Theory and Practice of GNSS Reflectometry (Navigation: Science and Technology, 9) 981160410X, 9789811604102

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Navigation: Science and Technology 9

Kegen Yu

Theory and Practice of GNSS Reflectometry

Navigation: Science and Technology Volume 9

This series Navigation: Science and Technology (NST) presents new developments and advances in various aspects of navigation - from land navigation, marine navigation, aeronautic navigation to space navigation; and from basic theories, mechanisms, to modern techniques. It publishes monographs, edited volumes, lecture notes and professional books on topics relevant to navigation - quickly, up to date and with a high quality. A special focus of the series is the technologies of the Global Navigation Satellite Systems (GNSSs), as well as the latest progress made in the existing systems (GPS, BDS, Galileo, GLONASS, etc.). To help readers keep abreast of the latest advances in the field, the key topics in NST include but are not limited to: – – – – – – – – – – –

Satellite Navigation Signal Systems GNSS Navigation Applications Position Determination Navigational instrument Atomic Clock Technique and Time-Frequency System X-ray pulsar-based navigation and timing Test and Evaluation User Terminal Technology Navigation in Space New theories and technologies of navigation Policies and Standards

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Kegen Yu

Theory and Practice of GNSS Reflectometry

Kegen Yu School of Environment Science and Spatial Informatics China University of Mining and Technology Xuzhou, China

ISSN 2522-0454 ISSN 2522-0462 (electronic) Navigation: Science and Technology ISBN 978-981-16-0410-2 ISBN 978-981-16-0411-9 (eBook) https://doi.org/10.1007/978-981-16-0411-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

GNSS reflectometry (GNSS-R) is an emerging remote sensing technology, which is a derivative technology from GNSS. GNSS-R makes use of the GNSS signals which are always available on the Earth’s surface, and hence it may be considered as a passive bistatic radar. Due to the fact that GNSS-R does not require any dedicated signal transmitter, it has the advantage of low cost. This technology also has the advantage of large coverage or higher spatial resolution because the GNSS receiver can process the signals of multiple satellites simultaneously, and the ground reflection tracks of the signals may have distances of a few hundred kilometers away from each other. Besides, the temporal resolution can be enhanced by using observations of dualfrequency and triple-frequency signals. GNSS-R can be a complement to traditional remote sensing technologies, and it may play a key role in some application scenarios such as in the measurement of Hurricane intensity. GNSS-R has achieved great advances over the past few decades, and it is still making fast progress. Thus, it is useful to have books with a focus on GNSS-R, so that audience especially the beginners can use them as a convenient reference to study the technology. However, there are currently only a few books related to GNSS-R, and most of them only have a few chapters related to the technology. This book completely focuses on GNSS-R and presents in-depth studies on a range of topics. In particular, extensive experimental results in this field are provided. I would appreciate very much the help from many people, without whose assistance the book would not be available. First, I would acknowledge that my Ph.D. and Master’s students made significant contributions especially the experimental results to Chaps. 6–9, who are Wei Ban, Yongchao Zhu, Yunwei Li, Xin Chang, Shuyao Wang, and Jinwei Bu. They did very nice work and their contributions are appreciated so much. I would also like to thank my former colleagues at UNSW for conducting the airborne experiments to obtain the data which were used in Chaps. 5, 6, and 9, who are Jason Middleton, Greg Nippard, Peter Mumford, Eamonn Glennon, Scott O’brien, and Bo Yang. Thanks also go to my colleagues and friends for useful discussions, who are Jiancheng Li, Jingui Zou, Xiaohong Zhang, Chris Rizos, Andrew Dempster, Eamonn Glennon, Peter Mumford, Kevin Parkinson, Taoyong Jin, Jens Wickert, Maximum Semmling, Valery Zonanvanty, Xinliang Niu, Wei Wan, Fan Gao, Lilong Liu, Zhiping Lu, Nanshan Zheng, Kefei Zhang, Adriano v

vi

Preface

Camps, and Chris Ruf. Finally, I would like to acknowledge the support of the Top-Notch Academic Programs Project of Jiangsu Higher Education Institutions (PPZY2015B144, Phase II), and the National Natural Science Foundation of China under Grants 41574031 and 41730109. Xuzhou, China

Kegen Yu

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Brief History of GNSS Reflectometry . . . . . . . . . . . . . . . . . . . . . . . 1.2 Challenges and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4 5 9

2

Navigation Satellite Constellations and Navigation Signals . . . . . . . . 2.1 Navigation Satellite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Global Positioning System . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Glonass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 BeiDou Navigation Satellite System . . . . . . . . . . . . . . . . . 2.1.4 Galileo Navigation Satellite System . . . . . . . . . . . . . . . . . 2.1.5 QZSS and IRNSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Satellite Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 PRN Codes and Multiple Access . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 m-Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Gold Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Multiple Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Carrier Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 BPSK Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 QPSK Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Variants of QPSK Modulation . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Binary-Offset-Carrier Modulation . . . . . . . . . . . . . . . . . . . 2.4 Composition of Navigation Signals . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Transmission of GNSS Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 14 15 15 16 17 19 19 20 21 22 22 23 25 26 29 32 33

3

Signal Scattering and Reception Schemes . . . . . . . . . . . . . . . . . . . . . . . . 3.1 First Fresnel Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Semi-major and Semi-minor Axes . . . . . . . . . . . . . . . . . . . 3.1.2 Simplified Elliptical Equation and Resultant Error . . . . . 3.2 Signal Power Based Estimation of Object Width in Forest . . . . . . 3.3 GNSS-R Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 35 37 39 42 vii

viii

4

5

Contents

3.3.1 Brief Description of Hardware Receiver . . . . . . . . . . . . . . 3.3.2 Signal Processor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Receiver Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Ground-Based Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Airborne Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Spaceborne Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 46 55 55 56 58 59 59

Theoretical Fundamentals of GNSS Reflectometry . . . . . . . . . . . . . . . 4.1 Interferometric Signal Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Signal Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Amplitude Attenuation Factor of Reflected Signal . . . . . 4.1.3 Signal-To-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Composite Excess Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Multipath-Induced Pseudorange Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Delay-Doppler Map and Delay Map . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Surface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Received Signal Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Wavelet Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Wavelet Transform Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 DWT Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Denoising Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Spectral Analysis of Unevenly Sampled Data . . . . . . . . . . . . . . . . . 4.4.1 Unevenly Sampled Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Lomb-Scargle Periodogram . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 61 63 66 69

Sea Surface Altimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Estimating Relative Delay of Reflected Signal . . . . . . . . . . . . . . . . 5.1.1 Relative Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Multipath Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Power Ratio Based Relative Delay Estimation . . . . . . . . 5.2 A Two-Loop Approach for Estimation of Sea Surface Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Geometrical Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Algorithm Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Calculating Total Path Length . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Updating Sea Surface Height Estimate . . . . . . . . . . . . . . . 5.2.5 Determining Power Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Algorithm Complexity Reduction . . . . . . . . . . . . . . . . . . . 5.2.7 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 An Airborne Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 90 91 93

70 72 72 75 78 78 80 82 83 83 84 86 86

94 94 95 96 97 98 100 102 104 104

Contents

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5.3.2

Lidar-Based Mean Sea Level and Wave Statistics Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Data Processing and Delay Waveform Generation . . . . . 5.3.4 Peak Power Based Relative Delay Measurements and Error Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Statistics of Ideal Power Ratio . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Joint Power-Ratio and SSH Estimation . . . . . . . . . . . . . . . 5.4 Spaceborne Altimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

107 109 110 113 116 120 122 122

Sea Surface Wind Speed Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Modeling of Sea Wave Spectrum and Received Signal Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Elfouhaily Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Theoretical Waveform of Received Signal Power . . . . . . 6.2 Near Sea Surface Wind Speed Retrieval . . . . . . . . . . . . . . . . . . . . . 6.3 An Airborne Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Experiment Campaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Wind Speed Estimation Results . . . . . . . . . . . . . . . . . . . . . 6.4 Spaceborne Wind Speed Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 DDM Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Empirical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 125 129 130 133 133 134 142 146 146 149 154 159 162

7

Sea Ice Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 DDM Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 DDM Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Sea Ice and Seawater Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Performance Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Detection Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Sec Ice Concentration Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Estimation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Estimation Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 166 174 174 176 182 184 186 187 188

8

Snow Depth and Snow Water Equivalent Estimation . . . . . . . . . . . . . 8.1 SNR-Based Snow Depth Estimation . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Basic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Data Fusion Based Improvement . . . . . . . . . . . . . . . . . . . . 8.2 Daul-Frequency Carrier Phase Combination Based Method . . . . .

191 191 192 194 198

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8.3

Triple-Frequency Carrier Phase Combination Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Triple-Frequency Carrier Phase Combination . . . . . . . . . 8.3.2 Property of Triple-Frequency Phase Combination . . . . . 8.3.3 An Example of Triple-Frequency Phase Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Theoretical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 A Practical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Single-Frequency Combination Based Method . . . . . . . . 8.4.2 Dual-Frequency Combination Based Method . . . . . . . . . 8.5 Dual Receiver System Based Snow Depth Estimation . . . . . . . . . . 8.5.1 Combination Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Spectral Peak Frequency Analysis . . . . . . . . . . . . . . . . . . . 8.5.3 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Snow Water Equivalent Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Empirical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Algorithm of SNR-Based SWE Estimation . . . . . . . . . . . 8.6.4 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Soil Moisture Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 A Classic Soil Moisture Estimation Method . . . . . . . . . . . . . . . . . . 9.2 Signal Power Based Soil Moisture Estimation . . . . . . . . . . . . . . . . 9.3 SNR Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Fundamental Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Definition of a Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Definition of Observation Variable . . . . . . . . . . . . . . . . . . 9.3.4 Development and Verification of Empirical Models . . . . 9.4 GEO Satellite Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 GEO-IR Based Soil Moisture Estimation . . . . . . . . . . . . . 9.4.2 GEO-R Based Soil Moisture Estimation . . . . . . . . . . . . . . 9.5 An Airborne Experiment for Soil Moisture Measurement . . . . . . 9.5.1 Selection of Experimental Site . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Design of Flight Trajectory and Actual Ground Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200 200 201 203 204 206 211 211 221 235 236 238 240 242 248 248 250 257 260 263 264 267 267 270 272 273 274 276 277 281 282 288 294 295 297 303 303

Contents

10 Tsunami Detection and Parameter Estimation . . . . . . . . . . . . . . . . . . . 10.1 Tsunami Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Examples of Tsunami Waveforms . . . . . . . . . . . . . . . . . . . 10.1.2 A Single Triangle Based Modeling . . . . . . . . . . . . . . . . . . 10.1.3 Two Triangles Based Modeling . . . . . . . . . . . . . . . . . . . . . 10.2 Average Bin Based Tsunami Detection . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Noise Corrupted SSH Measurement . . . . . . . . . . . . . . . . . 10.2.2 Method Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Signal-To-Noise Ratio and Bin Size . . . . . . . . . . . . . . . . . 10.2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Tsunami Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Wavelet Based Noise Reduction for Tsunami Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Cramer-Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Tsunami Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Tsunami Propagation Direction and Speed Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Tsunami Wavelength Estimation . . . . . . . . . . . . . . . . . . . . 10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

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Chapter 1

Introduction

Global navigation satellite system (GNSS) was originally designed and developed to provide positioning, navigation and timing (PNT) services. In addition to the fundamental PNT services, GNSS has been exploited for other applications and services such as surveillance, communications and remote sensing. As illustrated in Fig. 1.1 [63], the direct GNSS signals captured by the receiver are processed to obtain position and timing information, the reflected signals are processed to infer geophysical parameters of reflected areas of Earth surface, and the refracted signals are processed to measure the parameters of the atmosphere. Communication service is mainly related to two aspects. One is the two-way communications enabled by some constellations such as BeiDou Navigation Satillte System (BDS) which allows users to send short message upward to BDS satellites. Another aspect is that communications are enabled among the satellites of the same constellation. The application scenario of surveillance can be associated with maritime, in the air, or in the airport. The positions of the aircrafts and ships can be either determined by GNSS or by radar. GNSS remote sensing consists of two different technologies, GNSS reflectometry (GNSS-R) and GNSS radio occultation (GNSS-RO). GNSS-R mainly focuses on the sensing of Earth surface, while GNSS-OR concentrates on the sensing of the atmosphere. Comparatively, GNSS-OR is more developed and mature than GNSS-R. In this chapter, the history and a number of milestones of GNSS reflectometry (GNSS-R) over the past few decades are briefly studied. A number of challenges associated with GNSS-R are addressed. Also, the main contents of all the major chapters of the book are summarized.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_1

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1 Introduction

Fig. 1.1 Illustration of the geometry of the direct, reflected and refracted GNSS signals and their use for different applications

1.1 Brief History of GNSS Reflectometry GNSS-R is a low-cost and relatively new remote sensing technology, which relies on the processing and analysis of the GNSS signals reflected over the Earth surface to measure the geophysical parameters of the surface media and to detect the natural phenomena and manmade targets. The concept of GNSS bistatic radar was probably originally proposed by Hall and Cordey [16] when investigating multistatic scatterometry for estimating wind vector of ocean. Martin-Neira [31] proposed the concept of passive reflectometry and interferometry system (PARIS) and the use of reflected GPS signals for ocean altimetry. Auber et al. [3] captured the ocean-reflected GPS signals during an airborne experiment. Katzberg and Garrison [21] proposed to measure the ionospheric delay using ocean-reflected signals captured by LEO satellite to enhance the performance of single frequency altimeters. Garrison et al. [11] conducted a series of airborne experiments to collect direct GPS signals and reflections from water surface under a variety of sea states, to investigate the potential of GPS signal for measuring sea surface roughness. Kavak et al. [24] use a ground based GPS receiver to estimate surface permittivity and compared the measured and modeled signal power variation against satellite elevation angle. Komjathy et al. [25] investigated sea surface wind speed estimation using GPS signals reflected over seawater and made comparison between airborne experimental data and theoretical model. Komjathy et al. [26] predicted the use of surface-reflected GPS signals in various other remote sensing applications, such as wave height and salinity studies, delineation of wetlands, and monitoring ionospheric TEC above oceanic areas. Anderson [2] used GPS as a remote sensor to measure water levels and ocean tides. Based on a bistatic radar equation and the Kirchhoff approximation, Zavorotny and Voronovich [65] developed a theoretical model to describe the power of scattered GPS signals

1.1 Brief History of GNSS Reflectometry

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as a function of geometrical and environmental parameters such as sea surface mean square slope so that sea surface wind speed can be measured. Using GPS signals for soil moisture measurement was initially investigated with the basic observation of the reflected signal power which varies as surface soil moisture changes by Masters et al. [33], and Zavorotny and Voronovich [66]. In October 2000, an GPS system carried by an NOAA Hurricane Hunter research aircraft collected the GPS signals reflected from the sea surface inside Hurricane Michael off the South Carolina coast [23]. By fixing an GPS receiver at a lookout point about 480 m above the lake surface, Treuhaft et al. [51] reported high-precision lake surface height estimates with 2-cm precision for 1-s average. UK-DMC (Disaster Monitoring Constellation, or BNSCSAT-1) built by Surrey Satellite Technology Ltd. (SSTL) was launched on 27 September 2003, and retired in November 2011. Although there are rather limited reports on the use of UK-DMC data, ocean roughness was sensed by the system [8]. This is the first satellite mission related to GNSS-R, carrying four major payloads including one for conducting an experiment to demonstrate GNSS-R. UK TechDemoSat-1 (TDS-1, designed by SSTL) was launched from Baikonur Cosmodrome on 8 July 2014 [52, 53]. One of the eight payloads is SGR-RESI (Space GNSS Receiver—Remote Sensing Instrument) which was developed by UK SSTL (Surrey Satellite Technology Ltd) with a primary objective of sea state monitoring. On the 15th of December 2016, NASA funded CYGNSS project, led by Professor Chris Ruf at the University of Michigan, launched eight micro-satellites to orbit of 500 km with an initial objective of monitoring tropical cyclone intensity [9, 43]. Both TDS-1 and CYGNSS produced a large amount of data which have been exploited to sense a variety of geophysical parameters, in addition to the original targeted one or two parameters. On the 14th of July 2017, Japan launched the microsatellite WNISAT-1R developed by Weathernews Inc., using optical cameras to observe sea ice, tropical cyclones and volcanic ash clouds so as to improve the accuracy of meteorological and hydrographic forecasts [56] visited). One of the payloads is a GNSS receiving system, which has received GNSS-R signals and generated delay-Doppler maps. On the 5th of June 2019, China launched BuFeng-1 A/B twin satellites from Yellow Sea, carrying GNSS-R receiver developed by China Aerospace Science and Technology Corporation. This satellite mission is dedicated to conduct GNSS-R onorbit tests with a main focus on the monitoring of sea surface wind velocity field especially Typhoon [20, 36]. On the 5th of July 2019, UK launched DOT-1 satellite, also designed by SSTL, was launched. The GNSS-R payload onboard the DoT-1 satellite is intended to test an advanced design in Avionics which can be used in future satellite platforms and enable a nadir-looking antenna to be part of GNSSR small-sat constellation (GPSWorld [15] visited). It is expected that there will be more GNSS-R related satellite missions. For instance, one of them would be the GNSS REflectometry, Radio Occultation, and Scatterometry Onboard the International Space Station (GEROS-ISS, [57]). The GEROS-ISS project was successfully proposed to the European Space Agency by a group of scientists, which aims to conduct GNSS-R experiments on the ISS.

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Over the past two decades, GNSS-R has made great progress in both theory and practice. Making use of ground-based, airborne and space-borne data, researchers over the world have conducted in-depth studies on GNSS-R based detection of a number of natural phenomena and estimation of a range of geophysical parameters. They include ocean altimetry [6, 10, 17, 30, 32, 37, 39, 55], sea surface wind speed and direction estimation [12, 44, 67], ocean wave height estimation [1, 38, 46], snowfall estimation [27, 28, 35, 45, 62], soil moisture estimation [22, 34, 40, 60], sea ice detection [5, 13, 59, 68], vegetation estimation [4, 29, 41, 42, 48], object detection [7, 47, 50, 54 ], and potential Tsunami detection [49, 58, 61]. It is not intended to present a thorough review or list a large number of references on this innovative remote sensing technology, but mainly highlight some of the early pioneer works and a number of milestones. More references can be found in a number of review articles [19, 63, 64], books [14, 18], and particularly databases such as IEEE Xplore digital library. Also, each of the main chapters in this book provides a short list of references.

1.2 Challenges and Future Directions As mentioned above, GNSS-R has made tremendous progress since 1990s; so many ground based and airborne experiments have been conducted and five LEO satellite missions were successfully launched. Also, various theoretical and empirical formulas were derived and detection and estimation methods were proposed to detect objects and retrieve geophysical parameters. Indeed, it has also been demonstrated that GNSS-R based detection and estimation performance is satisfactory based on experimental results in many cases. For instance, the hurricane intensity measurement by CYGNSS is very close to the measurement by the stepped frequency microwave radiometer (SFMR) which is the primary instrument used to determine hurricane intensity. However, GNSS-R is still not yet a mature technology and it requires to devote more time and efforts to improve this remote sensing technology. There are a number of challenges in making GNSS-R to be commercial and practicable, including accuracy, reliability, resolution, and real time. Accuracy and reliability could be the two most important issues, which are related to a number of factors, such as receiver sensitivity and antenna directivity, theoretical and empirical modeling error, algorithm-induced error, calibration, raw data processing and selection. Receiver sensitivity may be improved by increasing receiver bandwidth and mitigating noise, while antenna directivity may be enhanced by use of smart antenna with dedicated design, advanced material, and beamforming technique. To reduce modeling error, significant parameters should all be taken into account to generate accurate theoretical and empirical models which are of general-purpose at least for specific scenarios. Algorithm performance usually depends on the accuracy of models, but in some cases the definition and selection of the observables and the threshold values also play a significant role. It is inevitable to have deterministic

1.2 Challenges and Future Directions

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measurement error such as bias, so careful calibration must be carried out to remove the deterministic errors. Raw data processing is also important to remove the effect of irrelevant factors and make the residual errors as small as possible. Low quality data should be excluded and reliable selection criteria should be developed. Temporal and spatial resolutions are also important issues especially when scrutinizing small areas of land and ocean surface. The initial temporal and spatial resolutions of raw TDS-1 data is 1 s and 1 km by 7 km. To increase SNR, incoherent integration may be applied, resulting in degraded temporal and spatial resolutions. Although it is possible to detect objects smaller than the spatial resolution, it is always desirable to have both resolutions improved. As the quality of the received signal increases, which involves receiver and antenna, the resolution would be improved without degrading other performance indexes such as accuracy. Although real time may not be required for many application scenarios, real time or nearly real time is often necessary for the monitoring of disasters such as Hurricane, Tsunami, and damaging swells. Prompt acquisition of the information on these disasters is important for timely warning to reduce the loss of economy and life. A main advantage of GNSS-R is the generation of sensing results over multiple surface trajectories, which is particularly useful for disaster monitoring; a disaster would more likely and more quickly be captured in the presence of more detection trajectories. The aim of the GNSS-R community would be to make the technology to become mature and deliver off-the-shelf products which have a good trade-off between cost and performance so as to satisfy a variety of customers. As the community continues to make advances for the technology, such an aim hopefully will be accomplished in the next five to ten years.

