters. However, the small flux rates reduced the sensitivity of the hydraulic conductivity, thereby making the continuous outflow method less suitable.
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10. Finally, measured drainage rate and matric head data can also be used to estimate the unsaturated hydraulic conductivity data directly. The concepts of the direct hydraulic conductivity estimation were discussed by Eching et al. (1994) and Liu et al. (1998). The procedure requires estimation of the matric head at the soil–membrane interface. Since outflow rates are relatively high at the beginning of each pressure step, it is preferred to use data from the time periods immediately following an increase in applied air pressure. If so required, the matric head at the soil–membrane interface is estimated from the saturated hydraulic conductivity and thickness of the saturated porous membrane in combination with known measured values of the water pressure at the bottom of the membrane and the measured drainage rate. The effective permeability of the soil can be estimated subsequently from the Darcy equation solving for K(Se), after substituting the average drainage rate and the assumed hm gradient in the soil core using the measured matric head values in the center of the core and at the soil–membrane interface (Wildenschild et al., 2001). 3.6.2.4 Evaporation Method 3.6.2.4.a Introduction The parameter estimation technique has also been successfully applied to the laboratory evaporation technique (Section 3.6.1.1.c). The evaporation method was first introduced by Gardner and Miklich (1962), who imposed a series of constant evaporation rates to one side of a soil sample after first allowing the sample to attain hydraulic equilibrium before the next evaporation rate was applied. They measured the matric head response of two tensiometers. Becher (1971) simplified the evaporation method by using a constant evaporation rate. Several other modifications of the evaporation method, with simultaneous measurements of evaporation rate and matric head values at different heights in the sample, have since been developed (Wind, 1968; Boels et al., 1978; Schindler, 1980; Tamari et al., 1993; Wendroth et al., 1993; Halbertsma & Veerman, 1994). Experimental data obtained with the evaporation method can be analyzed using either a simple approach introduced by Schindler (1980), the classical Wind analysis or its modifications (e.g., Wind, 1968; Wendroth et al., 1993; Halbertsma & Veerman, 1994), or increasingly more often from numerical inversion (Ciollaro & Romano, 1995; Santini et al., 1995; Šimçnek et al., 1998c,1999c). In their review, Feddes et al. (1988) obtained reasonable agreement between hydraulic conductivities determined from parameter estimation by the inverse procedure and using Wind’s method (Wind, 1968). As part of a study of the spatial variability of soil hydraulic properties, Ciollaro and Romano (1995) successfully used the inverse procedure to estimate parameter values, based on evaporation experiments, to determine the soil hydraulic parameters of a large number of samples. Santini et al. (1995) also used parameter estimation by the inverse procedure in connection with evaporation experiments. Their results compared favorably with independently measured retention and saturated hydraulic conductivity data. Šimçnek et al. (1998c) obtained excellent correspondence between retention curves and hydraulic conductivity functions obtained with parameter estimation by the inverse technique and the modified Wind’s method (Wendroth et al., 1993). They also
THE SOIL SOLUTION PHASE
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showed that, contrary to Wind’s method, which requires matric head measurements at several locations, comparable results could be obtained with tensiometer readings at just one location. Romano and Santini (1999) showed that the inversion results compared satisfactorily with soil hydraulic data as measured independently with the instantaneous profile method (Section 3.6.1.2.a). However, they concluded that in some cases unsaturated hydraulic conductivity functions with parameters independent of the soil water retention functions are needed for accurate hydraulic characterization.
3.6.2.4.b Experimental Procedures Detailed experimental procedures are outlined in Šimçnek et al. (1998c) and Romano and Santini (1999) and will only be summarized here. Initially, saturated 10-cm-high soil cores are placed on a ceramic plate to establish a static initial matric head distribution. Two to five miniature tensiometers are placed horizontally at different vertical locations in the soil core. After tensiometer equilibration, as evidenced from the tensiometer readings, the soil core is placed on an impermeable plate for the evaporation experiment. Total soil water storage changes are determined from weight measurements of the soil core. A strain-gauge load cell placed under the plate bearing the soil sample is used for soil sample weight measurements, while the matric head is monitored at the various soil depths connecting the tensiometers to pressure transducers. Calibrating each transducer across the working pressure range should be performed before and after the test. An evaporation rate can be induced using either natural laboratory conditions, or can be accelerated using a fan to blow air across the sample surface. A two-rate experiment can also be used in order to induce initially sufficiently large matric head gradients (Wendroth et al., 1993). Once the hm gradient is about 1.5 to 2.5 m m−1, evaporation is allowed to continue without the fan. The experiment is terminated after the matric head values become too low for reliable functioning of the tensiometers. Additional comments regarding the experimental setup are: 1. Extrapolation beyond the measurement range is associated with a high level of uncertainty. Inclusion of independently measured information beyond the measurement range, that is, additional soil water retention data or a residual water content value, could greatly decrease this uncertainty. 2. Using the inverse modeling approach, it has been shown that matric head readings from a single tensiometer in combination with a final soil water storage measurement may be adequate to guarantee precise estimation of the soil hydraulic characteristics within the range of measurements (Šimçnek et al., 1999c). Maximum sensitivities are generally attained by placing this single tensiometer near the evaporating soil surface where the largest soil water content changes occur. However, tensiometer placement very close to the soil surface is discouraged, as the soil surface remains dry for most of the latter part of the evaporation experiment, so that its information is limited. Moreover, to guarantee sufficient time measurements of changes in matric head, especially when hydraulic head gradients become large or as backup information if a tensiometer fails, at least two tensiometer locations are recommended.
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3.6.2.4.c Simulation and Optimization The governing flow equation for one-dimensional isothermal Darcian flow is given by Eq. [3.6.2–1]. Boundary and initial conditions applicable to an evaporation experiment are as follows: hm(z,t) = hm,i(z)
t = 0, 0 < z ≤ L
q(z,t) = qevap(t)
t > 0, z = L
q(z,t) = 0
t > 0, z = 0
[3.6.2–7]
where qevap(t) is the time-variable evaporation rate (L T−1) imposed at the soil surface, and all other variables are as defined in Eq. [3.6.2–6]. The objective function to be minimized includes all matric head measurements and a soil water storage value. This single water storage value provides a water content reference from which the saturated water content can be estimated using the evaporation rate data. The inclusion of more values of water storage in the objective function does not improve the optimization process, since the evaporation rate (upper boundary condition) is always enforced. A sensitivity analysis can be used to determine the optimum tensiometer location for the parameter estimation procedure. Additional comments include: 1. Parameter sensitivity is not affected by soil core height. However, duration of the evaporation experiment will be less for shorter soil cores. Šimçnek et al. (1999c) analyzed the sensitivity of the parameters of the van Genuchten relationship for a hypothetical two-rate evaporation experiment with tensiometer locations at depths of 1, 3, 5, 7, and 9 cm within a 10-cm-high sample. Their results showed that (i) the sensitivity increased as the experiment progressed in time and the soil core became drier, (ii) the sensitivity was highest for tensiometer 1, closest to the evaporating soil surface, and (iii) the matric head was most sensitive to n and θs, whereas the sensitivities of the parameters α, θr, and Ks were, by comparison, much smaller. 2. While a two-rate evaporation experiment has important advantages over a one-rate experiment when using the modified Wind method, the two-stage approach did not show an advantage in the parameter estimation approach by the inverse method, except that it will speed up the experiment. In addition, it was concluded by Romano and Santini (1999), who used a parameter uncertainty analysis of hypothetical numerical experiments for coarse-textured soils, that smaller parameter confidence intervals were obtained when evaporative fluxes increased. 3. When numerically solving the evaporation process, certain combinations of soil hydraulic parameters, selected by the parameter optimization procedure, may cause physically impossible water delivery rates towards the soil surface to satisfy the required evaporation flux. This may especially occur at the later stages of the evaporation experiment, when water contents near the soil surface are low. To eliminate convergence problems in such situations, the evaporation rate boundary condition must be replaced by a minimum allowable matric head value (about −100 to
THE SOIL SOLUTION PHASE
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−150 m) at the expense of disagreement between simulated and measured evaporation fluxes. Since this will usually result in larger residuals, selected hydraulic parameter values must be adjusted in the subsequent iterations so that imposed evaporation fluxes can be enforced. 4. In principle, the evaporation and the multistep outflow methods represent similar flow processes. In practice, however, there are some important differences. First, the multistep outflow method imposes abrupt step-wise changes of the pressure boundary condition, thereby inducing high flow rates, particularly under wet conditions. In contrast, the evaporation method imposes a smooth change of the boundary condition, which is more typical of a natural drying process. Second, as the soil drains, the outflow rates in the multistep method decrease due to the hydraulic head gradient approaching zero, thereby reducing the sensitivity of the hydraulic parameters. In the evaporation method, however, the hydraulic head gradients increase as the soil dries, thereby gaining parameter sensitivity within the operable range of tensiometric measurements (usually hm > −800 cm). Consequently, the evaporation method is preferable for hydraulic characterization in the intermediate water content range, whereas the multistep outflow method is recommended for estimation of soil hydraulic functions in the wet range. As stated above, the range of validity can be extended for both methods by including independently measured data in the objective function. 3.6.2.5 Tension Disc Infiltrometer 3.6.2.5.a Introduction Tension disc infiltrometers have recently become very popular devices for in situ measurements of the near-saturated soil hydraulic properties (Perroux & White, 1988; Ankeny et al., 1991; Reynolds & Elrick, 1991; Logsdon et al., 1993). Tension infiltration data have been used primarily for evaluating saturated and unsaturated hydraulic conductivities, and to quantify the effects of macropores and preferential flow paths on infiltration. Tension infiltration data are generally used to evaluate the saturated hydraulic conductivity, Ks, and the sorptivity parameter in Gardner’s (1958) exponential model of the unsaturated hydraulic conductivity using Wooding’s (1968) analytical solution (see Section 3.5.4). Adequate parameter estimation requires either infiltration measurements using two different disc diameters (Smettem & Clothier, 1989), or infiltration measurements using a single disc diameter but multiple tensions (e.g., Ankeny et al., 1991). Although early-time infiltration data from the tension disc infiltrometer can be used to estimate the sorptivity (White & Sully, 1987) and the matrix flux potential, only steady-state infiltration rates are usually used for Wooding-type analyses. Šimçnek and van Genuchten (1996, 1997) suggested using the entire cumulative infiltration curve in combination with parameter estimation to estimate additional soil hydraulic parameters. From an analysis of numerically generated data for one supply tension experiment they concluded that the cumulative infiltration curve by itself does not contain enough information to provide a unique inverse solution. Hence, additional information about the flow process, such as the water content and/or matric head measured at one or more locations in the soil profile is needed to successfully obtain unique inverse solutions for the soil hydraulic functions.
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3.6.2.5.b Experimental Procedures Details of the experimental procedure are outlined in Section 3.5.4. Usually, the soil surface is covered with a thin layer of sand to ensure hydraulic contact between the disc and the underlying soil. The sand should have a sufficiently large air-entry value to remain saturated at all applied tensions. It should also have a Ks value that is large enough to prevent extensive flow impedance effects (Reynolds & Zebchuk, 1996). The disc is connected with a tension-controlled water supply system, and transient infiltration rates can be accurately determined from pressure transducers connected to a data logging system (Ankeny et al., 1988). Water is supplied initially at the greatest suction and is decreased consecutively to a lower suction each time steady state has been attained. If desired, time domain reflectometry (TDR) or tensiometers can be installed below the supply disc to monitor water content and/or matric head changes (Wang et al, 1998; Šimçnek et al., 1999d). Alternatively, soil samples can be taken before and after an experiment (near and below the supply disc) and be used for initial and final volumetric water content determinations. Šimçnek and van Genuchten (1997) studied infiltration at several consecutive supply tensions. They considered several scenarios with different levels of information and concluded that the most practical experiment for inverse estimation of the hydraulic parameters is the cumulative infiltration curve measured at several consecutive tensions, augmented with the initial and final water content values directly below the disc. 3.6.2.5.c Simulation and Optimization Simulation of flow from the tension disc infiltrometer requires use of the radially symmetric two-dimensional flow equation (Eq. [3.6.2–2]) The following initial and boundary conditions apply θ(r,z,t) = θi
t = 0, 0 < r < ∞, 0 < z < ∞
hm(r,z,t) = hm,o(t)
t > 0, 0 < r < ro, z = 0
q(r,z,t) = 0
t > 0, r > ro, z = 0
hm(r,z,t) = hm,i
t > 0, r → ∞, z → ∞
[3.6.2–8]
where θi and hm,i denote constant initial water content and matric head (at zero time and far away from the infiltration source), respectively, ro is the infiltration disc radius (L), and hm,o is the time-variable water supply pressure at the infiltrometer membrane (L). Flow symmetry is assumed around the r = 0 axis, which is therefore also a no-radial-flux boundary. The objective function defined in Eq. [3.6.2–5] includes cumulative infiltration data for the successively supplied water pressures, and may include additional information such as matric head and/or water content data below the supply disc at various (r,z) positions as a function of time. In addition, the objective function could include a Bayesian estimate (Section 1.7) of the unsaturated hydraulic con-
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ductivity computed from Wooding’s (1968) analysis. We provide the following additional comments: 1. The inverse method was tested by Simçnek et al. (1998a) using data collected as part of the soil hydrology program of the HAPEX-Sahel regional-scale experiment (Cuenca et al., 1997). A tension disc diameter of 25 cm with supply tensions of 11.5, 9, 6, 3, 1, and 0.1 cm was used. Agreement between the measured and optimized cumulative infiltration curves was very good. The inverse method and Wooding’s (1968) analysis gave almost identical unsaturated hydraulic conductivities for matric head values in the interval between −2 and −10.25 cm. However, the hydraulic conductivity in the highest-matric head range (between −1 and −0.1cm) was overestimated by a factor of two using Wooding’s analysis. 2. The Šimçnek et al. (1998a) results also suggest that tension disc experiments may provide adequate information to estimate both the unsaturated hydraulic conductivity and the soil water retention properties, and that there is no need for additional tensiometer and TDR measurements to better define the inverse problem. Parameters of the soil water retention curve can be closely coupled with those of the unsaturated hydraulic conductivity function, as is the case with the van Genuchten–Mualem model. The fitting of the cumulative infiltration curves was improved by allowing the pore-connectivity parameter to be optimized as well, rather than assuming a constant value of l = 0.5. 3. The public domain program DISC (Šimçnek & van Genuchten, 2000) for analyzing tension disc infiltrometer data to estimate parameter values by the inverse procedure is available from the U.S. Salinity Laboratory, Riverside, CA (http://www.ussl.ars.usda.gov) and from Soil Measurement Systems, Tucson, AZ. 3.6.2.6 Field Drainage 3.6.2.6.a Introduction Certainly the most popular experiment to determine the soil water retention and hydraulic conductivity functions in the field (in situ) has been the instantaneous profile method (Hillel et al., 1972; Vachaud et al., 1978; Section 3.6.1.2.a). The first application of a parameter optimization technique to instantaneous profile field data was carried out by Dane and Hruska (1983). These authors questioned the uniqueness of the solution and concluded that the sensitivity of the optimized parameters depended on the prescribed boundary conditions. Kool et al. (1987) performed a successful inversion using lysimeter data and a slightly different approach than Dane and Hruska (1983). Kool and Parker (1988) investigated the numerical inversion of hypothetical in situ infiltration and redistribution flow events and showed the advantages of including simultaneous measurements of pressure heads and water contents in the optimization procedure. Santini and Romano (1992) provided guidelines for an optimal design of the field drainage experiment and explored the feasibility of simplifying the experimental procedures while maintaining acceptable parameter uncertainty. For example, they proposed reducing the number of θ and hm measurements within a vertical soil profile. To overcome limitations regarding the lower boundary condition, Romano (1993) investigated the effectiveness of defining a fixed hydraulic head gra-
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dient and the possibility of including it as an additional optimization parameter value. However, this approach may not apply to layered soils. Zijlstra and Dane (1996) applied the parameter optimization technique to layered soils, and concluded that the inverse problem may become ill-posed because of the increased number of optimized parameters. Their soil profiles consisted of one, two, or three distinct horizons, and they used water content measurements as a function of time and depth, and matric head values as a function of time at the bottom boundary, to minimize the objective function. The ill-posedness of the inverse problem depended on the size of the input data set. 3.6.2.6.b Experimental Procedures Parameter estimation of unsaturated soil hydraulic properties from an in situ transient drainage experiment involves soil water content and matric head measurements at multiple times and soil depths. The field setup is basically the same as for the classic instantaneous profile method (Section 3.6.1.2.a), but some simplifications are possible and will be discussed. First, a profile description is needed to determine the major horizon boundaries. The experiment should be conducted on a leveled, vegetation-free plot with an area not smaller than 12 m2. The plot should be instrumented in the center with measurement sensors located at depths depending on the soil profile description. The experiment is initiated by ponding water over the entire plot area until matric head values are approximately zero or do not change with time at any of the depth locations. After the water supply is stopped and the ponded water has infiltrated, the plot is covered with a plastic sheet and insulated to ensure a zero-flux top boundary condition and to reduce temperature variations. Subsequently, the soil is allowed to drain by gravity. The transient drainage process is monitored by simultaneous measurements of soil water content and matric head until no changes with time occur. Tensiometers are usually installed at 15-cm depth increments with at least one tensiometer in each soil horizon. Volumetric water contents can be conveniently monitored with a calibrated neutron probe or a TDR system for the same depth increments. As expected, the type of variables measured and their measurement locations, timing, and frequency can have a significant influence on the well-posedness and accuracy of the parameter estimation method. If the soil domain can be considered homogeneous, the unsaturated hydraulic parameters can be reasonably identified without much loss in parameter certainty by monitoring the change with time of matric head at a single depth and water content at two other depths close to the soil surface. It is most convenient to impose a constant total hydraulic head gradient (e.g., unit-gradient) at the lower boundary. The reduction in sampling depth, however, should not be at the expense of the measurement frequency. Data must be collected for the entire drainage experiment, especially since parameter sensitivities have shown to increase with drainage time. Generally, an accurate, simultaneous estimation of the hydraulic parameters requires between 6 and 10 measurement times during the drainage experiment. For a hypothetical single-layer soil profile, 90 cm in depth, Santini and Romano (1992) discussed the parameter estimation procedure, using perturbed drainage
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data to simulate random measurement errors. Successful results were obtained from measurements of a single tensiometer at the 30-cm depth in combination with water content values at depths of 30 and 60 cm. Other issues are discussed in the following comments: 1. The definition of the selected lower boundary condition of the draining soil requires careful consideration. To correctly select the appropriate boundary condition, accurate and numerous measurements of hm at the lower boundary are needed during the course of the drainage experiment. Taking these measurements for proper definition of the lower boundary condition undoubtedly increases field operations and makes the experiment more time-consuming, but it is required for the accurate solution of Eq. [3.6.2–1]. In this regard, the assumption of a time-invariant hydraulic gradient at the bottom of the flow domain significantly reduces the number of measurements needed, and can be especially worthwhile if many field measurements are needed, such as in spatial variability studies (Romano, 1993). However, this assumption will require field verification. Alternatively, if appropriate, the ∂H/∂z value at the lower boundary can be considered as an additional unknown constant parameter to be estimated by the inverse procedure together with the unsaturated soil hydraulic parameters. 2. Soil water content measurements should be taken frequently near the soil surface where their sensitivity to the hydraulic parameters is the greatest. The neutron probe is not recommended for near surface measurements, since its large measurement volume may extend to above the soil surface, especially under dry soil conditions. In contrast, TDR is especially suited for near surface measurements (Topp & Davis, 1985). Time domain reflectometry has the added advantage that it can provide automatic and real-time monitoring of soil water content (Baker & Allmaras, 1990). The depth location of the soil matric head sensors is less critical, as it has been shown that the sensitivity of the matric head to the hydraulic parameters is largely independent of soil depth in field drainage experiments. 3. The drainage method requires a large plot (≥12 m2) to ensure that lateral water movement at the plot boundary will not influence the water regime in the plot center. In that regard, soil horizon characterization is important, since the presence of impeding horizons may induce lateral water movement and affect the required plot size. Moreover, an impeding layer can prevent unsaturated conditions below the impeding soil layer, thereby limiting the valid water content range of the optimized hydraulic parameters. Also, multilayered soil profiles can create perched water tables. Lateral water movement can be reduced by constructing an impermeable boundary around the plot perimeter to a depth slightly greater than the lower boundary of the soil domain. 3.6.2.6.c Simulation and Optimization Equation [3.6.2–1] is solved for the following initial and boundary conditions hm(z) = hm,i
t = 0, 0 < z ≤ L
q(z,t) = 0
t > 0, z = L
[3.6.2–9]
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where z = L denotes the soil surface, with L equal to the depth of the measured soil domain. Several types of boundary conditions can be considered at the bottom of the soil profile (z = 0). Soil water content or pressure head values may be prescribed as a function of time, or a constant total hydraulic head gradient, ∂H/∂z, can be used. Representing the lower boundary condition by a time-invariant unit hydraulic gradient can accurately describe observed drainage field studies (Libardi et al., 1980; Ahuja et al., 1988; McCord, 1991). Unknown hydraulic parameters are estimated by minimizing Eq. [3.6.2–5] using all hm and θ measurements. Additional comments include: 1. A limitation of the drainage experiment approach is that the experiment takes a long time (Baker et al., 1974) and that the experimental water content range is rather small. The presence of a shallow water table or slow drainage largely reduces the water content range. Even though the method can be extended to include surface evaporation, success of the inverse method is quite sensitive to the upper boundary condition and would therefore require accurate evaporation rate measurements. 2. Parameter optimization using drainage data can be especially useful for hydraulic characterization of field soils with little horizon differentiation, so that the average hydraulic behavior of the entire soil profile can be characterized. The inverse technique can be further exploited to provide effective soil hydraulic parameters for simulation of hydrological responses of large-scale areas (Kabat et al., 1997). 3.6.2.7 Additional Applications Many more experimental techniques have demonstrated the potential benefits of inverse modeling to estimate soil hydraulic functions. Briefly, we summarize the upward infiltration method (Hudson et al., 1996), the sorptivity method (Section 3.5.3), the cone penetrometer method (Gribb, 1996), and the multistep extraction method (Inoue et al., 1998). Both outflow and evaporation experiments represent water extraction processes and provide parameter estimates for the draining branches of the soil water content and hydraulic conductivity relationships. Parameters for the wetting branches of these soil hydraulic relationships are typically obtained from infiltration processes. The upward infiltration experiment suggested by Hudson et al. (1996) and the sorptivity method represent typical infiltration applications in the laboratory. For the upward infiltration experiment, packed cores are placed in a flow cell, similar to a Tempe cell (Section 3.3.2). However, the imposed flux boundary condition does not require the presence of a porous membrane at the bottom end of the flow cell. Instead, nylon fabric between the bottom end cap and the soil core allows for unrestricted water flux into the soil core. The top end plate of the cell includes a porous stainless-steel plate to prevent soil swelling, minimize surface evaporation, and allow escape of displaced air through the top of the cell. Moreover, tensiometers are placed horizontally at different vertical positions in the soil core. A TDR probe can be installed horizontally in the center of the soil core at the same vertical position as one of the tensiometers. However, Hudson et al. (1996) showed that little is gained by including water content measurements in the objective func-
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tion. During the upward infiltration experiment, a constant water flux is imposed at the bottom of the soil sample and the matric head distribution within the soil sample is measured with one or more tensiometers. The latter should be installed only when the visible wetting front approaches each measurement location to prevent their malfunctioning at low initial matric head values. Upward infiltration rates can be controlled by a syringe pump to facilitate slow and constant movement of the one-dimensional wetting front through the soil core. In the sorptivity method of Bruce and Klute (Klute & Dirksen, 1986), water is allowed to infiltrate under tension into a horizontal soil column. Using a Boltzmann transformation, the one-dimensional water content form of the flow equation can be solved analytically for this particular boundary condition, so that the soil water diffusivity can be computed as a function of water content. The analytical solution requires measurement of the soil water content distribution along the soil column for a specific time and supply head. In Šimçnek et al. (2000), the experimental results of Nielsen et al. (1962) are compared with the inverse solution for water supply heads of −2, −50, and −100 cm. Excellent agreement was found between the inverse solution and the analytical solution. A modified cone penetrometer is instrumented with a porous filter close to the penetrometer tip and two tensiometer rings above the filter. The device is pushed into the soil to the desired depth, and a constant positive water head is applied to the filter for a short time period. While the volume of water imbibed by the soil is monitored, so is the response of both tensiometers to the advancing wetting front. Subsequently, after the water supply is cut off, the tensiometer measurements record the soil drying during water redistribution. The inverse method was used to estimate the hydraulic parameters of both the wetting and drying branches of the soil hydraulic characteristics by simultaneously analyzing the infiltration and redistribution data (Kodešová et al., 1999; Šimçnek et al., 1999a). A multistep extraction device consists of a ceramic soil solution sampler inserted into a wetted soil and subjected to a series of vacuum extraction pressures. The cumulative amount of soil solution extracted, as well as the response of various tensiometers near the extractor are included in the objective function to estimate the soil hydraulic functions. The method was tested both in the laboratory and in the field (Inoue et al., 1998). The cone penetrometer and the extraction method require solution of a two-dimensional radial flow equation (Eq. [3.6.2–2]), and are typically applied in the field, potentially to large depths. 3.6.2.8 Example We demonstrate the application of inverse modeling by comparing one-step and multistep outflow experiments for estimation of the hydraulic properties of a Columbia fine sandy loam (Wildenschild et al., 2001). We will illustrate the procedure in 12 consecutive steps. Practical comments are added where applicable to assist in solving the inverse problem. Although this example is specific to the outflow method, the same steps are required for all inverse methods. Step 1: Problem Definition. This example determines the drainage curve of a Columbia fine sandy loam using inverse modeling. It is assumed that the underlying flow process is described by the Richards equation, (Eq. [3.6.2–1]), and that
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the soil hydraulic properties are adequately represented by the coupled van Genuchten–Mualem model (Eq. [3.6.2–3] and [3.6.2–4]). Note that although these are common assumptions, they may not be appropriate in any specific case and may even cause unacceptable model errors. The analysis was conducted for a disturbed soil sample that was packed to a predetermined bulk density after sieving the soil through a 0.5-mm sieve. Consequently, the hydraulic behavior is determined mostly by its texture. If the focus of the analysis were on the estimation of hydraulic properties near saturation, which are largely controlled by soil structure, an undisturbed soil sample should be used. In that case, however, the measurement of additional replicates, using larger soil cores, is recommended to account for the inherent field soil spatial variability. For the purpose of this example, inverse modeling results are compared with the classical one-step and multistep experiments. Step 2: Selection of Measurement Types. As discussed in Section 3.6.2.3, outflow methods require a record of cumulative outflow as a function of time. Additionally, tensiometric data from the draining soil core will improve the well-posedness of the inverse solution. Although for this purpose measurements from a single tensiometer would be adequate, the presented example uses data from two tensiometers. The second tensiometer was included to allow simultaneous estimation of the unsaturated hydraulic conductivity. Moreover, the extra data may “save” the experiment if the other tensiometer fails. Step 3: Experimental Setup. The basic requirements for outflow methods are described in Section 3.6.2.3. We selected the experimental setup of Wildenschild et al. (2001). A diagram of the flow cell (3.5 cm high and 7.62-cm i.d.) with gas and water flow controls is shown in Fig. 3.6.2–2. All connections consisted of quickdisconnect fittings (Cole-Parmer, Delrin, 1/4” NPT, 06359-72; Cole-Parmer Instrument Co., Vernon Hills, IL; www.coleparmer.com) so that the cell could be periodically detached for weighing to determine the water content at various times during drainage. Two tensiometers were inserted 1.1 cm (Tensiometer 1) and 2.4 cm (Tensiometer 2) from the top of the soil core, respectively. The tensiometer ports were offset laterally to minimize flow disturbance caused by the presence of the tensiometers. The tensiometers were made from 0.72-cm o.d., 0.1-MPa (1-bar), highflow tensiometer cups (model 652X03-BIM3; Soilmoisture Corp., Santa Barbara, CA), glued to ~6-mm (1/4-inch) o.d. acrylic tubing. The tensiometers extended approximately 2 cm into the sample. Two 0.1-MPa (1-bar) transducers (model 136PC15G2 Honeywell, Minneapolis, MN) were used to monitor the matric head during the outflow experiments. Two additional ports were added on opposite sides of the cell to allow flushing the sample with CO2 prior to wetting, so that complete saturation of the sample is ensured at the start of the experiment. The outlet was connected to a burette for measurement of outflow as a function of time. The burette was mounted such that water drained at atmospheric pressure and was loosely covered with plastic to prevent evaporation. A 70-cm water pressure (1-psi) transducer (model 136PC01G2; Honeywell) was attached to the bottom of the burette to measure the cumulative drainage volume. The upper boundary condition was controlled using regulated pressurized N2. The N2 was bubbled through a distilled water reservoir before entering the pressure cell to minimize evap-
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oration losses from the soil core. Two layers of 1.2-µm, 0.1-mm-thick nylon filters (MSI Magna nylon disc filters, Osmonic Laboratory Products; www.osmolabstore.com) were used as a porous membrane at the bottom of the sample. We combined two nylon filters to minimize puncturing, thereby ensuring a bubbling pressure of at least 700 cm during the outflow experiments. With a saturated hydraulic conductivity of approximately 0.025 cm h−1 (Table 3.6.2–1), the hydraulic resistance of the thin nylon membrane was low compared with other commonly used but thicker porous membranes with comparable bubbling pressure. Water pressure differences across our porous membrane are thus minimized during drainage of the soil core. All outflow experiments were conducted on the same soil core. The measured average saturated water content was 0.445 cm3 cm−3. Time intervals for cumulative outflow and matric head measurements were approximately 10 s, which was sufficient to ensure that the rapid outflow after a gas pressure change was recorded with adequate temporal resolution. Step 4: Selection of Initial and Boundary Conditions. The soil sample was initially fully saturated and then partially drained before the start of the outflow experiment so that the matric head at the bottom of the soil core (hbottom) was −2 cm, corresponding with the following initial condition: hm,i(z) = −2 − z
t = 0, 0 < z < L
[3.6.2–10]
where z = L = 3.5 cm is the height of the soil surface above a datum set at z = 0 = base of sample. Since the same sample was used for both the one-step and multistep outflow experiments, the soil sample was resaturated prior to each drainage experiment, using the same procedure every time, including flushing with CO2 to ensure complete resaturation. This procedure ensured approximately identical initial saturation values between drainage experiments. Full saturation at the onset of the experiment may be desired to provide reproducible initial conditions. However, for undisturbed soils, maximum saturation is usually 0
hm = −500 cm
0 < t < te, z = 0 cm
[3.6.2–11]
where te denotes the duration of the one-step outflow experiment, q(0,te) ≈ 0. The lower boundary condition was experimentally realized by increasing the applied N2
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pressure at the top of the soil core to ha = +500 cm of water pressure, as expressed by Eq. [3.6.2–6]. For the multistep experiment, four pressure steps were applied in sequence, which covered the same pressure range as the one-step experiment. For each successive pressure step, time was allowed for the sample to equilibrate, as determined from a near zero drainage rate. The corresponding boundary conditions were q(L,t) = 0 cm d−1
t>0
hm = −125 cm
0 > t > t1, z = 0 cm
hm = −250 cm
t1 > t > t2, z = 0 cm
hm = −375 cm
t2 > t > t3, z = 0 cm
hm = −500 cm
t3 > t > t4, z = 0 cm
[3.6.2–12]
Step 5: Experimental Data. The cumulative outflow (Q) and matric head (hm) measurements are presented in Fig. 3.6.2–3a and 3.6.2–3b, for the one-step (O) and multistep (M) experiments, respectively. Although many more data points were collected than are shown in Fig. 3.6.2–3, we recommend not exceeding 100 data points per measurement type. The selected data must be representative of the whole set and include those measurements that describe the flow dynamics of the outflow experiment. Specifically, measurements prior to a pressure step increase must be included, and data should generally describe continuously increasing (Q) or decreasing hm values. Quality control of the raw data includes removal of spurious data caused by failure of the monitoring and data logging equipment, and comparison of hm values with their expected equilibrium values at the end of each pressure step. Moreover, measurement uncertainty can be inferred from the time series of measurements. Step 6: Definition of the Objective Function. The definition of the objective function is one of the most important steps in the inverse procedure. For the purpose of this example, we will compare optimizations with (+) and without (−) tensiometric data. As discussed in Section 3.6.2–2, selected weighting factors should be equal to the inverse of the expected measurement variances. However, in practice, this information may not be available. Moreover, for cases that include more than one measurement type, using measurement errors as weighting factors often resulted in local minima. It is, therefore, recommended to set all individual weighing factors (wi,j) in Eq. [3.6.2–5] equal to one and to compute weighing factors for the measurement types, vj, that are reciprocal to their average magnitude multiplied with the number of data, nj. For Measurement Type 1 (cumulative outflow), 2 (Tensiometer 1) and 3 (Tensiometer 2), the corresponding weights in Eq. [3.6.2–5] are then given by: –) v1 = 0.5/(n1Q
– v2 = 0.25/(n2h m,1)
– v3 = 0.25/(n3h m,2)
– denotes the time-averaged cumulative outflow value, and h– and h– repwhere Q m,1 m,2 resent the time-averaged tensiometric data of the top (Tensiometer 1) and bottom
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Fig. 3.6.2–3a. Measured (symbols) and optimized cumulative outflow and matric head curves for the one-step outflow experiment. In the notation of O4+ and O4−, the O stands for one-step outflow, the digit refers to the number of parameters to be optimized, and the + or − indicates the use or lack of use of tensiometric data, respectively. The 1 and 2 in the graph for the matric head data refer to the top and bottom tensiometer, respectively.
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Fig. 3.6.2–3b. Measured (symbols) and optimized cumulative outflow and matric head curves for the multistep outflow experiment. In the notation of M4+ and M4−, the M stands for multistep outflow, the digit refers to the number of parameters to be optimized, and the + or − indicates the use or lack of use of tensiometric data, respectively. The 1 and 2 in the graph for the matric head data refer to the top and bottom tensiometer, respectively.
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tensiometer (Tensiometer 2), respectively. In this example, the value of the objective function was calculated with vj = 1/(njσj2) and wi,j = 1, where σj2 denotes the variance of all fitted data of measurement type j (Clausnitzer & Hopmans, 1995). Step 7: Selection of Fitting Parameters. As stated in Section 3.6.1–3, the “classical” strategy in obtaining the van Genuchten–Mualem parameters is (i) to optimize α, n, θr, and Ks and to constrain the parameter m by the relation m = 1 − 1/n; (ii) to determine θs from an independent measurement; and (iii) to set the tortuosity parameter l to a fixed value of 0.5. However, since it has been increasingly demonstrated that improved fitting of the unsaturated hydraulic conductivity function is achieved with including l as a fitting parameter, we compare optimizations using l as a fitting parameter and as a fixed parameter set to 0.5 (Mualem, 1976). The saturated water content was fixed to its average value of 0.455 cm3 cm−3. Step 8: Inverse Simulations. Inverse simulations were conducted with the HYDRUS-1D code (Šimçnek et al., 1998b). Inverse simulation requires the same information as the forward problem, that is, time and geometric information, initial conditions, and boundary conditions, plus initial parameter estimates, position of observation points, and measurement times with corresponding data and weighing factors. Additionally, some convergence parameters need to be defined. It is, however, recommended to apply the default values as provided with HYDRUS-1D, which were determined from experience. The soil sample was discretized using a variable grid spacing with 50 nodes, using finer spacings at the bottom of the soil column. The porous nylon membrane was not considered, since its resistance to water flow was negligible. If a ceramic porous plate is used, it must be explicitly represented in the model, as its hydraulic resistance will underestimate the soil’s hydraulic conductivity near saturation. The optimizations were repeated three times with different initial estimates for the parameters to be optimized. Conveniently, initial estimates were taken from the database that is included in HYDRUS-1D. In total, we completed eight sets of inverse simulations (Table 3.6.2–2). The optimization results of the one-step (O) experiment will be compared with the multistep (M) experimental results for cases with tensiometric data (O4+, O5+, M4+, M5+) and without tensiometric data (O4−, O5−, M4−, M5−), using either four fitting parameters (l is fixed; cases denoted by 4) or five fitting parameters (l is an additional fitting parameter; cases denoted by 5). For all multistep simulations, the inverse solution converged to similar optimization results, indicating that solutions were unique without local minima. This was different for the one-step simulations without tensiometric measurements (O4− and O5−), which converged towards various local minima depending on the initial parameter values. Although impractical, improved optimization results for these cases were obtained using initial parameter estimates equal to their optimized values of the one-step experiments with hm measurements. Step 9: Comparison of Observed and Simulated Data. Residual Analysis. After evaluation of the uniqueness of an inverse solution, the next logical step is to compare simulated results with the corresponding observations. Figures 3.6.2–3a and 3.6.2–3b show this comparison for the optimizations of the one-step (O4+ and
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Table 3.6.2–2. Optimization results for one-step (O) and multistep (M) outflow experiments to estimate either four (O4+, O4−, M4+, and M4−) or five (O5+, O5−, M5+, and M5−) parameters with (+) or without (−) the help of tensiometric data. Values for the objective function and the average residuals were calculated according to Eq. [3.6.2–5] and [3.6.2–13], respectively. The values in parentheses in the last five columns refer to standard errors. Case
Data
φ
AR, Q/hm
R2
θr
α
n
cm−1
cm O5+
Q, hm
0.0112
0.0172/6.54 0.99
O4+
Q, hm
0.0385
0.0135/23.7 0.99
O5−
Q
0.00262
0.00917
0.99
O4−
Q
0.00881
0.0123
0.99
M5+
Q, hm 0.00960 0.0143/5.00 0.99
M4+
Q, hm
0.0249
M5−
Q
0.00625
0.0126
0.99
M4−
Q
0.00835
0.0167
0.99
0.0273/8.56 0.99
0.191 (0.00156) 0.0245 (0.0238) 0.0948 (0.0189) 0.0000 (0.00018) 0.194 (0.00270) 0.117 (0.01074) 0.192 (0.00475) 0.185 (0.00451)
0.00724 (0.00021) 0.0101 (0.00115) 0.00520 (0.00942) 0.00970 (0.00394) 0.00921 (0.00013) 0.0106 (0.00039) 0.00920 (0.00023) 0.00897 (0.00026)
Ks cm
4.05 (0.170) 1.57 (0.0787) 2.29 (2.28) 1.52 (0.780) 3.02 (0.0110) 1.86 (0.0755) 3.06 (0.208) 3.11 (0.258)
l
h−1
0.126 (0.0081) 1.08 (0.288) 0.219 (0.485) 1.08 (0.780) 0.278 (0.0224) 1.09 (0.09917) 0.358 (0.0488) 0.534 (0.580)
−1.23 (0.0167) 0.5 (fixed) −0.186 (1.05) 0.5 (fixed) −0.849 (0.0539) 0.5 (fixed) −0.318 (0.176) 0.5 (fixed)
O4−) and the multistep (M4+ and M4−) simulations, respectively. The respective optimization results are presented in Table 3.6.2–2. The comparison shows that the general dynamics of the measurements is matched by all simulations. This is a necessary, but not sufficient condition to evaluate the accuracy of the optimized hydraulic parameters. Upon closer inspection, we notice the presence of small systematic deviations between measured and predicted values, indicating that errors are autocorrelated. Moreover, it is apparent that not all four models fit the measured data equally well. Regarding the systematic deviations, outflow data for the multistep experiment show that during the first pressure step the simulated outflow reaches equilibrium much faster than the observed data (Fig. 3.6.2–3b). This is not uncommon and indicates that the process model (Richards’ equation) is unable to accurately describe the true flow behavior in the near-saturated water content region. This effect is most likely caused by the presence of a discontinuous air phase at matric head values near the soil’s air-entry value (Schultze et al., 1999; Wildenschild et al., 2001). Improved fitting of the measured data can be achieved by either applying the first pressure step after the soil is first slightly desaturated or by solving the multiphase flow problem. Other deviations of the simulated multistep outflow curves can be related to the assumption of a fixed l parameter in the unsaturated conductivity model (Durner et al., 1999b). When comparing cumulative outflow (Q) residuals (Column 4 in Table 3.6.2–2) between the one-step and the multistep method, it appears that the onestep method gives better results (i.e., the fit is better). However, it must be intuitively clear that as the number and complexity of different measurement types increase, the difficulty of fitting all combined data will increase as well, especially if the op-
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timization problem contains model and/or measurement errors. Because of this, case O5− (one-step method without tensiometric measurements and five fitting parameters) produces the lowest φ value of all eight cases (Table 3.6.2–2, third row). Inspection of Table 3.6.2–2 also clearly shows that the fitting of the cumulative outflow data is much improved if the connectivity parameter, l, in the conductivity expression is allowed to vary: the φ values for the five-parameter cases (l fitted) are always much lower than for the four-parameter cases, where l is fixed at 0.5. The average residual (Column 4 in Table 3.6.2–2) includes the average deviation between the measured and fitted Q or hm values. These values can be compared with the measurement error of each measurement type for model adequacy testing. Conservative estimates of measurement errors for the tensiometer and burette measurements are 1 and 0.011 cm, respectively. The burette measurement error corresponds with 0.5 mL of drainage for a soil core radius of 3.81 cm (sample area of 45.6 cm2). By inspection of Table 3.6.2–2, it is clear that the time-averaged residual (AR) of the matric head (measurement type j = 2 and 3), defined as: 3
nj
AR(hm) = j=2 Σ (1/nj) Σ [h*m,j−1(ti) − hm,j−1(ti,ββ)]2 i=1
[3.6.2–13]
is larger than the conservative measurement error in all cases (for notation, see Eq. [3.6.2–5]). Specifically, the lowest average residual is 5 cm (case M5+, Table 3.6.2–2), whereas the assumed measurement error was only 1 cm. When considering the average residual for cumulative outflow (measurement type j = 1 only; replace hm by Q in Eq. [3.6.2–13]), the AR of almost all cases is about equal to or slightly larger than the assumed measurement error of 0.011 cm. This is specifically so for the one-step experiments without hm measurements (cases O4− and O5−), since these are the easiest to fit. These considerations, in addition to the observations of the residual analysis, which showed that the deviations between measurements and observations are not randomly distributed, allow us to assess the adequacy of the flow model. We conclude that the combined measurement and model error for the outflow experiment using the described experimental conditions is larger than the precision of the measurement devices. The larger errors are caused by assuming that (i) the Richards’ type water flow equation, in combination with its numerical solution and the selected hydraulic model, is the correct physical model for simulating drainage of the one-step and multistep experiments and/or (ii) the experimental conditions are such that they exactly conform to the assumptions of the flow model. Clearly, the divergence between observed and measured outflow during the first two steps of the multistep method indicate that these assumptions are not fully met (Fig. 3.6.2–3b). For example, although analytical models have proven useful in integrating knowledge of soil hydraulic properties for numerical models and other applications, they rarely fit the measured retention data exactly, thereby introducing AR values larger than the measurement error. For a more thorough discussion of the influence of model errors on the interpretation of inverse modeling results see Durner et al. (1999a, b). Correlation of Simulated and Observed Data. The fifth column of Table 3.6.2–2 lists the R2 values that quantify the correlation between measured and sim-
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ulated flow variables (for an optimum parameter set β). As all correlation values are very high, their magnitude is of little value for model adequacy testing. Summarizing the results, we conclude that the residual errors are small enough to accept the assumed flow and hydraulic models for soil hydrological applications. Accordingly, we proceed further. Step 10: Analysis of Parameter Values and Hydraulic Functions. Parameter Uncertainty. Columns 6 through 10 of Table 3.6.2–2 list the optimized hydraulic parameters. The values in parentheses denote the standard error of estimation as determined from the 95% confidence intervals around the optimized value. The standard errors of all fitted parameters are computed from (Cii)1/2, where Cii is the diagonal element of the parameter covariance matrix C (see Section 1.7, Eq. [1.7–23]). It was already concluded from the residual analysis that these standard error values are not very useful statistically, since the requirement of independent residuals is violated. However, their relative magnitude may be useful when comparing parameters and hydraulic models. Parameter Correlation. Table 3.6.2–3 shows correlation coefficients between the optimized parameters for all eight cases. Values larger than ± 0.90 are bolded. Obviously, high correlation between parameters unnecessarily increases the number of optimized parameters. High correlation causes underestimation of parameter uncertainty, slows down convergence rate, and increases nonuniqueness. It is expected that the number of highly correlated parameters increases as the number of fitted parameters increases. As a result, the available information in the objective function is reduced. Table 3.6.2–3 confirms that the highest number of correlations occur for cases O4− and O5−, that is, the one-step outflow experiment without matric head measurements. As discussed in Section 3.6.2–3, the information content of these experiments is not sufficient to obtain unique parameters. The data show large correlations between n and θr for the multistep experiments, a result that is common when fitting retention data to the van Genuchten relationship if information for the dry range is missing. Hydraulic Functions. The optimized soil water retention and unsaturated hydraulic conductivity functions for all four cases using four fitting parameters are presented in Fig. 3.6.2–4. The optimized functions are accurate only for the experimental water content range of the outflow experiment (i.e., hm > −500 cm), and care must be exercised in their extrapolation to drier soil conditions. Independent soil water retention data (thick solid line) were obtained by a syringe pump procedure (Wildenschild et al., 1997) in a separate experiment by which the soil sample was drained at a constant, low flow rate of 0.5 mL h−1, simulating quasistatic conditions at any time during the drainage experiment. The conditions of the steady-state experiment are considered optimal; therefore, the optimized retention curves are compared with the retention data from the syringe experiment using cumulative drainage with corresponding tensiometric data. Unfortunately, it was impossible to estimate the hydraulic conductivity from the syringe pump experiments because of errors resulting from the extremely small hydraulic head gradients. Consequently, the optimized curves are compared with independently estimated unsaturated hydraulic conductivity data reported in Wildenschild et al. (2001).
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Table 3.6.2–3. Correlation matrices for the van Genuchten parameters for one-step (O) and multistep (M) outflow experiments to estimate either four (O4+, O4−, M4+, and M4−) or five (O5+, O5−, M5+, and M5−) parameters with (+) or without (−) the help of tensiometric data. Parameter θr
α
n
Ks
l
θr
α
n
Ks
l
Case
θr
O5+ O4+ O5− O4− O5+ O4+ O5− O4− O5+ O4+ O5− O4− O5+ O4+ O5− O4− O5+ O4+ O5− O4− M5+ M4+ M5− M4− M5+ M4+ M5− M4− M5+ M4+ M5− M4− M5+ M4+ M5− M4− M5+ M4+ M5− M4−
1
−0.162 −0.363 −0.282 −0.683 0.581 0.864 0.346 −0.662 −0.104 −0.352 −0.323 −0.661 0.083 -0.161 -1
−0.499 −0.481 −0.547 −0.650 0.913 0.950 0.887 0.912 −0.368 −0.267 −0.347 −0.006 −0.330 -−0.235 --
α
n
Ks
l
1
−0.761 −0.774 −0.998 −0.999 0.673 0.992 0.998 0.998 0.037 -0.981 --
1
0.715 −0.765 −0.998 −0.992 −0.063 -0.971 --
1
0.715 -−0.966 --
1
1
−0.737 −0.715 −0.804 −0.840 0.255 0.621 −0.056 0.158 0.111 -−0.172 --
1
−0.408 −0.412 −0.080 −0.170 0.322 -0.077 --
1
0.811 -0.751 --
1
When comparing the functions, the results are surprisingly close. Note, however, that the optimization results of the one-step experiment without tensiometric measurements (O4−) were obtained by using initial, optimized values from the optimization with hm measurements (O4+) as initial parameter values. We also find that the tensiometric measurements in the multistep experiments gave results similar to multistep experiments without tensiometric data, especially for cases with an optimized l. This may be caused partly by our experimental strategy to wait for static equilibrium before increasing the applied gas pressure. Generally speaking, it shows that for the conditions of our experiment, the use of outflow data alone from a multistep experiment may be sufficient to obtain the correct hydraulic properties
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Fig. 3.6.2–4. Comparison of optimized and independently measured retention and unsaturated hydraulic conductivity data for the four parameter optimizations (l = 0.5). For an explanation of the notation of O4+, O4−, M4+, and M4−, see the figure captions of Fig. 3.6.2–3a,b.
by inverse simulation. The unsaturated hydraulic conductivity comparison in Fig. 3.6.2–4 also supports the general conclusion that multistep experiments are preferred over one-step experiments. Step 11: Response Surface Analysis. The evaluation of response surfaces is usually done a priori to design the experimental conditions for inverse modeling using forward simulations. We like to illustrate these analysis in this final step to support the concepts introduced in Section 3.6.2–2. Response Surface Analysis. Response surface analysis can be used to investigate the posedness of the optimization problem. The behavior of the objective function within a multiparameter space can only be visualized by limiting the number of variable parameters to two. Consequently, the value of the objective function, φ, can be shown for combinations of two parameters, while keeping all other param-
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eters at their “true” (i.e., optimized) values. It thus follows that response surfaces can only be calculated posteriori if the true optimum is known as determined from forward modeling results. The case with four adjustable parameters requires analysis of six possible parameter pairs. Figure 3.6.2–5a shows response surfaces for (n,α), (Ks,α), (θr,α), (θr,Ks), (Ks,n), and (θr,n) for the one-step experiment without hm measurements (case O4− in Table 3.6.2–3). Along a response surface, the optimum two-parameter combination is determined by a φ valley or minimum enclosed by contour lines. The shape of the valley indicates the rate of convergence, degree of parameter correlation, parameter sensitivity, and presence of local minima. When comparing response surfaces, the parameter sensitivity decreases as the φ interval increases. The ideal response surface shows a narrow minimum area with circular shape, indicating no correlation between fitting parameters. Response surfaces that are parallel to one of the axes are insensitive to that respective parameter, indicating high parameter uncertainty and a large confidence interval. Unfavorable response surfaces consist of long narrow valleys with an approximate angle of 45°, indicating a high correlation between parameters, while L-shaped valleys indicate slow convergence. The response surfaces calculated from real data, contrary to those obtained from numerically generated data, are much more difficult to interpret since the residuals are not necessarily normally distributed and may exhibit some systematic bias. The response surfaces of the ill-posed one-step experiment in Fig. 3.6.2–5a are unfavorable because of (i) multiple minima within a low φ value valley (response surface θr,n), (ii) extremely narrow valleys for parameter combinations θr,n and Ks,n, and (iii) wide interval spacings. When comparing the results of Fig. 3.6.2–5a with those of Fig. 3.6.2–5b (multistep with hm data), we note that interval spacings are smaller and that most minima are elliptical in shape. Still, the Ks parameter appears to be the least sensitive, which is not surprising when considering that almost all data in the objective function are related to unsaturated flow. Response surfaces are usually calculated to describe the behavior of the objective function in the cross sections of the two-parameter plane for the multiparameter space. However, they cannot illustrate the shape of the objective function in other parts of the parameter space. Thus, if response surfaces do not display welldefined minima, the parameter optimization problem is certainly ill-posed (Figure 3.6.2–5a, parameter pair θr,n). There is, however, no guarantee that the problem is well-posed if well-defined minima are demonstrated for all parameter pairs. This is because minimization (especially for gradient-type methods; see Section 1.7) can lead to optimized parameters for any local minimum of the objective function, far away from the global minimum. For example, although the response surfaces for the one-step and multistep experiments look fairly similar (Fig. 3.6.2–5a and 3.6.2–5b), the response surfaces for the one-step method were obtained after using initial parameter estimates that were close to the actual global minimum. Thus, response surface analyses can show that a specific experimental setup will result in an ill-posed optimization problem, but it cannot guarantee a well-posed optimization problem by itself. Summary and Conclusions. Analysis of the one-step data without considering tensiometric information suffered from large uncertainties and cross corre-
Fig. 3.6.2–5a. Response surfaces for the one-step experiment without tensiometric data (O4-).
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Fig. 3.6.2–5b. Response surfaces for the multistep experiment with tensiometric data (M4+).