1.3 Overview of the Book The book consists of two main parts. The first part consists of the first three major chapters, studying the fundamental concepts and theory of GNSS-R. The second part comprises the last six chapters, dealing with the applications of the technology in six different application areas. Each major chapter is summarized, so that readers may quickly understand the main contents presented. Note that a number of important applications such as vegetation sensing are not studied because the author does not have the adequate materials to write the chapters. Chapter 2 first introduces the four global navigation satellite constellations which are China’s BDS, EU’s Galileo, Russia’s GLONASS), and U.S.’s GPS, Japan’s QZSS which is the space-based navigation augmentation system for the GPS, and India’s NAVIC which is a regional navigation satellite system. Satellite visibility is then briefly discussed with a focus on the numbers of visible GPS and BDS satellites in some areas of China. Pseudorandom noise (PRN) code is also studied, including the description of the generation of two types of PRN codes, which are m-sequences and Gold codes. Another significant part of this chapter focuses on the generation of

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GNSS signals. In particular, relevant carrier modulation schemes are described with details, which include BPSK, QPSK and binary-offset modulations. Chapter 3 first studies the first Fresnel zone related to the effective scattering area when coherent scattering is dominant. Mathematical expressions of the semi-major and semi-minor axes are presented. The widely used theoretical formulas for calculating the two axes are evaluated and the relative area error due to the simplification in the calculation of the axes can be significant for the case of ground-based receiver platform. Then, received reflected signal power is considered for surface change detection with a focus on the estimation of the length of signal reflection track over a river in a forest so that the width of the river can be estimated with the knowledge of the relative orientation between the river and the reflection track. The second part of the chapter studies the basic structure of GNSS-R receiver with a focus on hardware receiver. The signal processor for the generation of delay-Doppler map (DDM) is studied and both FPGA and DSP are considered. The selection of DDM parameters is discussed, and so are the issues related to front-ends. Finally, the three receiver platforms are briefly discussed. Chapter 4 focuses on theoretical fundamentals of GNSS-R. Signal reception is first studied for scenarios where only a single antenna is used such as in the case of CORS (continuously operating reference stations). Both the direct and reflected signals arrive at the same antenna, producing an interferometric signal. This is well suited for ground based applications especially by use of the existing infrastructure like CORS. Formulas of the amplitude, phase, SNR of the received interferometric signal, multipath-induced carrier phase error and pseudorange error are derived, and the amplitude attenuation factor (AAF) of reflected signal is introduced. The delayDoppler map (DDM) is described with some details, which is probably the most important concept of spaceborne GNSS-R. Then, wavelet theory is studied with a focus on noise mitigation and waveform reconstruction, which is particularly suited for processing transient signals. Finally, spectrum analysis for unevenly sampled signals is studied, with a focus on the Lomb-Scargle periodogram. Although GNSS data are evenly sampled with time in general, the data are not evenly sampled with respect to satellite elevation angle or the sine of the elevation angle. Chapter 5 studies ocean altimetry with a focus on altimetry with airborne experimental data. Relative delay of the reflected signal is determined by measuring the arrival time of the direct signal and reflected signal through cross-correlation. Due to the fact that the peak power of the reflected signal is shifted when the sea surface is rough, it is a challenge to precisely determine the relative delay. This chapter studies the power ratio method, which is one of the techniques to deal with the problem and is suited when measurements related to multiple satellites are available. An airborne experiment, conducted off the Sydney coast in Australia, was described with details and some experimental results are presented. Because the receiver, the nadir-looking and zenith-looking antennas are usually not in the same position, calibration of the relative delay is required, which is also studied. Then, spaceborne ocean altimetry is reviewed and some details of a few techniques are presented. Chapter 6 focuses on the wind speed estimation by use of airborne experimental data. The Elfouhaily model, one of the widely studied sea wave spectrum model, is

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described in detail. A simple wind speed estimation algorithm is studied, which makes use of the central delay waveforms normalized and truncated above a power value to remove the effect of noise. Then, the measured delay waveform is compared with theoretical delay ones related to different wind speeds in terms of the area size covered by each waveform. The theoretical waveform with the minimum difference in area size is selected and the corresponding wind speed is the estimate of the actual wind speed. The second part of the chapter focuses on spaceborne wind speed estimation with CYGNSS data. By using a large amount of real data, three normalized bistatic radar cross section (NBRCS) based empirical models and two leading edge slope (LES) of integrated delay waveform (IDW) based empirical models are developed to establish the relationship between the NBRC and LES of IDW and the sea surface wind speed. The performance of the models is also extensively evaluated. A data fusion method based on the determination coefficient is also studied, which shows some performance gain. Chapter 7 studies sea ice and sea water detection based on data collected by spaceborne receiver especially the TDS-1 receiver. The original raw DDM data are first processed through background noise subtraction, incoherent integration if needed, and power normalization to mitigate the effect of noise, atmosphere and instrument hardware. The definitions of a total of 17 DDM based observables are given and examples of the probability density functions of a few observables are presented. The detection performance indexes are defined and the overall detection performance may be evaluated with the overall false alarm rate which is defined as the average of ice detection false alarm rate and seawater detection one. Alternatively, the average of sea ice detection probability and seawater detection probability may be defined as the overall detection performance. A large amount of TDS-1 DDM data were used to evaluate the detection performance of the detection methods based on a number of the observables. Also, the estimation of sea ice concentration (SIC) is studied, which would be a more important parameter. This is because in many cases, the effective scattering surface area contains both sea ice and seawater. Chapter 8 studies snow depth estimation with a focus on the use of data collected with ground-based GNSS receiver with interferometric reception. SNR time series data were first considered for snow depth estimation. The raw SNR data are detrended to remove the effect of the SNR component of direct signal to generate time series of detrended SNR, which has a main frequency proportional to the antenna height relative to the reflection surface. Then, the chapter studies the use of dual-frequency and triple-frequency carrier phase combinations for snow depth estimation. The triplefrequency scheme can remove the effect of both ionoshperic delay and geometry, while the dual-frequency scheme can remove only one of the effects and the other one needs to be removed by another technique. Next, the chapter studies the combination of signal-frequency pseudorange and dual-frequency carrier phases as well as the combination of signal-frequency pseudorange and dual-frequency carrier phases. The latter also needs to use another technique to remove the effect of geometry or ionospheric delay, similar to the dual-frequency carrier phase combination. Data collected simultaneously from multiple receivers can also be used to increase data usage. Finally, snow water equivalent estimation is studied. The typical snow season

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in cold regions is divided into three different periods (accumulation, transition and melting). After preprocessing of the snow depth with such as a simple mean filtering, the cleaned snow depth data are produced. Empirical models are then developed to describe the direct relationship between the snow depth and snow water equivalent (SWE) for the three periods individually. The models are evaluated extensively and the SNR based SWE estimation is also studied. Chapter 9 studies soil moisture estimation based on data collected by groundbased GNSS receiver. It is still a challenging problem to estimate soil moisture using radio signals due to the complex soil composition, surface roughness and vegetation. Signal power was first used to estimate soil moisture and the notch position on the power time series is linked to soil moisture. Multiple notches would occur when soil is covered by vegetation and soil moisture and vegetation height may be estimated simultaneously. In the absence of vegetation or its effect can be ignored, SNR time series can be used to estimate soil moisture by developing an empirical model between soil moisture and observation variable. Development and verification of the model is described using two groups of data collected from two geographical locations. Then, GEO satellite based soil moisture estimation is studied, which has the advantage of much longer observation period. Both linear and second-order modeling between SNR and soil moisture is studied for GEO-IR based method. In the case of GEO-R, SNR and signal power are respectively used as observations to develop empirical models. Chapter 10 focuses on the potential application of GNSS-R on Tsunami detection and parameter estimation with a focus on the sea surface height (SSH) estimation based methods. Since the wavelength of a Tsunami can be a few hundred kilometers, it is impractical to detect the Tsunami for timely warning with a ground-based receiver, so spaceborne approach is preferable. Since the SSH data can be very noisy, statistical hypothesis testing can be used to decide whether or not a Tsunami is present. A bin average technique and a sliding window moving average method are studied, which are used to generate a decision variable to be compared with a predefined threshold. Simulations with real Tsunami waveforms observed by buoys and altimetry satellites are conducted to evaluate the detection performance. Tsunami parameters (e.g. wavelength, propagation direction and speed) can be estimated with SSH estimates along several surface reflection trajectories, and the moving directions of satellite and Tsunami are taken into account. Tsunami lead wave modeling is also studied with the lead wave modeled as a single triangle or two triangles, and the model parameters are determined with least-squares fitting. Finally, waveform reconstruction is studied, which makes use of wavelet denoising technique.

References

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23. Katzberg SJ, Walker RA, Roles JH, Terry L, Black PG (2001) First GPS signals reflected from the interior of a tropical storm: Preliminary results from Hurricane Michael. Geophys Res Lett 28(10):1981–1984 24. Kavak A, Vogel W, Xu G (1998) Using GPS to measure ground complex permittivity. Elec Lett 34(3):254–255 25. Komjathy A, Zavorotny V, Axelrad P, Born G, Garrison JL (1998) GPS signal scattering from sea surface: comparison between experimental data and theoretical model. In: Proceedings of the international conference on remote sensing for marine and coastal environments, San Diego, California 26. Komjathy A, Garrison JL, Zavorotny V (1999) GPS: A new tool for ocean science. GPS World 10(4):50–56 27. Larson KM, Gutmann ED, Zavorotny VU, Braun JJ, Williams MW, Nievinski FG (2009) Can we measure snow depth with GPS receivers? Geophys Res Lett 36:1–5 28. Larson KM, Nievinski FG (2013) GPS snow sensing: results from the EarthScope plate boundary observatory. GPS Solut 17(1):41–52 29. Larson KM, Small EE (2014) Normalized microwave reflection index: a vegetation measurement derived from GPS data. IEEE J Sel Top Appl Earth Obs Remote Sens 7(5):1501–1511 30. Li W, Yang D, D’Addio S, Martín-Neira M (2014) Partial interferometric processing of reflected GNSS signals for ocean altimetry. IEEE Geosci Remote Sens Lett 11(9):1509–1513 31. Martin-Neira M (1993) (1993) A passive reflectometry and interferometry system (PARIS): application to ocean altimetry. ESA J 17(4):331–355 32. Martin-Neira M, Caparrini M, Font-Rossello J, Lannelongue S, Vallmitjana CS (2001) The PARIS concept: an experimental demonstration of sea surface altimetry using GPS reflected signals. IEEE Trans Geosci Remote Sens 39(1):142–150 33. Masters D, Zavorotny V, Katzberg S, Emery W (2000) GPS signal scattering from land for moisture content determination. In: Proceedings of the IEEE international geoscience and remote sensing symposium (IGARSS), Honolulu, USA, pp 3090–3092 34. Masters D, PAxelrad P, Katzberg SJ, (2004) Initial results of land-reflected GPS bistatic radar measurements in SMEX02. Remote Sens Environ 92:507–520 35. Nievinski FG, Larson KM (2014) Inverse modeling of GPS multipath for snow depth estimation—Part II: application and validation. IEEE Trans Geosci Remote Sens 52(10):6564–6573 36. Niu X, Lu F, Liu Y, Jing C, Wan B (2020) Application and technology of Bufeng-1 GNSS-R demonstration satellites on sea surface wind speed detection. In: Proceedings of China satellite navigation conference (CSNC), Chengdu, China, pp 206–213 37. Park H, Valencia E, Camps A, Rius A, Ribo S, Martin-Neira M (2013) Delay tracking in spaceborne GNSS-R ocean altimetry. IEEE Geosci Remote Sens Lett 10(1):57–61 38. Peng Q, Jin S (2019) Significant wave height estimation from space-borne cyclone-GNSS reflectometry. Remote Sens 11(584):1–13 39. Rius A, Cardellach E, Martín-Neira M (2010) Altimetric analysis of the seasurface GPSreflected signals. IEEE Trans Geosci Remote Sens. 48(4):2119–2127 40. Rodriguez-Alvarez N, Bosch-Lluis X, Camps A et al (2009) Soil moisture retrieval using GNDSS-R techniques: experimental results over a bare soil field. IEEE Trans Geosci Remote Sens 47(11):3616–3624 41. Rodriguez-Alvarez N, Bosch-Lluis X, Adriano Camps A et al (2012) Vegetation water content estimation using GNSS measurements. IEEE Geosci Remote Sens Lett 9(2):282–286 42. Rodriguez-Alvarez N, Camps A, Vall-llossera M et al (2011) Land geophysical parameters retrieval using the interference pattern GNSS-R technique. IEEE Trans Geosci Remote Sens 49(1):71–84 43. Ruf C, Unwin M, Dickinson J et al (2013) CYGNSS: enabling the future of hurricane prediction. IEEE Geosci Remote Sens Mag 1(2):52–67 44. Ruf CS, Gleason S, McKague DS (2019) Assessment of CYGNSS wind speed retrieval uncertainty. IEEE J Sel Top Appl Earth Obs Remote Sens 12(1):87–97 45. Tabibi S, Geremia-Nievinski F, Dam T (2017) Statistical comparison and combination of GPS, GLONASS, and multi-GNSS multipath reflectometry applied to snow depth retrieval. IEEE Trans Geosci Remote Sens 55(7):3773–3785

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46. Shah R, Garrison JL (2014) Application of the ICF coherence time method for ocean remote sensing using digital communication satellite signals. IEEE J Sel Top Appl Earth Obs Remote Sens 7(5):1584–1591 47. Simone AD, Park H, Camps A (2017) Sea target detection using spaceborne GNSS-R delayDopplar maps: theory and experimental proof of concert using TDS-1 data. IEEE J Sel Top Appl Earth Obs Remote Sens 10(9):4237–4255 48. Small EE, Larson KM, Braun JJ (2010) Sensing vegetation growth with reflected GPS signals. Geophys Res Lett 37(L12401):1–5 49. Stosius R, Beyerle G, Helm A, Hoechner A, Wickert J (2007) Simulation of space-borne tsunami detection using GNSS-reflectometry applied to tsunamis in the Indian Ocean. Nat Hazards Earth Syst Sci 10(6):1359–1372 50. Suberviola I, Mayordomo I, Mendizabal J (2012) Experimental results of air target detection with a GPS forward-scattering radar. IEEE Geosci Remote Sens Lett 9(1):47–51 51. Treuhaft R, Lowe ST, Zuffada C, Chao Y (2001) 2-cm GPS altimetry over crater lake. Geophys. Res. Lett. 28(23):4343–4346 52. Tye J, Jales P, Unwin M et al (2016) The first application of stare processing to retrieve mean square slope using the SGR-ReSI GNSS-R experiment on TDS-1. IEEE J Sel Top Appl Earth Obs Remote Sens 9(10):4669–4677 53. Unwin M, Jales P, Tye J, Gommenginger C, Foti G, Rosello J (2016) Spaceborne GNSSReflectometry on TechDemoSat-1: Early mission operations and exploitation. IEEE J Sel Top Appl Earth Obs Remote Sens 9(10):4525–4539 54. Valencia E, Camps A, Rodriguez-Alvarez N et al (2013) Using GNSS-R imaging of the ocean surface for oil slick detection. IEEE J Sel Top Appl Earth Obs Remote Sens 6(1):217–223 55. Wang X, He X, Zhang Q (2019) Evaluation and combination of quad-constellation multi-GNSS multipath reflectometry applied to sea level retrieval. Remote Sens Environ 231:111229 56. Weathernews (2020, visited) https://global.weathernews.com/infrastructure/wnisat-1/ 57. Wickert J, Cardellach E, Martin-Neira M et al (2016) GEROS-ISS: GNSS reflectometry, radio occultation, and scatterometry onboard the international space station. IEEE J Sel Top Appl Earth Obs Remote Sens 9(10):4552–4581 58. Yan Q, Huang W (2016) Tsunami detection and parameter estimation from GNSS-R delayDoppler map. IEEE J Sel Top Appl Earth Obs Remote Sens 9(10):4650–4659 59. Yan Q, Huang W (2016) Spaceborne GNSS-R sea ice detection using delay-Doppler maps: First results from the UK TechDemoSat-1 mission. IEEE J Sel Top Appl Earth Obs Remote Sens 9(10):4795–4801. 60. Yang T, Wan W, Chen X, Chu T, Hong Y (2017) Using BDS SNR observations to measure nearsurface soil moisture fluctuations: results from low vegetated surface. IEEE Geosci Remote Sens Lett 14(8):1308–1312 61. Yu K (2015) Tsunami wave parameter estimation using GNSS-based sea surface height measurement. IEEE Trans Geosci Remote Sens 53(5):2603–2611 62. Yu K, Ban W, Zhang X, Yu X (2015) Snow depth estimation based on multipath phase combination of GPS triple-frequency signals. IEEE Trans Geosci Remote Sens 53(9):5100–5109 63. Yu K, Rizos C, Burrage D, Dempster A, Zhang K, Markgraf M (2014) An overview of GNSS remote sensing. EURASIP J Adv Signal Process 2014(134):1–14. 64. Zavorotny VU, Gleason S, Cardellach E, Camps A (2014) Tutorial on remote sensing using GNSS bistatic radar of opportunity. IEEE Geosci Remote Sens Mag 2(4):8–45 65. Zavorotny VU, Voronovich AG (2000) Scattering of GPS signals from the ocean with wind remote sensing application. IEEE Trans Geosci Remote Sens. 38(2):951–964 66. Zavorotny V, Voronovich AG (2000) Bistatic GPS signal reflections at various polarizations from rough land surface with moisture content. In: Proceedings of the IEEE international geoscience and remote sensing symposium (IGARSS), Honolulu, USA, pp 2852–2854. 67. Zhang G, Yang D, Yu Y, Wang F (2000) Wind direction retrieval using spaceborne GNSS-R in nonspecular geometry. IEEE J Sel Top Appl Earth Obs Remote Sens 13:649–658 68. Zhu Y, Tao T, Yu K, Li Z, Qu X, Ye Z, Geng J, Semmling M, Wickert J (2020) Sensing sea ice based on Doppler spread analysis of spaceborne GNSS-R data. IEEE J Sel Top Appl Earth Obs Remote Sens 13:217–226

Chapter 2

Navigation Satellite Constellations and Navigation Signals

Ground-based radio navigation had been used for more than half a century before the first GPS satellite was launched in the late 1970s. Many radio navigation systems have been developed and put into use since the early of 20th century, including Telefunken Kompass Sender (radio direction finder), Decca (Decca Navigator System), (VOR) VHF Omnidirectional Range, TACAN (Tactical Air Navigation System), LORAN (LOng RAnge Navigation), and OMEGA. Most of the systems ceased to provide navigation service, although some of them have been used for other purposes. Over the development history of around one century, radio navigation has achieved dramatic progress such as from ground-based system to satellite based system, from single functional system to multi-function system, from military-oriented to both military-oriented and civilian-oriented, from low positional accuracy to high positional accuracy, and from local coverage to global coverage. The remainder of the chapter first gives a brief introduction to the four global navigation satellite constellations, a space-based augmentation system, and a regional navigation satellite system. Then, the focus is on how GNSS signals are generated and transmitted. In particular, a number of data modulation schemes are discussed with details.

2.1 Navigation Satellite Systems Before introducing the four global navigation satellite system (GNSS) constellations, it is worth mentioning that the Transit 1B navigation satellite was successfully launched in 1960 and started to provide service in 1964, about 18 years earlier than the successful launch of the first GPS satellite. The last Transit satellite was launched in 1988 and the Transit system ceased navigation service in 1996, but it has been used for ionospheric monitoring since 1996.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_2

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2.1.1 Global Positioning System The U.S. Department of Defense started the global positioning system (GPS) project in 1973. The first satellite was launched in 1978 and a constellation of 24 satellites was operational with global coverage in 1993. Thus, GPS achieved the full orbital constellation ahead of three other constellations. GPS is a U.S.-owned utility that provides users with positioning, navigation, and timing (PNT) services. The system consists of three segments: the space segment, the control segment, and the user segment [7]. The space segment consists of a constellation of usually at least 24 operational GPS satellites transmitting radio signals to users. GPS satellites are all medium Earth orbit (MEO) satellites, flying at an altitude of about 20180 km around a circular orbit and circling the Earth twice a day. In the basic constellation, the GPS satellites are evenly distributed in six orbital planes such that each orbit has four operational satellites. The control segment consists of a network of facilities to track the GPS satellites, maintain the satellites in proper orbits, monitor their transmissions, upload navigation data, and so on. The facilities include a master control station, an alternative master control station, monitor stations, remote tracking stations, and ground antennas, located in about 20 countries and regions. The user segment consists of the GPS receiver and antenna(s) to receive and process the GPS signals for PNT services as well as for required surveillance and remote sensing applications.

2.1.2 Glonass The GLONASS (GLObal Navigation Satellite System) was initially developed by the former Soviet Union and owned and operated by Russia later. The first GLONASS satellite was launched in late 1982, and the constellation was completed in 1995. Over the history of GLONASS, three generations of satellites were developed, with the first second generation satellite (GLONASS-M) launched in 2003 and the first third generation satellite (GLONASS-K) launched in 2011 [5]. The system experienced a decline in capacity during the late 1990s, but a full global coverage was enabled by restoring a full orbital constellation of 24 satellites in 2011. Similar to GPS satellites, GLONASS satellites are also circling around the MEOs at an altitude of about 19,100 km, a bit lower than that of GPS. There are three nearly circular orbits or MEOs, on each of which, eight operational satellites are uniformly distributed. The GLONASS satellites circle the Earth around the orbital planes at a period of about 11 h and 16 min. The GLONASS orbits are particularly suited for the reception of the satellite signals in the regions of high latitudes including Russia. The GLONASS is also composed of space segment, ground control segment and user segment. Space segment consists of 24 operational satellites as mentioned earlier. The ground control segment monitors the status of satellites, determines the ephemerides and satellite clock offsets and uploads the navigation data to the

2.1 Navigation Satellite Systems

15

satellites. The GLONASS control and monitoring stations are almost entirely located within former Soviet Union territory, except for several in Brazil. The GLONASS user segment also consists of L-band radio receiver/processors and antennas which receive and process GLONASS signals for PNT services and other applications.

2.1.3 BeiDou Navigation Satellite System The BeiDou navigation satellite system (BDS) has been independently developed, operated and owned by China. The first BeiDou satellite was launched in October 2000 and the last BeiDou satellite, the 55th satellite in the BeiDou family, was launched in June 2020. Over the history of the BDS development, China has constructed three generations of BDS. The first-generation BDS, also termed BeiDou-1, consisted of only three satellites and thus only offered limited coverage and navigation services. The second-generation BDS, also called COMPASS or BeiDou-2, consists of ten satellites, has provided navigation services for customers in Asia-Pacific region since December 2012. The first BeiDou-3 satellite was launched in March 2015 and the final (i.e. 35th) BeiDou-3 satellite was launched in June 2020, providing global services since December 2019. Similar to other global navigation satellite constellations, BDS mainly consists of three segments: space segment, ground segment and user segment [1]. The BDS space segment consists of a number of satellites distributed on three different types of orbits: geostationary Earth orbit (GEO) with an altitude of 35,786 km, inclined geosynchronous orbit (IGSO) with an altitude of 35,786 km and MEO with an altitude of 21,528 km, which is rather different from other three satellite constellations. In the BeiDou-3 constellation, there are 3 GEO satellites, 3 IGSO satellites, and 24 MEO satellites. The BDS ground segment consists of master control stations, time synchronization/uplink stations, monitoring stations, and operation and management facilities of the inter-satellite link. The BDS user segment consists of receiver chips, modules and antennas as well as application systems and devices.

2.1.4 Galileo Navigation Satellite System Galileo is the European global satellite-based navigation system, operated by the European GNSS Agency (GSA) and the European Space Agency (ESA) [4]. The first Galileo test satellite was launched in December 2005, but it was not part of the operational system. The first two operational Galileo In-Orbit Validation satellites were launched in October 2011 and another two in October 2012, providing limited navigation services. Up to early 2020 there were 26 Galileo satellites in the constellation, 22 satellites operational, two satellites in “testing” state and two not available to users. The complete Galileo satellite constellation will consist of 24 operational

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2 Navigation Satellite Constellations and Navigation Signals

satellites and 6 active spares. The satellites are distributed in three circular MEO planes at an altitude of 23,222 km above the Earth. Similar to other satellite constellations, the Galileo system consists of three major segments: space segment, ground segment and user segment. The space segment consists of 24 operational satellites evenly distributed on three MEO orbital planes like GLONASS and a number of spares, as mentioned earlier, and two more operational satellites would be launched in a few years. The Galileo user segment consists of all the compatible receivers and devices which collect the Galileo signals for PNT services and other applications. The Galileo ground segment consists of two control centers, a global network of monitoring and control stations, and a series of service facilities which support the provision of the Galileo services.

2.1.5 QZSS and IRNSS Quasi-zenith satellite system (QZSS) is the Japanese satellite positioning system, which is sometimes called the “Japanese GPS” [14]. The first quasi-zenith satellite (QZS) was launched in 2010, followed by the launch of three other QZSs in 2017. QZSS has been operated as a four-satellite constellation in November 2018. Among the four QZSs, one is on the geostationary orbit, while the other three are on the quasizenith orbit which is a figure-eight shaped orbit with north-south asymmetry. Those three satellites are in periodic highly elliptical orbit with the perigee altitude about 32,000 km and apogee altitude about 40,000 km. The QZSs stays in the northern hemisphere for about 13 h and in the southern hemisphere for about 11 h. QZSS is intended to complement GPS as an augmentation system to enhance positioning for the Asia-Oceania regions especially Japan, instead of an independent positioning system. Indian regional navigation satellite system (IRNSS) is an independent regional navigation satellite system being developed by India [9]. It is intended to provide navigation service to users in India as well as the neighboring regions within up to 1500 km around it, which is its primary service area. Like the four GNSS systems, IRNSS will provide two types of services, namely, standard positioning service available to all the users and restricted service which is an encrypted service provided only to the authorized users. The first IRNSS satellite (IRNSS-1A) was launched in July 2013 and the eighth IRNSS satellite (IRNss-1I) was launched in April 2018. Three of the eight IRNSS satellites are located in geostationary orbit approximately 36,000 km above the Earth’s surface and the other five satellites are in inclined geosynchronous orbit.

2.1 Navigation Satellite Systems

17

2.1.6 Satellite Visibility Due to the four global navigation satellite constellations, a ground-based, airborne, or spaceborne receiver is able to receive signals transmitted from a number of GNSS satellites anytime and anywhere on the Earth’s surface. The number of visible satellites of the same constellation can be significantly different at a different geographical location. Figure 2.1 shows the skyplot (i.e. elevation and azimuth tracks) of GPS satellites observed during a day by the receiver of a MGEX (multi-GNSS experiment) station in Wuhan China. In this case more GPS satellites were observed in the south than in the north. Except for a particular area in the north, the tracks are roughly evenly distributed over the whole azimuth range and a range of elevation angle. Similar skyplot was observed for BDS satellites. Figure 2.2 shows the numbers of visible BDS satellite at four MGEX stations in China over a week in April 2020 [13]. The numbers of visible BDS satellites at each station varies from day to day with the minimum number being eight and maximum number being 30. Note that all the three generations of BDS satellites are considered. The numbers of visible BDS satellites are similar among the three stations, but the one at WUH2 station is much larger due to its specific location. Table 2.1 shows the average numbers of GPS and BDS satellites visible over one week at four MGEX stations. It can be seen that the average number of visible GPS satellites is almost the same at four different stations, while the average number of BDS satellites can be rather different at a different observation station. This may be due to the fact that BDS constellation consists of three different types of orbits (MEO, IGSO, GEO), while GPS constellation has only one type of orbit (MEO). Table 2.2 shows the visibility periods of GPS and BDS satellites. Since BDS has three different types of satellites, the visibility periods are calculated in terms of satellite type. As expected, the GEO satellites have the longest visibility period, and the MEO Fig. 2.1 Skyplot of GPS satellites observed at WUH2 MGEX station. The receiver is located at the center of the plot

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Fig. 2.2 Number of the BDS visible satellites at the 4 MGEX station

Table 2.1 Average numbers of visible GPS and BDS satellites

Table 2.2 Visibility periods of visible GPS and BDS satellites

Station

GPS average satellite number

BDS average satellite number

JFNG

9.19

14.36

WUH2

8.32

23.79

URUM

9.72

16.28

LHAZ

9.01

14.01

Aver

9.06

17.11

Constellation

Satellite type

Visibility period (h)

Mean visibility period (h)

GPS

MEO

6.17–7.63

BDS

MEO

6.45–8.04

IGSO

17.26 – 19.72

18.52

GEO

15.81–23.99

22.44

6.76 7.28

satellites have the shortest visibility period. The GPS and BDS MEO satellites have similar visibility periods, although the latter have a bit longer visibility period. Both the number and visibility period of satellites are important factors for position determination and GNSS-related applications. Typically, more visible satellites will enable more accurate position fix and this is one main reason why Japan developed the QZSS, as mentioned earlier. This is because more satellites or more base stations not only provide more information to achieve a diversity gain, but they could also improve the distribution of the satellites or base stations to reduce the geometric dilution of precision (GDOP). To some degree, a longer visibility period means more

2.1 Navigation Satellite Systems

19

visible satellites. Also, a long visibility of a satellite would be useful to generate more data of the direct and reflected signals of a satellite. As studied in Chap. 9, more soil moisture estimates can be obtained with the observed data of a GEO satellite over a day, compared with a MEO satellite.

2.2 PRN Codes and Multiple Access The navigation satellites in a specific constellation are synchronized and continuously transmit signals towards the Earth’s surface. A ground-based, airborne or spaceborne antenna and receiver can capture the signals of a group of visible satellites of the constellation. The receiver which needs to extract the signals of individual satellites one by one using the pseudorandom noise (PRN) code of each satellite. A PRN code possesses the properties of a random noise code, but it has a deterministic pattern with a given period and can be repeatedly generated by a device. The PRN codes of the satellites are different from each other, so that the signal of a satellite will be extracted without much interference from other satellites. This is similar to spreadspectrum communications, where the PRN codes play the role of spreading codes. To minimize the interference between the satellite signals, the spreading codes must be selected carefully. In the remainder of this section, more information on the PRN codes and multiple access will be presented.

2.2.1 m-Sequence A m-sequence of length N = 2m − 1 can be generated by using a m-stage linear feedback shift register (LFSR), as shown in Fig. 2.3. m-sequences are the maximal length sequences, which by definition are the longest sequences that can be generated by an LFSR of given length. At the output of the LFSR, a sequence of length N repeats continuously. The contents of the m-stage shift register go through all possible N mtuple values or stages. The all-zero state is the only forbidden m-tuple, since the LFSR would lock in this state. The feedback gain gi is either one or zero and a zero gain means that no data goes through the feedback branch or the connection does not exist.

SR

SR

Fig. 2.3 Linear feedback shift-register generator

SR

SR

output

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2 Navigation Satellite Constellations and Navigation Signals

A m-sequence has a number of remarkable properties, among which are: (a) (b)

A m-sequence contains one more one than zero and the number of ones in the sequence is equal to (N + 1)/2 The periodic autocorrelation function is two-valued and given by  Rs (k) =

1.0, k = l N − N1 , k = l N

(2.1)

where l is a positive integer. (c)

The modulo-2 sum of an m-sequence and any phase shift of the same sequence is another phase shift of the same m-sequence.

m-sequences also have undesirable properties. There are relatively few msequences for a given shift register length. In general, m-sequences do not have good cross-correlation properties except a few pairs of m-sequences with low full period cross-correlation. To achieve better interference cancellation, better spreading codes should be used.

2.2.2 Gold Codes Gold codes were invented for multiple-access applications [6] originally in wireless communications and later in satellite navigation. All the navigation satellites make use of Gold codes for multiple access except the early generations of GLONASS satellites. For instance, GPS used a set of 32 Gold codes as the spreading codes to modulate the navigation data transmitted from the 32 satellites, respectively. Later another 31 Gold codes were generated and allocated to other 31 current and future GPS satellites. The specific amount of interference from a user using a different spreading code is related to the cross-correlation between the two spreading codes. Gold codes are generated with the goal to minimize the cross-correlation. A set of 2m + 1 Gold codes with a period of 2m − 1 are generated by using a preferred pair of m-sequences, which are also used as spreading codes. As shown in Fig. 2.4, Fig. 2.4 Gold code generator

m-sequence generator A clock

bit-by-bit modulo-2 adder

Gold sequence m-sequence generator B

2.2 PRN Codes and Multiple Access

21

the outputs of two preferred pair of m-sequences are bit-by-bit modulo-2 added to generate the Gold sequences. The off-peak autocorrelation and cross-correlation functions of Gold codes are three-valued: − N1 t (m)

1 N 1 (t (m) N

(2.2) − 2)

where  t (m) =

1 + 20.5(m+1) , m is odd 1 + 20.5(m+2) , m is even

(2.3)

The most important property of Gold codes is that their cross-correlation peaks are not greater than the minimum possible cross-correlation peaks between any pair of m-sequences of the same length.

2.2.3 Multiple Access To enable multiple access, three basic multiple access schemes can be used to transmit data of multiple users simultaneously, which are code division multiple access (CDMA), frequency-division-multiple-access (FDMA), and time-divisionmultiple-access (TDMA). In some cases, a hybrid version may be used, such as CDMA + TDMA and TDMA + FDMA. In theory, very similar spectral efficiency is achieved by CDMA, FDMA and TDMA, but each multiple access technique has its own challenging issues. For instance, strict power control is required for CDMA, because any unnecessary increase in transmission power of one user would increase the interference to all other users. The received power depends on both transmission power and channel conditions, so that the transmission power may be adjusted based on channel conditions. Strict timing synchronization is needed for TDMA scheme and each time slot is assigned with a guard time to tolerate clock drift and possible multipath interference. In the case of FDMA, precise sub-band frequency generation is required and a guard band is typically inserted between adjacent channels to prevent Doppler-induced interference and possible frequency error. All navigation satellite constellations make use of CDMA, except the early generations of GLONASS satellites which completely used FDMA. Before modulating the carrier wave for signal transmission, the navigation data are first modulated by a PRN code and each satellite has a unique PRN code to enable the simultaneous transmission of signals from multiple navigation satellites and the signals can be distinguished from each other at the receiver. This is also called spectrum-spread modulation technique, which makes the overall GNSS signal bandwidth much wider than the bandwidth of the original navigation data.

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2 Navigation Satellite Constellations and Navigation Signals

When propagation channels are shared using spread-spectrum techniques, all users (satellites in this case) are permitted to transmit signals simultaneously using the same band of frequency. A different spreading code or spreading sequence is assigned to each satellite, each data bit of which is multiplied by the spreading sequence, which means spreading. The signals from different transmitters can be separated at the receiver through despreading. In a multiple access scenario, a goal is to find a set of spreading codes such that as many users as possible can use the same band of frequency with as little interference as possible.

2.3 Carrier Modulation In its basic form, a carrier wave can be described as s(t) = a(t) cos(2π f (t)t + φ(t))

(2.4)

where a(t), f (t) and φ(t) are amplitude, frequency and phase offset, respectively. If the three components are always constant prior to transmission, the carrier wave is unmodulated. As GNSS is concerned, phase modulation is typically applied to the carrier wave. This section briefly studies a number of carrier phase modulation schemes considered or used for the generation of navigation signals.

2.3.1 BPSK Modulation Binary phase shift keying (BPSK) just uses two phase offsets to represent the two binary symbols or digits such as zero and one. In order to best distinguish the two symbols from each other, the difference between the two phase offsets should be maximized. Since the phase ranges between 0◦ and 360◦ , the maximal difference is 180◦ . That is, the simple way to assign the two phase offsets is to assign them with 0◦ and 180◦ , respectively, as illustrated in Fig. 2.5. Accordingly, (2.3) can be written as Fig. 2.5 Constellation diagram of binary phase shift keying

Q 0

1

I

2.3 Carrier Modulation

23

Fig. 2.6 Modulated carrier wave in the absence of phase jump (a) and in the presence of phase jump (b)

(a)

(b)

sκ (t) = A cos(2π f t + (κ − 1)π ), κ = 0, 1

(2.5)

where the amplitude and frequency are constants. The binary data bits, zero and one, can be transmitted as sinusoidal signals: 

s1 (t) = A cos(2π f t) s0 (t) = −A cos(2π f t)

(2.6)

The PRN code-modulated navigation data sequence and the carrier wave are synchronized and the PRN code chip period is equal to multiple complete carrier cycles. For instance, GPS signals are transmitted on three carrier frequencies which are multiples of C/A code chip rate ( f 0, C/A = 1.023 MHz) and P(Y) code chip rate ( f 0,P(Y) = 10.23 MHz), [3]: ⎧ ⎪ ⎨ f L1 = 1540 × f 0, C/A = 154 × f 0, P(Y) = 1575.42 MHz f L2 = 1200 × f 0, C/A = 120 × f 0, P(Y) = 1227.60 MHz ⎪ ⎩ f L5 = 1150 × f 0, C/A = 115 × f 0, P(Y) = 11176.45 MHz

(2.7)

As a result, when the modulated navigation data bit changes from one value to the other, the carrier wave will change smoothly as illustrated in Fig. 2.6a. On the other hand, if they are not synchronized or the chip period is not equal to multiple carrier cycles, a phase jump may occur as illustrated in Fig. 2.6b, which is not desirable. Such a high slope change (90◦ in this case) is impractical to generate by a realistic device. Also, the phase jump would result in a rather high frequency change, causing interference to other radio systems.

2.3.2 QPSK Modulation In order to increase transmission data rate, Mary PSK may be applied, such as quadrature phase shift keying (QPSK), as illustrated in Fig. 2.7. Only one data bit per symbol is transferred with BPSK modulation, while two data bits per symbol is transmitted with QPSK. Unlike BPSK which can completely avoid phase jump, the phase jump is unavoidable with QPSK. For instance, in the standard QPSK, the

24

2 Navigation Satellite Constellations and Navigation Signals

Q

Fig. 2.7 Constellation diagram of quadrature phase shift keying

01

11 I

00

10

phase jump is ±90◦ if one bit changes in the symbol such as from 10 to 11 or 00 as shown by the solid black arrows in Fig. 2.7, but the phase jump can be 180◦ if the two bits change simultaneously such as from 00 to 11 or from 10 to 01, as indicated by the dashed red arrows in Fig. 2.7. The carrier wave is modulated by the four different symbols as   π sκ (t) = A cos 2π f t + (2κ − 1) , κ = 1, 2, 3, 4 4 which can be further written as √ √ ⎧

⎪ s1 (t) = A cos 2π f t + π4 = A 2 2 √cos(2π f t) − A 2 2 √sin(2π f t) ⎪ ⎪

⎨ 2 2 = − A√ cos(2π f t) − A√ sin(2π f t) s2 (t) = A cos 2π f t + 3π 4 2 2

5π A 2 A 2 ⎪ ⎪ ⎪ s3 (t) = A cos 2π f t + 4 = −√ 2 cos(2π f t) + √ 2 sin(2π f t) ⎩ = A 2 2 cos(2π f t) + A 2 2 sin(2π f t) s4 (t) = A cos 2π f t + 7π 4

(2.8)

(2.9)

where s1 (t), s2 (t), s3 (t) and s4 (t) are for symbols 11, 01, 00, and 10, respectively. It can be seen that the in-phase data bits (first data bits) are represented by cos(2π f t), while the quadrature data bits (second data bits) are represented by sin(2π f t). That is, the signal is constructed as sκ (t) = I(t) + Q(t) √ √ A 2 A 2 cos(2π f t) + b Q (t) cos(2π f t + π/2) = b I (t) 2 2 √ √ A 2 A 2 = b I (t) cos(2π f t) − b Q (t) sin(2π f t) 2 2

(2.10)

where b I (t) and b Q (t) are the in-phase and quadrature data bit sequences, respectively. Figure 2.8 shows the basic block diagram of the QPSK modulator. The original data bit series are first separated into in-phase and quadrature bit sequences. The NRZ (non-return to zero) encoding then converts digital data bits (zeros and ones) into rectangular pulse waveform of positive and negative voltages. The locally generated cos(2π f t) is multiplied with the in-phase data bits, while − sin(2π f t) yielded from

2.3 Carrier Modulation

In-phase bits 01101

bit sequence

25

NRZ Encoding

Sine wave generator

QPSK Signal

0110110010 phase Quadrature bits 10100 NRZ Encoding

Fig. 2.8 Block diagram of generation of QPSK signal

90◦ phase shift of the original sinusoidal signal is multiplied with the quadrature data bits. Finally, the two products are added to generate the QPSK signal.

2.3.3 Variants of QPSK Modulation To limit the maximum phase jump, i.e., to avoid the phase change of 180◦ , two of the QPSK variants, namely offset QPSK and π/4-QPSK, can be used. In the offset QPSK, the in-phase and quadrature symbol bits are offset by a bit period or half a symbol period as illustrated in Fig. 2.9. Such a simple operation does guarantee the maximum phase change is limited to ±90◦ , and completely remove the phase change of 180◦ . The phase change of 180◦ related to QPSK is also illustrated in Fig. 2.9. π/4-QPSK, another variant of QPSK has the constellation diagram as shown in Fig. 2.10. It consists of two constellations, the original and the rotated version of QPSK constellation with the rotation angle being π/4. One constellation is used to modulate odd symbol numbers, while the other constellation is used to modulate even symbol numbers. Thus, a stream of identical data bits will always yield a phase change. The maximum phase jump is 135◦ , which ranges between 90◦ of offset PSK and 180◦ of QPSK. Although the maximum phase jump of π/4-QPSK is greater than offset QPSK, π/4-QPSK can be used for differential detection.