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lation among parameters, with optimized parameters strongly dependent on their initial estimates. Results for the multistep experiment (both with and without additional tensiometric information) and the one-step experiment with measured matric head values were fairly similar. This was especially the case when the l parameter was allowed to be optimized as well. However, significant smaller confidence intervals were obtained when combining matric head and cumulative outflow data in the objective function. Thus, although matric head data are not required for inverse modeling of a multistep outflow experiment, their inclusion may lead to smaller confidence intervals of the optimized parameters. Tensiometric data are required when conducting a one-step outflow experiment to avoid nonuniqueness problems. Finally, the analysis has shown that the inverse modeling of multistep outflow can provide accurate and reliable estimates of both the retention and the unsaturated hydraulic conductivity curve parameters for the intermediate soil water content range. 3.6.2.9 Discussion Soil hydraulic parameter estimation by inverse modeling is a relatively complex procedure that provides a quick method for soil hydraulic characterization, yielding parameters for both the soil water retention and unsaturated hydraulic conductivity function from a single experiment. Its successful application requires suitable experimental procedures as well as advanced numerical flow codes and optimization algorithms. Numerical codes with user-friendly interfaces are becoming available that can be used for both inverse and direct simulations (e.g., Šimçnek et al., 1998b, 1999b). However, since the method as a whole is not fully developed yet, both experimental and numerical modeling expertise is required for successful application of the methodology and correct interpretation of the results. When compared with other measurement methods, the inverse modeling approach renders a suite of benefits. First and foremost, it mandates the combination of experimentation with numerical modeling. Since the optimized hydraulic functions are mostly needed as input to numerical flow and transport models for prediction purposes, it is an added advantage that the hydraulic parameters are estimated using similar numerical models as used for predictive forward modeling. An additional benefit of the inverse procedure is their application to transient experiments, thereby providing relatively fast results. Finally, the parameter optimization procedure computes confidence intervals of the optimized parameters, although their interpretation can be misleading. Inverse problems for parameter estimation of soil hydraulic functions can be ill-posed because of inadequate experimental design, measurement errors, and model errors. Analysis of such flow problems must include a search for the optimum number of flow variables required in the objective function. For example, as the number of optimized parameters increases, increased information content of the measurements is required, for example, by including observations of different types, or by using time-variable boundary conditions. Sensitivity analysis can largely optimize the need for type, number, and spatial location of the measurements. For example, it has been shown in a variety of applications that measured transient flow data, as induced by multiple step changes at the domain boundary, are more sensitive to the estimated parameters than using a single step. Also, an increase in
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the number of parameters to be optimized will generally lead to a reduction in model error, but will increase the parameter uncertainty. A well-posed problem requires a priori testing for nonuniqueness using response surface analysis and parameter sensitivity and correlation. Insensitive parameters should be measured independently, whereas highly correlated parameters will affect uniqueness of the inverse problem, requiring independent measurement of one of the correlated parameters. To reduce nonuniqueness and to extend the range of application beyond the experimental range of measurements, independently measured information on the soil hydraulic functions can be included in the objective function. This prior information can reduce parameter uncertainty, but may reduce the goodness of fit between model and data. As with all other laboratory and field methods to estimate soil hydraulic functions, it is assumed that the functional forms used are capable of accurately describing the soil hydraulic data. It is therefore essential to perform a model adequacy test by comparing objective function residuals with measurement errors, to identify model errors. Obviously, if the parametric models (Section 3.3.4) are not adequate for a tested soil, the resulting fitting parameters will not be valid. Especially, if the coupled Mualem approach is used, their improper selection may compromise the accuracy of both hydraulic functions. When comparing the optimized hydraulic functions with the results of other methods, one must consider differences in model assumptions and experimental range. Laboratory measurements, although accurate, provide hydraulic information for a relatively small soil core, detached from its surroundings. On the other hand, field experiments will generally include the continuum of soil horizons that will influence water flow and the estimated soil hydraulic functions. Moreover, as is the case for any method, the parameter estimates are only valid for the range of the experimental conditions, and care must be exercised in their extrapolation. This chapter must be regarded as a work in progress, since the mathematical, analytical, and experimental procedures that constitute the inverse method as a whole are still an area of intensive research. Improvements in parameter estimation methods in combination with experimental requirements and optimization algorithms continue to appear in a steady stream of publications. Nevertheless, the general inverse method has demonstrated to be an excellent new tool that allows for soil hydraulic characterization using a wide spectrum of transient laboratory and field experiments. To date, the application of the inverse parameter estimation method in the vadose zone has been limited to the estimation of soil hydraulic properties. This is not surprising because hydraulic parameters are required in most flow and transport models and their direct measurements are time-consuming. We pose that inverse modeling can be used to estimate other soil properties as well, such as solute transport, heat flow, and gaseous transport parameters. Moreover, the methodology can be applied to better understand processes, such as differentiation between matrix and macropore contributions to water flow for two-domain simulation models (Šimnek et al., 2001), or to infer root water uptake parameters in crop growth simulation models (Vrugt et al., 2001a). Summarizing, we conclude that parameter estimation by inverse modeling has tremendous potential in characterizing vadose flow and transport processes, while simultaneously presenting us with an additional tool to better understand their fundamental mechanisms.
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3.6.2.10 References Ahuja, L.R., B.B. Barnes, D.K. Cassel, R.R. Bruce, and D.L. Nofziger. 1988. Effect of assumed unit gradient during drainage on the determination of unsaturated hydraulic conductivity and infiltration parameters. Soil Sci. 145:235–243. Ankeny, M.D., M. Ahmed, T.C. Kaspar, and R. Horton.1991. Simple field method for determining unsaturated hydraulic conductivity. Soil Sci. Soc. Am. J. 55:467–470. Baker, F.G., P.L.M. Veneman, and J. Bouma. 1974. Limitations of the instantaneous profile method for field measurement of unsaturated hydraulic conductivity. Soil Sci. Soc. Am. J. 38:885–888. Baker, J.M., and R.R. Allmaras. 1990. System for automating and multiplexing soil moisture measurement by time-domain reflectometry. Soil Sci. Soc. Am. J. 54:1–6. Bear, J. 1972. Dynamics of fluids in porous media. American Elsevier, New York, NY. Becher, H.H. 1971. Ein Verfahren zur Messung der ungesättigten Wasserleitfähigkeit. Z. Pflanzenernaehr. Bodenkd. 128:1–12. Boels, D., J.B.H.M. van Gils, G.J. Veerman, and K.E. Wit. 1978. Theory and system of automatic determination of soil moisture characteristics and unsaturated hydraulic conductivities. Soil Sci. 126:191–199. Brooks, R.H., and A.T. Corey. 1966. Properties of porous media affecting fluid flow. J. Irrig. Drain. Div. Am. Soc. Civ. Eng. 92:61–88. Celia, M.A., E.T. Bouloutas, and R.L. Zarba. 1990. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res. 26:1483–1496. Chen, J., J.W. Hopmans, and M.E. Grismer. 1999. Parameter estimation of two-fluid capillary pressuresaturation and permeability functions. Adv. Water Resour. 22:479–493. Ciollaro, G., and N. Romano. 1995. Spatial variability of the soil hydraulic properties of a volcanic soil. Geoderma 65:263–282. Clausnitzer, V., and J.W. Hopmans. 1995. Nonlinear Parameter estimation. LM-OPT. General purpose optimization code based on the Levenberg–Marquardt algorithm. LAWR Report 100032. University of California, Davis, CA. Cuenca, R.H., J. Brouwer, A. Chanzy, P. Droogers, S. Galle, S.R. Gaze, M. Sicot, M., H. Stricker, R. Angulo-Jaramillo, S.A. Boyle, J. Bromly, A.G. Chebhouni, J.D. Cooper, A.J. Dixon, J.-C. Fies, M. Gandah, J.-C. Gaudu, L. Laguerre, M. Soet, H.J. Steward, J.-P. Vandervaere, and M. Vauclin. 1997. Soil measurements during HAPEX-Sahel intensive observation period. J. Hydrol. 188–189:224–266. Dane, J.H., and S. Hruska. 1983. In-situ determination of soil hydraulic properties during drainage. Soil Sci. Soc. Am. J. 47:619–624. Dirksen, C. 1991. Unsaturated hydraulic conductivity. p. 209–269. In K.A. Smith and C.E. Mullins (ed.) Soil analysis: Physical methods. Marcel Dekker, New York, NY. Doering, E.J. 1965. Soil water diffusivity by the one-step method. Soil Sci. 99:322–326. Durner, W. 1994. Hydraulic conductivity estimation for soils with heterogeneous pore structure. Water Resour. Res. 30:211–233. Durner, W., E. Priesack, H.-J. Vogel, and T. Zurmühl. 1999a. Determination of parameters for flexible hydraulic functions for inverse modeling. p. 817–827. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, CA. Durner, W., E.B. Schultze and T. Zurmühl. 1999b. State-of-the-art in inverse modeling of inflow/outflow experiments. p. 661–681. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, CA. Eching, S.O., and J.W. Hopmans. 1993a. Optimization of hydraulic functions from transient outflow and soil water pressure data. Soil Sci. Soc. Am. J. 57:1167–1175. Eching, S.O., and J.W. Hopmans. 1993b. Inverse solution of unsaturated soil hydraulic functions from transient outflow and soil water pressure data. Land, Air and Water Resources Paper No. 100021. Univ. of California, Davis, CA. Eching, S.O., J.W. Hopmans, and O. Wendroth. 1994. Unsaturated hydraulic conductivity from transient multi-step outflow and soil water pressure data. Soil Sci. Soc. Am. J. 58:687–695. Feddes, R.A., P. Kabat, P.J.T. van Bakel, J.J. B. Bronswijk, and J. Habertsma. 1988. Modeling soil water dynamics in the unsaturated zone—State of the art. J. Hydrol. 100:69–111. Gardner, W.R. 1956. Calculation of capillary conductivity from pressure plate outflow data. Soil Sci. Soc. Am. Proc. 20:317–320. Gardner, W.R. 1958. Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci. 85:228–232.
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Gardner, W.R. 1962. Note on the separation and solution of diffusion type equations. Soil Sci. Soc. Am. Proc. 26:404. Gardner, W.R., and F.J. Miklich. 1962. Unsaturated conductivity and diffusivity measurements by a constant flux method. Soil Sci. 93:271–274. Gribb, M.M. 1996. Parameter estimation for determining hydraulic properties of a fine sand from transient flow measurement. Water Resour. Res. 32:1965–1974. Gupta, S.C., D.A. Farrell, and W.E. Larson. 1974. Determining effective soil water diffusivities from one-step outflow experiments. Soil Sci. Soc. Am. Proc. 38:710–716. Halbertsma, J.M., and G.J. Veerman. 1994. A new calculation procedure and simple set-up for the evaporation method to determine soil hydraulic functions. Report 88. DLO Winand Staring Centre, Wageningen, The Netherlands. Hillel, D., V.D. Krentos, and Y. Stylianou. 1972. Procedure and test of an internal drainage method for measuring soil hydraulic characteristics in situ. Soil Sci. 114:395–400 Hollenbeck, K.J., and K.H. Jensen. 1998a. Maximum-likelihood estimation of unsaturated hydraulic parameters. J. Hydrol. 210:192–205. Hollenbeck, K.J., and K.H. Jensen. 1998b. Experimental evidence of randomness and nonuniqueness in unsaturated outflow experiments designed for hydraulic parameter estimation. Water Resour. Res. 34:595–602. Hollenbeck, K.J., J. Šimçnek, and M.Th. van Genuchten. 2000. RETCML: Incorporating maximumlikelihood estimation principles in the soil hydraulic parameter estimation code RETC. Comput. Geosci. 26:319–327. Hopmans, J.W., and J. Šimçnek. 1999. Review of inverse estimation of soil hydraulic properties. p. 643–659. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, CA. Hopmans, J.W., T. Vogel, and P.D. Koblik. 1992. X-ray tomography of soil water distribution in onestep outflow experiments. Soil Sci. Soc. Am. J. 56:355–362. Hudson, D.B., P.J. Wierenga, and R.G. Hills. 1996. Unsaturated hydraulic properties from upward flow into soils cores. Soil Sci. Soc. Am. J. 60:388–396. Inoue, M., J. Šimçnek, J.W. Hopmans, and V. Clausnitzer. 1998. In situ estimation of soil hydraulic functions using a multi-step extraction technique. Water Resour. Res. 34:1035–1050. Kabat, P., R.W.A. Hutjes, and R.A. Feddes. 1997. The scaling characteristics of soil parameters: From plot scale heterogeneity to subgrid parameterization. J. Hydrol. 190:363–396. Klute, A., and C. Dirksen. 1986. Conductivities and diffusivities of unsaturated soils. p. 687–734. In A. Klute (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron. Monogr. 9. ASA and SSSA, Madison, WI. Kodešová, R., S.E. Ordway, M.M. Gribb, and J. Šimçnek. 1999. Estimation of soil hydraulic properties with the cone permeameter: Field studies. Soil Sci. 164:527–541. Kool, J.B., and J.C. Parker. 1988. Analysis of the inverse problem for transient unsaturated flow. Water Resour. Res. 24:817–830. Kool, J.B., J.C. Parker, and M.Th. van Genuchten. 1987. Parameter estimation for unsaturated flow and transport models—A review. J. Hydrol. 91:255–293. Kool, J.B., J.C. Parker, and M.Th. van Genuchten. 1985. Determining soil hydraulic properties for onestep outflow experiments by parameter estimation. I. Theory and numerical studies. Soil Sci. Soc. Am. J. 49:1348–1354. Kulkarni, R., A.T. Watson, J.E. Nordtvedt, and A. Sylte. 1998. Two-phase flow in porous media: Property identification and model validation. AICHE J. 44:2337–2350. Libardi, P.L., K. Reichardt, D.R. Nielsen, and J.W. Biggar. 1980. Simple field methods for estimating soil hydraulic conductivity. Soil Sci. Soc. Am. J. 44:3–7. Liu, Y.P., J.W. Hopmans, M.E. Grismer, and J.Y. Chen. 1998. Direct estimation of air–oil and oil–water capillary pressure and permeability relations from multi-step outflow experiments. J. Contam. Hydrol. 32:223–245. Logsdon, S.D., E.L. McCoy, R.R. Allmaras, and D.R. Linden. 1993. Macropore characterization by indirect methods. Soil Sci. 155:316–324. McCord, J.T. 1991. Application of second-type boundaries in unsaturated flow modeling. Water Resour. Res. 27:3257–3260. Mortensen, A.P., R.J. Glass, and K. Hollenbeck. 1998. Visualization of quasi-2D unsaturated flow during dynamic outflow experiments. Am. Geophys. Union. Eos Trans. 79:368. Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513–522. Nielsen, D.R., J.W. Biggar, and J.M. Davidson. 1962. Experimental consideration of diffusion analysis in unsaturated flow problems. Soil Sci. Soc. Am. Proc. 26:107–111.
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Nielsen, D.R., M.Th. van Genuchten, and J.W. Biggar. 1986. Water flow and solute transport processes in the unsaturated zone. Water Resour. Res. 22:89S–108S. Parker, J.C., J.B. Kool, and M.Th. van Genuchten. 1985. Determining soil hydraulic properties from one-step outflow experiments by parameter estimation. II. Experimental studies. Soil Sci. Soc. Am. J. 49:1354–1359. Passioura, J.B. 1976. Determining soil water diffusivities from one-step outflow experiments. Austr. J. Soil Res. 15:1–8. Perroux, K.M., and I. White. 1968. Design for disc permeameters. Soil Sci. Soc. Am. J. 52:1205–1215. Quintard, M., and S. Whitaker. 1999. Fundamentals of transport equation formulation for two-phase flow in homogeneous and heterogeneous porous media. p. 3–57. In M.B. Parlange and J.W. Hopmans (ed.) Vadose zone hydrology: Cutting across disciplines. Oxford University Press, Oxford, UK. Reynolds, W.D., and D.E. Elrick. 1991. Determination of hydraulic conductivity using a tension infiltrometer. Soil Sci. Soc. Am. J. 55:633–639. Reynolds, W.D., and W.D. Zebehuk. 1996. Use of contact material in tension infiltrometer measurements. Soil Technol. 9:141–159. Reeve, M.J., and A.D. Carter. 1991. Water release characteristic. p. 111–160. In K.A. Smith and C.E. Mullins (ed.) Soil analysis: Physical methods. Marcel Dekker, New York, NY. Romano, N. 1993. Use of an inverse method and geostatistics to estimate soil hydraulic conductivity for spatial variability analysis. Geoderma 60:169–186. Romano, N., and A. Santini. 1999. Determining soil hydraulic functions from evaporation experiments by a parameter estimation approach: Experimental verifications and numerical studies. Water Resour. Res. 35:3343–3359. Russo, D. 1988. Determining soil hydraulic properties by parameter estimation: On the selection of a model for the hydraulic properties. Water Resour. Res. 24:453–459. Russo, D., E. Bresler, U. Shani, and J.C. Parker. 1991. Analysis of infiltration events in relation to determining soil hydraulic properties by inverse problem methodology. Water Resour. Res. 27:1361–1373. Santini, A., and N. Romano. 1992. A field method for determining soil hydraulic properties. Proceedings of XXIII Congress of Hydraulics and Hydraulic Constructions. Florence, Italy. 31 Aug.–4 Sept. Tecnoprint S.n.c., Bologna, I:B.117–B.139 (in Italian; available also in English). Santini, A., N. Romano, G. Ciollaro, and V. Comegna. 1995. Evaluation of a laboratory inverse method for determining unsaturated hydraulic properties of a soil under different tillage practices. Soil Sci. 160:340–351. Schaap, M.G., and F.J. Leij. 2000. Improved prediction of unsaturated hydraulic conductivity with the Mualem–van Genuchten model. Soil Sci. Soc. Am. J. 64:843–851. Schindler, U. 1980. Ein Schnellverfahren zur Messung der Wasserleitfähigkeit im teilgesättigten Boden an Stechzylinderproben. Arch. Acker-Pflanzenbau u. Bodenkd. 24:1–7. Schultze, B., O. Ippisch, B. Huwe, and W. Durner. 1999. Dynamic nonequilibrium in unsaturated water flow. p. 877–892. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, CA. Schultze, B., T. Zurmühl, and W. Durner 1996. Untersuchung der Hysterese hydraulischer Funktionen von Böden mittels inverser Simulation. Mitteilungen der Deutschen Bodenkundlichen Gesellschaft 80:319–322. Šimçnek, J., R. Angulo-Jaramillo, M.G. Schaap, J.-P. Vandervaere, and M.Th. van Genuchten. 1998a. Using an inverse method to estimate the hydraulic properties of crusted soils from tension disc infiltrometer data. Geoderma 86:61–81. Šimçnek, J., J.W. Hopmans, D.R. Nielsen, and M.Th. van Genuchten. 2000. Horizontal infiltration revisited using parameter optimization. Soil Sci.165:708–717. Šimçnek, J., R. Kodešová, M.M. Gribb, and M.Th. van Genuchten. 1999a. Estimating hysteresis in the soil water retention function from modified cone penetrometer test. Water Resour. Res. 35:1329–1345. Šimçnek, J., M. Šejna, and M.Th. van Genuchten. 1998b. The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media. Version 2.0. IGWMC-TPS-70. International Ground Water Modeling Center, Colorado School of Mines, Golden, CO. Šimçnek, J., M. Šejna, and M.Th. van Genuchten. 1999b. The HYDRUS-2D software package for simulating two-dimensional movement of water, heat, and multiple solutes in variably-saturated media. Version 2.0. IGWMC-TPS-53. International Ground Water Modeling Center, Colorado School of Mines, Golden, CO. Šimçnek, J., and M.Th. van Genuchten. 1996. Estimating unsaturated soil hydraulic properties from tension disc infiltrometer data by numerical inversion. Water Resour. Res. 32:2683–2696.
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Šimçnek, J., and M.Th. van Genuchten. 1997. Estimating unsaturated soil hydraulic properties from multiple tension disc infiltrometer data. Soil Sci. 162:383–398. Šimçnek, J., and M.Th. van Genuchten. 2000. The DISC computer software for analyzing tension disc infiltrometer data by parameter estimation. Versions 1.0. Research Rep. 145. U.S. Salinity Laboratory, USDA-ARS, Riverside, CA. Šimçnek, J., O. Wendroth, and M.Th. van Genuchten. 1998c. A parameter estimation analysis of the evaporation method for determining soil hydraulic properties. Soil Sci. Soc. Am. J. 62:894–905. Šimçnek, J., O. Wendroth, and M.Th. van Genuchten. 1999c. Estimating soil hydraulic properties from laboratory evaporation experiments by parameter estimation. p. 713–724. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, CA, Šimçnek, J., O. Wendroth, and M.Th. van Genuchten. 1999d. A parameter estimation analysis of the laboratory tension disc experiment for determining soil hydraulic properties. Water Resour. Res. 35:2965–2979. Šimçnek, J., O. Wendroth, N. Wypler, and M.Th. van Genuchten. 2001. Nonequilibrium water flow characterized from an upward infiltration experiment. Eur. J. Soil Sci. 52:13–24. Smettem, K.R.J., and B.E. Clothier. 1989. Measuring unsaturated sorptivity and hydraulic conductivity using multiple disk permeameters. J. Soil Sci. 40:563–568. Tamari, S., L. Bruckler, J. Halbertsma, and J. Chadoeuf. 1993. A simple method for determining soil hydraulic properties in the laboratory. Soil Sci. Soc. Am. J. 57:642–651. Toorman, A.F., P.J. Wierenga, and R.G. Hills. 1992. Parameter estimation of soil hydraulic properties from one-step outflow data. Water Resour. Res. 28:3021–3028. Topp, G.C., and J.L. Davis. 1985. Measurement of soil water content using time domain reflectometry (TDR). Soil Sci. Soc. Am. J. 46:19–24. Vachaud, G., C. Dancette, S. Sonko, and J.L. Thony. 1978. Méthodes de caractérisation hydrodynamique in situ d’un sol non saturé. Application à deux types de sol du Sénégal en veu de la détermination des termes du bilan hydrique. Ann. Agron. 29:1–36. Valiantzas, J.D., and D.G. Kerkides. 1990. A simple iterative method vor the simultaneous determination of soil hydaulic properties from one-step outflow data. Water Resour. Res. 26:143–152. van Dam, J.C., J.N.M. Stricker, and P. Droogers. 1992. Evaluation of the inverse method for determining soil hydraulic functions from one-step outflow experiments. Soil Sci. Soc. Am. J. 56:1042–1050. van Dam, J.C., J.N.M. Stricker, and P. Droogers. 1994. Inverse method for determining soil hydraulic functions from multi-step outflow experiments. Soil Sci. Soc. Am. J. 58:647–652. van Genuchten, M.Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892–898. van Genuchten, M.Th., and E.A. Sudicky. 1999. Recent advances in vadose zone flow and transport modeling. p. 155–193. In M.B. Parlange and J.W. Hopmans (ed.) Vadose zone hydrology: Cutting across disciplines. Oxford University Press, Oxford, UK. Vogel, T., M. Nahaei, and M. Císlerová. 1999. Description of soil hydraulic properties near saturation from the point of view of inverse modeling. p. 693–703. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, CA. Vrugt, J., J.W. Hopmans, and J. Šimçnek. 2001a. Calibration of a two-dimensional root water uptake model. Soil Sci. Soc. Am. J. 65:1027–1037. Vrugt, J., A. Weerts, and W. Bouten. 2001b. Information content of data for identifying soil hydraulic properties from outflow experiments. Soil Sci. Soc. Am. J. 65:19–27. Wang, D., S.R. Yates, and F. F. Ernst. 1998. Determining soil hydraulic properties using tension infiltrometers, time domain reflectometry, and tensiometers. Soil Sci. Soc. Am. J. 62:318–325. Wendroth, O., W. Ehlers, J.W. Hopmans, H. Kage, J. Halbertsma, and J.H.M. Wösten. 1993. Reevaluation of the evaporation method for determining hydraulic functions in unsaturated soils. Soil Sci. Soc. Am. J. 57:1436–1443. Whisler, F.D., and K.K. Watson. 1968. One-dimensional gravity drainage of uniform columns of porous materials. J. Hydrol. 6:277–296. White, I., and M.J. Sully. 1987. Macroscopic and microscopic capillary length and time scales from field infiltration, Water Resour. Res. 23:1514–1522. Wildenschild, D., J.W. Hopmans, and J. Šimçnek. 2001. Flow rate dependence of soil hydraulic characteristics. Soil Sci. Soc. Am. J. 65:35–48. Wildenschild, D., K.H. Jensen, K.J. Hollenbeck, T.H. Illangasekare, D. Znidarcic, T. Sonnenborg, and M.B. Butts. 1997. A two-stage procedure for determining unsaturated hydraulic characteristics using a syringe pump and outflow observations. Soil Sci. Soc. Am. J. 61:347–359.
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Wind, G. P. 1968. Capillary conductivity data estimated by a simple method. p. 181–191. In P.E. Rijtema and H. Wassink (ed.) Water in the unsaturated zone. Proc. Wageningen Symp. June 1966. Vol. 1. IASAH, Gentbrugge, Belgium. Wooding, R.A. 1968. Steady infiltration from large shallow circular pond. Water Resour. Res. 4:1259–1273. Zachmann, D.W., P.C. Duchateau, and A. Klute. 1981. The calibration of Richards flow equation for a draining column by parameter identification. Soil Sci. Soc. Am. J. 45:1012–1015. Zachmann, D.W., P.C. Duchateau, and A. Klute. 1982. Simultaneous approximation of water capacity and soil hydraulic conductivity by parameter identification. Soil Sci. 134:157–163. Zijlstra, J., and J.H. Dane. 1996. Identification of hydraulic parameters in layered soils based on a quasiNewton method. J. Hydrol. 181:233–250. Zurmühl, T. 1998. Capability of convection dispersion transport models to predict transient water and solute movement in undisturbed soil columns. J. Contam. Hydrol. 30:99–126. Zurmühl, T., and W. Durner. 1998. Determination of parameters for bimodal hydraulic functions by inverse modeling. Soil Sci. Soc. Am. J. 62:874–880.