26

2 Navigation Satellite Constellations and Navigation Signals Data bit sequence 011011001110 I(t)

Q(t) QPSK Signal phase QPSK phase jump

Time

Q(t-Tb) OQPSK Signal phase OQPSK phase jump

Time

Fig. 2.9 Illustration of signal phase of QPSK and offset QPSK

Q

Fig. 2.10 Constellation diagram of quadrature phase shift keying

01

01

1 11

00 00

I

10 10

2.3.4 Binary-Offset-Carrier Modulation Binary offset carrier (BOC) modulation, originally developed by John Betz [2], is a square sub-carrier modulation, where a signal is multiplied by a rectangular subcarrier of frequency which is equal to or greater than the chip rate. The spectral energy of BPSK-modulated signals is concentrated around the carrier frequency, while the

2.3 Carrier Modulation

27

spectral energy of the BOC-modulated signal is low around the carrier frequency. Therefore, the interference between the BOC-modulated and BPSK-modulated navigation signals would be reduced. This would be the main reason for the introduction of BOC modulation for GNSS signals, especially for GPS L2C and Galileo signals. A BOC-modulated signal has two main spectral lobes located on both sides of the carrier frequency, so BOC modulation is also termed split-spectrum modulation. BOC modulation has a number of variants, including sine (or sine-phased) BOC (SinBOC), cosine (or cosine-phased) BOC (CosBOC), alternative BOC (AltBOC), multiplexed BOC (MBOC), and double BOC (DBOC) [12].

2.3.4.1

Sine and Cosine Binary-Offset-Carrier Modulation

The SinBOC waveform is defined as   NBOC1 × π t , 0 ≤ t ≤ Tc sSinBOC (t) = sign sin Tc

(2.11)

where sign(t) is the sign function of t, Tc is the chip period and NBOC1 is the BOC modulation order which is typically designed as a positive integer and defined as NBOC1 = 2 ×

f sc m =2× n fc

(2.12)

Here f sc is the frequency of rectangular sub-carrier; f c = 1/Tc is the chip rate; both f sc and f c have units of MHz; m = f sc /1.023; n = f c /1.023; Both m and n are positive integers, which are the two parameters of SinBOC, usually denoted as BOC(m, n). Figure 2.11 shows two examples of BOC waveform, BOC(1,1) and BOC(6,1), which are rectangular waveforms in time domain. Similarly, the CosBOC waveform is defined as Fig. 2.11 Examples of SinBOC waveform in time domain

BOC(1,1)

1 0 -1

0

0.2

0.4

0.6

0.8

1

0.8

1

BOC(6,1)

1 0 -1

0

0.2

0.6 0.4 Time (chip)

28

2 Navigation Satellite Constellations and Navigation Signals

  NBOC1 sCosBOC (t) = sign cos × π t , 0 ≤ t ≤ Tc Tc

2.3.4.2

(2.13)

Double Binary-Offset-Carrier Modulation

The DBOC waveform can be described by NBOC2 −1 NBOC1 −1

sDBOC (t) = pTB (t) 



k=0

i=0

(−1)i+k δ(t − i TB1 − kTB )

(2.14)

where pTB (t) is a rectangular pulse of amplitude 1,  is the convolution operator, TB =Tc /(NBOC1 NBOC2 ) with NBOC2 being treated as the BOC-modulation order of the second stage, δ(t) is the Dirac delta function, and TB1 =Tc /NBOC1 . The expression of the DBOC waveform was derived based on the expressions of the SinBOC and CosBOC waveforms [12]. In fact, BPSK, SinBOC and CosBOC are special cases of DBOC defined by (2.12): ⎧ ⎪ ⎪ ⎨

BPSK, SinBOC, DBOC ≡ ⎪ CosBOC, ⎪ ⎩ Higher order DBOC,

2.3.4.3

NBOC1 = 1,NBOC2 NBOC1 > 1,NBOC2 NBOC1 > 1,NBOC2 NBOC1 > 1,NBOC2

=1 =1 =2 >2

(2.15)

Multiplexed Binary-Offset-Carrier Modulation

BOC(1, 1) was originally proposed as the modulation for the open service signal of Galileo navigation system as well as for GPS L1C signal. It was observed that the system performance could be enhanced if another BOC component with a wider bandwidth is combined with BOC(1, 1), which was chosen as BOC(6, 1) [8]. As a consequence, the so-called Multiplexed-BOC (MBOC) modulation signal is generated. There are two basic ways to combine the two modulation signals; one is the composite BOC (CBOC) and the other is the Time-Multiplexed BOC TMBOC), as illustrated in Fig. 2.12. The CBOC scheme simply adds the two signals in the time domain together, while the TMBOC scheme treats the BOC(6, 1) signal in a similar way to a pilot signal, allocating it to the Nth chip with N being the number of chips. In both schemes, the power of BOC(6,1) signal is just 1/N of the total power. It is worth mentioning that an initial comparison of the effect of BOC and BPSK modulation on GNSS-R was conducted by Juang et al. [10] with airborne data. The comparison makes use of the channel response function to model the surface scattering and a commutative diagram to show the relationship between direct and

2.3 Carrier Modulation Fig. 2.12 Illustration of CBOC and TMBOC

29 Power

BOC(1,1)

BOC(6,1)

1 (N-1)/N 1

2

…

i

…

N-1 N time

CBOC

Power 1 1

2

…

i TMBOC

…

N-1 N time

reflected signals under different modulations. It would be useful to do more investigations on the use of BOC modulated signals for specific GNSS-R applications in the future.

2.4 Composition of Navigation Signals GNSS signals are designed to enable several functions [11], including (1) precise distance measurement by the user equipment (GNSS receiver); (2) Delivery of navigation message including the locations of the GNSS satellites, clock errors, and satellite health condition; (3) Simultaneous broadcasting among multiple satellites using a common carrier frequency. Thus the properties of the carrier wave must be varied with time using spread-spectrum multiple access and phase modulation applied to the carrier wave. Accordingly, GNSS signals basically consist of three parts: navigation message or navigation data, carrier wave, and PRN code. In some cases, a fourth part exists, which is the rectangular sub-carrier. The purpose of the signal transmission is to send the navigation data to the end users to realize PNT services. The navigation message includes the satellite orbital parameters, the clock and time parameters, the ionospheric parameters, and the satellite health parameters. In the case of GPS, the navigation message is modulated by both C/A codes and P codes at a rate of 50 bit/s. Note that the encrypted P code is called P(Y) code. The navigation message is divided into frames, each of which contains 1500 data bits, or 30 s long. Each frame consists of five 6-second subframes which contain specific relevant information. Similar to typical radio communication technology, a navigation satellite makes use of sinusoidal waves of specific frequencies as the carrier to send the navigation message to users. Probably due to the availability of frequency bands and the consideration of signal power loss over the propagation path including ionosphere and troposphere from the navigation satellite to the receiver, all the frequency bands of GNSS signals were selected from the L band of radios signal, which is defined as

30

2 Navigation Satellite Constellations and Navigation Signals

Phase modulation Modulated carrier waveform Carrier waveform

Spreading sequence (Gold codes)

Spread-spread modulated navigation data Spreading by modulo-2 addition

Data sequence of navigation message Fig. 2.13 Illustration of navigation data sequence modulated by spreading codes, followed by carrier phase modulation

a frequency range between 1 and 2 GHz (i.e. wavelength between 15 and 30 cm) by IEEE. Radio stations usually use amplitude modulation (AM) or frequency modulation (FM) to modulate the carrier wave with the voice data, so that the voice signals can be received via an AM or FM radio device. Differently, GNSS uses phase modulation (PM), such as BPSK as mentioned earlier, to modulate the carrier waves with the navigation data. Figure 2.13 illustrates the operations among the three different parts of the signal: carrier wave, navigation data bits, and PRN code or spreading sequence. First, the navigation data bits are dispersed by the spreading sequence through modulo-2 addition, which may be called spread-spectrum modulation. Each data bit is dispersed into a sequence of data bits of length being equal to the PRN code length such as 1023 for GPS C/A code. Due to modulo-2 addition, in a binary navigation data sequence, a bit of zero means that the data bit is simply replaced with the spreading sequence. Equivalently, the data bit is divided into chips filled with the spreading sequence. On the other hand, if the navigation data bit is one, then it is replaced with the reversal of the spreading sequence, that is the spreading waveform is turned upside down. The modulated navigation data sequence is then used to modulate the carrier waveform. The carrier frequency, which is 1.57542 GHz for GPS L1 signal, is much higher than the modulated navigation data sequence, which has a chip rate of 1.023 MHz for C/A code and 10.23 MHz for P(Y) code. Due to the tremendous increase in frequency, this process is called up-conversion and the device is called up-converter. After the up-conversion or phase modulation, the GNSS signal is generated and ready for transmission through a right-hand circularly polarized (RHCP) antenna. Figure 2.14 shows an example of the modulated carrier waveform and the BPSK modulation is used, but BOC modulation is not considered in this case. Note that this is really an illustration of how a carrier waveform is modulated by the PRN

2.4 Composition of Navigation Signals Fig. 2.14 Example of modulated carrier waveform

31

Modulated navigation data bit sequence Original Carrier waveform

Modulated carrier waveform

code-modulated navigation data bit sequence. Here, the period of a code chip only corresponds to two cycles of the carrier waveform, but in reality it contains nearly one thousand cycles of carrier waveform for GPS C/A code. Regarding a single carrier frequency and a single spreading scheme, the GNSS signal may be described by sGNSS (t) =

√ 2PsBOC (t) · (g(t) ⊕ d(t)) · cos(2π f L t)

(2.16)

where P is the signal power, sBOC (t) is the BOC modulation waveform, g(t) is the spreading code, d(t) is the navigation data series, and ⊕ is the modulo-2 addition operator. In the case of sBOC (t) = 1, i.e. the first condition in (2.15) satisfies, only BPSK modulation is applied. Single-frequency and dual-frequency signals were designed for navigation satellites during the initial development of satellite constellations. Currently, many navigation satellites transmit triple-frequency signals, so that frequency diversity can be utilized to achieve a performance gain and to enable some specific applications such as triple-frequency carrier phase combination based snow depth estimation to be studied in Chap. 8. In the case where multiple signals of multiple carrier frequencies and/or multiple spreading schemes, (2.16) becomes the summation of multiple signals which are transmitted from the same satellite and probably the same antenna simultaneously. As mentioned earlier, except that both L band and S band are used for IRNSS, the frequencies of currently all navigation satellite signals as listed in Table 2.3 belong to the L band (1–2 GHz) of radio spectrum. In particular, they are mostly around 1.57 and 1.2 GHz. Although many of the frequency bands are close to each other, they are well separated because the signal bandwidths are just about a few dozen megahertzs. In the case of some overlapping between neighboring signal bands of different constellations, the signals can still be well distinguished from each other by dispreading, although minor interference would exist. The PRN code lengths of the four global constellations of satellites are rather different from each other, but the period of each code is one millisecond or multiple milliseconds, which may be convenient for data processing.

32

2 Navigation Satellite Constellations and Navigation Signals

Table 2.3 Signal parameters of navigation satellites System

Coding scheme

Carrier frequency (GHz)

Signal BW (MHz)

Code length (chips)

Chip rate (Mcps)

BeiDou

CDMA

1.561098 (B1)

32.736

2046

2.046

1.20714 (B2)

20.46

1.26852 (B3)

20.46

1.57542 (E1)

24.552

4092

1.023

1.191795 (E5)

51.15

1.17645(E5a)

20.46

1.20714(E5b)

20.46

Galileo

GLONASS

GPS

IRNSS QZSS

CDMA

FDMA CDMA

CDMA

CDMA CDMA

1.27875 (E6)

40.92

1.593–1.610 (G1)

17

511 (ST)

0.511 (ST)

1.237–1.254 (G2)

17

33554432 (VT)

5.11 (VT)

1.189–1.214 (G3)

25

1.57542 (L1)

24

1023 (C/A)

1.023 (C/A)

1.226 (L2)

24.6

6.18704 × 1012 (P)

10.23 (P)

1023

1.023 1.023

1.17645 (L5)

23

1.17645 (L5)

24

2.492028 (S)

16.5

1.57542 (L1)

12

1023 (L)

1.22760 (L2C)

11

4092 (E6)

1.17645 (L5)

24

1.27875 (E6)

20

2.5 Transmission of GNSS Signals Electromagnetic waves including radio waves consist of both electric and magnetic fields which oscillate in perpendicular planes. Polarization of radio waves actually refers to the direction of the electric field. A transmit antenna with a specific polarization is used to radiate the radio wave of a signal which is polarized by the antenna with the same polarization. At the receiver, typically, antennas with the same types of polarization are used to receive the radio waves, except that additional antenna with different polarization is used to capture the radio wave with changed polarization such as due to reflection. Two types of polarization are widely used in wireless communications as well as in satellite navigation, which are linear and circular polarizations. Linear polarization consists of three different schemes, vertical, horizontal, and slant polarization. When the electric field vibrates in a plane perpendicular to the Earth’s surface or a different reference plane, the antenna is vertically polarized. Alternatively, the antenna is horizontally polarized if the electric field vibrates in a plane parallel to the reference

2.5 Transmission of GNSS Signals

33

plane. In the case of slant polarization, there is an angle of 45° between the electric field vibration plane and the reference plane. In circular polarization, the electric field rotates in corkscrew pattern and completes one resolution during each wavelength. The signal energy is radiated out in all planes including the horizontal and vertical planes. There are two types of circular polarization, left-hand circular polarization (LHCP) and RHCP. Note that RHCP also refers to right-hand circularly polarized, while LHCP refers to left-hand circularly polarized too, when antenna is concerned with. Whether a radio wave is RHCP or LHCP can be determined by the hand rules. That is, curling one’s right hand fingers in the same direction of the electric field rotation, if one’s right thumb is in the direction of the wave’s propagation, it is RHCP radio wave; otherwise, the radio wave is LHCP. After transmission from the satellite, the GNSS signals propagate through the atmosphere including the ionosphere and troposphere, and finally arrive at the antenna of the receiver. During the process of propagation and reception, a number of issues would occur, which include multipath interference around the receive antenna, and rain and snow caused reflection, absorption, and phasing. Compared with linear polarization, circular polarization is more resistant to these issues mainly due to the fact that circular polarization allows signals to transmit on all planes instead of a single plane with linear polarization. For instance, regarding reflection, the polarization of some reflected signals would be changed, so that the effect of reflection would be reduced. Circularly-polarized signals would better penetrate and bend around obstructions. This is probably why antennas with circular polarization are used to transmit and receive GNSS signals.

References 1. BeiDou (2020, visited) http://en.beidou.gov.cn/ 2. Betz J (1999) The offset carrier modulation for GPS modernization. In: Proceedings of ION technical meeting. pp 639–648 3. Borre K, Akos DM, Bertelsen N, Rinder P, Jensen SH (2006) A software-defined GPS and Galileo receiver: a single-frequency approach. In: Applied and numerical harmonic analysis. Birkhauser, Boston 4. Galileo (2020, visited) https://www.gsa.europa.eu/ 5. GLONASS (2020, visited) https://www.glonass-iac.ru/en/ 6. Gold R (1967) Optimal binary sequences for spread spectrum multiplexing. IEEE Trans Inf Theory 13(4):619–621 7. GPS (2020, visited) https://www.gps.gov/ 8. Hoult N, Aguado LE, Xia P (2008) MBOC and BOC(1,1) performance comparison. J Navig 61(4):613–627 9. IRNSS (2020, visited) https://www.isro.gov.in/irnss-programme 10. Juang J-C, Lin C-T, Tsai Y-F (2020) Comparison and synergy of BPSK and BOC modulations in GNSS reflectometry. IEEE J Sel Top Appl Earth Obs Remote Sens 13:1959–1971 11. Kaplan ED, Hegarty C (2017) Understanding GPS/GNSS: principles and applications, 3rd edn. Artech House 12. Lohan ES, Lakhzouri A, Renfors M (2007) Binary-Offset-Carrier modulation techniques with applications in satellite navigation systems. Wirel Commun Mob Comput 7:767–779

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2 Navigation Satellite Constellations and Navigation Signals

13. Ma X, Yu K, He X, Montillet JP, Zhang Q (2020) Positioning performance comparison between GPS and BDS with data recorded at four MGEX stations in 2020. IEEE Access 8:147422– 147438 14. QZSS (2020, visited) https://qzss.go.jp/en/

Chapter 3

Signal Scattering and Reception Schemes

After emitted from the antenna onboard a GNSS satellite with a transmission power such as about 25 W, the GNSS signal propagates towards the Earth. After traveling over a distance of about 20,000 km in the case of GEO satellites, the signal power has dropped about 182 dB, making the signal power being rather small, even below the background noise level. Fortunately, the weak GNSS signals can be reliably received and detected partly due to the large spreading gain. When traveling through the ionosphere and the troposphere, extra propagation delays compared with propagation in free space are produced. The extra delay due to the ionospheric effect is particularly significant, which is different for different signal propagation path. Nice models and techniques have been developed to compensate those delays, enabling accurate pseudorange and carrier phase measurement. When GNSS signals touch the Earth’s surface which can be soil, water, ice, snow, vegetation, or forest, signal scattering occurs. Depending on the scattering surface, the scattering can be coherent, diffuse, or hybrid. This chapter studies the first Fresnel zone of signal scattering and potential application of signal power variation for surface change detection. Another focus of the chapter is on the basic structure of GNSS-R receiver. Furthermore, the three GNSS-R receiver platforms are briefly introduced.

3.1 First Fresnel Zone 3.1.1 Semi-major and Semi-minor Axes Spatial resolution is an important concept/specification for every remote sensing technique. Unlike many other remote sensing techniques, the spatial resolution of GNSS reflectometry may be approximated by the first Fresnel zone (FFZ) on the ground surface, which is the scattering surface contributing to most of the received signal power when the coherent scattering is dominant. The Fresnel zones are ellipses whose © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_3

35

36

3 Signal Scattering and Reception Schemes

Fig. 3.1 Geometry relating receiver, GNSS satellite and specular reflection point

semi-minor axis and semi-major axis can be determined with the aid of Fig. 3.1. The path length from the transmitter T, through the specular point S, and to the receiver R is equal to TS + SR, while the path length from the transmitter, through another scattering point P, and to the receiver is TP + PR. The path length difference is then given by

=



δ1 (x, y) = (T P + P R) − (T S + S R)  r 2 2 2 (x − r ) + y + h 2 + x 2 + y 2 + h 21 − cos θ

(3.1)

where θ is the grazing angle which is usually approximated as the elevation angle especially for ground-based receivers provided that reflection surface is flat without slope. Setting the path difference to be half a wavelength, (3.1) can be rewritten as   r λ − x 2 + y 2 + h 21 (x − r )2 + y 2 + h 22 = + 2 cos θ Squaring both sides of (3.2) produces 

2r λ+ cos θ



= 2r x + h 21 − h 22 − r 2 + λr + (h 1 + h 2 )(h 1 − h 2 ) + cos θ λr = 2r x + + r h 1 tan θ − r h 2 tan θ + cos θ = 2r x +



x 2 + y 2 + h 21 r λ + 2 cos θ

2

λ2 + (r tan θ )2 4 λ2 + (r tan θ )2 4

(3.2)

3.1 First Fresnel Zone

37

= 2r x +

λr λ2 + 2r h 1 tan θ + cos θ 4

(3.3)

Since cosr θ = T S + S R  λ, with some mathematical manipulations (3.3) can be simplified as 

x 2 + y 2 + h 21 = x cos θ + h 1 sin θ +

λ 2

(3.4)

Squaring both sides of (3.4) and then rearranging the equation produce the standard form of the ellipse equation: (x − xc )2 y2 + =1 a2 b2

(3.5)

where (xc , 0) is the centre of the ellipse and a and b are the semi-major axis and semi-minor axis, respectively, given by 

 λ cos θ xc = + h 1 sin θ 2 (sin θ )2  λh 1 sin θ + λ2 /4 a= (sin θ )2  λh 1 sin θ + λ2 /4 b= sin θ

(3.6)

The area covered by such an ellipse is called the FFZ where any reflected signal will arrive at the receiver with a delay of up to half a wavelength relative to the signal reflected at the specular point. Figures 3.2 and 3.3 show the semi-major and semi-minor axes, respectively, with the grazing angle ranging from 10° to 90° and with six receiver altitudes which may represent ground-based, airborne and spaceborne receiver platforms. To show clearly the small semi-major or semi-minor axis in the case of a ground-based platform, the vertical axis uses logarithmic scaling.

3.1.2 Simplified Elliptical Equation and Resultant Error In the case where the receiver platform is an aircraft or a satellite, (3.6) can be simplified as xc = h 1 cot θ √ λh 1 sin θ a= (sin θ )2

38

3 Signal Scattering and Reception Schemes 4

Fig. 3.2 Semi-major axis versus grazing angle and receiver altitude

10

h=600km

3

Semi-major Axis (m)

10

2

10

h=300km

h=3km

1

h=200m

0

h=2m

10

10

h=10m

-1

10

10

20

30

40 50 60 Grazing Angle (deg)

70

80

90

80

90

3

Fig. 3.3 Semi-minor axis versus grazing angle and receiver altitude

10

h=600km h=300km

Semi-minor Axis (m)

2

10

h=3km 1

h=200m

0

h=2m

10

h=10m 10

-1

10

10

20

30

√ λh 1 sin θ b= sin θ

40 50 60 70 Grazing Angle (deg)

(3.7)

The formulas for the semi-major and semi-minor axes in (3.7) are in accordance with those given in [5, 6]. It is worth mentioning that significant error may be produced if (3.7) instead of (3.6) is used to calculate the dimensions of the FFZ when the receiver is ground-based with a low height and the grazing angle is small. Figure 3.4 shows the error of the FFZ area normalised to the true area size versus the grazing angle with four antenna/receiver altitudes. The maximum area error can be 35% for the case of antenna altitude 1 m and grazing angle 5°. However, in the airborne and spaceborne cases, such error is completely negligible.

3.2 Signal Power Based Estimation of Object Width in Forest 0.4

1m 2m 4m 6m

0.35 Normalised Area Error

Fig. 3.4 FFZ area error by (3.7) versus small grazing angles and low antenna heights. The error is normalised to the exact zone area with semi-major and semi-minor axes calculated from (3.6)

39

0.3 0.25 0.2 0.15 0.1 0.05 0 5

10 Grazing Angle (deg)

15

3.2 Signal Power Based Estimation of Object Width in Forest

5 Sate#31 0 -5 -10 0

Peak Power (dB)

Fig. 3.5 Reflected signal peak power associated with two satellites

Peak Power (dB)

It is useful to measure the dimensions of surface changes such as forest change. Here, a data set collected when an aircraft flew over a specific area in a forest was processed to gain some insight into this issue. In this case the non-coherent integration time is set to be 100 ms, equivalent to about a 6 m distance. Figure 3.5 shows the peak power of the reflected signal associated with two GPS satelites (PRN#31 and PRN#1). The corresponding two reflection tracks are colourised by signal power as shown in Fig. 3.6. These two tracks cross a river three times and the crossing sections can be clearly identified by the signal power spikes in Fig. 3.5. By carefully checking Fig. 3.6, one notices that there are some mismatches between the river

0.2

0.4

0.6

1.2 0.8 1 Distance (km)

1.4

1.6

1.4

1.6

5 Sate#1 0 -5 -10 0

0.2

0.4

0.6

1.2 1 0.8 Distance (km)

40

3 Signal Scattering and Reception Schemes -4

-5

-6

-7

-8

-9

-10

Fig. 3.6 Two colourised reflection track segments, upper for satellite #31 and lower for satellite #1

crossing sections (RCS) and the power spike locations. Here the RCS is defined as the segment of reflection track on the river, which is not equal to the width of the river except for the case where the reflection track is perpendicular to the river. This may be caused by the reflection track position error due to receiver position determination error, ground elevation error, or error in the georegistration of Google Earth. Is it possible to use the width of the power spikes to estimate the width of the RCS? To answer this question, some preliminary studies and discussions are provided below. The power spike width can be determined by finding the approximate start point of the leading edge and end point of the trailing edge. The width of the first spike in the second figure of Fig. 3.5 may not be determined accurately because of the presence of extra smaller spikes caused by the rather small crossing angle. From the Google Earth image the RCS width (along track direction) can be manually determined using Google Earth Ruler. The measured power spike widths and the RCS widths are listed in Table 3.1. The power spike width is wider than the RCS width by a value between 9 and 19 m. These width differences would be associated with several factors. The power threshold was set empirically, so the threshold-related two points on the leading and trailing edges do not correspond to the start and end points of the crossing section in general. The dimensions and orientations of FFZ and the effective scattering area will affect the difference. The 100 ms non-coherent integration will Table 3.1 Actual river crossing section width and power spike width RCS-1 Power spike width (m)

RCS-2

RCS-3 30

Sate#31

75

50

Sate#1

136

54

33

RCS width (m)

Sate#31

60

34

21

Sate#1

120

35

20

Width difference (m)

Sate#31

15

16

9

Sate#1

16

19

13

3.2 Signal Power Based Estimation of Object Width in Forest

41

also increase the width of the power spike. It is interesting to conduct further research to derive the appropriate power threshold by considering all the major factors so that the power spike width approaches the length of the crossing section. Note that the major axis or minor axis of the FFZ and the effective scattering area are greatly affected by the geometry of the satellite, the receiver and the specular reflection point. For instance, the major axis is on the projection of the straight line connecting the satellite and receiver on the ground surface. The reflection track segment crosses the FFZ at an interior angle denoted by ϑ which satisfies 0 ≤ ϑ ≤ π/2. In the case where the major axis is in the along-track direction, ϑ = 0. The direction of the major axis is equal to the azimuth angle of the satellite, while the reflection track direction can be measured by Google Earth Ruler. Thus, the interior angle can be readily determined. With some mathematical manipulations, it can be shown that the length of such a cross segment is given by L cs = 

2ab

(3.8)

(a sin ϑ)2 + (b cos ϑ)2

The satellite azimuth, the track direction, the interior angle, and the cross section width (L cs ) are all listed in Table 3.2. Because of the small interior angles and relatively large elevation angles, the FFZ cross section length is close to the major axis. The FFZ can only be used as the approximate pixel size (spatial resolution) of the GNSS reflectometry technology when coherent scattering is dominant. For instance, it would be suited for the case where the river surface is calm and flat. However, when approaching the river bank, the scattering is rather complex since the scattering surface includes both river surface and the forest for Fig. 3.6. It is thus interesting to do more investigations on this important issue through careful design and conduct of airborne or ground-based experiments. Table 3.2 Dimensions of the FFZ of the signal reflected along the tracks in Fig. 3.6 Satellite no

#31

#1

RCS

1st

2nd

3rd

1st

2nd

3rd

Flight height (m)

230

270

200

273

285

200

Sat elevation (deg)

69

69

69

53

53

53

Sat azimuth (deg)

97.1

97.1

97.1

267.7

267.7

267.7

Track direction (deg)

110

106.6

104.3

109

104

102

Interior angle (deg)

12.9

9.5

7.2

21.3

16.3

14.3

Major axis (m)

14.6

15.8

13.6

20.2

20.6

17.2

Minor axis (m)

13.6

14.8

12.8

16.2

16.4

13.8

(m)

14.5

15.7

13.6

19.5

20.2

16.9

42

3 Signal Scattering and Reception Schemes

3.3 GNSS-R Receiver Originally, GNSS-R experiments were conducted using software receivers which have multiple RF front-ends. At the present time, some software receivers are still often used for conducting research in GNSS-R. The front-ends process the received signals to produce intermediate frequency (IF) signals. By sampling the analogue signal followed by quantization during the analogue-to-digital conversion (ADC), the raw IF data bits are stored typically on a hard disk or a laptop for early ground-based and airborne experiments. There are many different software receivers; for instance, more than a decade ago, nine different software receivers were listed in (NoguesCorreig et al. [10]). One of such software receivers, the NordNav receiver, which has four front-ends, has been used in the experiments for the Garada Project (2010– 2013) managed by Australian Center for Space Engineering Research (ACSER), University of New South Wales. When using a software receiver, the raw IF data are usually processed through cross-correlating the raw data with the local signal replica of GNSS pseudorandom noise codes and using dedicated software. As a result, samples of cross-correlation function are produced over a range of Doppler frequencies and a number of code chips (i.e. a waveform is produced). Clearly, such a procedure requires intensive cross-correlation computation which involves the use of software routines, good knowledge on how to use the software to process the raw data, and a lot of time to run these routines to generate the results. These waveform data are just the basis for further analysis to remotely sense geophysical parameters. In addition, such receivers usually utilize a large number of slaved correlators, which is not accommodated with traditional ASIC design. This issue is even severer for two particular correlation schemes, the so-called interferometric processing and the direct-signal enhanced semi-codeless processing [8], which require a substantially increased signal processing effort. Both methods do not correlate the reflected signal with a clean replica signal, but directly cross-correlate the line-of-sight signal received from a zenith-looking antenna either with the reflected signal from a nadir-looking antenna for the first method or with a modelled replica of the encrypted signal obtained in a semi-codeless way for the second method. Nevertheless, these schemes have the advantages that no prior knowledge of the incoming signal is required and there is a significant improvement in the measurement precision and the carrier-to-noise ratio. For altimetric applications in particular, which are probably the most demanding applications in terms of the required instrument hardware and processing capabilities, this is of crucial benefit [16]. To resolve such issues a better instrument, based on a hardware receiver, needs to be developed so that the cross-correlation can be performed in real time. As a consequence, the scientific end users and industry can directly analyze the data obtained from either airborne or spaceborne experiments or missions to infer the geophysical parameters. Due to the importance of such a hardware receiver, about one decade ago, a number of universities and research institutions were already developing hardware receivers. They include the early instruments developed by NASA Langley Research

3.3 GNSS-R Receiver

43

Center [2], CSIC-IEEC (Nogues-Correig et al. [10]), Beihang [7], SSTL [11], UPC [13], and UNSW [3]. In the past decade, many other research institutions have also developed their own GNSS-R receivers, but the structures and fundamental principles would be similar. The basic design of the bistatic receiver reported in this section is based on the information of some of those early existing hardware receivers. Although the work presented was mainly performed in 2013, the information may still be usable in the design of future hardware receiver. The focus of this section is first on the basic description of the bistatic receiver, the architecture and signal processing of the back-end of the receiver. Also, detailed discussions on the number of front-ends and the utilisation of multi-frequency and multi-GNSS constellations are provided. In addition, the issue of antenna selection is also discussed for the sake of system simplicity as well as good performance.

3.3.1 Brief Description of Hardware Receiver Figure 3.7 shows the block diagram of the first scheme of the hardware receiver. It consists of two main parts, the multiple RF front-ends and the signal processing

GNSS receiver

RHCP antenna

Navigation message

Reference oscillator

LNA RF front-end 1

n-bit ADC

RF front-end 2

n-bit ADC

Analogue IF signal

Digital IF Signal

LNA High-gain LHCP antenna

RF Front-End

Signal processor

USB interface

Waveform output Signal Processing Back-End

Fig. 3.7 Block diagram of the first scheme of the hardware receiver

44

3 Signal Scattering and Reception Schemes

back-end. Typically, external antennas are required to receive the GNSS signals. That is, the RHCP antenna is used to receive the direct GNSS signals, whereas the LHCP antenna (and RHCP antenna in some cases) is (are) used to receive the reflected GNSS signals. Although there is a low noise amplifier (LNA) or even more in the RF frontends, an external LNA may be necessary to amplify the reflected signals when passive antennas are used. In the case of a spaceborne receiver, to reliably detect the really weak reflected signal, some special strategies are required. One strategy is to utilize a high-gain antenna. The RF front-end consists of two identical front-ends which would be similar to most of the existing front-ends. These front-ends produce analogue IF signals with a specific central frequency which can be selected to be around 20 MHz. Digital IF signals are produced by sampling the analogue signals at a frequency of around 40 MHz through an ADC. In some hardware receiver designs, the resolution of the ADC is significantly different from others. For instance, one receiver uses 1-bit ADC, whereas another uses 8-bit ADC. Clearly, the computational complexity is the minimum when using the 1-bit ADC. As the resolution (i.e. the number of bits) increases, the computational complexity increases. On the other hand, the SNR of the waveform produced by the signal processor will improve with the number of bits as observed in (Intel [4]). However, the improvement would be negligible when the number of bits is greater than some value. At the moment, an ADC with 3-6 bits can be considered optimal although the most suitable value still needs to be determined. These digital samples from the ADC are the input to the signal processor to generate the two-dimensional delay waveform or three-dimensional delay-Doppler waveform. A reference oscillator is used to maintain synchronization between the GNSS receiver clock, the signal processor clock, and the local oscillators in the two RF frontends. This system reference oscillator operates at a frequency of around 40 MHz, which may also be adjustable. An oven controlled crystal oscillator (OCXO) can be used as the reference oscillator. The accuracy of the OCXO is moderate or even high; for instance, the accuracy of a 5–10 MHz OCXO is 20 ppb (parts per billion), whereas the long-term (e.g. 10 years) stability is around 20-200 ppb per year. The GNSS receiver can be a commercial GNSS receiver card which takes the direct signal received by the RHCP antenna as input to calculate the navigation solution which is then forwarded to the signal processor. USB interface is required to output the waveform data (preferably, and the navigation data as well) to an external device such as a laptop where the data are logged for further processing, or a flash drive as a buffer in order to forward the data to the ground through a wireless downlink channel. Only two front-ends are considered in the initial design of the receiver; however, such a design can be extended to the case of multiple front-ends capturing signals from different GNSS constellations (e.g. GPS, GLONASS, BDS, and Galileo) or different frequency bands (e.g. GPS L1, L2 and L5; BDS B1, B2 and B3). The state-of-theart GNSS-R receivers are usually capable of processing GNSS signals of multiple frequencies and multiple constellations. In some cases, the manufacture may build a receiver which can processing signals of specific constellations and frequencies based on the requirements of customers. The second design scheme is shown in Fig. 3.8. It can be seen that the extra GNSS receiver is removed. In this scheme the front-end (or front-ends) of the receiver can

3.3 GNSS-R Receiver

45

RHCP antenna LNA

Analogue IF signal

Digital IF Signal

RF front-end 1

n-bit ADC

Reference oscillator RF front-end 2 LNA High-gain LHCP antenna

RF Front-End

Signal processor n-bit ADC Waveform output

USB interface

Signal Processing Back-End

Fig. 3.8 Block diagram of the second scheme of the hardware receiver

be used as the front-ends of the hardware receiver subject to some modifications. In particular, new facilities and features need to be added to the existing platform such as the Namuru platform used by UNSW to realize the function of delay-Doppler map (DDM) generation. The digital IF data bits from the ADC connected to front-end 1 are used to track the code phase and the carrier frequency of the direct signal as well as to decode the navigation message. The navigation message retrieval, the code and frequency tracking, as well as the cross-correlation computation are all realized in the signal processor. The advantage of the first scheme is that any modification of the two RF frontends and the two ADCs will not affect the GNSS receiver. That is, the two parts can be treated independently. As a result, such a scheme is more flexible. The disadvantage of the scheme is that an extra GNSS receiver is required. On the other hand, the advantage of the second scheme is the avoidance of the extra GNSS receiver. However, in the presence of any modifications in the frond-ends or in the ADCs, both the navigation message detector and the cross-correlator in the signal processor may need to be modified accordingly.

46

3 Signal Scattering and Reception Schemes

3.3.2 Signal Processor As mentioned earlier there are two main parts in the hardware receiver, the RF front-end and the signal processing back-end. In this subsection the key hardware components, the architecture, and the flow of the signal processor are described. Also, the selection of the DDM parameters is discussed.

3.3.2.1

Architecture of Signal Processor

Figure 3.9 shows the block diagram of the signal processor. The processor board consists of two main chips, the FPGA (field-programmable gate array) chip and the DSP (digital signal processing) chip. The functions implemented on a FPGA chip include the acquisition of the code phase and carrier frequency of the direct signal, the local code and carrier generation, the cross-correlation of the reflected signal with the local replica, the waveform data and other data buffer, and the control functions. Note that the functions of the DSP chip can also be realized on a FPGA chip. The FPGA chip can be selected from the Altera (now acquired by Intel) FPGA series (Intel [4]). The DSP chip performs functions including code and carrier tracking, and navigation data/message decoding. The DSP design can be implemented in a DSP chip which can be selected from the TI (Texas Instruments) processor series [12]. Namuru v3.3 of UNSW has a Cyclone IV FPGA and retains the use of the NIOS-2 softcore processor,

FPGA

Digital IF data (direct)

Control unit Coarse code phase & frequency estimates Digital IF data

Acquisition loop

Code & carrier generators

Digital IF data (reflected)

Navigation & control data Waveform data

Correlator array

Waveform data

Fig. 3.9 Signal-processor block diagram

Data buffer

USB interface

DSP

3.3 GNSS-R Receiver

47

although future versions may switch to Cyclone V, which includes hard-coded ARM Cortex cores on board. The FPGA and DSP communicate via an interface. More details about the signal processor are provided in the following subsections.

3.3.2.2

Basic Principle of the Signal Processor

Figure 3.10 shows the structure of the signal processor with a detailed illustration of the waveform-generation correlator array. Although a number of different functions are performed in the processor, the key product of this signal processor is the delayDoppler waveform of the reflected signal. Note that the waveform of the direct signal is not of interest since it does not provide any useful information about the characteristics of the scattering surface. Nevertheless, the processing of the direct signal provides the estimates of the code phase and Doppler frequency of the direct

Digital IF data (direct)

Acquisition & tracking

Navigation solution

Code phase & Doppler frequency (reflected)

Multicarrier generator

Waveform generation with 1st Doppler frequency O I {−mδτ } {−mδτ }

I {nδτ }

Q {nδτ }

Σ

( )2

Σ

()

2

Σ

( )2

Σ

Σ Digital IF data (reflected)

Σ

( )2

Fig. 3.10 Architecture of the signal processing back-end. Only the part with the generation of DDM data associated with one Doppler frequency is shown and the structures of the other parts related to other Doppler frequencies are the same

48

3 Signal Scattering and Reception Schemes

signal, which are used to infer those of the reflected signal. It is also worth mentioning that the GNSS receiver of interest is actually equivalent to a reflectometric sensor where the direct and reflected signals are received and processed separately. In the case of a GNSS interferometer, the direct signal and the reflected signal are received by one antenna simultaneously so that the two signals are superimposed. Alternatively, the two signals are received by different antennas (the antenna for receiving the reflected signal must have a high gain), but the two signals are then cross-correlated for a certain period of time; the amplitude is squared and then accumulated over a number of samples. The advantage of interferometric processing is that accurate estimation of the relative delay can be obtained without the need to generate any replica of the modulating codes onboard. Also, all embedded codes in a given GNSS frequency band would contribute to the cross-correlation shape, including the high-chip rate restricted access codes such as the GPS P(Y) code which can be used to improve the ranging performance. The detection of these embedded codes may require the use of a high-gain nadir-looking antenna. Since the aim of the signal processor is to generate a 3D DDM, a range of code delays and a range of Doppler frequencies need to be defined. With the aid of the acquisition and tracking of the direct signal, the estimates of the central Doppler frequency and the code phase of the reflected signal at the specular reflection point can be obtained and thus the ranges of the code phases and Doppler frequencies can be selected. Note that the specular reflection point is the point where the total path length of the reflected signal travelling from the transmitter to the receiver is the minimum. As shown in Fig. 3.10, there are (m +n +1) code phases, ranging between −mδτ and nδτ where m and n are positive integers and δτ is the code phase spacing. A zero code phase corresponds to the code phase of the signal reflected at the specular reflection point. The Doppler frequency ranges between − f and  f where  is an integer and f is the Doppler frequency spacing. Each Doppler frequency corresponds to a specific carrier frequency of a carrier generated by a multi-carrier generator. More discussion about these code phase and frequency parameters will be provided later. For each pair of a Doppler frequency and code phase, there is a pair of correlators for cross-correlating the in-phase and the quadrature components of the reflected signal with those of the modelled signal. The amplitude of the correlation output of the in-phase component and that of the quadrature one are squared and then added, followed by non-coherent summation. The code sequences with different code phases can be generated by using a serialin and parallel-out shift register which is driven by a clock as shown in Fig. 3.11. There are N parallel outputs from the shift register at each sampling instant, resulting in (N + 1) C/A code sequences. Note that, to match the number of code phases in Fig. 3.10, N = m + n. The sampling period of the clock is equal to the code phase spacing or resolution, i.e. δτ . The generation of different Doppler frequencies is realized using a multi-carrier generator with each carrier corresponding to a specific Doppler frequency. Each of the carriers is generated using a Numerically Controlled Oscillator (NCO) and thus the implementation of such a multi-carrier generator is not a difficult issue.