Published 2002
3.6.3 Indirect Methods FEIKE J. LEIJ AND MARCEL G. SCHAAP, George E. Brown, Jr. Salinity Laboratory, Riverside, California LALIT M. ARYA, Soil Consultant, Oceanside, California
3.6.3.1 Introduction Knowledge of the unsaturated soil hydraulic properties, which are the constitutive relations in the Richards equation for water flow in unsaturated soils, is essential for many problems involving subsurface flow and transport. A large number of laboratory and field methods have been developed over the years to determine the relationships between the volumetric water content, θ, the soil water matric head, hm, and the hydraulic conductivity, K, or soil water diffusivity, D. These methods are often perceived as too costly and laborious for many applications. This is particularly true for the field, where reliable in situ methods are difficult to implement, while experimental results may not be all that useful because of inherent experimental errors, spatial variability, and discrepancies between the observation and application scales. Alternatively, hydraulic properties may be estimated conveniently with indirect methods from surrogate data that can be more easily measured or are already available. Pedotransfer functions (PTFs)—a term coined by Bouma and van Lanen (1987)—attempt to estimate desired hydraulic properties from already existing data. Textural data are often used as input data. The measurement of the particle-size distribution (PSD) is the topic of Section 2.4. The output of a PTF may consist of conductivity or retention values at particular water content or matric head values, or parameter values in closed-form expressions for retention and conductivity functions. These PTFs will depend on the type of hydraulic data to be predicted, the available input data, the assumptions made to relate input and output parameters, and the algorithm to implement these relationships. In the following sections we will distinguish between semiempirical and empirical methods to predict the soil water retention and hydraulic conductivity. We will only be concerned with drying curves, since indirect methods for imbibition curves are virtually nonexistent due to a limited amount of experimental data. Many parametric models have been used to describe water retention and conductivity data (Leij et al., 1997). Several functions for the water retention are also presented in Section 3.3.4. Two popular functions are those by Brooks and Corey (1964): θ−θ (α|hm|)−λ Se = _____r = θs − θ r 9 1 1009
(α|hm| > 1) (α|hm| ≤ 1)
[3.6.3–1]
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and by van Genuchten (1980): θ−θ Se = _____r = [1 + (α|hm|n)]−m θs − θr
[3.6.3–2]
where Se is the effective degree of saturation or the reduced water content (0 ≤ Se ≤ 1); θr and θs are the residual and saturated water content, respectively; α is a parameter inversely related to the air entry value (L−1); and λ, m, and n are parameters that affect the shape of the retention curve. Note that the soil water matric head, hm, is negative for partially saturated soils. To avoid the use of negative values, however, we will use capillary head (a term first introduced by Buckingham (1907), which is somewhat deceptive because water is also retained by mechanisms other than capillarity), hc = −hm, instead of matric head. Closed-form expressions may also be formulated for the hydraulic conductivity (L T−1), which may be expressed as a function of capillary head, hc, or water content, θ. 3.6.3.2 Semiempirical Approaches In these approaches, the hydraulic properties are estimated based on simplifications regarding the medium and the physical mechanisms for flow and retention in the soil. To account for the simplifications, empirical parameters are used to reconcile differences between experimental and predicted results. Two scenarios are prevalent for semiempirical predictions. First, PSD models have been widely used to predict the hydraulic conductivity curve, whose measurement is more cumbersome than the retention curve, from experimental θ(hc) results. Secondly, PSD models can be employed to predict the retention curve from the particle-size distribution, bulk and particle densities, and other soil properties. The latter approach involves additional assumptions and is not as accurate, because its estimation of the pore-size distribution is less accurate than its direct inference from water retention data. Both types of models relate the pore-size distribution with water retention by using the pressure drop across a capillary interface to obtain the corresponding water pressure. The equation of Young and Laplace defines the pressure drop across a curved surface (Adamson & Gast, 1997): ∆P = Pn − Pw = σ[(1/r1) + (1/r2)]
[3.6.3–3]
where ∆P (M L−1 T−2) is the pressure difference across a curved surface separating a wetting (w) and a nonwetting (n) fluid, σ is the surface tension (M T−2), and r1 and r2 are the two main radii of curvature (L) for the plane. If a capillary tube is placed in a water reservoir, water will rise in the tube above the reservoir level until gravitational and capillary forces are in equilibrium. For a cylindrical tube with radius, r, the capillary rise (∆hc) can be written as ∆hc = (2σcosΘ)/(ρwgr)
[3.6.3–4]
where Θ is the contact angle between liquid and solid surface, g is gravitational acceleration (L T−2), and ρw is the density of water (M L−3). To simplify the geome-
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try, the complex soil pore space has commonly been represented by a bundle of cylindrical tubes. Pressures are usually with respect to atmospheric pressure and the air phase is ignored. The soil water capillary head is defined from the capillary pressure according to hc = ∆P/(ρwg). Before proceeding, we note that the above approach has some shortcomings. Only water retention by capillary forces is considered; other retention mechanisms such as osmotic and adsorptive forces are ignored (Parker, 1986). Osmotic forces only play a role at high electrolyte levels of the soil solution, for example, in saline soils, but even glass beads retain considerably more water than predicted by the capillary theory, as was already observed by Waldron et al. (1961). Furthermore, the pore geometry may not be well described by a bundle of capillary tubes; for example, see Morrow and Xie (1999), who reviewed water retention in triangular pores. These shortcomings point to the need to include empirical corrections in indirect methods, especially at greater capillary head values (e.g., hc >10 m). 3.6.3.2.a Pore-Size Distribution Analytical models for the conductivity function can be derived by representing the soil as an idealized medium, which consists of well-defined pores with a uniform pore size or a known pore-size distribution. The water retention curve (WRC), obtained from measurements or estimation methods, is used to define the pore-size distribution for the idealized medium. An exact description of the flow may be given at the microscopic level (i.e., the pore scale) with the Navier–Stokes equation or for cylindrical pores with the Hagen–Poiseuille equation. The macroscopic velocity can readily be obtained via summation or integration of the microscopic velocities over all water filled pores. This macroscopic velocity yields the Darcy velocity, and an expression for the hydraulic conductivity can be determined. Since the pore system of a real porous medium is not as simple as these models assume, empirical parameters are included in the models to improve the fit between experimental and theoretical conductivity functions. Brutsaert (1967) and Mualem (1986) presented fairly detailed accounts of predictive equations for K based on pore-size distributions. As a starting point, consider the hydraulic conductivity at saturation. Kozeny (1927) derived the following expression for the saturated conductivity by conceptualizing the medium as an assembly of spherical particles: Ks = (gθs3)/(CvA2)
[3.6.3–5]
where θs is the saturated water content, v is the kinematic viscosity (L2 T−1), C is a dimensionless flow-configuration constant, and A is the solid surface area per volume of porous medium. The last two parameters are typically unknown and should be viewed as empirical constants. The unsaturated hydraulic conductivity can be described in a similar manner according to Kr = Sen
[3.6.3–6]
where the relative conductivity is defined as Kr = K/Ks and the aforementioned empirical factors are embedded in Ks. In analogy with Eq. [3.6.3–5], Irmay (1954) set
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the exponent n equal to 3. Other values have also been used (see Mualem, 1986). Although Eq. [3.6.3–6] is convenient for practical applications, it does not explicitly account for the effect of the pore-size distribution on the conductivity. Mualem (1978) attempted to attach a conceptual underpinning to Eq. [3.6.3–6] by setting n = 3 + 0.015w, where w is the energy per unit volume of soil required to drain the saturated soil to the wilting point (hc = 150 m): θ
w = ∫θs γwhcdθ
[3.6.3–7]
150
where γw is the specific weight of water. Optimization of the conductivity function for 50 soils yielded n values ranging from 1.0 up to 24.5. Most classical hydraulic conductivity models conceptualize the porous medium as an assembly of parallel tubes (Purcell, 1949; Burdine et al., 1950; Wyllie & Spangler, 1952). The tubes are completely filled with either water or air. The distribution of the radii of these tubes can be inferred from the WRC using the equation of Young and Laplace. The hydraulic conductivity of the medium is obtained by summing the contribution of individual pores, as calculated with Poiseuille’s law for flow in a cylindrical tube. The models typically contain a tortuosity factor to account for deviations of the pores from the main flow direction and an experimental conductivity value to match the model to an observation, usually the saturated conductivity. Fatt and Dijkstra (1951) assumed the tortuosity to be inversely related to the radius of the largest water-filled pore. Burdine (1953) considered the tortuosity independent of pore radius and obtained the following model: Kr =
Se2
‰
θ ∫0 hc−2dθ ___________ θs ∫0 hc−2dθ
[3.6.3–8]
The conductivity is determined by integrating hc−2(θ). Obviously, the assumption made for the “parallel” models that the radius of a particular tube remains constant in the direction of flow is not realistic. To account for changes in pore size in both the direction of flow (i.e., for one pore) as well as in a plane normal to this direction (i.e., for different pores), parallel-series models were introduced. Childs and Collis-George (1950) were the first to present such a model, followed by Wyllie and Gardner (1958) and Marshall (1958), among many others. The model by Mualem (1976a), which has been particularly popular, describes the unsaturated hydraulic conductivity according to: Kr = SeL
‰
θ ∫0 hc−1dθ ___________ θs ∫0 hc−1dθ
2
[3.6.3–9]
Although there are no restrictions on the value of L, Mualem suggested that L = 0.5 on the basis of the optimization of conductivity data for 45 soils. The retention data needed for this and other conductivity models is usually given in the form of a closedform expression. The parametric model used for the retention should: (i) express capillary head as a function of water content, that is, hc(Se) or hc(θ), and (ii) prefer-
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ably be of a mathematical form that can be integrated analytically. Finally, it should be pointed out that many of the previously mentioned models are very similar. Mualem and Dagan (1978), Raats (1992), and Hoffmann-Riem et al. (1999) have discussed the commonality of many conductivity models based on the pore-size distribution. Mualem–van Genuchten Model. Although there are many different functions that can be used to describe and predict unsaturated hydraulic properties, the models by Mualem (1976a) and van Genuchten (1980) have been particularly popular and appear to provide an adequate description of the hydraulic properties of most soils (Leij et al., 1997). Inserting retention Eq. [3.6.3–2] into Eq. [3.6.3–9] yields the following expression for the relative hydraulic conductivity (van Genuchten, 1980): Kr = SeL[Iζ(p,q)]2
[3.6.3–10]
with ζ = Se1/m, p = m + 1/n, q = 1 − 1/n, and n ≥ 1, and where Iζ is the incomplete beta function. The expressions may be simplified by imposing the constraint m = 1 − 1/n: Kr = SeL[1 − (1 − Se1/m)m]2,
(m = 1 − 1/n, 0 < m 0) _________ =0 ∂x
for −L/2 ≤ x < 0 for x = 0 for 0 < x ≤ L/2 for x = ± L/2
[6.2–36a]
[6.2–36b]
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The solution of Eq. [6.2–29] and [6.2–36] is given in Crank (1975, p. 63) and Shackelford (1991). The total amount of material transferred from the spiked cell to the other can be derived in the same way as in the infinite case, and the solution is given in Barrer (1951) and Shackelford (1991). The two solutions for the finite problem are infinite series of exponentials, sine and cosine functions, and as such are considerably more complicated than for the infinite case. Procedures. Two half cells, sealed at one end and open at the other end, are filled with the porous material. If undisturbed samples are used, the half cells can be obtained by cutting a column containing the porous material into two halves. One of the cells is uniformly spiked with the solute. The spiking is most conveniently made by saturating the cell in solution containing the solute for a certain time. The time required to obtain uniform spiking is proportional to (L/2)2 and inversely proportional to Deff/R. If diffusion in unsaturated soil is desired, the cells can be drained after saturation with standard hanging water columns or a pressure plate apparatus. After these preconditioning steps, the two half cells are joined, fastened together with clamps, and sealed with electrician tape or equivalent material to prevent evaporation. A thin filter paper may be placed between the cells (Fig. 6.2–2). After a certain time t, the two half cells are separated. The time t should be chosen to obtain a reasonable resolution of the concentration profile within the half cells or an accurate measurement of the mass transferred between the cells. If Eq. [6.2–32] and [6.2–35] are used to determine the diffusion coefficient, the concentrations at x = ± L/2 should be equal to Ci or C0 at all times during the experiment. This requirement limits the maximum time of the diffusion experiment. Figure 6.2–3 gives an indication under which conditions the assumption of an infinite system is valid. The figure shows dimensionless concentration profiles in the nonspiked half cell for various dimensionless values %D && t/&/L. Note that %D && t/&/L is the square eff&R eff&R root of the dimensionless time Defft/(RL2). As long as the scaled concentrations are
Fig. 6.2–3. Concentration distributions in the nonspiked half of the cell for an infinite system at different times. The numbers indicated in the graph are dimensionless values of %D && t/&/L, corresponding eff&R to the square root of the dimensionless time. Curves are calculated with a dimensionless form of Eq. [6.2–32].
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zero at the border x/L, the system can be adequately described as infinite. The graphs indicate that this is true when %D && t/& %t& 0.12 ______ eff&R < 0.12 or __ < _______ L L %D && /R eff&&
[6.2–37]
The value of 0.12 corresponds to an approximate error of 0.02% of the normalized concentration (C − Ci)/(C0 − Ci) at the boundary x = ± L/2. Given a ballpark estimate of Deff/R, the duration t of the experiment and the length L of the diffusion cell can be chosen as to satisfy Eq. [6.2–37]. When using the concentration profile method (Eq. [6.2–32]), the half cells are sectioned into increments and solute concentrations are determined by an extraction and analysis procedure appropriate for the specific solute. Soil samples may be frozen after termination of the diffusion experiment to prevent continuing diffusion and to facilitate sectioning (Brown et al., 1964). One has to be careful with freezing, however, since this can induce solute redistribution (Stähli & Stadler, 1997). Alternatively, solute concentrations can be measured within the soil at different locations and as a function of time with suitable electrodes or with TDR probes (Section 6.1). The effective diffusion coefficient Deff or Deff/R (depending on whether R is known or not) is then determined by fitting Eq. [6.2–32] to the measured concentration profile or time sequence. If the column has been sectioned, a second estimate of Deff or Deff/R is easily obtained by using Eq. [6.2–35]. An illustration of a concentration profile obtained with the half-cell method is shown in Fig. 6.2–4. This example shows diffusion of tritium in a saturated freshwater sediment from an experiment performed by van Rees et al. (1991). The half cells were each 8 cm long with a diameter of 3.2 cm, and were made of Plexiglas. Tritium concentrations were determined 2 d after joining the two half cells. Retardation of tritium to sediment material was assumed zero. It is obvious from the concentration profiles that the infinite system describes the experiment adequately. The estimated effective diffusion coefficient for tritium is Deff =1.18 × 10−9 m2 s−1.
Fig. 6.2–4. Experimental concentration distribution and fitted model calculation (Eq. [6.2–32]) for two half cells. Data are taken from van Rees et al. (1991).
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Klute and Letey (1958) used the half-cell method to measure diffusion coefficients of rubidium in unsaturated porous media. Different water contents were adjusted by placing each half cell on a tension plate before joining them together. Comments. The half-cell method is simple to perform and has been used frequently. Examples of the application of the infinite problem (Eq. [6.2–32] and [6.2–35]) are given in, for example, Klute and Letey (1958), Porter et al. (1960), Phillips and Brown (1964), van Schaik and Kemper (1966), Bhadoria et al. (1991), Sawatsky and Oscarson (1991), and van Rees et al. (1991). The analysis for a finite column has been used by Robin et al. (1987). The biggest concern in the halfcell method is the establishment of good contact and the prevention of advective flow between the two cells, particularly under unsaturated conditions (Tinker, 1970). When measuring a concentration-distance profile, these errors are likely to be noticed, because poor contact between the half cells leads to a discontinuity in the concentration profile (Tinker, 1970; Olesen et al., 2000) and mass flow causes a displacement of the diffusion front (Tinker, 1970). To prevent the problem of insufficient contact, particularly when using undisturbed soil samples, Olesen et al. (2000) proposed use of a disturbed soil sample in one of the half cells, combined with the undisturbed sample in the other half cell. Since the diffusion coefficients in the two half cells will then be different from each other, the diffusion coefficient in the disturbed sample has to be determined in a separate experiment. The experimental data in this case will have to be analyzed with a numerical solution of the diffusion equation. The half-cell method allows determination of Deff or Deff/R, depending on whether the retardation factor R is known. If R is not known and knowledge of Deff is desired, an independent sorption experiment has to be carried out to determine R. 6.2.3.2.b The Ion Exchange Resin Paper Method Principle. A special case of the half-cell method is the ion exchange resin paper method. In this method a cell uniformly spiked with ions is brought into contact with an ion exchange resin (Vaidyanathan & Nye, 1966). This ion exchange resin adsorbs ions very rapidly and acts as a sink for ions. At least for short times, the resin maintains a zero concentration boundary. The total mass of ions Mt transferred from the spiked cell to the resin, assuming a semi-infinite cell, is given by (Crank, 1975, p. 32) Mt = 2AθC0%D &&& && eff t/π
[6.2–38]
The diffusion coefficient can then be obtained from Eq. [6.2–38] as Deff =
2 π Mt _____ __ ‰ 2AθC0 t
[6.2–39]
Procedures. A cell is filled with porous material and uniformly spiked with ions of interest. An ion exchange resin paper is then placed in contact with the surface of the porous material. A glass or Plexiglas lid may ensure good contact between paper and porous medium (Vaidyanathan & Nye, 1966; Warncke & Barber,
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1973). To maintain the moisture content of the soil, the experimental setup can be placed in a humidifier. After a short period of time (on the order of a few hours), the resin paper is removed. If measurements are needed for a longer period of time, the paper has to be replaced to maintain the zero concentration boundary condition (Barraclough & Tinker, 1982). The ions of interest are extracted from the resin and quantified with an appropriate analytical procedure. The measured quantity and time are then used to calculate the effective diffusion coefficient with Eq. [6.2–39]. Comments. The ion exchange resin paper method is simple and fast. The method has often been used to measure diffusion coefficients of plant nutrients in soils (Vaidyanathan & Nye, 1966; Warncke & Barber, 1973; Mullins & Summers, 1986). An advantage of the resin method is that the soil water content or the water potential can be adjusted prior to placing the resin paper onto the surface. Thus, the resin method is useful to determine diffusion coefficients under unsaturated conditions. An example of such an unsaturated diffusion experiment is given in Mullins and Summers (1986). 6.2.3.2.c The Reservoir Method Principle. The reservoir method comes in various modifications, each of which has its own initial and boundary conditions. The methods differ primarily in the number of reservoirs used, namely one or two, in the geometry of the soil samples, and the location of the tracer spiking. In general, soil or sediment samples are brought into contact with a reservoir containing a uniform aqueous solution. The solutes will diffuse from the reservoir into the soil or vice versa, depending on which part has been spiked. The effective diffusion coefficient is determined from either the concentration profile within the porous medium or the temporal change of the concentrations in the aqueous solution reservoir. Consider a soil sample void of solutes except for a residual concentration Ci and the aqueous reservoir on top of the sample spiked with concentration C0 (Fig. 6.2–5a). Assume that the reservoir is well stirred; that is, the mixing of solutes is instantaneous and complete. The initial and boundary conditions for an infinite system are then given as (Crank, 1975; Stoessell & Hanor, 1975) C(x,t = 0) =
for x ≤ 0 for x > 0
[6.2–40a]
Deffθ ∂C(x,t > 0) ____ ________ a ∂x /x=0
[6.2–40b]
C0
8 Ci
∂C(x,t > 0) ________ = ∂t /x=0 ∂C(x,t > 0) = 0 _________ ∂x C(x,t > 0) = Ci
for −a < x < 0
[6.2–40c]
for x → +∞
[6.2–40d]
where a is the depth of the solution reservoir (Fig. 6.2–5a). The solution of Eq. [6.2–29] and [6.2–40] is given by (Shackelford, 1991; van Rees et al., 1991) C(x,t) − Ci θRx θ2RDeff t x θ%R && Deff &t & ________ = exp ___ + _______ erfc ________ + _______ C0 − Ci ‰ a a2 ‰2%D &&& t/R & & a eff
[6.2–41]
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Fig. 6.2–5. Schematic of experimental setup for the reservoir method and coordinate selection. (a) Reservoir spiked, soil nonspiked; (b) reservoir nonspiked, soil spiked.
The concentration in the reservoir is obtained from Eq. [6.2–41] by setting x = 0 (Shackelford, 1991; van Rees et al., 1991) C(x = 0,t) − Ci θ2RDeff t θ%R && Deff &t & ___________ = exp _______ erfc _______ 2 C0 − Ci ‰ a ‰ a
[6.2–42]
Equation [6.2–41] can then be used to analyze the concentration distributions in a soil sample, and Eq. [6.2–42] can be used to analyze the temporal change of concentrations in the reservoir. When the spiking is reversed—that is, the soil sample is spiked with the solute at a concentration C0 and the reservoir is initially devoid of solutes or has a background concentration Ci (Fig. 6.2–5b)—C0 and Ci have to be exchanged in the boundary conditions Eq. [6.2–40] for the infinite system. The left-hand term of Eq. [6.2–41] is then equal to [C(x,t) − C0]/(Ci − C0), and the solution for the scaled concentration profile in the soil sample is obtained as C(x,t) − Ci θRx θ2RDeff t x θ%R && Deff &t & ________ = 1 − exp ___ + _______ erfc ________ + _______ C0 − Ci ‰ a a2 ‰2%D &&& t/R & & a eff
[6.2–43]
The temporal change of concentrations in the reservoir is in this case C(x = 0,t) − Ci θ2RDeff t θ%R && Deff &t & ___________ = 1 − exp _______ erfc _______ C0 − Ci ‰ a2 ‰ a
[6.2–44]
Equations [6.2–43] and [6.2–44] can be fitted to measurements of concentrations in the soil column and the reservoir, respectively.
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For both setups depicted in Fig. 6.2–5, there is an analogous mathematical solution for the case when the system cannot be assumed to be infinite. These solutions are more complicated than the solutions for the infinite situations and are not presented here. The reader is referred to Stoessell and Hanor (1975) and Shackelford (1991). A slightly different geometry of the setup is used if diffusion coefficients of soil aggregates are to be determined. Instead of a single core, a given number k of soil aggregates are placed into a well-stirred solution (Rao et al., 1980). Again, either the reservoir solution or the soil aggregates are initially spiked with a tracer. If the aggregates can be considered more or less spherical and sorption is assumed to be linear and instantaneous, the diffusion process can be described by ∂C Deff __ ∂ r2 ∂C __ = ___ __ ∂t Rr2 ∂r ‰ ∂r
[6.2–45]
where r is the radial coordinate. The initial and boundary conditions for a well-stirred reservoir spiked with an initial concentration C0 and for particles extending from r = 0 to r = b are C(r,t = 0) =
Ci
8 C0
for r < b for r = b
∂C(r,t > 0) 4πb2k ∂C(r,t > 0) ________ ________ = −Deffθ _____ ∂t / r=b ‰ Vr ∂r / r=b ∂C(r,t > 0) _________ =0 ∂r
for r = 0
[6.2–46a]
[6.2–46b]
[6.2–46c]
where Vr is the volume of the liquid in the surrounding reservoir. The solution of Eq. [6.2–45] and [6.2–46] for r ≤ b is (after Crank, 1975, p. 93–95) C(r,t) − Ci α 6αb ________ = ____ + ___ C0 − Ci α+1 r
∞ exp(−D q 2t/b2) sin(q r/b) _____________ eff n _______ n Σ 2 2 n=1
9 + 9α + qn α
sin(qn)
[6.2–47]
which yields for the reservoir concentration at r = b ∞ 2 2 C(t) − Ci = ____ α + 6α Σ exp(−D _______ effqn t/b ) __________________ n=1 C0 − Ci α+1 9 + 9α + q2nα2
[6.2–48]
In Eq. [6.2–47] and [6.2–48], α is the ratio of the volume of the reservoir and the capacity (i.e., fluid volume times retardation coefficient) of the aggregates, 3Vr α = _______ 4πb3kθR
[6.2–49]
and k is the number of aggregates with radius r and water content θ, and the qn are the nonzero roots of
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3qn tanqn = ______ 3 + αqn2
[6.2–50]
The diffusion coefficient is obtained from fitting of Eq. [6.2–48] for a given α to the measured temporal change of the reservoir concentration. If α or the aggregate radius b are not known a priori, they can be estimated from the equilibrium reservoir concentration, which is reached at infinite time, α αC0 + Ci C∞ = ____ (C0 − Ci) + Ci = _______ α+1 α+1
[6.2–51]
which leads to α = (C∞ − Ci)/(C0 − C∞)
[6.2–52]
and b=
1/3 3Vr ______ ‰ 4πkθRα
[6.2–53]
If the aggregates, rather than of the reservoir, are spiked with a concentration C0, then C0 and Ci have to be exchanged in Eq. [6.2–46a]. The corresponding solution is then C(r,t) − Ci 1 6αb ________ = ____ − ___ C0 − Ci α+1 r
∞
2
2
effqn t/b ) sin(q nr/b) _____________ _______ Σ exp(−D 2 2 n=1
9 + 9α + qn α
sin(qn)
[6.2–54]
from which the reservoir concentration at r = b can be calculated as described above. The capacity ratio α is now related to C∞, the concentration at equilibrium, in the following way α = (C0 − C∞)/(C∞ − Ci)
[6.2–55]
whereas b can also be estimated from Eq. [6.2–53]. Figure 6.2–6 shows scaled reservoir concentrations (C − Ci)/(C0 − Ci) as a function of dimensionless time T = Defft/(Rb2) for spiking of the reservoir or the aggregates and a capacity ratio of α = 9. Analogous to the soil aggregate method, a block of rock material can be placed into a well-stirred solution reservoir. The mathematical solution of the three-dimensional diffusion problem for rectangular rock blocks is rather complicated and can only be solved numerically, as illustrated by Ibaraki (2001) for an initially spiked reservoir. Procedures. If soil or sediment cores are investigated, they are packed into a column. The top portion of the column is left empty, or a separate column is later sealed on top, to hold the reservoir (Fig. 6.2–5). The soil or sediment material is saturated with spiked solute solution or water alone, depending on the specific method that is used. After the column is saturated and uniformly spiked, a filter paper may be placed on top of the soil to minimize disturbances during filling of the reser-
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Fig. 6.2–6. Scaled reservoir and average aggregate concentrations (C − Ci)/(C0 − Ci) as a function of dimensionless time T = Defft/(Rb2) for a reservoir in contact with spherical aggregates. Average aggregate concentrations refer to total mass of chemical inside the aggregate divided by the total amount of liquid inside the aggregate. Either the reservoir or the aggregates were spiked with the tracer, and the capacity ratio was α = 9. Curves are calculated with Eq. [6.2–47] and [6.2–54].
voir. The reservoir solution, either spiked or not, is then filled on top of the soil material. A lid is placed on the column to prevent evaporation. The reservoir can be gently stirred to assure uniform and instantaneous mixing. If stirring is not possible or desired, the reservoir height needs to be sufficiently small so that the assumption of the boundary conditions are still fulfilled. Samples are then periodically taken from the reservoir solution to measure solute concentrations over time. Alternatively, the reservoir concentrations can be monitored continuously with electrodes. At the end of the diffusion experiment, the reservoir solution is removed and the soil column can be sectioned. Solutes are extracted and quantified with an appropriate analytical procedure. Again, instead of sectioning the column at the end of the experiment, concentration profiles at different times can be measured with electrodes or TDR probes installed in the soil sample. The effective diffusion is finally estimated by fitting Eq. [6.2–41], [6.2–42], [6.2–43], or [6.2–44] to the experimental data. For soil aggregates, the procedure is similar, except that Eq. [6.2–48] or [6.2–54] is used for the analysis. Aggregates saturated with a possibly spiked solution are kept in an initially empty reservoir. The experiment is then started by adding the reservoir solution. Care has to be taken that the reservoir solution is well mixed and is in contact with all aggregate surfaces. Figure 6.2–7 shows an application of the reservoir methods for sediment cores, reported by van Rees et al. (1991). The tracer used was tritium, and the retardation factor R was assumed to be one. For the case of the spiked reservoir (Fig. 6.2–7a, b), the sediment columns were 8 cm long and 3.2 cm in diameter. The water content of the sediments was θ = 0.42 m3 m−3, and the depth of the reservoir was a = 1.24 cm. Figure 6.2–7a indicates that the assumption of a semi-infinite soil column, used to derive Eq. [6.2–41] and [6.2–42], is well justified for this experiment; that is, the concentration at the boundary x = 8 cm is zero. For the case of a spiked soil (Fig. 6.2–7c, d), the sediment column was 12.9 cm long and 4.4 cm in diameter.
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Fig. 6.2–7. Experimental concentration distributions and fitted model calculations for the reservoir method. (a,b) Spiked reservoir; (c,d) spiked sediment material. (a) Tritium concentration in sediment column, (b) tritium concentration in spiked reservoir, (c) tritium concentration in sediment column, no data were reported, solid line is Eq. [6.2–43] using the effective diffusion coefficient from example (d), (d) tritium concentration in nonspiked reservoir. Data and model parameters are taken from van Rees et al. (1991).
The water content was θ = 0.45 m3 m−3, and the depth of the reservoir was a = 1.97 cm. No data were reported for concentrations within the sediment, but a simulation shows that concentrations at the border x = 12.9 cm were not affected by the diffusion process and the assumption of a semi-infinite system is justified (Fig. 6.2–7c). In all cases shown in Fig. 6.2–7, the model simulations can reasonably describe the experimental data, and the fitted effective diffusion coefficients are indicated in each graph. The mathematical solutions presented here for the reservoir methods are derived for a semi-infinite soil column. In order to satisfy this condition experimentally, the initial solute concentration at the boundary x = L should remain constant. Figure 6.2–8a shows dimensionless plots of Eq. [6.2–41] for various dimensiont/&/L and dimensionless values Γ = θL/a. The parameter Γ repreless values %D && eff&R sents the ratio of the effective column length θL to the depth a of the reservoir. From Fig. 6.2–8a it can be seen that the assumption of a semi-infinite system is justified when %D &&&t/ & ______ eff &R < 0.24 or L
%t & 0.24 __ < _______ L %D & &&/ & eff&R
[6.2–56]
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Fig. 6.2–8. Concentration distributions in soil column for an infinite system with a spiked reservoir. (a) t/&/L; (b) the numbers are diThe numbers indicated in the graph are dimensionless values of %D && eff&R mensionless values of Γ = θL/a. Curves are calculated with a dimensionless form of Eq. [6.2–41].
This is valid independent of the value of Γ, since Γ does not affect the penetration distance of diffusion into the soil column very much (Fig. 6.2–8b). Note that Eq. [6.2–56] is identical to Eq. [6.2–37] when considering that L in Eq. [6.2–56] is the length of one half cell in the half-cell method. Given an estimate of Deff/R , Eq. [6.2–56] can be used to design the experiment to satisfy the semi-infinite condition. Figure 6.2–9 shows an example of the determination of diffusion coefficients of aggregates with the reservoir method (Gimmi, 1992, unpublished data). Porous glass particles with an average diameter of 2.5 mm and a porosity of 0.61 were saturated with a KCl solution. At time t = 0 they were brought into contact with a reservoir solution with lower salt concentration. A diffusion coefficient of Deff = 1.13 × 10−9 m2 s−1 was estimated from the temporal change of the reservoir concentration measured with electrodes, indicating a tortuosity τ = 0.565. The tortuosity value was then used in the description of the movement of volatile chlorinated hydrocarbons in a partly saturated, structured medium consisting of porous glass beads (Gimmi
Fig. 6.2–9. Experimental reservoir concentrations and fitted model calculations for a reservoir in contact with aggregates where the aggregates were spiked with KCl. α = 9.07, estimated diffusion coefficient Deff = 1.13 × 10−9 m2 s−1. Data are taken from Gimmi (unpublished data, 1992).
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et al., 1993). Other examples for diffusion in spherical porous aggregates can be found in Rao et al. (1980) and Grathwohl (1998). Comments. The reservoir method is one of the simplest methods to determine the diffusion coefficients in porous media. If the effective diffusion coefficient is determined from the temporal change of concentrations in the solution reservoir, soil sectioning and extraction is not necessary, making the method very efficient. This is especially true if continuous concentration measurements using devices such as electrodes are possible. The reservoir methods presented in this chapter require the reservoir to be completely mixed at all times. This mixing can be assured by mechanical stirring or by using a small reservoir depth a or reservoir volume Vr. For aggregates, the amount of reservoir solution should be large enough to ensure full contact between the solution and all aggregate surfaces. However, using a small reservoir volume has the drawback that its volume might significantly decrease when samples are taken for periodic measurements. This decrease in reservoir volume can be considered in the mathematical formulation of the diffusion problem. Van Rees et al. (1991) presented an analytical solution for the core setup when the sample taken from the reservoir is large and replaced with tracer-free solution. Inspection of the mathematical solutions of the reservoir problems reveals that the analysis of the diffusion experiment does not allow estimation of the apparent diffusion coefficient Deff/R, as was the case for the half-cell method. The reservoir method requires the retardation factor R to be known. There exist several specific experimental procedures with which both Deff and R can be determined simultaneously (Rowe et al., 1988; Glass et al., 1998; Kau et al., 1999). In a modification of the reservoir method for cores, two reservoirs are used at each side of the soil column (e.g., Skagius & Neretnieks, 1986). Steady-state setups are a special case of such a reservoir method, where the concentrations in the reservoirs are kept constant during the diffusion experiment (see previous section). The reservoir methods can also be used to measure effective diffusion coefficients in unsaturated soil. In this case, advective water flow between soil and solution reservoir caused by differences in water potential needs to be prevented. This can be achieved by adjusting the osmotic potential of the reservoir solution to match the matric potential of water in the soil sample. A semipermeable membrane needs to be used to separate soil and reservoir. An example of such an experiment is given by Sulaiman and Kay (1972). 6.2.3.3 Other Methods 6.2.3.3.a Nuclear Magnetic Resonance Methods Nuclear magnetic resonance (NMR) techniques allow measurements of diffusion coefficients in liquids and porous media. Since NMR methods are nondestructive and noninvasive, they provide a powerful and elegant tool to measure diffusion coefficients, particularly in microporous media. Diffusion processes have been studied in various porous media, such as porous pellets (Hollewand & Gladden, 1995a), silica beads (Hallmann et al., 1996; Stapf et al., 1998), sandstone (Tessier et al., 1997), and granular sludge (Lens et al., 1997). With NMR, especially
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small-scale processes can be studied, such as intraparticle and interparticle diffusion (Tallarek et al., 1999; Lens et al., 1999). We do not discuss the methodology of NMR-based measurements of diffusion coefficients here. Hollewand and Gladden (1995a, b) provide an excellent description of how diffusion coefficients can be determined from NMR measurements. A review of NMR methods related to transport processes in porous media is given by van As and van Dusschoten (1997). 6.2.3.3.b X-ray Methods X-ray absorption measurements are widely used to examine the three-dimensional spatial distribution of differently absorbing materials, like solid or water and air, or to quantify the elemental composition. If diffusing solutes with a large absorption contrast with respect to water are used, like KI, the advancement of solutes within the porous medium can be studied. Based on these observations, diffusion coefficients for the porous material can be estimated from Fick’s second law (Dowker et al., 1996; Kozul et al., 1999). Since these techniques are still under development, we refer the reader to the specific literature dealing with x-ray absorption and x-ray tomography for a detailed discussion of these procedures. 6.2.4 Conclusions The measurement of diffusion coefficients is based on Fick’s first or second law. In general, the diffusion coefficient is determined by analyzing experimental data with an analytical expression of the diffusion coefficient or by fitting an appropriate mathematical solution to experimental data. In this paper, we focused on some more common experimental methods, and demonstrated procedures needed to determine the diffusion coefficient. However, many more experimental setups are conceivable as long as a proper mathematical representation and a solution to the experiment can be formulated. It is also evident that concentration measurements can be obtained with a variety of techniques, such as electrodes, TDR sensors, extraction and quantification with an analytical procedure, NMR, or x-ray absorption. The choice of the appropriate measurement method depends on the type of tracer, on the porous medium used, and on the spatial scale of interest.
6.2.5 References Ampadu, K.O., K. Torii, and M. Kawamura. 1999. Beneficial effect of fly ash on chloride diffusivity of hardened cement paste. Cement Concrete Res. 29:585–590. Barraclough, P.B., and P.B. Tinker. 1982. The determination of ionic diffusion coefficients in field soils. II. Diffusion of bromide ions in undisturbed soil cores. J. Soil Sci. 33:13–24. Barrer, R.M. 1951. Diffusion in and through solids. Cambridge University Press, Cambridge, UK. Bhadoria, P.B.S., J. Kaselowsky, N. Claassen, and A. Jungk. 1991. Soil phosphate diffusion coefficients: Their dependence on phosphate concentration and buffer power. Soil Sci. Soc. Am. J. 55:56–60. Brown, D.A., B.E. Fulton, and R.E. Phillips. 1964. Ion diffusion: I. A quick-freeze method for the measurement of ion diffusion in soil and clay systems. Soil Sci. Soc. Am. Proc. 28:628–632. Cho, W.J., D.W. Oscarson, and P.S. Hahn. 1993. The measurement of apparent diffusion coefficients in compacted clays: And assessment of methods. Appl. Clay Sci. 8:283–294. Conkling, B.L., and R.W. Blanchar. 1986. Estimation of calcium diffusion coefficients from electrical conductivities. Soil Sci. Soc. Am. J. 50:1455–1459.
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Crank, J. 1975. The mathematics of diffusion. 2nd ed. Clarendon Press, Oxford University Press, Oxford, UK. Cussler, E.L. 1984. Diffusion. Mass transfer in fluid systems. Cambridge University Press, Cambridge, UK. Dowker, S.E.P., P. Anderson, and J.C. Elliott. 1996. X-ray microradiographic determination of diffusion coefficients within porous media based on Fick’s second law: Theory and error analysis. Microsc. Microanal. Microstruct. 7:49–63. Fick, A. 1855. Über Diffusion. Annal. Physik Chem. 94:59–86. Gimmi, T., H. Flühler, B. Studer, and A. Rasmuson. 1993. Transport of volatile chlorinated hydrocarbons in unsaturated aggregated media. Water Air Soil Pollut. 68:291–305. Glass, G.K., and N.R. Buenfeld. 1992. Theoretical assessment of the steady state diffusion cell test. J. Mater. Sci. 33:5111–5118. Glass, G.K., G.M. Stevenson, and N.R. Buenfeld. 1998. Chloride-binding isotherms from the diffusion cell test. Cement Concrete Res. 28:939–945. Graham, T. 1850. On the diffusion of liquids. Philos. Trans. R. Soc. London 140:1–46. Grathwohl, P. 1998. Diffusion in natural porous media; Contaminant transport, sorption/desorption and dissolution kinetics. Kluwer Academic Publisher, Boston, MA. Hallmann, M., K.K. Unger, M. Appel, G. Fletscher, and J. Kärger. 1996. Evaluation of transport properties of packed beds of microparticulate porous and nonporous silica beads by means of pulsed field gradient NMR spectroscopy. J. Phys. Chem. 100:7729–7734. Hollewand, M.P., and L.F. Gladden. 1995a. Transport heterogeneity in porous pellets—1. PGSE NMR studies. Chem. Eng. Sci. 50:309–326. Hollewand, M.P., and L.F. Gladden. 1995b. Transport heterogeneity in porous pellets—2. NMR imaging studies under transient and steady-state conditions. Chem. Eng. Sci. 50:327–344. Ibaraki, M. 2001. A simplified technique for measuring diffusion coefficients in rock blocks. Water Resour. Res. 35:1519–1523. Jost, W. 1952. Diffusion in solids, liquids, gases. Academic Press, New York, NY. Jungk, A., and N. Claassen. 1997. Ion diffusion in the soil-root system. Adv. Agron. 61:53–110. Jurinak, J.J., S.S. Sandhu, and L.M. Dudley. 1986. Ionic diffusion coefficients as predicted by conductometric techniques. Soil Sci. Soc. Am. J. 51:625–630. Kau, P.M.H., P.J. Binning, P.W. Hitchcock, and D.W. Smith. 1999. Experimental analysis of fluorid diffusion and sorption in clays. J. Contam. Hydrol. 36:131–151. Kemper, W.D. 1986. Solute diffusivity. p. 1007–1024. In A. Klute (ed.) Methods of soil analysis. Part 1. 2nd ed. ASA and SSSA, Madison, WI. Klute, A., and J. Letey. 1958. The dependence of ionic diffusion on the moisture content of nonadsorbing porous media. Soil. Sci. Soc. Am. Proc. 22:213–215. Kozul, N., G.R. Davis, P. Anderson, and J.C. Elliot. 1999. Elemental quantification using multiple-energy x-ray absorptiometry. Meas. Sci. Technol. 10:252–259. Lens, P., F. Vergeldt, G. Lettinga, and H. van As. 1999. 1H NMR characterization of the diffusional properties of methanogenic granular sludge. Water Sci. Technol. 39:187–194. Lens, P., L.H. Pol, G. Lettinga, and H. van As. 1997. Use of 1H NMR to study transport processes in sulfidogenic granular sludge. Water Sci. Technol. 36:157–163. Li, Y., and S. Gregory. 1974. Diffusion of ions in sea water and in deep-sea sediments. Geochim. Cosmochim. Acta 38:703–714. Lide, D.R. 1994. CRC handbook of chemistry and physics. 75th ed. CRC Press, Boca Raton, FL. Millington, R.J., and J.P. Quirk. 1961. Permeability of porous solids. Trans. Faraday Soc. 57:1200–1207. Moldrup, P., T. Olesen, D.E. Rolston, and T. Yamaguchi. 1997. Modelling diffusion and reacting in soils. 7. Predicting gas and ion diffusivity in undisturbed and sieved soils. Soil Sci. 162:632–640. Mullins, G.L., and L.E. Summers. 1986. Characterization of cadmium and zinc in four soil treated with sewage sludge. J. Environ. Qual. 15:382–387. Neretnieks, I. 1980. Diffusion in the rock matrix: An important factor in radionuclide retardation? J. Geophys. Res. 85:4379–4397. Nye, P.H. 1979. Diffusion of ions and uncharged solutes in soils and soil clays. Adv. Agron. 31:225–272. Olesen, T., P. Moldrup, and J. Gamst. 1999. Solute diffusion and adsorption in six soil along a soil texture gradient. Soil Sci. Soc. Am. J. 63:519–524. Olesen, T., P. Moldrup, K. Henriksen, and L.W. Petersen. 1996. Modeling diffusion and reaction in soils: IV. New models for predicting ion diffusivity. Soil Sci. 161:633–645.
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Olesen, T., P. Moldrup, Y. Yamaguchi, H.H. Nissen, and D.E. Rolston. 2000. Modified half-cell method for measuring the solute diffusion coefficient in undisturbed, unsaturated soil. Soil Sci. 165:835–840. Olesen, T., P. Moldrup, Y. Yamaguchi, and D.E. Rolston. 2001. Constant slope impedance factor model for predicting the solute diffusion coefficient in unsaturated soil. Soil Sci. 166:89–96. Olsen, S.R., and W.D. Kemper. 1968. Movement of nutrients to plant roots. Adv. Agron. 20:91–151. Oscarson, D.W., H.B. Hume, N.G. Sawatsky, and S.C.H. Cheung. 1992. Diffusion of iodide in compacted bentonite. Soil Sci. Soc. Am. J. 56:1400–1406. Palmer, C.J., and R.W. Blanchar. 1980. Prediction of diffusion coefficients from electrical conductance of soil. Soil Sci. Soc. Am. J. 44:925–929. Park, I.-S., D.D. Do, and A.E. Rodrigues. 1996. Measurement of the effective diffusivity in porous media by the diffusion cell method. Catal. Rev. Sci. Eng. 38:189–247. Pfankuch, H.O. 1963. Contibution à létude des déplacement de fluides miscible dans un milieu poreux. Rev. Inst. Fr. Pet. 2:215–270. Phillips, R.E., and D.A. Brown. 1964. Ion diffusion: II. Comparison of apparent self and counter diffusion coefficients. Soil Sci. Soc. Am. Proc. 28:758–763. Pearson, F.J. 1999. What is the porosity of a mudrock? p. 9–21. In A.C. Aplin et al. (ed.) Muds and mudstones: Physical and fluid flow properties. Special Publ. 158. Geological Society, London, UK. Porter, L.K., W.D. Kemper, R.D. Jackson, and B.A. Stewart. 1960. Chloride diffusion in soils as influenced by moisture content. Soil Sci. Soc. Am. Proc. 24:460–463. Rao, P.S.C., R.E. Jessup, J.M. Davidson, and D.P. Kilcrease. 1980. Experimental and mathematical description of nonadsorbed solute transfer by diffusion in spherical aggregates. Soil Sci. Soc. Am. J. 44:684–688. Reid, R.C., J.M.. Prausnitz, and B.E. Poling. 1987. The properties of liquids and gases. 4th ed. McGrawHill, New York, NY. Robin, M.J.L., R.W. Gillham, and D.W. Oscarson. 1987. Diffusion of strontium and chloride in compacted clay-based materials. Soil Sci. Soc. Am. J. 51:1102–1108. Robinson, R.A., and R.H. Stokes. 1959. Electrolyte solutions. 2nd ed. Academic Press, New York, NY. Rowe, R.K., C.J. Caers, and F. Barone. 1988. Laboratory determination of diffusion and distribution coefficients of contaminants using undisturbed clayey soil. Can. Geotech. J. 25:108–118. Saffman, P.G. 1960. A theory of dispersion in porous medium. J. Fluid Mech. 2:194–208. Sawatsky, N.G., and D.W. Oscarsen. 1991. Diffusion of technetium in dense bentonite under oxidizing and reducing conditions. Soil Sci. Soc. Am. J. 55:1261–1267. Shackelford, C.D. 1991. Laboratory diffusion testing for waste disposal—A review. J. Contam. Hydrol. 7:177–217. Skagius, K., and I. Neretnieks. 1986. Porosities and diffusivities of some nonsorbing species in crystalline rocks. Water Resour. Res. 22:389–398. Stähli, M., and D. Stadler. 1997. Measurement of water and solute dynamics in freezing soil columns with time domain reflectometry. J. Hydrol. (Amsterdam) 195:352–369. Stapf, S., R.J. Packer, R.G. Graham, J.-F. Thovert, and P.M. Adler. 1998. Spatial correlations and dispersion for fluid transport through packed glass beads studied by pulsed field-gradient NMR. Phys. Rev. E 16:463–469. Stoessell, R.K., and J.S. Hanor. 1975. A nonsteady state method for determining diffusion coefficients in porous media. J. Geophys. Res. 80:4979–4982. Sulaiman, W., and B.D. Kay. 1972. Measurement of the diffusion coefficient of boron in soil using a single cell techinque. Soil Sci. Soc. Am. Proc. 36:746–752. Tallarek, U., F.J. Vergeldt, and H. van As. 1999. Stagnant-mobile phase mass transfer in chromatography media: Intraparticle diffusion and exchange kinetics. J. Phys. Chem. B 103:7654–7664. Tang, N., and L.O. Nilson. 1992. Rapid determination of chloride diffusivity in concrete by applying electrical field. ACI Mater. J. 89:49–53. Tessier, J.J., K.J. Packer, J.-F. Thovert, and P.M. Adler. 1997. NMR measurements and numerical simulation of fluid transport porous solids. AIChE J. 43:1653–1661. Tinker, P.B. 1970. Some problems in the diffusion of ions in soils. Soc. Chem. Ind. Monogr. 37:120–134. Vaidyanathan, L.V., and P.H. Nye. 1966. The measurement and mechanism of ion diffusion in soils. II. An ion exchange resin paper method for measurement of the diffusive flux and diffusion coefficient of nutrient ions in soils. J. Soil Sci. 17:175–183. van As, H., and D. van Dusschoten. 1997. NMR methods for imaging of transport processes in microporous systems. Geoderma 80:389–403. van Brakel, J., and P.M. Heertjes. 1974. Analysis of diffusion in macroporous media in terms of a porosity, a tortuosity and a constrictivity factor. Int. J. Heat Mass Transfer 17:1093–1103.
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Published 2002
6.3 Solute Transport: Theoretical Background TODD H. SKAGGS AND FEIKE J. LEIJ, USDA-ARS, George E. Brown, Jr. Salinity Laboratory, Riverside, California
The transport of solutes in soils has always been of interest in agronomy because of the impact that nutrient and salt concentrations have on conditions for plant growth. During the last few decades, interest in solute transport has broadened due to concerns about the fate of chemicals in the subsurface environment, particularly with regard to the possible contamination of soil and groundwater by agricultural and industrial chemicals. Although significant progress has been made, the quantitative description of solute transport in soils remains a challenging and active area of research. In this section we review basic theoretical concepts and models for solute transport in soils (miscible displacement). Later sections discuss experimental procedures (Section 6.4) and methods of data analysis (Section 6.5). 6.3.1 Elementary Concepts 6.3.1.1 Solute Transport Experiments It is useful to first consider the general character of solute transport experiments before delving into specific theoretical transport models. Our discussion follows that of Nielsen and Biggar (1962) and Danckwerts (1953). Suppose a column packed with soil is attached to a water reservoir such that water flows through the column at a steady volumetric rate Q (Fig. 6.3–1). The volume of water in the column, V, is referred to as the pore volume. This terminology is used for both watersaturated and water-unsaturated columns, even though in the unsaturated case V is not equal to the total pore space as the name suggests (pore water volume is a more precise term, but it is not used). Now suppose that the column’s inlet reservoir, which originally contains no solute, is suddenly switched to an identical reservoir with solute concentration c0. Solute flows into the column and is transported through the soil, eventually exiting the soil and column in the column’s effluent (outflow). If we monitor the solute content of the effluent, we find that the sharp increase in solute concentration we imposed at the inlet does not materialize at the outlet. Instead, the effluent concentration increases gradually with time. The sharp front is diffused because individual solute particles take a variety of tortuous pathways through the soil, with some pathways being faster or slower on average. The time required for an individual particle to traverse the column is called the residence time. A plot of the effluent concentration as a function of time, ce(t), reflects the distribution of residence times in a column. The plot is called the breakthrough curve. Breakthrough curves are commonly represented using dimensionless variables. The normalized 1353
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Fig. 6.3–1. Schematic diagram of the apparatus used in laboratory column outflow experiments. During the experiment, the influent solute concentration is switched from c = 0 to c = c0 and the effluent concentration ce is measured as a function of time (after Jury et al., 1991).
cumulative outflow volume, T = Qt/V, is a dimensionless time variable that corresponds physically to the number of pore volumes eluted. At any time T, the dimensionless solute content of the column outflow may be expressed as the ratio of the effluent concentration to the influent concentration, ce(T)/c0. A plot of ce(T)/c0 vs. T constitutes the dimensionless breakthrough curve. 6.3.1.2 Breakthrough Curves Figure 6.3–2 shows some examples of typical breakthrough curves (Nielsen & Biggar, 1962; Danckwerts, 1953). Figure 6.3–2a corresponds to the case where all solute particles move through the column at the same rate, a situation that is referred to as piston flow. Piston flow never actually occurs, but is a useful point of reference when describing or comparing other breakthrough curves. The curves in Fig. 6.3–2b and 6.3–2c are typical of solutes that do not react with the soil solid phase or undergo other chemical reactions, with Fig. 6.3–2c showing a wider distribution of residence times than Fig. 6.3–2b. Note that in Fig. 6.3–2b and 6.3–2c the areas labeled “A” and “B” are equal. More precisely, the area below the breakthrough curve between zero and one pore volume is equal to the area above the curve greater than one pore volume (Nielsen & Biggar, 1962):
Fig. 6.3–2. Types of breakthrough curves observed in column outflow experiments. In each plot the ordinate is the ratio of the effluent concentration to the influent concentration, ce(T)/c0, and the abscissa is cumulative number of pore volumes eluted, T = Qt/V (after Nielsen & Biggar 1962; see also Danckwerts, 1953).
MISCIBLE SOLUTE TRANSPORT 1355
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1
∫0 ce(T)/c0 dT = ∫1 [1 − ce(T)/c0] dT
[6.3–1]
This relationship holds whenever there is no solute sorption or other chemical reaction, regardless of the shape of the breakthrough curve. Thus, a lack of solute reaction can be inferred from a measured breakthrough curve whenever Eq. [6.3–1] holds approximately. The integral on the left side of Eq. [6.3–1] is defined as holdback (Danckwerts, 1953), which has a value of zero in the case of piston flow and, assuming no sorption, a value between zero and one for other cases. Whenever there is minimal solute sorption or reaction, holdback is a useful qualitative measure of the ease or efficiency with which the fluid in the column is displaced (a larger value indicates displacement is more difficult). The holdback has often been found to be greater for (water) unsaturated soils than in saturated soils (Kutílek & Nielsen, 1994), except for highly structured (macroporous) soils. A similar breakthrough curve attribute is called the holdup, ∞
H = ∫0 [1 − ce(T)/c0] dT
[6.3–2]
The holdup is equal to the area above the curve and is related to the amount of solute that can be stored in the column (van Genuchten & Parker, 1984). The breakthrough curve in Fig. 6.3–2d is typical of a solute that undergoes sorption or exchange reactions. The chemical reaction causes the solute to appear at the column outlet later than in Fig. 6.3–2a, 2b, and 2c. Conversely, Fig. 6.3–2e shows solute arriving earlier, a situation that arises when not all of the pore water is accessible to the solute. For example, electrostatic forces may cause anionic solutes to be excluded from water near the surfaces of negatively charged soil particles. The breakthrough curve in Fig. 6.3–2f is characteristic of nonequilibrium transport. In equilibrium transport, solute sorption, exchange, and diffusion processes occur instantaneously, whereas in nonequilibrium transport these processes are time-dependent. Measured breakthrough curves are generally some combination of the curves shown in Fig. 6.3–2. Any plot of solute concentration vs. time at a fixed location in the soil may be referred to as a breakthrough curve. Thus, breakthrough curves are not limited to column outflow experiments and step changes in influent solute concentrations. They may be obtained, for example, in the field by applying a short solute pulse to the soil surface and monitoring the solute content at a fixed depth using solute sensors or other instrumentation (Section 6.1). Figure 6.3–3 shows breakthrough curves corresponding to those in Fig. 6.3–2 for the case of a short solute pulse applied to a laboratory column. Noteworthy is the long, thin tail on the nonequilibrium breakthrough curve (Fig. 6.3–3f). Plots of solute concentration vs. depth at a fixed time are called concentration profiles or depth profiles, and are also used to describe and analyze solute transport. 6.3.1.3 Moments When analyzing concentration distributions (vs. either time or depth), it is often desirable to quantify various attributes of the distributions. For example, in the case of breakthrough curves, we may wish to evaluate the average residence time or the width of the residence time distribution. These and similar attributes can be
Fig. 6.3–3. Breakthrough curves corresponding to those in Fig. 6.3–2 for the case of a short pulse of solute applied at the inlet.