3.3 GNSS-R Receiver

49

Fig. 3.11 Generation of C/A codes with different code phases

In addition to the code phase and Doppler frequency parameters to be discussed later, two other correlation-related parameters, the coherent integration time and the non-coherent integration time, need to be selected. For a spaceborne receiver, the coherent integration is typically set at 1 ms which is the GPS C/A code chip length. For a receiver dedicated to airborne experiments, the coherent integration can be selected to be equal to or greater than 1 ms depending on the coherence time of the signal scattered from the surface. The selection of the non-coherent integration time may be more flexible, but typically greater than 1 s. Due to the divergent selection of these two parameters, they should be readily adjustable so that the user can conveniently select the parameter values of their interest. For convenience the default values of the coherent and non-coherent times can be specified as 1 ms and 1 s, respectively. The dashed box in Fig. 3.10 corresponds to the generation of waveform data related to one specific Doppler frequency. In fact, there are 2 other similar dashed boxes related to different Doppler frequencies, but they are omitted to save space. That is, a delay waveform is produced by each of the boxes. In each dashed box, there are 2(m + n + 1) parallel correlators. Since there are 2 + 1 Doppler frequencies, there are totally 2(2 + 1)(m + n + 1) parallel correlators. Although in some cases only the two-dimensional delay waveform is used to remotely sense the geophysical parameters, the generation of three-dimensional waveforms should be attempted. This is because the 3-D waveform contains more information than the 2-D waveform. However, the use of 3-D waveform or 2-D waveform is the choice of the user. The 2-D waveform is simply a part of the 3-D waveform when the Doppler frequency is set at its central value.

3.3.2.3

Signal Processing Flow Chart

Figure 3.12 shows the signal processing flow chart in the signal processor which uses both the direct digital signal and the reflected digital signal to generate the delay-Doppler waveform. Specifically, the processor consists of two parts, the direct signal channel (DSC) and the reflected signal channel (RSC). The DSC generates both the reference Doppler frequency and the reference code phase for the RSC.

50

3 Signal Scattering and Reception Schemes

Digital IF data (direct)

Navigation solution

Acquisition & tracking

Doppler (direct)

Code phase (direct)

Relative delay (reflected)

Code phase (reflected) Digital IF data (reflected)

Cross-correlation (reflected)

Waveform (reflected)

Fig. 3.12 Illustration of signal processing flow chart

In fact, the DSC performs the functions of a typical software receiver, as described below. The acquisition and tracking is first realized to estimate the code phase and Doppler frequency of the direct signals associated with specific GNSS satellites. The navigation data decoding and extraction are then carried out to generate the navigation message and to determine the relative delay of the reflected signal (with respect to the direct signal). Next, the code phase (C PD ) of the direct signal and the relative delay (δτ R ) of the reflected signal are used to estimate the code phase of the reflected signal using the formulas C PR = C PD + δτ R

(3.9)

δτ R ≈ 2h sin(θ )

(3.10)

where

Here h is the altitude of the receiver and θ is the elevation angle of the GNSS satellite. Since the Doppler frequency of the reflected signal is similar to that of the direct signal, both the code phase and the Doppler frequency of the reflected signal can be estimated and the accuracy is sufficiently good for the generation of the delayDoppler waveform. The RSC makes use of these Doppler and code phase estimates to perform 2D cross-correlation to produce the delay-Doppler waveform. It is expected that a certain processing time will be consumed in generating the reference code phase and Doppler frequency estimates of the reflected signal even though the processing is hardware-based. However, such a processing delay is negligible when considering the code phase and Doppler frequency variation over such a

3.3 GNSS-R Receiver

51

delay. Airborne experimental results such as reported in Chaps. 5 and 6 have demonstrated that the variation of the Doppler frequency over duration of one second is rather small, just up to a few dozen Hertz. The variation of the code phase in one second is between a few samples and around 40 samples. Note that the sampling frequency is 16.3676 MHz and there are about 16 samples per code chip for the receiver used for the experiments. Except that at the starting point, dozens of 1 ms data are required to perform signal acquisition, the update period of the code and frequency tracking would be of the order of milliseconds. Also, at the starting point a delay of 30 s is inevitable in order to detect the navigation message to obtain the GNSS satellite data.

3.3.2.4

Selection of DDM Parameters

As mentioned earlier four important parameters associated with the DDM need to be defined in the design of the bistatic receiver. They are the number of code pixels, number of Doppler pixels, both of which define the DDM size, code phase resolution or spacing (δτ ), and the Doppler frequency resolution ( f ). The knowledge of these parameters is required to determine the total number of correlators which will be implemented on the FPGA chip. It is important to appropriately choose these parameters so that the observed DDMs can be utilized to reliably retrieve the geophysical parameters. To enable the accuracy requirement, the code phase and Doppler frequency spacing should not be large and the DDM size should not be small. On the other hand, to reduce system complexity and speed up the DDM generation, the DDM size should not be large. Thus, a trade-off is required. In [13] each DDM has 24 × 32 complex points and the resolutions are configurable with ( f min = 20 Hz, δτmin = 0.05 chips). In addition, the coherent time (minimum = 1 ms, maximum = 100 ms) and non-coherent integration time (minimum of one coherent integration time period and maximum unlimited but typically less than 1 s) are selectable. Such a design was focused on building a GNSS receiver for conducting airborne experiments. In [11] the design objective was to build a receiver suited for a spaceborne platform. The implemented DDM processor has 52 Doppler frequencies by 128 code delays and the resolution can be configurable. It is a fact that the spread over both time and frequency of the reflected signal received via a spaceborne receiver is much larger than that of the signal received via a land-based or an airborne receiver. Thus, the DDM size for a spaceborne receiver should typically be larger than that of an airborne receiver. Based on the above discussions and the consideration of a receiver suited for both airborne and spaceborne applications, the DDM parameters of the bistatic receiver of interest are initially specified in Table 3.3. The DDM parameters of TDS-1, CYGNSS, and Bufeng-1 A/B are basically within the parameter ranges specified in Table 3.3.

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Table 3.3 Possible specification of DDM parameters Parameter

Min

Max

Characteristic

Delay resolution

0.03125 chips

Less than 0.5 chips

Configurable

Doppler resolution

15.625 Hz

Less than 2 kHz

Configurable

128

Fixed

60

Fixed

Number of code delays 64 Number of Doppler frequencies

20

Coherent Integration time

1 ms

Less than 0.1 s

Selectable

Non-coherent Integration time

Larger than coherent integration time

Unlimited but typically Less than 10 s

Selectable

3.3.2.5

Issues Related to Front-Ends

In the preceding subsections although the receiver structure and signal processing back-end parameters were defined, a number of issues were not mentioned. In this subsection the issues of number of front-ends, multiple frequencies and multi-GNSS systems are discussed. If the receiver is based on the UNSW-developed Namuru receivers, it would have the facility support for multiple GNSS systems. At the early 2010s, the 2-frontends Namuru receiver was designed to acquire and track the GPS L1 and L5 signals and the Galileo E1 and E5 signals with one front-end for handling the L1/E1 signals and the other front-end for the L5/E5 signals. In fact, the design of GNSS receivers typically considered using multiple front-ends and multiple frequencies. The reason for using such a GNSS receiver structure is rather obvious, since there are four different operational GNSS systems, namely U.S.’s GPS, Russia’s GLONASS, China’s BeiDou, and EU’s Galileo. Furthermore, a number of different frequencies are used by these systems, including L1/E1, L2/L2C, L5/E5, E6, B1, B2, and B3. It is desirable to make use of the multi-frequency and multi-constellation to achieve a diversity gain in estimating the geophysical parameters with regards to both estimation accuracy and coverage. Based on the above considerations a number of different options of the bistatic receiver front-ends and frequencies can be considered. In terms of the number of front-ends there are four different options: three, four, five and six. As shown in Fig. 3.13a, the 3-front-end structure is the simplest one with one front-end for dealing with the direct signal captured via the zenith-looking RHCP antenna, another frontend for the reflected signal received via the nadir-looking LHCP antenna, and the third front-end for the reflected signal via the nadir-looking RHCP antenna. In this case both dual constellations and dual polarization are exploited. Note that GPS and Galileo are used as an example, but any other combination of two constellations or more constellations and relevant frequency bands can be considered. The second option is that there are four front-ends with two front-ends for L1/E1 signals and the other two for L5/E5 signals, as shown in Fig. 3.13b. A pair of L1/E1

3.3 GNSS-R Receiver Fig. 3.13 a Three front-ends and two frequency bands. b Four front-ends and four frequency bands. c Five front-ends and four frequency bands. d Six front-ends and four frequency bands

53

a

Direct signal RHCP antenna L1/E1

L1/E1

L1/E1

1st Frontend

2nd Frontend

3rd Frontend

Reflected signal

Reflected signal

b

Direct signal Direct signal RHCP antenna RHCP antenna

c

d

L1/E1

L5/E5

L1/E1

L5/E5

1st Frontend

2nd Frontend

3rd Frontend

4th Frontend

Reflected signal

Reflected signal

Direct signal RHCP antenna L1/E1

L5/E5

1st Frontend

2nd Frontend L1/E1

L5/E5

L1/E1 or L5/E5

3rd Frontend

4th Frontend

5th Frontend

Reflected signal

Reflected signal

Reflected signal

Direct signal RHCP antenna L1/E1

L5/E5

1st Frontend

2nd Frontend

L1/E1

L1/E1

L5/E5

L5/E5

3rd Frontend

4th Frontend

5th Frontend

6th Frontend

Reflected signal

Reflected signal

Reflected signal

Reflected signal

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3 Signal Scattering and Reception Schemes

and L5/E5 front-ends is connected for dealing with the direct signal, while the other pair for the reflected signals. The downward antenna is assumed to have the left hand circular polarization unless the surface has a better sensitivity or higher reflectivity to the RHCP signals, depending on the satellite elevation angle and the surface roughness. This option is under the assumption that the frequency diversity gain is greater than the polarization gains. In the literature there were no reports on the diversity gains associated with different parameters (based on literature review up to early 2010s). However, results on these issues can now be found in the literature. Nevertheless, it is still desirable to carry out more investigations for comparing the performance gain achieved by using multiple polarizations and the gain obtained by using multiple constellations or multiple frequencies. In the option of five front-ends, the only difference to the option of four frontends is that an additional front-end can be used for handling either L1/E1 or L5/E5 signals but not both as shown in Fig. 3.13c. The selection would depend on which signals have a better sensitivity and/or higher reflectivity to the surface of interest. If the additional front-end is L1/E1, then a pair of L1/E1 front-ends would be used to handle the reflected signals via the LHCP antenna and the RHCP one, respectively. In the option of six front-ends the simple combinations would be three L1/E1 frontends and three L5/E5 front-ends, as shown in Fig. 3.13d. A pair of L1/E1 and L5/E5 front-ends is used for processing the direct signal, while the other four front-ends for handling the reflected signals via a L1/E1 RHCP, a L1/E1 LHCP, a L5/E5 RHCP, and a L5/E5 LHCP antenna, respectively. Alternatively, three front-ends may be used to receive the direct signal with the third one for a third satellite constellation of either GLONASS or BeiDou. The three other front-ends are assigned to process the reflected signals of the three different constellations. In this case, only the LHCP antennas are used to capture the reflected signals. The advantage of such an option is that the spatial coverage is increased since the elevation or azimuth angles of the satellites of a constellation typically would not be the same as those of the satellites of another constellation. Further analysis and evaluation are required to make a final decision on which of the above options will be used when implementing the design and building the bistatic receiver. Another issue is that the bistatic receiver should provide users with the flexibility that they can obtain the raw IF digital data, the delay-Doppler map data, or both. That is, the receiver should be able to output both digital IF data and the processed DDM data. As mentioned earlier, using the DDM data will save the end-users much time in retrieving the geophysical parameters. On the other hand, in addition to DDM data, some end-users may be interested in using the digital IF samples to extract more useful information. As a fact, TDS-1, CYGNSS and Bufeng-1 all provide IF digital data for a limited period of time, while the DDM data are continually generated. Finally, it is worth mentioning one more issue related to antennas. A number of off-the-shelf multi-GNSS antennas are available. For instance, the 3G + C antenna built by Navxperience is able to capture nearly all the GNSS signals. However, only RHCP polarization is used and the dimensions are relatively large, 72mmx172mm. More information about this antenna can be found in [9] visited). Thus, a single multiGNSS antenna can be used to capture the signals from the different constellations.

3.3 GNSS-R Receiver

55

On the front panel of the receiver, only one input connection may be used to a nadirlooking multi-GNSS RHCP antenna, but an internal splitter is needed to forward the signal to different front-ends. Otherwise, an external splitter is required to split the signal to different ports on the receiver front panel. In the absence of a multiGNSS LHCP antenna, multiple individual LHCP antennas are needed to receive the reflected signals related to different constellations. Another point is that it is desirable to use high-gain antennas especially for capturing the reflected signals.

3.4 Receiver Platforms Depending on the application scenarios, a GNSS receiver can be placed on three different platforms, which are ground-based, airborne, and spaceborne. This section presents a brief discussion on these three platforms.

3.4.1 Ground-Based Platform Figure 3.14 shows two examples of ground-based placement of GNSS receivers and antennas. Figure 3.14a shows the experimental setup for soil moisture estimation (Yan et al. [14]), while Fig. 3.14b shows the setup of the receiver and antenna for snow depth estimation [15]. Ground-based applications mainly use a single antenna to receive both direct and reflected signal. The antenna may be facing horizontally such as in Fig. 3.14a or it is up-looking such as in Fig. 3.14b. The latter is similar to the antenna placement of CORS, which is intended to maximize the reception of the direct signal and minimize the effect of multipath interference. Clearly, compared with the zenith-looking scheme, the horizontal facing option is much better for the reception of the reflected signals. Nevertheless, CORS receivers and antennas are

Fig. 3.14 Reception of direct and reflected signals with vertical and horizontal polarization antennas placed in the field (a); Reception of direct and reflected signals with a RHCP GNSS antenna fixed on top of a pole and a receiver in the yellow box (b)

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3 Signal Scattering and Reception Schemes

existing infrastructure, so the relevant data can be directly used without extra cost. In such interferometric receptions, it is necessary to remove or greatly suppress the effect of the direct signal through detrending as will be studied in Chap. 4. Interferometric reception is widely used in ground-based experiments for the studies of retrieving geophysical parameters. This is not only because the experiment can be more easily set up, but the interferometric signal contains the information of the geophysical parameter such as snow depth, soil moisture, and sea/lake surface height. This corresponds to the concept of GNSS-IR, GNSS-MR, or iGNSS-R. Of course, in some cases, a dual-antenna scheme is used to receive the direct and reflected signals separately. For instance, one down-look antenna and another zenith-looking are packed together for lake surface height measurement. One end of a long narrow wooden or metal plate holds the antennas,while the other end is fixed on a pole or a similar supporting structure. This is associated with the concept of GNSS-R or conventional GNSS-R. Note that, in most cases, GNSS-R is considered to include all the mentioned reception schemes. In the dual-antenna case, one important thing is to isolate the mutual interference between the direct and reflected signals. Regarding the interferometric case, the selection of antenna height and facing direction would not be trivial issue, which may significantly affect the accuracy of parameter retrieval.

3.4.2 Airborne Platform Over the GNSS-R history of few decades, a large number of airborne experiments have been conducted to collect data for scientific research and demonstrate the feasibility of GNSS-R. The top of Fig. 3.15 shows the aircraft which was used for the airborne experiments reported in Chaps. 5, 6, and 9. Typically, a zenith-looking RHCP antenna is fixed on the top of the aircraft to receive direct signal for navigation purpose as shown in the bottom left of Fig. 3.15, while one or multiple down-looking antennas are attached on the bottom of the aircraft as shown in the bottom right of Fig. 3.15, which is a composite antenna and actually consists of a LPCP antenna and a RHCP antenna. After RHCP GNSS signals are reflected by a surface, the polarization of part of the signals is changed to LHCP. The power of the LHCP component of the reflected signal increases as the satellite elevation angle increases, while the power of the RHCP component of the reflected signal decreases as the satellite elevation angle increases. That is, at low elevation angle, a RHCP antenna should be used for reflected signal reception, while at high elevation angle, a LHCP antenna should be used. More discussions on this issue are provided in Chap. 4. Compared with the ground-based platform, airborne platform can be used to carry the receiver to collect data over almost any places especially those where it is difficult to approach by pedestrians or vehicles. For instance, Fig. 3.16a shows the ground trajectory of an aircraft for an airborne experiment conducted in New South Wales in 2013, which covers forest, reservoir, grassland and residential area [17]. Figure 3.16b shows the ground reflection tracks of the signals of four GPS satellites related to the short segment AB of the trajectory in Fig. 3.16a; some details of the surroundings

3.4 Receiver Platforms

57

Fig. 3.15 A small aircraft used for airborne experiments (top); a zenith-looking RHCP antenna for navigation (bottom left) and a nadir-looking composite RHCP/LHCP antenna for capturing reflected signals (bottom right)

can be easily observed. This experiment was intended to make use the reflected GPS signals for forest monitoring. In fact, airborne GNSS-R can be used to collect data over any areas to sense a wide range of geophysical parameters such as those to be studied in the following chapters. The main drawback of airborne platform is that one experiment usually lasts a few hours so it cannot be used for continuous sensing of the reflection surface. Similar to aircraft, unmanned aerial vehicle (UAV) can be used as receiver platform. UAV or drone is currently very popular in both civilian and military applications since it has a number of advantages such as low cost, easy implementation, more flexible in maneuver, and no pilot on board, compared with aircraft. A number of issues need to be considered if an UAV is used to perform the remote sensing task, to satisfy the technical and customer requirements. For instance, the UAV needs to be able to fly at the desired flight height over the required flight range. Also, the receiver and antennas as well as other relevant devices need to be reliably starched to UAV. This looks simple, but it may be difficult to do in practice.

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Fig. 3.16 a Aircraft flight ground route during the experiment. The dimensions of the image are 110 km by 95 km. b Four ground reflection tracks of four GPS signals associated with a segment of the aircraft flight trajectory in Fig. 3.16a

3.4.3 Spaceborne Platform As mentioned in Chap. 1 there have been five satellite missions with successful satellite launches since 2003 in the world and two of them have the main payloads dedicated for GNSS-R remote sensing. The satellites were launched from a groundbased vehicle, from a ship on the ocean, or from an aircraft in the air. The main advantage of spaceborne GNSS-R is the large-scale coverage and fast scanning of the surface. For instance, the specular scattering trajectories of the GPS signals captured

3.4 Receiver Platforms

59

by the eight CYGNSS micro-satellites have a good coverage of the whole ocean surface of interest over a full day to monitor the Hurricane intensity (CYGSS [1]). Although TDS-1 mission has only one satellite, a full day would collect data over 96 trajectories if signals of six GPS satellites are used and one trajectory is equivalent to one satellite pass. The International Space Station may also be used as a potential platform for GNSS-R receiver and antennas as indicated by the GEROS-ISS project.

3.5 Summary Four global navigation satellite constellations, one regional navigation satellite system and one space-based augmentation system were introduced, the signals of which can all be used to sense the geophysical parameters. One advantage of GNSSR is the simultaneous use of the signals of different satellites from the same and different satellite constellations to increase the spatial and temporal resolution of the measurement. It is necessary to decide the signals of which satellites should be used to perform the sensing task. This would first depend on the receiver used for signal collection and data processing, which is designed to receive and process signals of one satellite constellation or multiple satellite constellations. Given the signals or data collected by the receiver, the data quality should be evaluated so that the high quality data are utilized to enable high quality retrieval of the parameters. Although only navigation satellite constellations are described, the signals of other types of satellites such as communication satellites can be exploited, which is referred to the use of signals of opportunities. Signal modulation has been studied for so many decades especially in wireless communications, so there are many mature techniques. Nevertheless, it still requires careful consideration in the design of new GNSS signals to achieve desired performance in system complexity, data rate, data detection, power consumption, etc.