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quantified with the moments of the distribution. The pth time moment, mp, is defined as ∞
mp = ∫0 t pc(t) dt
(p = 0,1,2,...)
[6.3–3]
where the breakthrough curve c(t) is the concentration history at a fixed location. In a column outflow experiment, c(t) ≡ ce(t). The zeroth and first moments are ∞
m0 = ∫0 c(t) dt
[6.3–4]
and ∞
m1 = ∫0 tc(t) dt
[6.3–5]
As will be seen later, the zeroth moment is related to the mass of solute contained in the breakthrough curve and the first moment is related to the mean residence time. Normalized moments are defined as Mp = mp/m0
[6.3–6]
and central moments as ∞
Mp′ = (1/m0)∫0 (t − M1)pc(t) dt
[6.3–7]
The second central moment, ∞
M2′ = (1/m0)∫0 (t − M1)2c(t) dt = M2 − (M1)2
[6.3–8]
relates to the width of the residence time distribution, and hence to the degree of solute spreading. Analogous expressions may be written for the depth moments of concentration profiles; for example, ∞
μp = ∫0 x pc(x) dx
[6.3–9]
and so forth (x is the space coordinate). For step changes in the inlet solute concentration (e.g., Fig. 6.3–2), these moment definitions are not useful because mp is infinite. In this case, moments can be defined using the compliment of the breakthrough curve (Leij & Dane, 1991; Yu et al., 1999); that is, ∞
Mp = ∫0 t p[1 − c(t)/c0] dt
[6.3–10]
Theoretical models that predict c(t) or c(x), or more generally c(x,t), usually contain parameters that correspond in some fashion to the moments of solute distributions. Before reviewing specific models, we present basic theoretical concepts that are common to each of them. 6.3.1.4 Mass Balance Consider one-dimensional transport of a solute in a soil–water–air system. The soil solid phase is assumed to be stationary and rigid. For a macroscopic con-
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trol volume of soil that contains no solute sinks or sources, conservation of mass requires ∂ct/∂t + ∂Js/∂x = 0
[6.3–11]
where ct is the total solute concentration (mass of solute per volume of soil), Js is the solute mass flux, and x and t are again the space and time coordinates, respectively. Assuming a negligible amount of solute in the vapor phase, the total solute concentration is ct = ρbS + θcr
[6.3–12]
where ρb is the soil bulk density, S is the sorbed solute concentration (mass of sorbed solute per unit mass of dry soil), θ is the volumetric water content, and cr is the dissolved solute concentration (mass of solute per unit volume of soil water).
6.3.1.5 Flux and Resident Concentrations The solution concentration cr is a resident concentration. A second type of concentration that is relevant to solute transport is the flux concentration, defined for one-dimensional transport as the ratio of the solute flux to the water flux (Jw): cf = Js/Jw
[6.3–13]
The flux concentration cf is the mass of solute per unit volume of fluid passing through a soil cross section during an elementary time interval. In the case of steady water flow (∂θ/∂t = 0 and ∂Jw/∂x = 0) and no solute sorption (S = 0), substituting Eq. [6.3–12] and [6.3–13] into Eq. [6.3–11] shows that the liquid phase resident and flux concentrations are related by ∂cr/∂t = −v( ∂cf/∂x)
[6.3–14]
where v = Jw/θ is the mean pore water velocity. In principle, distinguishing between flux and resident concentrations is important in the analysis of solute transport because different experimental procedures may yield data for different concentration types. For example, given the definition of cf, the concentration of column effluent is logically interpreted as a flux concentration, whereas concentrations measured by soil coring are resident concentrations. Other methods of solute detection may be conceptually ambiguous as to the type of concentration they measure (e.g., porous cup solution samplers). Similarly, flux and resident solute application modes may be distinguished (Kreft & Zuber, 1978). In practice, the difference between flux and resident concentrations is frequently negligible and the distinction is immaterial. Nevertheless, the distinction may be important in some instances (such as for short laboratory columns), so we present the theoretical models hereafter in terms of both flux and resident concentrations.
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6.3.2 Convection–Dispersion Model The convection–dispersion (also referred to as advection–dispersion) model of solute transport specifies that the total solute flux is Js = Jwcr − θDd(∂cr/∂x) − θDm(∂cr/∂x)
[6.3–15]
where Dm(θ,v) is the coefficient of mechanical dispersion and Dd is the soil liquid diffusion coefficient (see Section 6.2 for a discussion of Dd). The first term on the right side of Eq. [6.3–15] is the convective solute flux, where convection is defined as the passive movement of solute by flowing water. The second and third terms are the diffusive and dispersive fluxes, respectively. Diffusion is driven by concentration gradients according to Fick’s Law, while mechanical dispersion occurs because of local variability in the water velocity within and among pores. The key assumption in Eq. [6.3–15] is that the dispersive flux can be represented as a Fickian process with a constant transport coefficient, Dm. As shown by Taylor’s (1953) analysis of dispersion during laminar pipe flow, this assumption is reasonable only after some start-up time has passed and solute particles have had an opportunity to mix and experience the entire range of pore-scale velocities. For this reason, the convection–dispersion model is sometimes referred to as an asymptotic or long-time model. In general, Dm depends on the pore structure, the solute velocity, and the water content. The velocity dependence is often expressed as Dm = λv
[6.3–16]
where λ is the dispersivity (units of length). Typical values of λ are 0.5 to 2 cm in packed soil columns and 5 to 20 cm in the field (Jury et al., 1991). A rule of thumb is that λ is often found to be roughly equal to one-tenth of the transport distance. Except at very low flow velocities, Dm is much larger than Dd, and the diffusive flux is therefore usually negligible. Defining D = Dm + Dd as the effective diffusion–dispersion coefficient (subsequently referred to simply as the dispersion coefficient), Eq. [6.3–15] becomes Js = Jwcr − θD(∂cr/∂x)
[6.3–17]
Substituting Eq. [6.3–12] and [6.3–17] into Eq. [6.3–11] yields the convection–dispersion equation (CDE), r ∂(ρ ∂ ∂cr ∂(Jwcr) _________ bS + θc ) = __ θD ___ − ______ ∂t ∂x ‰ ∂x ∂x
[6.3–18]
The sorbed concentration S is related to cr through the sorption isotherm Γ, S = Γ(cr)
[6.3–19]
This relationship defines the sorbed concentration at all times if sorption is instantaneous. Although Γ is nonlinear for most solutes and soils, the linear isotherm
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S = Γ(cr) = Kd cr
[6.3–20]
may be a reasonable approximation at low solute concentrations. The proportionality constant Kd is referred to as the distribution coefficient. The linear approximation is important because it reduces Eq. [6.3–18] to a form that may be solved analytically. In the case of steady water flow (∂θ/∂t = 0 and ∂Jw/∂x = 0) and a uniform water content (∂θ/∂x = 0), substituting Eq. [6.3–20] into Eq. [6.3–18] yields R(∂cr/∂t) = D(∂2cr/∂x2) − v(∂cr/∂x)
[6.3–21]
where R is the retardation factor, R = 1 + ρbKd/θ
[6.3–22]
For a nonsorbing solute, Kd is zero and R is equal to one. In practice, R is commonly regarded as an apparent (empirical) retardation factor rather than the quantity defined explicitly by Eq. [6.3–22]. In that case, R may take on values less than one if a portion of the soil water is not accessible to the solute (anion exclusion). Equation [6.3–21] is the form of the CDE most commonly used for estimating transport parameters from laboratory displacement experiments. Hereafter we refer to Eq. [6.3–21], rather than Eq. [6.3–18], as the CDE. 6.3.2.1 Dimensionless Parameters It is sometimes desirable to express Eq. [6.3–21] in terms of the following dimensionless variables: T = vt/L, X = x/L, Pe = vL/D,
Cr = cr/c0
[6.3–23]
where L is the column length or some other characteristic transport distance, T is the number of pore volumes, Pe is the column Peclet number, and c0 is a constant reference concentration. Because Eq. [6.3–21] requires steady water flow and uniform water content, our definition of T is consistent with our prior definition, T = Qt/V = Qt/ALθ = vt/L, where A is the cross-sectional area of the column or transport volume and v = Jw/θ = Q/Aθ. Inserting these variables into Eq. [6.3–21] yields the following dimensionless form of the CDE: R(∂Cr/∂T) = (1/Pe)(∂2Cr/∂X2) − (∂Cr/∂X)
[6.3–24]
6.3.2.2 Flux Concentrations Substituting Eq. [6.3–13] into Eq. [6.3–17] shows that for convective–dispersive transport the flux and resident concentrations are related as (Kreft & Zuber, 1978) cf = cr − (D/v)(∂cr/∂x)
[6.3–25]
It can also be shown that in the case of a linearly and instantaneously sorbing solute, the CDE has the identical mathematical form when expressed in terms of cf (Kreft & Zuber, 1978; Parker & van Genuchten, 1984):
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R(∂cf/∂t) = D(∂2cf/∂x2) − v(∂cf/∂x)
[6.3–26]
Solutions of the CDE in terms of cf may be obtained by solving Eq. [6.3–26] subject to boundary and initial conditions by using the flux concentration as the dependent variable, or by using Eq. [6.3–25] to transform solutions for the resident concentration. Parker and van Genuchten (1984) give several additional expressions that relate resident and flux concentrations. 6.3.2.3 Boundary Conditions Solutions to the CDE may be obtained after specifying boundary and initial conditions. The proper form of boundary conditions for different experimental conditions and different concentration types has been the subject of much discussion in the literature. 6.3.2.3.a Inlet Boundary Condition When a solute is applied at a specified rate to a soil surface, the requirement of a continuous solute flux at the surface boundary leads to a flux-type (or thirdtype) boundary condition ‰
∂cr Jwcr − θD ___ = J c (t) ∂x / x=0 w 0
[6.3–27]
where the left side of Eq. [6.3–27] specifies the solute flux just inside the soil surface, the right side specifies the flux just outside, and c0 is the concentration of the invading fluid. The right side of Eq. [6.3–27] correctly quantifies the applied solute flux only when the inlet reservoir is physically disconnected from the soil surface (such as when the applied solution is sprinkled onto the column), or when a physically connected reservoir is perfectly mixed (implying no diffusion or dispersion occurs within the reservoir) and no diffusion occurs across the inlet boundary (van Genuchten & Parker, 1984). Equation [6.3–27] commonly results in a discontinuous macroscopic concentration across the inlet boundary, cr(x = 0,t) ≠ c0(t), with the size of the discontinuity at any instant depending on the magnitude of the dispersion term in Eq. [6.3–27] (Barry & Sposito, 1988). Dagan (1989, p.152) and others question the correctness of the left side of Eq. [6.3–27] and the applicability of the CDE in the vicinity of the inlet boundary, noting that D obtains its constant asymptotic value only after solute has traveled some distance inside the column (a distance of the order of a few pore scales). An alternative inlet condition is the concentration-type (or first-type) boundary condition, cr(0,t) = c0(t)
[6.3–28]
Equation [6.3–28] signifies a continuous macroscopic concentration at the inlet boundary rather than a continuous solute flux. Equation [6.3–28] would be appropriate for an experiment that controlled the solute concentration at the inlet boundary instead of the rate of solute application. It is generally considered that it is more
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difficult to control the boundary solute concentration than the boundary solute flux, particularly in an unsaturated flow experiment. Analytical solutions of the CDE subject to Eq. [6.3–28] have been shown to have a mass balance error, with the error being largest at early values of the dimensionless time vt/λR and diminishing rapidly at later times (van Genuchten & Parker, 1984). However, we may reason along with Dagan and Bresler (1985) and argue that the constant coefficient CDE is not applicable during the early times when errors are observed. In summary, because it is usually easier to control the rate of solute application than the boundary concentration, and because of possible mass balance errors associated with the concentration-type boundary condition, the flux-type (third-type) inlet condition (Eq. [6.3–27]) is ordinarily recommended when the CDE is written in terms of the resident concentration (van Genuchten & Parker, 1984). The corresponding recommended condition for the flux concentration is obtained by substituting Eq. [6.3–25] into Eq. [6.3–27], giving cf(0,t) = c0(t)
[6.3–29]
Thus, the concentration-type (first-type) inlet condition is preferred when the CDE is written in terms of cf (Eq. [6.3–26]). It is worth keeping in mind that for sufficiently long transport distances (L o D/v), solutions obtained with first- and thirdtype inlet conditions are similar and differences between flux and resident concentrations may be ignored. In this case, the first-type inlet condition may be preferred because it leads to slightly simpler mathematical expressions. 6.3.2.3.b Exit Boundary Condition For a semi-infinite or infinite system, the boundary condition at x → ∞ can be specified as c(∞,t) = 0
[6.3–30]
∂c(∞,t)/∂x = 0
[6.3–31]
or
where c can be either a flux or a resident concentration. For a finite system, formulating the exit condition is more complicated. One possibility, in analogy with Eq. [6.3–27], is to require a continuous solute flux across the exit boundary. This requirement leads to ∂cr Jwcr − θD ___ = J c (t) ‰ ∂x /x=L w e
[6.3–32]
where the left side of Eq. [6.3–32] specifies the solute flux just inside the exit surface and ce is the effluent concentration. An additional relationship is needed to determine the new unknown ce. If we assume that the macroscopic concentration is continuous at x = L, the necessary additional equation is cr(L,t) = ce(t)
[6.3–33]
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Substituting Eq. [6.3–33] into [6.3–32] yields the exit condition (Danckwerts, 1953) ∂cr(L,t)/∂x = 0
[6.3–34]
Equation [6.3–34] specifies that a concentration gradient does not exist at any time in the soil very close to the outlet. According to Eq. [6.3–17], this absence of a concentration gradient means that the solute flux across the exit boundary is purely convective, with no dispersion or diffusion occurring across the boundary. An alternative exit condition for the finite system is obtained by assuming that the exit boundary does not affect solute concentrations inside the column (van Genuchten & Parker, 1984). With this premise we may envision a fictitious semiinfinite soil column extending beyond the end of the finite column, and calculate cr(x ≤ L,t) using the semi-infinite exit condition Eq. [6.3–30]. Since we have not assumed that Eq. [6.3–33] is true, the column effluent concentration ce must be derived from cr(L,t) using Eq. [6.3–32]. When ce is calculated this way (assuming the third-type inlet condition), it can be shown that ce(t) ≡ cf(x = L,t), where cf(x,t) is the solution of Eq. [6.3–26] subject to Eq. [6.3–29] and [6.3–30] (van Genuchten & Parker, 1984). In other words, the effluent concentration is the same as the flux concentration at x = L calculated for a semi-infinite profile. It is unlikely that either of the two exit conditions for finite systems is strictly correct (Parlange et al., 1992). However, it turns out that for most systems of practical importance the two exit conditions predict breakthrough curves that are reasonably similar. Specifically, for Pe = vL/D greater than about 4 or 5, the predicted breakthrough curves are in good agreement, and for Pe greater than about 20 they are essentially identical (Parlange & Starr, 1975; van Genuchten & Alves, 1982; van Genuchten & Parker, 1984; Parlange et al., 1992). Predicted resident concentrations inside the column are then also in good agreement except near the exit boundary. The distance over which there is a discrepancy also decreases as Pe increases. Thus, for moderately large values of Pe, the predicted concentrations are the same and the choice between the two exit conditions is arbitrary. Because the analytical solution for the semi-infinite system condition is simpler than the solution for the finite system, the semi-infinite condition, Eq. [6.3–30], is preferable (van Genuchten & Parker, 1984). 6.3.2.4 Analytical Solutions When the soil initially (at time zero) contains no solute, c(x,0) = 0
[6.3–35]
the general form of the solution to both Eq. [6.3–21] and Eq. [6.3–26] for an arbitrary solute input c0(t) is t
c(x,t) = ∫0 c0(t − τ)k(x,τ) dτ
[6.3–36]
where the integral kernel k depends on the imposed boundary conditions, and the absence of a superscript on c indicates that the concentration may be either the flux
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or resident type. In light of the above discussion of boundary conditions, we recommend that the flux concentration be predicted by solving Eq. [6.3–26] subject to the first-type inlet condition Eq. [6.3–29] and semi-infinite exit condition Eq. [6.3–30], and that the resident concentration be predicted by solving Eq. [6.3–21] subject to the third-type inlet condition Eq. [6.3–27] and semi-infinite exit condition Eq. [6.3–30]. The integral kernels for these cases are, respectively, x − vt)2 _______ kf(x,t) = ___________ exp − (Rx 3 1/2 (4πDt /R) — 4DRt
[6.3–37]
and v (Rx − vt)2 ________ k r(x,t) = ________ exp − (πDRt)1/2 — 4DRt v2 exp __ vx erfc ______ Rx + vt − ____ 2DR ‰ D — (4DRt)1/2
[6.3–38]
For many commonly used solute inputs c0(t), the integral in Eq. [6.3–36] can be evaluated analytically. The solutions for a Dirac delta input, a step input, and a pulse input are given in terms of dimensional variables in Table 6.3–1, and in terms of dimensionless variables in Table 6.3–2. Figure 6.3–4 illustrates the predicted dimensionless resident and flux concentrations for a step input of solute. 6.3.2.5 Moments Moments of the CDE can be determined by integrating the appropriate analytical solution according to Eq. [6.3–3]. However, it is usually easier to obtain moments using the identity pc$(x,s) mp = (−1)plim d________ s→0 dsp
[6.3–39]
where c$ is the concentration in the Laplace domain and s is the Laplace variable. For example, in the case of a Dirac delta input of solute (Table 6.3–1), the Laplace transform of cf is (Jury & Roth, 1990) m exp __ vx 1 − 1 + _____ 4sDR c$f = __ Jw — 2D ‰ r v2
[6.3–40]
where m is the applied solute mass per unit area of soil (do not confuse m with the time moment mp). It is then straightforward to show, using Eq. [6.3–39], that the first two normalized moments are M1 = m1/m0 = Rx/v
[6.3–41]
M2 = m2/m0 = R2(2Dx + vx2)/v3
[6.3–42]
and
Selected moments for the CDE are given in Table 6.3–3.
0 < t < t0 t > t0
c a (x,t) cr(x,t) = 8 c0 a2 (x,t) − c a (x,t − t ) 0 2 0 2 0
cr(x,t) = c0 a2 (x,t)
m Kr (x,t) cr(x,t) = __ Jw
Resident concentration, cr(x,t)
1 erfc _______ Rx − vt + ____ v2t 1/2 exp − ________ (Rx − vt)2 − __ 1 1 + __ vx + ___ v2t exp __ vx erfc _______ Rx + vt a2 (x,t) = __ 2 — (4DRt)1/2 ‰ πDR — 4DRt 2 ‰ D DR ‰D — (4DRt)1/2
1 erfc _______ Rx − vt + __ 1 exp __ vx erfc ________ Rx + vt a1 (x,t) = __ 2 —(4DRt)1/2 2 ‰D — (4DRt)1/2
v (Rx − vt)2 − ____ v2 exp __ vx erfc _______ Rx + vt k r (x,t) = _______ exp − ________ (πDRt)1/2 ‰ 4DRt 2DR ‰D — (4DRt)1/2
2
0 < t < t0 t > t0
x − vt) _______ k f (x,t) = __________ exp − (Rx (4πDt 3/R)1/2 ‰ 4DRt
c a (x,t) cf(x,t) = 8 c0 a1 (x,t) − c a (x,t − t ) 0 1 0 1 0
cf(x,t) = c0 a1 (x,t)
m Kf (x,t) cf(x,t) = __ Jw
Flux concentration, cf(x,t)
† m = mass of solute per unit surface area of soil; δ (t ) = Dirac delta function; H(t) = Heaviside step function.
Definitions:
c c0(t) = 8 00
Pulse
c0(t) = c0H(t)†
Step function
m δ(t)† c0(t) = __ Jw
Dirac delta
Input, c0(t)
Table 6.3–1. Solutions to the convection–dispersion equation for various input functions (van Genuchten & Alves, 1982).
0 < t < t0 t > t0
1366 CHAPTER 6
0 < T < T0 T > T0
A (X,T) Cr(X,T) = 8 A2 (X,T) − A (X,T − T ) 2 2 0
1 erfc _______ RX − T + __ 1 exp (PeX) erfc ________ RX + T A1 (X,T) = __ 2 — (4RT/Pe)1/2 2 — (4RT/Pe)1/2
Pe 1/2 exp − ________ (RX − T)2 − ___ Pe exp (PeX) erfc _________ RX + T K r (X,T) = ____ ‰ πRT ‰ 4RT/Pe 2R — (4RT/Pe)1/2
RPeX 2 1/2 exp − (RX − T)2 _______ K f (X,T) = ______ ‰ 4πT 3 ‰ 4RT/Pe
A (X,T) C f(X,T) = 8 A1 (X,T) − A (X,T − T ) 1 1 0
C r(X,T) = A2 (X,T)
m K r (X,T) Cr(X,T) = ____ θLc0
0 < T < T0 T > T0
Dimensionless resident concentration, Cr = cr/c0
1 erfc _________ RX − T PeT 1/2 exp − ________ (RX − T)2 − __ 1 1 + PeX + PeT RX + T ___ exp (PeX) erfc _________ A2 (X,T) =__ + ____ 2 — (4RT/Pe)1/2 ‰ πR — 4RT/Pe 2 ‰ R — (4RT/Pe)1/2
0 < T < T0 T > T0
Cf(X,T) = A1 (X,T)
m K f (X,T) Cf(X,T) = ____ θLc0
Dimensionless flux concentration, Cf = cf/c0
† m = mass of solute per unit surface area of soil; δ (T) = Dirac delta function; H(T) = Heaviside step function; all other parameters defined in the text.
Definitions:
1 C0(T) = 8 0
Pulse
C0(T) = H(T)†
Step function
m δ (T)† C0(T) = ____ θLc0
Dirac delta
Dimensionless input, C0
Table 6.3–2. Solutions to the convection–dispersion equation in terms of dimensionless parameters (van Genuchten & Alves, 1982).
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Fig. 6.3–4. Dimensionless flux (C f) and resident (C r) concentration breakthrough curves as modeled by the convection–dispersion equation for two different values of the column Peclet number (Pe) and R = 1. The difference between C f and C r decreases as Pe increases.
6.3.3 Nonequilibrium Models Experimentally measured breakthrough curves and depth profiles often exhibit asymmetries and tailing that are not consistent with the CDE (Eq. [6.3–21]). Experiments have shown that the CDE provides a reasonably accurate description of nonreactive solute transport in uniformly packed, saturated laboratory columns and—less conclusively—uniformly packed, unsaturated columns (see, e.g., the literature review by Khan, 1986). For undisturbed soil columns, field soils, and reactive transport, CDE predictions are often less satisfactory. A variety of mechanisms may cause transport to differ from the CDE, including nonlinear sorption, rate-limited sorption, hysteretic desorption, immobile water, and the presence of soil chemical and physical heterogeneities. In this section we review modifications to the CDE that account for some of the above mechanisms. The resulting models are called nonequilibrium transport models. In Section 6.3.4, we discuss a stochastic–convective transfer function model that provides another alternative to the classical CDE. Nonequilibrium mechanisms can be classified as being either physical or chemical. Physical nonequilibrium arises when macroscopic heterogeneities exist in the flow field, such as for structured soils. Chemical nonequilibrium occurs when time-dependent sorption or exchange reactions are present. We review two of the most popular nonequilibrium models: the two-region physical nonequilibrium model (TRM) and the two-site chemical nonequilibrium model (TSM). The book edited by Selim and Ma (1998) provides a broader overview of nonequilibrium processes and models than is possible here.
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Table 6.3–3. Normalized (M1) and central (M2’) time moments of transport models for a Dirac delta input solute. Moment Model†
M1
M2′
CDE
Rx/v
2DR2x/v3
TRM
Rx/v
2 2θ 2θ(1 − β)2 R2x mDmR x ________ + ___________ θv3 αv
TSM
Rx/v
2x 2DR 2(1 − b)2R2x _____ + __________ 3 v αv
CLT
(x/L)exp(μL + σL2/2)
(x2/L2)exp(2μL + σL2)[exp(μL2) − 1]
† CDE, convection–dispersion equation; TRM, two-region model; TSM, two-site model; CLT, convective lognormal transfer function model.