References 1. CYGSS (2020, visited) https://clasp-research.engin.umich.edu/missions/cygnss/sci 2. Garrison JL, A. Komjathy A, Zavorotny VU, Katzberg SJ (2002) Wind speed measurements using forward scattered GPS signals. IEEE Trans Geosci Remote Sens 40(1):50–65 3. Glennon EP, Munford PJ, Parkinson K (2011) Aquarius firmware for UNSW Namuru GPS receivers. In: Proceedings of international global navigation satellite systems society ignss symposium, Sydney, NSW, Australia, pp 1–12, 15–17 Nov 2011 4. Interl (2020, visited) https://www.intel.com/content/www/us/en/products/programmable/fpga. html 5. Katzberg SJ, Torres O, Grant MS, Masters D (2006) Utilizing calibrated GPS reflected signals to estimate soil reflectivity and dielectric constant: results from SMEX02. Remote Sens Environ 100(1):17–28

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6. Larson KM, Braun JJ, Small EE, Zavorotny VU, Gutmann E, Bilich AL (2010) GPS multipath and its relation to near-surface soil moisture content. IEEE J Sel Top Appl Earth Obs Remote Sens 3(1):91–99 7. Li W, Yang D, Zhang M, Zhang Q (2009) Design of a delay/Doppler-mapping receiver for GPS remote sensing. In: Proceedings of international conference on space information technology, Beijing, China, pp 1–6, 26–27 Nov 2009 8. Lowe S, Meehan T, Young L (2014) Direct-signal enhanced semi-codeless processing of GNSS surface-reflected signals. IEEE J Sel Top Appl Earth Obs Remote Sens 7(5):1269–1472 9. Navxperience (2020, visited) https://navxperience.com/ 10. Nogués-Correig O, Gali EC, Campderrós JS, Rius A (2007) A GPS-reflections receiver that computes Doppler/delay maps in real time. IEEE Trans Geosci Remote Sens 45(1):156–174 11. Steenwijk RV, Unwin M, Jales P (2010) Introducing the SGR-ReSI: a next generation spaceborne GNSS receiver for navigation and remote-sensing. In: Proceedings of ESA workshop on satellite navigation technologies and european workshop on GNSS signals and signal processing (NAVITEC), Noordwijk, Netherlands, pp 1–7, 8–10 Dec 2010 12. TI (2020, visited) https://www.ti.com/processors/overview.html?paramCriteria=no%2525253f 13. Valencia E, Camps A, Marchan-Hernandez JF, Bosch-Lluis X, Rodriguez-Alvarez N, RamosPerez I (2010) Advanced architectures for real-time delay-Doppler map GNSS-reflectometers: The GPS reflectometer instrument for PAU (griPAU). Adv Space Res 46(2):196–207 14. Yan S, Li Z, Yu K, Zhang K (2018) GPS-R L1 interference signal processing for soil moisture estimation: an experimental study. EURASIP J Adv Signal Process 2014(107):1–13, 2014-08 15. Yu K, Li Y, Chang X, Wang S, Zhang K (2018) Snow depth estimation using pseudorange and carrier phase of GNSS signals. In: Proceedings of international conference on electromagnetics in advanced applications (ICEAA), Cartagena De Indias, Colombia, pp 1–4, 10–14 Sept 2018 16. Yu K, Rizos C, Burrage D, Dempster A, Zhang K, Markgraf M (2014) An overview of GNSS remote sensing. EURASIP J Adv Signal Process 134:1–14 17. Yu K, Rizos C, Dempster A (2013) Forest change detection using GNSS signal strength measurements. In: Proceedings of International geoscience and remote sensing symposium (IGARSS), Melbourne, Australia, pp 1–4, 21–26 July 2013

Chapter 4

Theoretical Fundamentals of GNSS Reflectometry

GNSS reflectometry usually relies on a model or a criterion which is established in advance, to measure geophysical parameters or detect natural phenomena or manmade objects. The models and criteria can be theoretical or empirical depending on the application scenario and the observation platform. A theoretical model or criterion is convenient to use and it often can be widely applied. However, in the case where it is difficult to derive a theoretical model or criterion, an empirical one may be developed. In this chapter, the focus is on the theoretical fundamentals of GNSS reflectometry, which are the basis for the development of a range of models and criteria. Also, wavelet transform theory is briefly studied, which is one of the methods often considered for noise mitigation. In addition, a brief study is provided for spectral analysis of unevenly sampled data.

4.1 Interferometric Signal Reception 4.1.1 Signal Modeling Consider the case where the data recorded by a geodetic GNSS receiver such as a CORS receiver is used to estimate the snow depth or soil moisture around a faceup antenna, as illustrated in Fig. 4.1. For simplicity, only one reflection path is considered, which is the specular scattering path with the reflection point occurs at the specular point. This corresponds to the ideal case where the reflection surface is perfectly smooth. In the general case where the surface is rough, signals from many different reflection paths will arrive at the antenna, resulting in a modeling error. When only considering the specular reflection path, the sum of the direct signal and the reflected signal is given by s(t) = Ad (t) sin ψ(t) + Am (t) sin(ψ(t) + δφ (t)) © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_4

(4.1) 61

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Direct signal Zenith-looking antenna h

Reflected signal

h

Fig. 4.1 Geometry of direct and reflected signals with a single reflection path

where Ad (t) and Am (t) are the amplitudes of direct and reflected signal. The direct carrier phase in radian is defined as ψ(t) = 2π(φ(t) + N )

(4.2)

Here N is the integer ambiguity and φ(t) is the recorded carrier phase (unit in cycle) which does not include the integer ambiguity. The term δφ (t) is the excess phase incurred by the propagation delay of the reflected signal compared to the direct signal. The CORS antenna height is usually less than 10 m, while the GNSS MEO (medium Earth orbit) satellites are about 20,000 km (about 36,000 km for GEO satellites) above the Earth surface. Thus, the arrival and departure angle of the GNSS signal at the reflection point can be well approximated to be the satellite elevation angle θ (t) which is a time-varying parameter, and the approximation error is negligible. The approximation is also suited for airborne and space-borne receivers. From the geometric relationship, the propagation delay of the reflected signal equals m (t) = 2h sin θ (t)

(4.3)

where h is the antenna height which is the distance from the antenna phase center to the reflection surface. Accordingly, the excess phase, also known as the interferometric phase in radians, is given by

4.1 Interferometric Signal Reception

δφ (t) = 2π f

63

4π h m (t) = sin θ (t) c λ

(4.4)

where c and λ are the propagation speed and wavelength of the signal, and the important thing is the excess phase is a linear function of the antenna height. Equation (4.1) can be written as s(t) = (Ad (t) + Am (t) cos δφ (t)) sin ψ(t) + Am (t) sin δφ (t) cos ψ(t) ˜ = A(t) sin ψ(t)

(4.5)

where A(t) =



A2d (t) + A2m (t) + 2 Ad (t)Am (t) cos δφ (t)

˜ ψ(t) = ψ(t) + β(t) α(t) sin δφ (t) β(t) = tan−1 1 + α(t) cos δφ (t)

(4.6)

˜ Note that ψ(t) is the composite phase and β(t) is the composite excess phase with respect to the direct phase, which is also known as the error phase. α(t) = Am (t)/Ad (t) is termed as the amplitude attenuation factor (AAF) of reflected signal. The resulting signal is a single quasi-sinusoid and both amplitude and frequency of the signal vary as satellite elevation changes. It is worth mentioning that in the derivation of the above formulas, an approximation is made. In fact, there is an additional term in the composite phase, which describes the effects of the antenna radiation pattern and the surface reflectivity [16]. In the case where precise modeling is required, this additional phase may be taken into account.

4.1.2 Amplitude Attenuation Factor of Reflected Signal The AAF α(t) is a time-varying parameter, which is a function of the complex-valued Fresnel reflection coefficient of the scattering surface and the antenna gain pattern at the receiver. Some details are provided below to calculate the AAF. Since the Fresnel reflection coefficient (FRC) for linear polarization is well developed, it is convenient to consider the FRC for the linear polarization at first. Let R H H be the FRC for a wave with an incident horizontal polarization and scattered horizontal polarization, and let RV V be the FRC for a wave with an incident vertical polarization and scattered vertical polarization. Then, for two adjacent homogeneous half-spaces or propagation media, the two Fresnel reflection coefficients are described by the classic Fresnel equations [6]:

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 ε − cos2 θ (t)  R H H (t) = sin θ (t) + ε − cos2 θ (t)  ε sin θ (t) − ε − cos2 θ (t)  RV V (t) = ε sin θ (t) + ε − cos2 θ (t) sin θ (t) −

(4.7)

(4.8)

where ε is the complex dielectric constant of the medium of lower half-space. When the medium of upper half-space is air, the complex dielectric constant can be approximated by [11]: ε ≈ εb + j

cλμσb 2π

(4.9)

where εb and σb are respectively the relative permittivity and conductivity of the medium of lower half-space; μ is the permeability in a vacuum. All GNSS signals are transmitted as Right-Hand Circularly Polarised (RHCP) signals. When a RHCP incident signal is scattered by a rough surface, some scattered signal components are RHCP, while the others are LHCP. Accordingly, there are two reflection coefficients, a co-polarization reflection coefficient (R R R , incident RHCP and scattered RHCP) and a cross-polarization reflection coefficient (R R L , incident RHCP and scattered LHCP). They are related to the linear polarization reflection coefficients by [8]: R R R (t) = (R H H (t) + RV V (t))/2

(4.10)

R RL (t) = (R H H (t) − RV V (t))/2

(4.11)

It should be noticed that all the reflection coefficients are complex-valued. The AAF mainly depends on the characteristics of scattering surface and antenna gain pattern as mentioned earlier. Assuming a purely RHCP incident electric field, the received direct power Pd and reflected power Pr can be written as 

Pd (t) = P G d (t)Wd2 = A2d (t) Pr (t) = P|X (t)E Wr |2 = (α(t)Ad (t))2

(4.12)

where P is the power of direct signal; G d is the antenna gain for direct signal; Wd and Wr are the real-valued Woodward ambiguity function for direct and reflected signals respectively, which is the product of the code auto-correlation function (dependent on the multipath relative time delay and the code chipping period) and a normalized sinc function. E denotes a loss of coherent power which is related to surface roughness; X is the complex-valued sum of the reflection coefficients including the effect of the antenna gain. Thus, the AAF of the reflected signal received by antenna can be

4.1 Interferometric Signal Reception

65

written as:  α(t) =

|X (t)E Wr |2 G d (t)Wd2

(4.13)

The following provides some details about the AAF for the case where a groundbased receiver is involved.

4.1.2.1

A Case Study

In order to avoid errors caused by antenna shaking, the antenna height is usually installed to be smaller than 6 m for the typical GNSS station fixed on the ground in general. That is, the time delays of the reflected signal are no more than 20 ns when the elevation angle is smaller than 30°. If the antenna phase contribution to the reflected signal is ignored, the complex-valued sum of the reflection coefficients in (4.13) can be described as:   X (t) = R R R (t) G rR (t) + R R L (t) G rL (t)

(4.14)

where G rR and G rL are the co-polarization and cross-polarization antenna gain for reflected signal, respectively. Figure 4.2 shows the radiation patterns of a GNSS antenna, which is designed to maximize the reception of the direct signal and minimize the effect of multipath interference. GNSS signal is typically transmitted as RHCP and a RHCP antenna is used to capture the GNSS signal for positioning, navigation and timing. Such an antenna radiation pattern design is for the usual situation where a RHCP antenna is used to receive RHCP signals. It can be seen that the antenna gains at negative elevation angles are significantly smaller than those at large positive elevation angles. Reflected signals arrive at the antenna along negative elevation angles, so that they are greatly suppressed. As the navigation satellite moves in the sky, the elevation angle continuously increases in the rising period or decreases in the falling period. Due to the antenna gain variation, the amplitude Ad (t) and Am (t) vary slowly with time. Although the antenna is designed for the reception of RHCP signals, it also captures LHCP signals but with significantly reduced gains. Thus, when analyzing the received signals, it is useful to consider the LHCP signal component. For ground-based receivers, the approximation Wr ≈ Wd is acceptable, as the multipath relative time delay is far less than the code chipping period (i.e. 980 ns for GPS C/A-code and 490 ns for BDS I-code). The loss of coherent power in the case of snow-covered surface is negligible, so the parameter E is unity. Consequently, (4.13) can be simplified as:

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Fig. 4.2 An example of radiation patterns of a geodetic antenna for RHCP and LHCP GNSS signals

α(t) =

   R (t)G R (t) + R (t)G L (t) 2 R R R L  r r G d (t)

(4.15)

Figure 4.3 shows an example of AAF (upper plot) for BDS B2 band signal that is RHCP and scattered by the surface of dry snow. At the elevation angle of 0°, the AAF is 1 for the co-polarized signals and 0 for the cross-polarized signals. With the increase of elevation angle, the AAF decreases for the co-polarized component, but increases for the cross-polarized component. When the elevation angle is around 35°, the AAF is the same for both components and this angle is referred to as the Brewster angle for the reflector of dry snow. When the elevation angle is greater than the Brewster angle, the AAF of cross-polarized signals is higher than that of co-polarized signals. The former gradually approaches zero, while the latter gradually approaches the maximum, close to 0.2. When the co-polarized and the cross-polarized signals arrive at the typical RHCP antenna, a combined signal is produced.

4.1.3 Signal-To-Noise Ratio From the first equation in (4.6), the SNR can be expressed as S N R(t) =

1 A2 (t) = (A2 (t) + A2m (t) + 2 Ad (t)Am (t) cos δφ (t)) PN PN d

(4.16)

4.1 Interferometric Signal Reception

67

Fig. 4.3 The amplitude attenuation factor of BDS B2 band signal for dry snow with density and temperature being 0.5 g/cm3 and −10 °C, respectively. Dashed and dotted lines are for co- and cross-polarized reflected signals, respectively. Black, fuchsia and cyan solid lines are the AAFs for the scattered signals received by three different antennas (TRM29659.00, TRM55971.00 and TRM59800.00)

where PN is the total noise power. The amplitude, frequency, phase, or combinations of the three parameters may be utilized in the sensing of the geophysical parameters. In the case where the antenna height is of interest such as in the estimation of snow depth to be discussed in Chap. 8, the third term in (4.16) is retained, while the first two terms need to be removed since they are not directly related to the antenna height. One way to remove the two terms is to use a low-order polynomial to fit the two terms. Alternatively, a low-pass filter is applied to extract the two terms. Then, the SNR time series is subtracted by the low-order polynomial or the output of the low-pass filter, producing S N Rd (t) =

2 Ad (t)Am (t) cos δφ (t) PN

(4.17)

As a result, the detrended SNR is a quasi-sinusoidal signal, oscillating as the elevation angle changes with time. More information about the real SNR data and the detrending is provided in Chap. 8. From (4.4), the reflection excess phase term can be simplified if sin θ (t) is treated as an independent time variable. Then, differentiating the reflection excess phase with respect to sin θ (t) yields 2h d(δφ (t)) = 2π × d(sin θ (t)) λ

(4.18)

That is, the reflection excess phase is a linear function of variable sin θ (t) with a constant frequency given by

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4 Theoretical Fundamentals of GNSS Reflectometry

f =

2h λ

(4.19)

It should be noted that, although the phase rate in (4.18) might resemble an ordinary frequency, it should not be confused with its temporal rate of change: d(δφ (t)) 4π h dθ (t) = cos θ (t) dt λ dt

(4.20)

where the instantaneous frequency is a function of the elevation angle and its rate, in addition to the wavelength and antenna height. Figure 4.4 shows the time series and the power spectral density of the detrended SNR given by (4.17). The direct signal SNR is set to be 25 dBHz, and the antenna height is 1.5 m, 2.5 m and 3.5 m, respectively. Due to the effect of amplitude attenuation of the reflected signal discussed in Sect. 4.1.2, the vibration is attenuated as the elevation angle increases and hence it is a damping quasi-sinusoidal signal. As indicated by (4.19) and shown in Fig. 4.4, the spectral peak frequency increases with the increased antenna height. Such a relationship is useful in the estimation of the height of reflection surface.

Fig. 4.4 (upper) Detrended SNR series with respect to sin θ(t) under three different antenna heights; (lower) power spectral density of the three SNR time series

4.1 Interferometric Signal Reception

69

4.1.4 Composite Excess Phase The composite excess phase (or error phase) given by the third equation in (4.6) is a function of the AAF α(t) = Am (t)/Ad (t) and the reflection excess phase (or interferometric phase)δφ (t). Figure 4.5 shows an example of the L1 signal error phase which is a function of sin θ (t) when α(t) is 0.1, 0.2, and 0.5 respectively and antenna height is 1 m (upper) and 2 m (lower) respectively. All three waves exhibit the same periodicity under the same antenna height, while the wave shapes become progressively more skewed as amplitude increases. In fact, in the case ofα(t)  1, the composite excess phase is approximated by β(t) ≈ tan−1 (α(t) sin δφ (t))

(4.21)

Also, if tan x = ε andε TC P

(4.24)

where TC P is the code chipping period (the reciprocal of ranging code chipping rate, e.g. 977.6 ns for GPS C/A code and 488.8 ns for BDS I-code), and x is code phase offset. Assuming the time delay (δm (t) = m (t)/c) of the reflected signal for groundbased GNSS-receivers satisfies, δm (t) < D E L

(4.25)

the multipath-induced pseudorange error can be explicitly derived as, τ P (t) =

δm (t)α(t) cos δϕ (t) 1 + α(t) cos δϕ (t)

(4.26)

By substituting (4.3) into (4.26), the multipath-induced pseudorange error (t) in meters can be obtained as, (t) = c · τ P (t) =

m (t)α(t) cos δϕ (t) 1 + α(t) cos δϕ (t)

(4.27)

Note that different signal modulation type (e. g. BPSK for GPS signals and AltBOC for Galileo signals) could produce different curve of auto-correlation function, and thus different multipath-induced pseudorange error; however, as studied in [3], the difference caused by different modulation type is very small in the case of short-delay multipath especially when the time delay is smaller than 20 ns.

4.1 Interferometric Signal Reception

71

Fig. 4.6 Multipath-induced pseudorange error for BDS B2 band signal oscillates with the sine of elevation angle when antenna height is 2.5 m (upper) and 3.5 m (lower) and three AAFs are tested. Solid, dashed and dotted lines are for AAF equal to 0.1, 0.3 and 0.5, respectively

Figure 4.6 shows an example of the multipath-induced pseudorange error for BDS B2 band signal described by (4.27) in the case of two different antenna heights, when AAF is 0.1, 0.3 and 0.5, respectively. Similar to multipath-induced carrier-phase error which periodically oscillates with respect tosin θ (t), all three waves for multipathinduced pseudorange error also exhibit the same periodicity under the same antenna height, and the periodicities are significantly different in the case of different antenna height. The amplitude of the waves becomes bigger as elevation angle increases. This is because the multipath-induced pseudorange error is proportional to the relative time delay, which increases withsin θ (t). For the same reason, the amplitude of the wave is bigger under the higher antenna height, as shown in Fig. 4.6, whereas this is not true for the multipath-induced carrier-phase error. This implies that fixing the receiver antenna on the top of a taller pole above a reflective ground surface will have a higher ratio of multipath-induced pseudorange error to noise.

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4 Theoretical Fundamentals of GNSS Reflectometry

4.2 Delay-Doppler Map and Delay Map In the case where the GNSS-R receiver is onboard an aircraft or carried by a low Earth orbit (LEO) satellite, two or more separate GNSS antennas are often used. For instance, as shown in Fig. 4.7, a face-up RHCP antenna is fixed on the top of the aircraft for positioning, navigation and timing services and a composite RHCP/LHCP antenna is attached on the bottom of the aircraft for collection of reflected signals. That is, the direct signals and the reflected ones are recorded and processed separately.

4.2.1 Surface Scattering As the GNSS signals arrive at a surface of land or water, some of the signal energy is absorbed by the medium, while the other energy is reflected. The reflected signals that travel towards the receiver will be captured by the nadir-looking antenna. Meanwhile, the GNSS signal will arrive at the zenith-looking antenna directly via the atmosphere as illustrated in Fig. 4.8. Let the positions of the transmitter (on the GNSS satellite) and the receiver (on an aircraft, LEO satellite, or land-based) be (xt , yt , z t ) and (xr , yr , zr ) respectively. Also define the position of the scattering point on the surface as (xs , ys , z s ). Then the distance from the transmitter through the scattering point to the receiver is given by  dtsr (xs , ys , z s ) = (xt − xs )2 + (yt − ys )2 + (z t − z s )2  + (xr − xs )2 + (yr − ys )2 + (zr − z s )2

(4.28)

The specular point is the scattering point (x S P , yS P , z S P ) on the surface where the distance dtsr is minimal. With respect to the signal reflected at the specular point, the signals reflected at other scattering points arrive at the receiver with a delay given

Fig. 4.7 An example of two antennas mounted separately on an aircraft, a RHCP antenna for direct signals and b composite RHCP/LHCP antenna for reflected signals

4.2 Delay-Doppler Map and Delay Map

73

Fig. 4.8 Illustration of the GNSS scattering and reception. The ellipses are the iso-delay lines, while the hyperbolas are the iso-Doppler lines

by δτ = dtsr (xs , ys , z s )/c − τ S P

(4.29)

where c is the speed of light and τ S P = dtsr (x S P , yS P , z S P )/c. Given the transmitter and receiver positions and the delay δτ , the scattering points define an ellipse on the surface. That is, at each specific delay the signals reflected on the ellipse will arrive at the receiver at the same time, supposing that they travel towards the nadir-looking antenna. Due to the relative movement between the transmitter and the receiver, Doppler frequencies are produced, resulting in the increase or decrease of the signal carrier frequency. Let the velocity vectors of the transmitter and the receiver be Vt and Vr respectively. The Doppler frequency is determined by  − Vr • n)/λ f D = (Vt • m

(4.30)

where m  and n are the unit vectors of the incident wave and the reflected wave respectively, and “•” denotes the vector dot product. For a given Doppler frequency, Eq. (4.30) represents a hyperbola. That is, the signals reflected on such a hyperbola will have the same Doppler frequency. The intersection of the iso-delay lines and the iso-Doppler lines forms a network of grids which will be used to determine the power of the reflected signals arriving at the receiver. Figures 4.9 and 4.10 show an example of the iso-Doppler map and the iso-delay map, respectively, when the satellite elevation angle is 63 deg. The three components of the velocity of the aircraft are 21.166946 m/s, − 52.149224 m/s, and − 2.527502 m/s, and the receiver position is (−33.693117°, 151.2745950°, 508.0884 m). The satellite position is (−52.3194°, 162.2910°, 1.9903e + 007 m) and the velocity vector is (−131.75275, − 2727.07856, − 573.77554) (m/s).