6.3.3.1 Two-Region Model The two-region physical nonequilibrium transport model (alternatively referred to as the mobile–immobile model or the dual-porosity model) modifies the convection–dispersion model by stipulating that soil water is partitioned into mobile and immobile regions. Solute transport occurs by convection and diffusion–dispersion processes in the mobile region, and solute is transferred between the mobile and immobile regions by means of a first-order diffusion process (Coats & Smith, 1964). The two-region model is (van Genuchten & Wierenga, 1976) (θM + fρbKd)(∂cM/∂t) = θMDM(∂2cM/∂x2) − θMvM(∂cM/∂x) − α(cM − cIM)
[6.3–43a]
[θIM + (1 − f)ρbKd](∂cIM/∂t) = α(cM − cIM)
[6.3–43b]
where α is the first-order mass transfer coefficient and f is the fraction of sorption sites in the mobile region. The remaining terms in Eq. [6.3–43] have been defined above except that the subscripts “M” and “IM” have been added to some to indicate quantities defined for the mobile and immobile regions. For example, cM is the liquid phase solute concentration in the mobile region, and so forth. The volumetric water content is equal to the sum of the mobile and immobile volumes, θ = θM + θIM, while the mobile pore water velocity is defined as vM = Jw/θM. Griffioen et al. (1998) reviewed a number of studies and reported experimental values for α in the range of 0.0001 to 10 h−1. For simple systems with well-defined aggregate geometry, it is possible to derive explicit expressions for the rate constant based on diffusion mechanisms. Otherwise, α may be regarded as an empirical parameter (van Genuchten & Dalton, 1986). Griffioen et al. (1998) also found reported values for the mobile water fraction φ = θM/θ to be >0.6 for partially saturated soil columns. Kung (1990) and others have demonstrated with dye tracer experiments that φ may be T0
A (X,T) C 1f(X,T) = 8 A1 (X,T) − A (X,T − T ) 1 1 0
C1f(X,T) = A1 (X,T)
m F (X,T) C1f(X,T) = ____ θLc0
C1f(X,T) = c1f/c0
2ω − τ) 1/2 __ τ(T ______ † — R ‰ β(1 − β)
0 < T < T0 T > T0
0 < T < T0 T > T0
ω(T − τ) b ≡ _______ (1 − β)R
y
J(y,z) = 1 − exp(−z)∫0 exp(−λ)I0 [2(zλ)1/2]dλ†
T
A2 (X,T) = ∫0 G(X,τ)[1 − J(b,a)]dτ
X _____ βRPe 1/2 exp − Pe(βRX − τ)2 __________ G(X,τ) = __ τ ‰ 4πτ — 4βRτ
A (X,T) C2(X,T) = 8 A2 (X,T) − A (X,T − T ) 2 2 0
C 2(X,T) = A2 (X,T)
T m ______ ω C2(X,T) = ____ ∫ G(X,τ)h0(T,τ)dτ θLc0 (1 − β)R 0
C2(X,T)
† m, mass of solute per unit surface area of soil; δ(T), Dirac delta function; H(T), Heaviside function; Ij (-), jth order modified Bessel function; J(y,z), Goldstein’s J function.
__ a ≡ ωτ βR
hj (T,τ) = exp[−a − b]Ij
A1 (X,T) = ∫0 G(X,τ)J(a,b)dτ
T
1/2 ωT ω T τ F(X,T) = G(X,T)exp − __ + __ ∫0 ___________ G(X,τ)h1(T,τ)dτ ‰ βR R ‰ β(1 − β)(T − τ)
Definitions:
1 C0(T) = 8 0
Pulse
C0(T) = H(T)†
Step function
m δ (T)† C0 = ____ θLc0
Dirac delta
C0(T)
Table 6.3–4. Solutions to generalized nonequilibrium transport model (Toride et al., 1993).
MISCIBLE SOLUTE TRANSPORT 1373
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the inlet boundary condition and solute breakthrough at a particular depth in the soil, but do not concern ourselves with the details of the transport processes occurring in the soil. Since these processes may be very complicated in field soils, not having to specify a process model is potentially very advantageous. Suppose a narrow pulse of nonreactive solute is applied to the soil surface, c0(t) = (m/Jw)δ(t)
[6.3–52]
where m is the mass of applied solute per unit surface area of soil, Jw is the steady water flux, and δ(t) is the Dirac delta function. The normalized breakthrough curve at depth x = L is called the impulse response function for depth L: fLf(t) = (Jw/m)cf(L,t)
[6.3–53]
If we perform such an experiment and determine fLf(t), we can use the principle of superposition to predict transport to x = L for any subsequent input of nonreactive solute (Jury et al., 1986): t
cf(L,t) = ∫0 c0(t − τ)fLf(t) dτ
[6.3–54]
where c0(t) is an arbitrary input function. Equation [6.3–54] is the transfer function model (TFM). It is valid only if transport is linear. The impulse response function fLf(t) may be interpreted as a probability density function (pdf). In this case, fLf(t) is called the travel-time pdf, and fLf(t) dt is the probability that a solute particle released at the soil surface at time zero will reach the depth x = L in the time interval t to t + dt (Jury et al., 1986). The strength of the transfer function approach is that transport is predicted based on fLf(t), a function that can be determined experimentally. There is no need to specify a process model. The main drawback is that transport can be predicted only to the depth x = L. Also, the linearity assumption may not always be valid. 6.3.4.2 Stochastic–Convective Transfer Function Model The transfer function approach can also be used in conjunction with a transport process model, thus permitting transport predictions to depths other than x = L. For example, the convection–dispersion model may be cast as a transfer function model (Sposito et al., 1986). Of course, with this approach we again face the problem of specifying the correct process model. In this case, the advantage of using the transfer function approach is that it provides a powerful and convenient conceptual and analytical framework for deriving, evaluating, and comparing transport models. The reader is referred to Jury and Roth (1990) for numerous examples. A process model that is closely associated with transfer function modeling is the stochastic–convective model. According to the stochastic–convective model, solute transport occurs by convection along isolated pathways, with no mixing of water or solute between pathways (Simmons, 1982). While the average linear velocity along an individual pathway is constant, a distribution of velocities exists among different pathways. It can be shown that the stochastic–convective model
MISCIBLE SOLUTE TRANSPORT
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implies that the travel-time pdf at an arbitrary depth x is related to the pdf at x = L by (Jury, 1982) f f(x,t;L) ≡(L/x)fLf(tL/x)
[6.3–55]
The stochastic–convective TFM is then defined as t
cf(x,t) = ∫0 c0(t − τ)f f(x,τ;L) dτ
[6.3–56]
Now if we determine fLf(t) experimentally as before, we can invoke the stochastic–convective process model and use Eq. [6.3–56] to predict transport to all depths and times for subsequent solute applications. 6.3.4.3 Convective Lognormal Transfer Function Model It is usually convenient to specify a parameterized model for the probability density function f f. For the special case of a lognormal travel-time pdf (Jury, 1982), 1 [ln(tL/tux) − μL]2 f f(x,t;L) = ______ exp − _____________ %2 &π &σLt 9 2σL2 A
[6.3–57]
the stochastic–convective TFM is called the convective lognormal transfer function model, or CLT (Jury & Roth, 1990). In Eq. [6.3–57], μL is the mean of lnt at x = L and σL2 is the corresponding variance. The CLT transport parameters μL and σL are constant for all x, but their values are referenced to the fixed calibration depth x = L. The constant tu is equal to unity and has dimensions of time, thereby making the argument of the logarithm in Eq. [6.3–56] dimensionless (Ellsworth et al., 1996). For many common solute source functions, the CLT reduces to a form that is computationally more convenient. Table 6.3–5 gives CLT solutions that correspond to the CDE solutions given in Table 6.3–1. Table 6.3–3 contains selected moments of the CLT. Other parameterizations of the travel time pdf are possible. Jury and Roth (1990) discuss Gaussian and gamma pdfs. Utermann et al. (1990) use the gamma distribution to construct a bimodal travel time pdf. 6.3.4.4 Field Applications The stochastic–convective model can also be incorporated into a stochastic stream tube model of field-scale solute transport (Dagan & Bresler, 1979; Jury & Roth, 1990). Stream tube models conceptualize the field as collection of noninteracting stream tubes (or parallel columns). It is assumed that the same local process model governs transport within each tube, but the values of the transport parameters are randomly distributed across the field. If the local process model is piston flow, then the result is a field-scale stochastic–convective process. Other local process models are possible (Toride & Leij, 1996). Jury and Roth (1990) and Jury and Scotter (1994) give additional details on field-scale stream tube modeling. Liu and Dane (1996) expanded on the stream tube model by introducing various degrees of horizontal mixing between the solutes in the tubes.
Jwtu ln(tL/tux) − μL + σL2 g2(x,t) = ____ 1 + erf ________________ 2θL 9 — %2 &σL A
c g (x,t) cr(x,t) = 8 c0 g2 (x,t) − c g (x,t − t ) 0 2 0 2 0
1 1 ln(tL/tux) − μL g1(x,t) = __ + __ erf ___________ 2 2 — %2 &σL
0 < t < t0 t > t0
1 (ln(tL/tux) − μL)2 fr(x,t;L) = ________ exp − _____________ 1/2 (2π) σLx — 2σL2
c g (x,t) cf(x,t) = 8 c0 g1 (x,t) − c g (x,t − t ) 0 1 0 1 0
cr(x,t) = c0 g2 (x,t)
m f r (x,t) cr(x,t) = __ θ
Resident concentration, cr(x,t)
1 (ln(tL/tux) − μL)2 f f(x,t;L) = ________ exp − ______________ 1/2 (2π) σLt — 2σL2
0 < t < t0 t > t0
cf(x,t) = c0 g1 (x,t)
m f f (x,t) cf(x,t) = __ Jw
Flux concentration, cf(x,t)
† m, mass of solute per unit surface area of soil; δ(T), Dirac delta function; H(T), Heaviside function.
Definitions:
c c0(t) = 8 00
Pulse
c0(t) = c0H(t)†
Step function
m δ(t)† c0(t) = __ Jw
Dirac delta
Input, c0(t)
Table 6.3–5. Predicted concentrations for the convective lognormal transfer function model (CLT).
0 < t < t0 t > t0
1376 CHAPTER 6
MISCIBLE SOLUTE TRANSPORT
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6.3.4.5 Comments The stochastic–convective TFM presented here is appropriate for steady water flow and nonreactive solutes. The reader is referred to Jury (1982), Jury et al. (1986), and Jury and Roth (1990) for TFM generalizations that are appropriate for unsteady water flow, solute sorption, and solutes undergoing physical, chemical, or biological transformations. The Taylor (1953) analysis of dispersion shows that solute transport in a macroscopically homogeneous medium starts out as a stochastic–convective process and evolves at later times to a convective–dispersive process. Thus, the stochastic–convective transfer function model is an early-time model that is appropriate in soils when the time required for solute mixing is long relative to the residence time. Jury and Roth (1990) noted that these time scales may be related to the geometry of the system. The cross-sectional area of laboratory columns is usually small relative to their length, whereas in the field the areal extent of the field or plot is often large relative to the transport depth. It is plausible, then, that the convective–dispersive model is more appropriate for columns and the stochastic–convective model more appropriate for the field. Of course, such a conclusion cannot be made without also considering the scales of the heterogeneities. Ellsworth and Jury (1991), for example, found that the CDE provided a good description of field average transport. Despite the differences between convective–dispersive and stochastic–convective transport, it is important to realize that the two models can be calibrated so that they predict breakthrough curves that are nearly identical at a given depth (Simmons, 1982; Jury & Roth, 1990). Differences in predicted transport become apparent only when models calibrated for one depth predict transport to other depths. Thus, the correctness of a transport model (and model parameters) cannot be established through the analysis of a single breakthrough curve.
6.3.5 Transport Equation Generalizations The CDE can be generalized to account for the presence of solute sinks and sources in the soil. Equation [6.3–18] becomes then ∂(ρbS + θcr)/∂t − (∂/∂x)[θD(∂cr/∂x)] + ∂(Jwcr)/∂x + r = 0
[6.3–58]
where r is a generalized solute sink (r > 0) or source (r < 0) term. In principle many forms can be used, but r is typically represented using a combination of zero- and first-order rate terms (Nielsen et al., 1986), namely, r = θμliqcr + ρbμsS − θγliq(x) − ρbγs(x)
[6.3–59]
where μliq and μs are first-order decay coefficients for solute degradation in the liquid and solid phases, respectively, and γliq and γs are zero-order production terms for the liquid and solid phases. For the case of linear equilibrium sorption (S = Kdcr), Eq. [6.3–58] then becomes
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R(∂cr/∂t) − D(∂2cr/∂x2) + v(∂cr/∂x) + μcr −γ(x) = 0
[6.3–60]
where μ and γ are combined first- and zero-order coefficients: μ = μliq + ρbμsKd/θ
[6.3–61]
γ = γliq(x) + ρbγs(x)/θ
[6.3–62]
The reader is referred to Jury and Roth (1990) for similar generalizations of the CLT, and to Toride et al. (1993) for generalizations of the TSM and TRM. 6.3.6 References Barry, D.A., and G. Sposito. 1988. Application of the convection–dispersion model to solute transport in finite soil columns. Soil Sci. Soc. Am. J. 52:3–9. Brusseau, M.L., R.E. Jessup, and P.S.C. Rao. 1989. Modeling the transport of solutes influenced by multiprocess nonequilibrium. Water Resour. Res. 25:1971–1988. Cameron, D.R., and A. Klute. 1977. Convective–dispersive solute transport with a combined equilibrium and kinetic adsorption model. Water Resour. Res. 13:183–188. Coats, K.H., and B.D. Smith. 1964. Dead-end pore volume and dispersion in porous media. Soc. Petrol. Eng. J. 4:73–84. Dagan, G. 1989. Flow and transport in porous formations. Springer-Verlag, New York, NY. Dagan, G., and E. Bresler. 1979. Solute dispersion in unsaturated heterogeneous soil at field scale. I. Experimental methodology. Soil Sci. Soc. Am. J. 43:461–467. Dagan, G., and E. Bresler. 1985. Comment on “Flux-averaged and volume averaged concentrations in continuum approaches to solute transport” by J.C. Parker and M.Th. van Genuchten. Water Resour. Res. 21:1299–1300. Danckwerts, P.V. 1953. Continuous flow systems. Chem. Eng. Sci. 2:1–13. Ellsworth, T.R., and W.A. Jury. 1991. A three-dimensional field study of solute transport through unsaturated, layered, porous media. 2. Characterization of vertical dispersion. Water Resour. Res. 27:967–981. Ellsworth, T.R., P.J. Shouse, T.H. Skaggs, J.A. Jobes, and J. Fargerlund. 1996. Solute transport in unsaturated soil: Experimental design, parameter estimation, and model discrimination. Soil Sci. Soc. Am. J. 60:397–407. Gerke, H.H., and M.Th. van Genuchten. 1993. A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour. Res. 29:305–319. Griffioen, J.W., D.A. Barry, and J.-Y. Parlange. 1998. Interpretation of two-region model parameters. Water Resour. Res. 34:373–384. Jarvis, N.J., P.E. Jansson, P.E. Dik, and I. Messing. 1991. Modelling water and solute transport in macroporous soils. I. Model description and sensitivity analysis. J. Soil Sci. 42:59–70. Jury, W.A. 1982. Simulation of solute transport using a transfer function model. Water Resour. Res. 18: 363–368. Jury, W.A., W.R. Gardner, and W.H. Gardner. 1991. Soil physics. John Wiley and Sons, Inc., New York, NY. Jury, W.A., and K. Roth. 1990. Transfer functions and solute movement through soil. Birkhauser Verlag, Boston, MA. Jury, W.A., and D.R. Scotter. 1994. A unified approach to stochastic–convective transport problems. Soil Sci. Soc. Am. J. 58:1327–1336. Jury, W.A., G. Sposito, and R.E. White. 1986. A transfer function model of solute transport through soil, 1, Fundamental concepts. Water Resour. Res. 22:243–247. Khan, A.-U.-H. 1986. A laboratory test of the validity of the convection–dispersion equation, Ph.D. diss. University of California, Riverside, CA. Kreft, A., and A. Zuber. 1978. On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions. Chem. Eng. Sci. 33:1471–1480. Kung, K.-J.S. 1990. Preferential flow in a sandy vadose zone. 1. Field observation. Geoderma. 46:51–58. Kutílek, M., and D.R. Nielsen. 1994. Soil hydrology. Catena Verlag, Reiskirchen, Germany.
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Leij, F.J., and J.H. Dane. 1991. Solute transport in a two-layer medium investigated with time moments. Soil Sci. Soc. Am. J. 55:1529–1535. Liu, H.H., and J.H. Dane. 1996. An extended transfer function model of field-scale solute transport: model development. Soil Sci. Soc. Am. J. 60:986–991. Ma, L., and H.M. Selim. 1998. Coupling of retention approaches to physical nonequilibrium models. p. 83–115. In H.M. Selim and L. Ma (ed.) Physical nonequilibrium in soils. Ann Arbor Press, Chelsea, MI. Nielsen, D.R., and J.W. Biggar. 1962. Miscible displacement in soils: III. Theoretical considerations. Soil Sci. Soc. Am. Proc. 26:216–221. Nielsen, D.R., M.Th. van Genuchten, and J.W. Biggar. 1986. Water flow and solute transport processes in the unsaturated zone. Water Resour. Res. 22:89S–108S. Nkedi-Kizza, P., J. W. Biggar, H.M. Selim, M.Th. van Genuchten, P.J. Wierenga, J.M. Davidson, and D.R. Nielsen. 1984. On the equivalence of two conceptual models for describing ion exchanges during transport through an aggregated oxisol. Water Resour. Res. 20:1123–1230. Parker, J.C., and M.Th. van Genuchten. 1984. Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport. Water Resour. Res. 20:866–872. Parlange, J.-Y., and J.L. Starr. 1975. Linear dispersion in finite columns. Soil Sci. Soc. Am. Proc. 39:817–819. Parlange, J.-Y., J.L. Starr, M.Th. van Genuchten, D.A. Barry, and J.C. Parker. 1992. Exit condition for miscible displacement experiments. Soil Sci. 153:165–171. Selim, H.M., and M.C. Amacher. 1988. A second-order kinetic approach for modeling solute retention and transport in soils. Water Resour. Res. 24:2061–2075. Selim, H.M., J.M. Davidson, and R.S. Mansell. 1976. Evaluation of a two-site adsorption–desorption model for describing solute transport in soils. p. 444–448. In Proceedings of the Summer Computer Simulation Conf. Washington, DC. 12–14 July 1976. Simulation Councils, La Jolla, CA. Selim, H.M., and L. Ma. 1995. Transport of reactive solute in soils: A modified two-region approach. Soil Sci. Soc. Am. J. 59:75–82. Selim, H.M., and L. Ma (ed.) 1998. Physical nonequilibrium in soils. Ann Arbor Press, Chelsea, MI. Seyfried, M.S., and P.S.C. Rao. 1987. Solute transport in undisturbed columns of an aggregated tropical soil: Preferential flow effects. Soil Sci. Soc. Am. J. 51:1434–1444. Simmons, C.S. 1982. A stochastic–convective transport representation of dispersion in one dimensional porous media systems. Water Resour. Res. 18:1193–1214. Skopp, J., W.R. Gardner, and E.J. Tyler. 1981. Miscible displacement in structured soils: Two region model with small interaction. Soil Sci. Soc. Am. J. 45:837–842. Sposito, G., R.E. White, P.R. Darrah, and W.A. Jury. 1986. A Transfer function model of solute transport through soil. 3. The convection–dispersion equation. Water Resour. Res. 22:255–262. Steenhuis, T.S., J.-Y. Parlange, and M.S. Andreini. 1990. A numerical model for preferential solute movement in structured soils. Geoderma 46:193–208. Taylor, G.I. 1953. Dispersion of soluble matter in solvent flowing slowly through a pipe. Proc. R. Soc. London. Ser. A. 219:186–203. Toride, N., F.J. Leij. 1996. Convective–dispersive stream tube model for field-scale transport. I. Moment analysis. Water Resour. Res. 60:342–352. Toride, N., F.J. Leij, and M.Th. van Genuchten. 1993. A comprehensive set of analytical solutions for nonequilibrium solute transport with first-order decay and zero-order production. Water Resour. Res. 29:2167–2182. Utermann, J., E.J. Kladivko, and W.A. Jury. 1990. Evaluating pesticide migration in tile-drained soils with a transfer function. J. Environ. Qual. 19:707–714. Valocchi, A.J. 1985. Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resour. Res. 21:808–820. van Genuchten, M.Th., and W.J. Alves. 1982. Analytical solutions of the one-dimensional convectivedispersive solute transport equation. USDA Tech. Bull. 1661. U.S. Gov. Print. Office, Washington, DC. van Genuchten, M.Th., and F.N. Dalton. 1986. Models for simulating salt movement in aggregated filed soils. Geoderma 38:165–183. van Genuchten, M.Th., and J.C. Parker. 1984. Boundary conditions for displacement experiments through short laboratory soil columns. Soil Sci. Soc. Am. J. 48:703–708. van Genuchten, M.Th., and E.A. Sudicky. 1999. Recent advances in vadose zone flow and transport modeling. p. 155–193. In M.B. Parlange and J.W. Hopmans (ed.) Vadose zone hydrology. Oxford University Press, New York, NY.
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van Genuchten, M.Th., and P.J. Wierenga. 1976. Mass transfer studies in sorbing porous media: I. Analytical solutions. Soil Sci. Soc. Am. J. 40:473–480. Yu, C., A.W. Warrick, and M.H. Conklin. 1999. A moment method for analyzing breakthrough curves of step inputs. Water Resour. Res. 35:3567–3572.
Published 2002
6.4 Solute Transport: Experimental Methods TODD H. SKAGGS, USDA-ARS, George E. Brown, Jr. Salinity Laboratory, Riverside, California G. V. WILSON, USDA National Sedimentation Laboratory, Oxford, Mississippi PETER J. SHOUSE, AND FEIKE J. LEIJ, USDA-ARS, George E. Brown, Jr. Salinity Laboratory, Riverside, California
Section 6.3 reviewed basic theoretical principles and terminology associated with solute transport experiments and discussed several theoretical transport models. Here we cover experimental procedures for measuring breakthrough curves and related data. Methods for analyzing transport data are presented in Section 6.5. 6.4.1 Laboratory 6.4.1.1 Introduction A typical laboratory column outflow experiment was described in Section 6.3.1. The experiment consists of establishing steady water flow conditions in a soil column and then introducing a solute at the inlet while maintaining the same flow rate. Outflow samples are collected and analyzed for solute content to determine the breakthrough curve. Several variations of this experiment are possible, and experimental procedures may vary considerably depending on the experimental objectives, the size of the column, and the soil properties. 6.4.1.2 Sample Collection and Preparation Soil columns are referred to as disturbed or undisturbed. In disturbed columns, soil is packed uniformly into the column, often after grinding and sieving. Soil is normally packed to the bulk density that was observed in the field. Packing can be done using either a wet or dry procedure. Klute (1986) recommended the following wet packing procedure (see Section 3.3.2.1). Place soil in a container (e.g., a bucket or a bag) and add sufficient water to the soil to make the soil slightly cohesive. A gravimetric water content of about 5% for sands and 8 to 10% for finer-textured soils is generally suitable. If the soil is not uniformly wet, cover or seal the container to minimize evaporative losses and allow the water to redistribute and equilibrate for 2 or 3 d, periodically mixing the soil. After equilibration, sample the soil and determine its gravimetric water content, θg. Calculate the mass of moist soil, Mwet, needed to pack a column with volume Vc to the desired dry bulk density, ρb. The required mass is given by Mwet = ρb(1 + θg)Vc 1381
[6.4–1]
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Lastly, weigh the calculated mass of soil and pack it into the column uniformly with a piston or similar device. For long columns, mark the column wall and pack the column in layers using weighed masses that are appropriate for the volume between markings. Lightly raking the top of one layer before packing the next helps assure uniform contact between layers. To use a dry packing procedure, weigh the mass of dry soil, Mdry, needed to obtain the desired dry bulk density, Mdry = ρbVc
[6.4–2]
Using a funnel or tube, add the weighed soil to the column slowly, while tapping the sides of the column and occasionally tamping the sample. Again, mark long columns and pack them in weighed layers. A special apparatus to uniformly pack soil columns was described by Dane et al. (1992). Grinding aggregates and packing columns destroys soil structure and macropores. Other packing procedures try to minimize soil disturbance by using natural aggregates in the packing procedure (Selim et al., 1987). Nevertheless, because of the importance of soil structure in flow and transport processes, transport in disturbed columns is frequently not representative of transport in the field. Experiments on undisturbed soil columns may yield results that are more representative of field conditions. Undisturbed columns are created by sampling the soil in situ using the column as the sample holder. For example, a sample may be collected by driving a thin-walled cylinder with a sharpened edge into the soil. It is better to drive the cylinder with a hydraulic press system than a hammering system because a press system causes less sample disturbance. Alternatively, undisturbed columns can be obtained by excavating around the sample and placing or constructing the column housing around the exposed surfaces. The sides of the exposed sample may be coated with an inert adhesive or material such as parafin wax (Jardine et al., 1988) before the housing is attached. The coating helps stabilize the exposed soil, secures the housing, and prevents water flow between the soil and the column wall during experiments. Several procedures for undisturbed sampling have been reported. The sampling procedures are often specific to the objectives of the study and the experimental site, so we will not provide details. We refer the reader to Tindall et al. (1992), who thoroughly describe one sampling procedure and discuss several others. 6.4.1.3 Apparatus Example laboratory column apparatuses are shown schematically in Fig. 6.4–1 and 6.4–2. These apparatuses consist of the soil column, a precise way of controlling the flow or head at the inlet boundary, and a fraction collector for sampling outflow. The column housing and tubing are made from nonreactive materials that do not affect solute transport. Materials used for the column housing include Plexiglas, PVC pipe, glass-coated steel, and stainless steel. In soils research, experiments are conventionally performed with the column vertical and flow downward. Horizontal columns are common in the chemical engineering literature. Some soils researchers have used a vertical column and upward flow (Ma & Selim, 1994a) because it is easier to maintain saturated conditions.
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Fig. 6.4–1. Schematic of the unsaturated flow system used by Seyfried and Rao (1987). The apparatus is an example of the head control method, with the head controlled by a Mariotte device at the inlet and by a hanging water column at the outlet. The other components are fritted glass endplates, manometers, air vents, and fraction collector (modified from Seyfried & Rao, 1987).
For unsaturated flow experiments, tensiometers or other instrumentation should be installed to monitor hydraulic conditions (matric potential, water content) in the soil. Sealable air vents should also be incorporated into the column housing so that unsaturated flow conditions can be established and maintained (Jardine et al.,1993; Wilson et al., 1998). The setup is similar to that used to measure unsaturated hydraulic conductivity. At the column entrance, a porous membrane, plate, or fabric covers the soil to help distribute inflowing solution over the entire soil surface. An end cap holds the membrane in place and connects the column to the inlet reservoir. Additional sealable openings in the cap are used to bleed entrapped air and solution. For undisturbed samples, hydraulic continuity may be improved by incorporating a thin layer of sand or similar material between the soil and end plate (Jardine et al., 1993; Ward et al., 1994). The column exit is fitted with a high flow porous plate made of an inert material such as fritted glass or stainless steel. Fritted glass is relatively inexpensive but is fragile and may react with some solutes such as phosphates. Stainless-steel plates are more rugged but are generally more expensive. If only saturated flow experiments are planned, it may be adequate to attach cheesecloth over the outflow end with a rubber band (this works if the cloth and end cap are sufficient to hold
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Fig. 6.4–2. Schematic of the flux-control system used by Wilson et al. (1998). The setup works for unsaturated and saturated flow. The tensiometers allow monitoring of the internal hydraulic gradient (modified from Wilson et al., 1998).
the sample in place during the experiment). Whatever the material, the plate or covering must be fitted into the end cap in a manner that eliminates solution leakage, usually with some type of O-ring configuration (as done, for example, in Tempe pressure cells; see Section 3.3.2.3). An opening in the end cap funnels column outflow to the fraction collector. The space between the end plate and end cap should be small so as to minimize solute mixing outside the soil (again, Tempe cells provide an excellent design example). Steady-state water flow conditions are maintained by applying solution from a constant head source (head control method) or by applying solution with a precision constant-volume pump (flux control method). Figure 6.4–1 illustrates the head control method with a Mariotte device controlling the inlet pressure head. The inlet pressure head can also be controlled with a tension infiltrometer (e.g., Langner et al., 1998). The flux control method is illustrated in Fig. 6.4–2. In saturated flow experiments the column outlet drains freely into the fraction collector. In unsaturated experiments the outlet is controlled by suction. Suction can be applied with a hanging water column (Fig. 6.4–1; Nielsen & Biggar, 1961; Seyfried & Rao, 1987) or a regulated vacuum chamber (Fig. 6.4–2; Wilson et al., 1998; Jardine et al., 1993). With the vacuum chamber method, the fraction collector is placed inside the chamber and aligned under the outlet. Alternatively, Langner et al. (1998) described a column setup in which vacuum is applied directly to the outlet and a chamber is not required. The vacuum chamber method is often the preferred method of applying suction because it provides relatively good con-
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trol of the lower boundary condition and permits sampling with minimal solute mixing outside the soil. The main drawback is that the fraction collector and effluent samples cannot be accessed without breaking the vacuum, meaning there is no way to prevent evaporative losses from the samples or to measure their solute content while the experiment is in progress. Lastly, we suggest incorporating sensors to monitor solute concentrations inside the soil at different depths. Measuring breakthrough curves at multiple depths allows for a better assessment of model fits and parameter estimates (Section 6.5). Studying transport between sensors also helps circumvent difficulties in specifying and controlling column boundary conditions. Time domain reflectometry (TDR) (Section 6.1) is a noteworthy technology in this regard because it can measure solute breakthrough curves while simultaneously monitoring water content (Nadler et al., 1991; Ward et al., 1994; Hart & Lowery, 1998). 6.4.1.4 Displacing and Resident Solutions Typical laboratory column experiments involve two solutions, a resident solution and a displacing solution. Either solution may contain the dissolved substances being studied, while the other one is free of these substances. The preferred chemical composition of these solutions depends on the objectives of a particular experiment. Some factors that need consideration are discussed in the following. Distilled and deionized water are generally not recommended as displacing or resident solutions because they cause clay dispersion. Freshly drawn tap water is also not recommended because it commonly contains high amounts of dissolved gases that may come out of solution and collect in the column and end plates. Additionally, tap water may contain measurable amounts of the solute being studied. For many column studies, the solution not containing the solutes being studied should itself be a dilute salt solution. The solution is prepared by dissolving a salt in deaerated, deionized water. The cation is usually Ca2+ in the form of CaSO4, CaCl2, or Ca(NO3)2, with a typical Ca2+ concentration of.005 M. The Ca2+ minimizes clay dispersion and, for reactive transport studies, can serve as an exchangeable cation of known charge and sorption affinity. Other salts can be used, although Na+ should be avoided because it tends to decrease the hydraulic conductivity of soil (Dane & Klute, 1977). For longer duration experiments where microbial growth (biofilms) may clog the soil or column end plates, a biological inhibitor such as thymol can be added to the solutions. Thus, a suitable solution is the CaSO4–thymol solution that has been recommended in the past for hydraulic conductivity measurements (Section 3.3.2.1; Dane & Klute, 1977). It is also useful to wrap the column in aluminum foil to minimize biological activity by keeping the light out. Differences in the physical and chemical properties of the displacing and resident solutions should be minimized (e.g., pH, density, viscosity). In particular, the displacing solution should have approximately the same ionic strength as the resident solution. A displacing solution with a significantly different ionic strength than the resident solution will lead to solute exchange and transport that is not consistent with the models presented in Section 6.3 (also see Starr & Parlange, 1979; Barry et al., 1983; Selim et al., 1992). The resident and displacing solutions should con-
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tain the same anion when studying cation transport, and the same cation when studying anion transport. For example, Mg2+ transport can be studied using a 0.005 M CaCl2 resident solution and a 0.005 M MgCl2 displacing solution, or Br− transport can be studied using a 0.005 M CaCl2 resident solution and a 0.005 M CaBr2 displacing solution. If a lower solute concentration is desired, the solute solution can be prepared by dissolving a small amount of a particular solute in the other solution. For example, to study Cd2+ transport, prepare two 0.005 M Ca(NO3)2 solutions and then make the solute solution by dissolving a small amount of Cd(NO3)2 into one of them. As noted, the experimental objectives ultimately dictate the composition of the resident and displacing solutes. For example, we may want to investigate solute transport using a particular irrigation water as the displacing solution. Or we may want to investigate the leaching of salts already present in the soil using a variety of displacing solutions. 6.4.1.5 Procedures 6.4.1.5.a Preliminaries The recommended first step in a column experiment is to saturate the column with the resident solution. The column is normally saturated by attaching a reservoir of resident solution to the outlet and slowly raising the reservoir such that water infiltrates upward into the soil and removes air from the soil through the top. To avoid air entrapment and slaking, raise the reservoir slowly and incrementally over the course of 24 h or longer, depending on the soil texture and column length. Alternatively, saturate the column by attaching a pump to the bottom and pumping solution in at a very low flow rate. In either case, applying a low vacuum to the top of the column may expedite the saturation process. Flushing the sample with CO2 prior to saturating can also help (CO2 being more soluble in water than O2). Even if an unsaturated experiment is planned, it is recommended that the column be initially saturated. A saturated column provides a well-defined and repeatable starting point from which the desired unsaturated water content is achieved (i.e., the soil water content is achieved on the initial drainage curve). Also, for reactive transport studies the desired initial chemical conditions in the soil can be established more quickly in a saturated column. For most reactive transport experiments, the soil column should be made homoionic so that the reactive solute is displacing an ion of known charge and sorption affinity rather than a mixture of unknown ions (Selim et al., 1987). Saturate the exchange sites, for example, with Ca2+ by leaching the column with several pore volumes of the resident solution. Periodically stopping flow and allowing for periods of equilibration may facilitate Ca2+ saturation. It can also be beneficial to use a higher molarity salt solution than will be used in the displacement experiment. In this case, the column should be thoroughly leached with the correct resident solution prior to initiating the displacement experiment. 6.4.1.5.b Saturated Flow Weigh the saturated column and determine the pore volume V ( L3) (volume of solution in the column) by subtracting the weight of the dry soil and column hous-
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ing from the weight of the saturated column. Attach the reservoir with the resident solution to the inlet and establish the desired flow rate by setting the pump rate (flux control method) or by adjusting the inlet pressure head until the desired rate is achieved (head control method). Avoid setting the flow rate too high with the flux control method because pressure can build up and cause soil compaction. The outlet is left open to the atmosphere for both flow control methods. Verify the flow rate and steady-state conditions by measuring the column’s outflow and/or inflow rates a few times. If instrumentation is available, verify that hydraulic conditions inside the soil are unchanging. When steady conditions are established, record the volumetric flow rate, Q (L3 T−1). This flow rate should be maintained throughout the experiment. Begin the experiment by switching the inlet from the resident solution to the displacing solution and record the time of the switch as time zero. Simultaneously with the switch, advance the fraction collector to a new sample tube and begin collecting effluent samples. The influent solutions need to be switched rapidly so that flow disruption and solution mixing outside the soil are minimized. Designing a minimal amount of the void space in the inlet assembly (cap and tubing) helps achieve a rapid switch. The switching can also be facilitated by, at the time of the switch, briefly opening the bleeder valve in the inlet cap and flushing resident solution out of the cap and tubing with the newly attached displacing solution. Adding a nonreactive dye (food color) to the displacing solution is helpful because it allows one to observe the displacing solution as it invades the inlet tubing and cap. The dye also makes it possible to observe when the displacing solution first begins to appear in the effluent. Displacing solution is applied until the effluent concentration stabilizes (ce/c0 = 1 for a conservative solute in the displacing solution; see Fig. 6.3–2). If a pulse input is desired instead of a step input, the inlet is switched from the displacing solution back to the initial resident solution after some period of time, t0, with the same considerations for rapid switching applying. The initial resident solution is then applied until the effluent solute concentration equals the initial resident solution concentration. Depending upon the solutes used, the column may be regenerated to the homoionic condition and a subsequent displacement experiment can be conducted. When done, disassemble the apparatus and reweigh the saturated column to verify the pore volume. 6.4.1.5.c Unsaturated Flow The procedures for unsaturated flow experiments are similar to the procedures for saturated experiments. However, controlling water flow is more complicated. For uniformly packed or otherwise homogeneous columns, we desire a uniform water content (∂θ/∂x = 0) and steady water flow (∂θ/∂t = 0). A uniform water content in a homogeneous, nonhysteretic column is achieved when the matric head hm is constant throughout the column (∂hm/∂x = 0). Assuming the air pressure head in the column to be zero, the pressure head (h) is equal to the matric head (hm). Water flow in a vertically oriented column is then driven by a unit hydraulic gradient; that is, dH/dx = [d(h + x)]/dx = dx/dx = 1
[6.4–3]
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The water flux in the column will be equal to the hydraulic conductivity for that water content, K(θ). Head Control Method. The head control method for homogeneous columns involves applying suction at the inlet and outlet so that ∂hm/∂x = 0. In principle, this is accomplished by imposing the same negative pressure head at the inlet and outlet. In practice, head losses associated with imperfect contact between soil and end plates and minor soil heterogeneities can cause the head inside the column to differ from that set at the boundary (Langner et al., 1998). It is therefore necessary to use tensiometry to verify the unit hydraulic gradient and/or TDR (or other instrumentation) to verify the uniform water content. Additionally, it has been noted that the hydraulic conductivity of a soil column often decreases with time. The decrease is usually attributed to clogging. Clogging mechanisms include (Vandevivere & Baveye, 1992) the release of trapped air bubbles, filtration of solid particles suspended in the percolating liquid, progressive disintegration of soil structure, soil swelling related to dispersion of colloidal particles, and various clogging processes associated with microbial organisms. Because of clogging it may be necessary to periodically adjust the applied suction to maintain a unit hydraulic gradient and constant flow rate (Langner et al., 1998). To use the head control method, saturate the column and make it homoionic as discussed above. Assemble the apparatus and set the pressure head at the inlet and outlet to the same tension using the Mariotte device (Fig. 6.4–1) or a tension infiltrometer (e.g., Langner et al., 1998) at the inlet and the hanging water column (Fig. 6.4–1) or regulated vacuum (Fig. 6.4–2) at the outlet. Simultaneously with applying suction, open the air vents to provide air entry during desaturation and to maintain a constant soil air pressure. Adjust the suction until steady flow and a unit hydraulic gradient (uniform water content) are achieved, as indicated by tensiometers or other instrumentation. Record the steady flow rate Q (and water content if TDR is used). Once the desired flow conditions are established, the experiment proceeds exactly as the saturated experiment. At the end of the experiment, weigh the column and determine the pore volume V. If a number of experiments are to be done under different flow conditions, it is usually convenient to start with a saturated experiment and perform subsequent experiments at progressively drier water contents. Flux Control Method. The flux control method involves setting the inlet water flux with a pump and then adjusting the suction at the bottom of the column so that a uniform water content (unit hydraulic gradient) is achieved. The magnitude of the required vacuum depends on the pumping rate and the hydraulic properties of the soil (van Genuchten & Wierenga, 1986). The flux control method has the advantage that a precise flow rate can be established. Tensiometry, TDR, or other instrumentation again needs to be used to establish and monitor hydraulic conditions in the soil. To use the flux control method, set the inlet pump to the desired flow rate and apply suction to the bottom of the column by adjusting the hanging water column (Fig. 6.4–1) or by applying vacuum (Fig. 6.4–2). Simultaneously with applying suction, open the air vents. Adjust the suction until steady flow and a unit hydraulic gradient (uniform water content) are achieved, as indicated by tensiometers or other
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instrumentation. Record the steady flow rate Q and proceed with the experiment as described for the head control method. 6.4.1.6 Calculations and Data Presentation Breakthrough curves are constructed by plotting the effluent solute concentration vs. either time, the cumulative volume of effluent, or the number of pore volumes eluted. Some ambiguity exists because effluent samples are actually collected for an interval of time rather than at a particular point in time. For plotting and analysis, the midpoint of the sampling interval should be used as the time t of the sample. The cumulative number of pore volumes eluted, T, is calculated as T = Qt/V
[6.4–4]
Using T as the independent variable is recommended since many important transport features can be deduced from a breakthrough curve plotted vs. T (see discussion of Fig. 6.3–2 and 6.3–3). For uniform water content θ and a cylindrical column with length L and cross-sectional area A, we also have T = vt/L, where v = Jw/θ is the average pore water velocity and Jw = Q/A is the water flux density (Darcy flux). 6.4.1.7 Additional Apparatuses The inlet assemblies in Fig. 6.4–1 and 6.4–2 show the top plate or fabric in contact with a constant head or pump device. Some alternatives are possible. Tindall et al. (1992) dripped solution onto the soil using a single emitter. Wierenga and van Genuchten (1989) applied solution using a rotating arrangement of syringe needles (Fig. 6.4–3). Yule and Gardner (1978) incorporated ceramic tubes into the soil near the top and bottom of a column and used the tubes to apply and extract solution. Drip-type application systems are particularly useful for soil block studies. Soil blocks are similar to undisturbed soil columns, except they have a much larger horizontal extent. Soil blocks are preferred over soil columns for studying discrete preferential transport pathways (Bejat et al., 2000; Shipitalo et al., 1990). Soil blocks are typically carved in situ and encased in a Plexiglas or plywood box with polyurethane foam filling the annulus (Shipitalo et al., 1990). The large horizontal extent of soil blocks permits the use of a spatially discrete drainage collection system (Fig. 6.4–3). The collection system consists of a grid of inner collection cells and an outer collection area for the soil–foam interface. Each collection cell is tapered to a drain hole that is covered with an inertporous material. Collection cells are partially filled with a porous medium such as glass beads or well-sorted sand to establish hydraulic continuity with the soil. The collection system is pressed into the bottom of the soil and sealed. Breakthrough curves can be constructed for each cell. Quisenberry et al. (1994) applied vacuum to the collection system, while Shipitalo et al. (1990) used free drainage. The amount of vacuum that is appropriate for identifying preferential flow paths is uncertain; Bejat et al. (2000) used 20 cm based upon the work of Phillips et al. (1995).
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6.4.1.8 Additional Procedures 6.4.1.8.a Identifying Physical Nonequilibrium Jardine et al. (1998) reviewed several experimental procedures for identifying and quantifying nonequilibrium solute transport. These procedures include varying pore water velocity or pressure-head, multiple tracers, and flow interruption. Pore water velocity and pressure head can be varied by performing the previously outlined steady-state displacement experiment for a range of velocities or pressure heads (e.g., Seyfried & Rao, 1987; Jardine et al., 1993). Multiple tracer techniques
Fig. 6.4–3. Soil block assembly featuring a rotating syringe pump for solution application. The soil block is encased in a box with polyurethane foam in the annulus. The drainage collection system consists of spatially discrete inner collection cells and an outer collection area. The collection system is attached to the soil block and possibly put under vacuum. The lower illustration depicts the design of an individual cell in the collection system.
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also use the steady-state procedure but apply several tracers with differing diffusion coefficients. If physical nonequilibrium is prevalent, the tracers will have different breakthrough curves. Jardine et al. (1998) stated that multiple tracing is “one of the most powerful” techniques for quantifying nonequilibrium. Maloszewski and Zuber (1993) looked at data from several published tracer tests on fractured rock and concluded that multiple tracing should be applied as a rule to distinguish the impact of matrix diffusion. Multiple tracing can also include different size tracers and combinations of dissolved tracers and colloids of various sizes. It is possible to select colloids that are physically excluded from specific pore sizes (McKay et al., 1993). Flow interruption deserves special mention because it has been used for quantifying both physical (Reedy et al., 1996) and chemical (Brusseau et al., 1989; Ma & Selim, 1994b) nonequilibrium. The technique involves temporarily stopping the steady-state flow experiment for a prescribed period of time and allowing for physical and chemical equilibration. Nonequilibrium is indicated by a perturbation in the effluent concentration when flow is resumed. Flow should be interrupted during the latter stages of tracer injection (ce/c0 ≅ 0.8) and/or during the latter stages of tracer displacement (ce/c0 ≅ 0.2). The size of the concentration perturbation depends upon chemical reaction kinetics (chemical nonequilibrium) and matrix diffusion (physical nonequilibrium), but can be manipulated experimentally by varying the pore water velocity and the duration of the flow interruption (Reedy et al., 1996). 6.4.1.8.b Flow Path Mapping Flow paths can be qualitatively and quantitatively identified through the use of dyes and fluorescent microspheres (Jardine et al., 1998). Dyes, in particular, have been used extensively (Ritchie et al., 1972; Starr et al., 1978; Jardine et al., 1988; Ghodrati & Jury, 1990; Kung, 1990) and are very useful for assessing the significance of nonequilibrium processes in soils (Jardine et al., 1998). Dyes are most effective when used in combination with other procedures. For example, at the end of a regular displacement experiment, switch the influent to a dye solution that will stain the flow paths. Solution should be flushed through the column until the dye concentration in the effluent stabilizes. It is important to maintain the same steady-state flow rate used in the displacement experiments. The column can then be dissected and the flow paths mapped. If we assume that a surface becomes more darkly stained the longer it is in contact with the dye (Seyfried & Rao, 1987), then we expect that preferential flow paths will be more darkly stained than the surrounding matrix (Jardine et al., 1998). Mapping can include image analyses that convert visual images of stained surfaces to digital images that provide information on the number, size, and shape of preferential flow paths (Murphy et al., 1977; Ewing & Horton, 1999). 6.4.1.8.c Destructive Sampling Instead of measuring breakthrough curves, a resident concentration profile for a column can be measured by destructively sampling the column. After applying solute and displacing the solution for an appropriate length of time, the flow is
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stopped and the column quickly disassembled. Dissecting the soil and analyzing solution extracts for solute content allows one to construct a plot showing the concentration distribution vs. depth. Smiles et al. (1978) used such a procedure to study solute transport during infiltration. 6.4.1.9 Comments We indicated above that time-averaged effluent concentration measurements should be treated as point measurements when constructing a breakthrough curve. One may question the effects of this procedure on breakthrough curve analysis and parameter estimation (Section 6.5). Leij and Toride (1995) found that the effects were negligible when the point measurements were assigned to the midpoint of the sampling interval and the sampling interval was not excessively large. For additional information see Leij and Toride (1995), Schnabel and Ritchie (1987, 1988), and Barry (1988). Another possible concern is the effect of the laboratory apparatus on the observed solute dispersion and retardation. As we have discussed, this effect is minimized by reducing void space in the inlet and outlet assemblies and by using thin, nonreactive end plates. Nevertheless, a quantitative assessment of the impact of a particular apparatus may be warranted. Solute mixing in the inlet and outlet assemblies can by analyzed using a multilayer representation of the apparatus (James & Rubin, 1972; Starr & Parlange, 1976; Schwartz et al., 1999). 6.4.2 Field 6.4.2.1 Introduction The general principles of laboratory and field experiments are analogous— a solute is applied to the soil and leached while solute concentrations are monitored in the soil water. However, field experiments are considerably more difficult to perform and pose significant challenges in terms of engineering, data collection, and data analysis. The large size of the field, soil spatial and temporal variability, bypass flow and transport, and fluctuating environmental conditions all make it difficult to control or even monitor basic experimental parameters such as water and solute fluxes. Historically, the number of field experiments performed has been quite small compared with the number of column experiments. The last decade has seen an increase in the number of field studies being performed (Flury, 1996). The procedures used in past field experiments are wide ranging and we do not attempt here to present details of any one particular method. Instead, we give an overview of some of the approaches and devices that have been employed. 6.4.2.2 Water and Solute Application Field experiments have been carried out using both quasi-steady and transient water flow conditions. Quasi-steady flow conditions have been achieved using sprinklers (Butters et al., 1989), drip lines (Wierenga et al., 1991; Ellsworth et al., 1991; Costa et al., 1994), and custom-made, traveling spray booms (Ghodrati et al., 1990; Ellsworth et al., 1996). The irrigation rate needed to maintain steady flow conditions changes during an experiment as the evapotranspiration (ET) rate varies. The
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net applied water (applied water minus ET) can be measured with a weighing lysimeter (Jury et al., 1982; Ellsworth et al., 1996), or estimated by calculating ET on the basis of weather data (Butters et al., 1989). Wierenga et al. (1991) covered their experimental plot with polyethylene to prevent ET losses. Although it is generally easier to analyze an experiment if steady water flow is maintained, transient flow conditions prevail in agricultural and natural environmental settings. It has been shown that the timing of irrigation events in transient experiments can significantly impact the retention and degradation of chemicals (Flury, 1996). Such effects may not be apparent in steady flow experiments. Transient flow experiments have been performed using intermittent irrigation (Roth et al., 1991; Wierenga et al., 1991) or natural precipitation (Jury et al., 1982; Jardine et al., 1990; Wilson et al., 1993). Advances in computer hardware and software have opened up new possibilities for analyzing transient field experiments, as discussed in Section 6.6. Whatever the irrigation method, water (and solute) should be applied as uniformly as possible. Catch-cans may be used to evaluate the uniformity of sprinkler and sprayer systems. The sprinkler system of Jury et al. (1982) had a coefficient of uniformity of 0.9, whereas Ellsworth et al. (1996) obtained a coefficient >0.95 with their sprayer system. Solute pulses can be applied with the same irrigation system that is used to apply water. Alternatively, Kachanoski et al. (1992) applied bromide with a hand sprayer but leached the soil with drip emitters. Roth et al. (1991) applied granular salts to the soil surface and used irrigation and rainfall to dissolve the salts and leach the soil. Ellsworth et al. (1991) excavated the top 5 cm of two plots, physically mixed solute into the soil, and then replaced the soil into the plots. Tracers have also been applied directly into the subsurface by injection from a line source buried in a trench (Hammermeister et al., 1982; Wilson et al., 1993). If natural precipitation is used to leach the soil, then a different system will be needed to apply solute. Jury et al. (1982) applied solute with a sprinkler, while Jardine et al. (1990) used a hand sprayer. 6.4.2.3 Solute Monitoring and Measurement There are a number of techniques and devices that can be used to quantify the solute content of field soils. Field experimentalists have historically used some form of soil or soil water extraction to measure solute concentrations in the vadose zone (e.g., Biggar & Nielsen, 1976), but indirect, in situ sensors such as TDR are gaining in popularity (e.g., Caron et al., 1999). For studies involving perched or shallow water tables, interflow wells, groundwater wells, and tile drains have been used to monitor subsurface solute concentrations. Section 6.1 gives procedures for installing and using the different forms of lysimetry and sensor technologies that are available for the vadose zone. We discuss here more general issues related to solute detection in field experiments and also briefly describe solute detection methods for perched and shallow water tables. An important consideration in field experiments is the mass balance, that is, the fraction of applied solute that is detected in the soil. The observed solute mass per unit area of soil can be calculated from a flux concentration breakthrough curve at any measurement location in the soil,
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m = Jw∫0 cf(L,τ)dτ
[6.4–5]
or from a resident concentration distribution, ∞
m = θ∫0 cr(χ,t)dχ
[6.4–6]
For stochastic–convective (Vanderborght et al., 1996) and convective–dispersive (Kreft & Zuber, 1978) transport, it is also true that ∞
m = Jw∫0 cr(L,τ)dτ
[6.4–7]
although Kreft and Zuber (1986) indicated that this latter relationship does not hold more generally. Plot or field averaged mass balances for nonreactive solutes tend to be in the range of 60 to 140%, although individual breakthrough curves usually exhibit high variability, with mass recoveries ranging from just a few percent to several hundred percent. The method of solute detection has been shown to affect mass recovery. Caron et al. (1999) found less bias in the estimated mass when using horizontally installed TDR probes than with porous cup samplers. Ellsworth et al. (1996) observed that vacuum porous cup samplers yielded a poor mass balance as compared with soil coring. At least two factors may simultaneously influence the observed mass balance. One is that measured breakthrough curves and concentration distributions can be in error because instruments may not correctly detect solute concentrations. As discussed in Section 6.1, all solute detection methods have their advantages and limitations. For example, solute concentrations measured with porous cup solution samplers are known to be affected by the magnitude, method, and duration of the applied suction (Section 6.1). Solute concentrations determined by soil coring are affected by the extraction ratio used to make soil solution extracts (Section 6.1). Indirect sensors such as TDR rely on calibrations that may not always be sufficiently accurate (Section 6.1). A second factor affecting observed mass balance is that much of the soil matrix may be bypassed during transport. Bypass has been demonstrated by the dye tracer studies of Kung (1990), Ghodrati and Jury (1990), and many others. A solute sampler or sensor located in a preferential pathway may detect a large percentage of the applied solute, whereas instruments located elsewhere may detect very little or no solute. The sample size of a sensor or sampler determines its likelihood of being bypassed, with instruments that make point measurements being the most susceptible (e.g., Shaffer et al., 1979). Porous cup solution samplers generally sample only a small volume of soil. Pan lysimeters used in the past have ranged in size from a few square centimeters (Jardine et al., 1990; Thompson & Scharf, 1994) to several square meters (Owens et al., 2000; Gee et al., 1992). The collection efficiency of pan lysimeters increases with the size of the pan, where collection efficiency is defined as the volume of water collected divided by the water flux predicted from a water balance model (Radulovich & Sollins, 1987; Jemison & Fox, 1992). Radulovich and Sollins (1987) and Jemison and Fox (1992) tested pans with collection areas from 162 cm2 to 4650 cm2 and observed efficiencies from