74

4 Theoretical Fundamentals of GNSS Reflectometry Doppler Frequency (Hz) -900 15 -1000

10

width (km)

5

-1100

0 -1200 -5 -1300 -10 -1400

-15 -20 -20

-15

-10

-5

5 0 Length (km)

10

15

Fig. 4.9 Example of iso-Doppler map Delay (chip) 100

15

90 10

80 70

width (km)

5

60 0 50 -5

40 30

-10

20 -15 10 -20 -20

-15

-10

Fig. 4.10 Example of iso-delay map

-5

0 5 Length (km)

10

15

4.2 Delay-Doppler Map and Delay Map

75

4.2.2 Received Signal Power The reflected signals received via the nadir-looking antenna are first down-converted to IF signals. The code phase offset and the Doppler frequency associated with the satellite of interest can be estimated based on processing the direct signal received via the zenith-looking antenna through code acquisition and tracking. The carrier frequency of the IF signals is then compensated for and the resulting baseband signal is correlated with a replica of the pseudorandom noise (PRN) code related to a specific satellite. At the central Doppler frequency, the cross-correlation with a sequence of code phases (or time delays) produces a delay map (correlation power versus code phase). A delay-Doppler map (DDM) is produced when both a sequence of Doppler frequencies and a sequence of code phases are considered. DDMs are the basic and probably the most useful data recorded by a GNSS-R receiver. Figure 4.11 shows an example of DDM produced by post-processing the IF data recorded by a software GNSS receiver onboard an aircraft. DDM can be calculated by theoretical formulas [17]. According to the model, after ignoring the noise, the DDM can be described as a function of delay offsets and Doppler shifts of reflected signals:



T 2 Pt η2 |Y(τ, f D )| = i 4π 2

¨ A

G 2 (ρ)  2 (τ − τ (ρ))  |S( f D − f D (ρ))|  2 σ0 d 2 ρ (4.28) 2 (ρ) Rts2 (ρ)R  sr 

Fig. 4.11 An example of DDM of reflected GNSS signals

76

4 Theoretical Fundamentals of GNSS Reflectometry



where |Y(τ, f D )|2 is the average power of the reflected signal over a time period of Ti and over the effective scattering zone A; ρ is the position vector at a scattering point  are the delay and the Doppler frequency of the reflected within A; τ (ρ)  and f D (ρ) signal at the positionρ;  Pt is the trasmission power of the GNSS signal of interest, taking transmit antenna radiation pattern into account; G(ρ)  is the antenna radiation pattern of the receiver; η is the propagation attenuation factor caused by propagation  and Rts (ρ)  are the distance from the transmitter to through the atmosphere; Rts (ρ) the scattering point on the scattering surface and that from the scattering point to the receiver; (τ − τ (ρ))  is the correlation function of the locally generated PRN code (a(t)) and the PRN code of the reflected signal (a(t)), ˜ defined by (τ )=

Ti

a(t)a(t ˜ − τ )dt

(4.29)

0

where τ = τ − τ (ρ)  is the delay offset. In practice, since discrete samples are used, the continuous integration is replaced with the dot product of the two PRN code seuqences: (τi )=

La

a(ti )a(t ˜ i − τi )

(4.30)

i=1

where there are L a discrete signal samples over the period ofTi . S( f D − f D (ρ))  is the sinc function, indicating the signal attenuation caused by the Doppler frequency shift between the reflected signals and locally generated signal, defined as: S( f D )=

sin(π  f D Ti ) π  f D Ti

(4.31)

where  f D = f D − f D (ρ)  is the Doppler frequency shift. In addition, σ0 is the normalized bistatic radar scattering section (NBRSC) which, in the case of scattering over an ocean surface, can be modeled as σ0 =

  π ||2 q 4 q⊥ P − qz4 qz

(4.32)

where  is the polarisation-dependent Fresnel reflection coefficient; q is the scattering vector defined as q = n − m  with m  and n being the unit incident and scattered vectors, respectively, as illustrated in Fig. 4.12; q⊥ and qz are the horiq |; zontal and vertical components of q , respectively, that is, q = (qz , q⊥ ) andq = | and p(− q⊥ /qz ) is the probability density function (PDF) of the scattering surface slope (− q⊥ /qz ), which may be simply assumed as omni-directional Gaussian distribution with the standard deviation being the mean square slope (MSS). Note that the

4.2 Delay-Doppler Map and Delay Map

77

Fig. 4.12 Illustration of the geometric relationship among scattering related vector

q

qz n

m

Scattering surface q

flat sea surface

included angle between the flat surface and scattering surface is denoted by ϑ and the angle satisfiestan ϑ = q⊥ /qz . In this case the scattering surface slope is given by q⊥ /qz . Given a Doppler frequency shift, a delay map (DM, signal power versus time delay) or delay waveform is produced from the DDM. At the central Doppler frequency shift, the DM (termed central DM) is the widest and the peak power is the maximum. Figure 4.13 shows an example of the central DM, produced from Fig. 4.11. There was a problem associated with the encoding of the GPS IF signals of the software receiver. The 2-bit quantization scheme was used so that the output data bits are within {3, 1, − 1, − 3}. But the encoding of the two data bits {−1} and {−3} was swapped in the receiver, such that a 2-bit quantization encoding scheme is actually equivalent to a 1.5-bit quantization scheme. As a result, certain performance degradation would be incurred due to the wrong encoding. As shown in Fig. 4.13, the noise floor in this case was reduced by about 0.8 dB after correcting the encoding 0 original after swapping

-2

Correlation Power (dB)

-4

-6

-8

-10

-12

-14 446

447

448

449

450

451

452

453

CA Code Phase (Chip)

Fig. 4.13 An example of Delay map of GNSS signal reflected over sea surface

454

78

4 Theoretical Fundamentals of GNSS Reflectometry

error. As will be discussed in Chap. 6, the central DM can be used to measure the sea surface wind speed.

4.3 Wavelet Denoising Wavelet has been widely used to mitigate measurement noise in signal and image processing. In this section wavelet transform theory, wavelet decomposition and reconstruction are briefly reviewed to provide a quick reference.

4.3.1 Wavelet Transform Theory A signal presented in time domain tells us precisely the spatial duration and amplitude of the signal at fixed time without any information about the signal frequency spectrum. On the other hand, when presenting the signal in frequency domain through Fourier transform, the signal bandwidth and the strength of each frequency component can be obtained, but neither signal spatial duration nor the time stamps information are available, although windowed fast Fourier transform can provide useful signal temporal information. The time–frequency localization, one of the prominent properties of wavelets, can provide the spectrum information with respect to time. Different signal features can be studied by choosing the appropriate scales; broad features on large scales, while fine features on small scales [7]. Fourier transform is a powerful tool for the analysis of stationary processes, whereas wavelet transform is effective for non-stationary signals which have features at different scales, have singularities, or have short-lived transient components. The continuous wavelet transform of any square integrable function f (t) is similar to the continuous Fourier transform, which is defined as +∞ W f (λ, t) =

∗ f (u)ψλ,t (u)du

(4.33)

−∞

where λ > 0 and   1 x −t ψλ,t (x) = √ ψ λ λ

(4.34)

which is a family of functions called wavelets. Here λ is a scale parameter having the effect of dilating (λ > 1) or contracting (λ < 1), while t is a location param∗ eter allowing the analyzing of the function at different locations, and ψλ,t (u) is the complex conjugate ofψλ,t (u).

4.3 Wavelet Denoising

79 Daubechies

Meyer

1 1 0.5

0.5

0

0

-0.5

-0.5

-1

0

5

10

15

20

25

-1

-5

Coiflet

0

5

BIOR

1.5 1

1 0.5

0 0 -0.5 -1

-1 0

5

10

15

0

5

10

Fig. 4.14 Typical patterns of four widely studied wavelets

There are a number of constraints on the waveletψ(t): (1) it has zero mean and may also have n vanishing moments if all moments up to nth one are zero; (2) it has compact support, or sufficiently fast decay and thus square integrable; and (3) it has unit energy. A number of wavelets have been studied extensively, which are the Haar wavelet, Daubechies wavelets, Symlets, Coiflets, Biorhtogonal wavelets, Meyer wavelets, Maxican hat wavelet, Morlet wavelet, and Reverse biorthogonal wavelets. Among these widely studied real-valued wavelets, only three wavelet families (Haar, Mexican Hat, and Morlet) have explicit expressions, while Meyer Wavelet has explicit expressions in frequency domain. Figure 4.14 shows four of the mentioned nine wavelets. In practice, when processing sampled signals, discrete wavelet transform is required. The scale and location parameters need to be discretized such as λ = λm 0 where λ0 is the step constant greater than one and m is an integer, and t = nt0 λm 0 where n is an integer andt0 > 0. Substituting discretized λ and t into (4.34) produces the discrete wavelets −m/2

ψm,n (t) = λ0

ψ(λ−m 0 t − nt0 )

(4.35)

In the case where λ0 = 2 andt0 = 1, ψm,n (t) = 2−m/2 ψ(2−m t − n)

(4.36)

80

4 Theoretical Fundamentals of GNSS Reflectometry

constitutes orthogonal bases. The function f (x) can be approximated by a linear combination of the wavelets as

wm,n ψm,n (t) f (t) = =

m

m

n

wm,n 2

−m/2

ψm,n (2−m t−n)

(4.37)

n

where wm,n is called the discrete wavelet transform (DWT) defined as wm,n =

f (x)ψm,n (x)d x

(4.38)

Note that as m decreases, the scale decreases and hence the resolution increases. That is, smaller-scale features appear when increasing the resolution by using a larger m, while larger-scale features are captured when decreasing the resolution by using a smaller m.

4.3.2 DWT Realization The DWT can be realized through fast wavelet transform algorithm proposed in [10]. As shown in Fig. 4.15, two types of filters are employed in signal decomposition and reconstruction. The low-pass filters are denoted by L_D (for decomposition) and L_R (for reconstruction), while the high-pass filters are denoted by H_D (for decomposition) and H_R (for reconstruction). All four filters have the same length of an even number (say 2 K). The decomposition filters L_D and H_D are quadrature mirror filters (QMR), that is, their coefficients are related byL_D( j) = (−1) j H _D(2K + 1 − j), j = 1, 2, · · · , 2K . The reconstruction filters L_R and H_R are also QMR, and the coefficient vectors of L_R and H_R are the flipped versions of those of L_D and H_D, respectively. The output sequences of L_D and H_D are down-sampled via dyadic decimation. The filters are associated with the wavelet function by ϕ(x) = 21/2

K −1

ld (k)ϕ(2x − k)

(4.39a)

h d (k)ϕ(2x − k)

(4.39b)

k=0

ψ(x) = 2

1/2

K −1

k=0

where {ld (k)} and {h d (k)} are the impulse responses of L_D and H_D, respectively, K is the filter order, ϕ(x −k) is the scaling function which is orthogonal to the wavelet

4.3 Wavelet Denoising

81

Fig. 4.15 Basic procedure of signal decomposition and reconstruction based on discrete wavelet transform and inverse discrete wavelet transform

functionψ(x − k). For instance, Fig. 4.16 shows {ϕ(x)} for the four wavelets shown in Fig. 4.14. Given {ld (k)}, the impulse responses of the other three filters can be readily obtained. Note that when Matlab software is used for simulation or data processing, the wavelet toolbox library function ‘wfilters’ can be used to produce the impulse responses of the four filters. For one-dimensional DWT, the down-sampled output of the low-pass filter can be treated in the same way as the original signal to undergo low-pass filtering and high-pass filtering. This is the second level or stage decomposition and a higher level decomposition can be further pursued in the same procedure to extract more noise components. The number of decomposition stage can be up to int(log2 N ) where int(x) is the operation of taking the integer part of x and N is the length of signal sample sequence. The procedure for the signal reconstruction using inverse DWT (IDWT) follows the reverse order of signal decomposition procedure and downsampling is replaced with up-sampling by inserting a zero between each neighboring pair of samples. The output of low-pass filter and that of high-pass filter are added and the sum is multiplied by two.

82

4 Theoretical Fundamentals of GNSS Reflectometry Meyer

Daubechies 1 1 0.5 0.5 0 0 -0.5

0

10

5

15

5

0

-5

25

20

BIOR

Coiflet 1.5 1 1 0.5

0.5 0

0 0

5

10

15

0

2

4

6

8

Fig. 4.16 Scaling functions of four wavelets

4.3.3 Denoising Procedure When the signal is corrupted by measurement noise, it is necessary to mitigate the noise so that the signal can be recovered as precisely as possible. Signal denoising procedure is the same as the DWT based signal decomposition and the IDWT based reconstruction mentioned above except for an additional step for threshold determination and thresholding of outputs of the high-pass filters in the decomposition stage. Due to the basic fact that the signal of interest is typically smooth except for some possible abrupt changes, most of the signal components go through the low-pass filters, whereas the noise components mainly pass through the high-pass filters. Thus, the low-pass filtered data is remained, while the high-pass filtered data needs to be handled such as through thresholding or shrinkage before reconstruction is carried out. There is a range of threshold selection methods proposed for signal processing and wireless communications. The fixed-form method simply sets the threshold to √ be a constant of 2 ln N where N is the signal length. Such a simple method does not consider the noise power so that it may not produce the desired performance. Stein’s Unbiased Risk Estimate (SURE) method is one of the widely investigated methods for threshold selection [5]. The basic idea is to choose a threshold which minimizes the estimate of the mean square error. Minimax principle is also often used for threshold selection to achieve the minimax performance, i.e. the minimum of the maximum mean square error [13]. To deal with nonstationary noise whose variance

4.3 Wavelet Denoising

83

is varying over time or space, adaptive techniques are needed to dynamically select the threshold to accommodate noise variance variation [4]. As mentioned earlier, Tsunami wavelength can be several hundred kilometres, over which the sea state may vary significantly such as from a calm surface to a very rough surface. Meanwhile, the SSH estimation performance would be considerably affected by the sea state, resulting in a space-dependent SSH estimation error variance and making it desirable to employ an adaptive threshold selection technique to denoise the SSH measurements. Two rules are generally employed for thresholding, which are soft and hard thresholding [15]. Hard thresholding is probably the simplest scheme, simply setting all the wavelet coefficients (high-pass filter output) to zeros if their absolute values are less than a specified threshold limit denoted by γ . On the other hand, the soft thresholding performs thresholding according to ⎧ ⎨ x − γ, x > γ y(x) = 0, |x| ≤ γ ⎩ x + γ , x < −γ

(4.40)

4.4 Spectral Analysis of Unevenly Sampled Data 4.4.1 Unevenly Sampled Data In a wide range of fields such as astronomy, medicine, and wireless communications, uneven data sampling rate often occurs. Figure 4.17 shows the sampling intervals of satellite elevation angle and its sinusoidal function. The elevation data were recorded by a ground-based GNSS receiver. The sampling frequency of the receiver is 1 Hz, but the data shown are down-sampled for better clarity. It can be seen that although the sampling time interval is constant (50 s), the elevation angle intervals are not the same but with basically three different levels over this period of observation. The sampling interval of sin(θ ) always changes with time. When treating sin(θ ) as a time variable such as for snow depth estimation to be discussed in Chap. 8, the issue of unevenly sampled data occurs. In scenarios where the data are sampled unevenly, the conventional spectral analysis techniques such as the conventional fast Fourier transform (FFT) cannot be used to perform spectral analysis. To deal with spectral analysis of unevenly sampled data, a variety of different signal processing methods have been proposed, which can be grouped into four broad categories: least-squares based, interpolation techniques based, slotted resampling based, and continuous time models based methods [2]. In particular, the LombScargle periodogram [9, 14], one of the least-squares based methods, is widely used, which is briefly studied below.

84

4 Theoretical Fundamentals of GNSS Reflectometry Sampling Interval of Elevation Angle

(Deg)

0.5 0.4 0.3 0.2

8

20

0 x 10

40

-3

100 120 140 80 60 Number of Sampling Interval

160

180

200

160

180

200

Sampling Interval of sin( )

sin( )

6 4 2 0

0

20

40

60 80 100 120 140 Number of Sampling Interval

Fig. 4.17 Calculated elevation intervals and sin intervals based on elevation angles recorded by a GNSS receiver; the sampling interval is 50 s and the selected observation period is 10000 s

4.4.2 Lomb-Scargle Periodogram Suppose that there are N signal samples,xi , measured at time instantsti , i = 1, 2, · · · , N . The sampling intervals may not be the same and there may exist missing samples. The Lomb-Scargle periodogram for the data sequence is defined as PL S ( f )

⎧ N 2 2 ⎫ N   ⎪ ⎪ ⎪ ⎪ ⎪ (xi − x) ¯ cos(ω(ti − τ )) (xi − x) ¯ sin(ω(ti − τ )) ⎪ ⎨ ⎬ 1 i=1 i=1 = + N N ⎪   2σ 2 ⎪ ⎪ ⎪ ⎪ ⎪ cos2 (ω(ti − τ )) sin2 (ω(ti − τ )) ⎩ ⎭ i=1

i=1

(4.41) where ω = 2π f is the angular frequency, x and σ 2 are the mean and variance, respectively, defined as x=

N N 1 1 xi , σ 2 = (xi − x)2 N i=1 N − 1 i=1

For each frequency f > 0, the time offset τ is calculated by

(4.42)

4.4 Spectral Analysis of Unevenly Sampled Data

85

N 

tan(2ωτ ) =

k=1 N 

sin(2ωtk ) (4.43) cos(2ωtk )

k=1

Such a constant time offset enables the spectrum to be completely independent of any shift tk → tk + δt since the same shift will occur to the time offsetτ → τ + δt. The Lomb-Scargle periodogram weights the data on a “per point” basis rather than on a “per time interval basis”, making it suited for handling spectral analysis on unevenly sampled data. By using approximations to (4.41) and (4.43) (but to any desired precision), fast Fourier transform can be used to greatly reduce the computational complexity of the Lomb-Scargle periodogram [12]. It is worth mentioning that Matlab has implemented the Lomb-Scargle periodogram and the library function is “plomb”. A simple example of Lomb-Scargle periodogram is given as follows. Normalizing the detreaned SNR in (4.17) produces  S N Rdn (t) = cos

 4π h sin θ (t) λ

(4.44)

where sin θ (t) is unevenly sampled as shown in Fig. 4.17. Figure 4.18 shows the normalized power spectral density (PSD) of the detrended SNR in (4.44). It can be seen that the Lomb-Scargle periodogram recovers the PSD of the signal better than the conventional FFT based method. In particular, the spectral peak frequency of the Lomb-Scargle periodogram is equal to 19.925, while the actual frequency of Lomb-Scargle Periodogram

0.8 0.6 0.4 0.2 0

10 20 30 Dimensionless Frequency

Conventional FFT Based

1

Power Spetral Density

Power Spectral Density

1

0.8 0.6 0.4 0.2 0

0

10 20 30 Dimensionless Frequency

Fig. 4.18 Normalized power spectral density of the unevenly sampled theoretical SNR with respect tosin θ(t); (left) PSD calculated by Lomb-Scargle periodogram, (right) PSD calculated by conventional fast Fourier transform

86

4 Theoretical Fundamentals of GNSS Reflectometry

the sinusoidal signal calculated by (4.19) is 19.936. On the other hand, the use of the conventional FFT method produces PSD with a spectral peak frequency of 24.0, significantly deviating from the true frequency. The spectral peak frequency is useful in the estimation of snow depth to be discussed in Chap. 8.

4.5 Summary The fundamental theory only focused on the composition of the signals captured by the antenna and processed by the receiver. Also studied include wavelet theory for signal denoising and spectrum analysis for unevenly sampled signals. These are just a few theoretical aspects which are involved in some GNSS-R related applications. More theoretical aspects will be presented in the following chapters. In addition, the book does not cover a range of GNSS-R applications such as vegetation sensing, ocean object detection, ocean brightness temperature estimation, and volcano ash sensing, which may involve other theoretical aspects. Readers can find more information in the literature.

References 1. Axelrad P, Larson K, Jones B (2005) Use of the correct satellite repeat period to characterize and reduce site-specific multipath errors. Proceedings of the 18th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS), Long Beach, CA, September 2005, pp. 2638–2648 2. Babu P, Stoica P (2010) Spectral analysis of nonuniformly sampled data—a review. Dig Signal Proc 20:359–378 3. Betz JW (2015) Code tracking, in engineering satellite-based navigation and time global navigation satellite systems, signals, and receivers. John Wiley & Sons Inc. 4. Chang SG, Yu B, Vetterli M (2000) Spatially adaptive wavelet thresholding with context modeling for image denoising. IEEE Trans Image Proc 9(9):1522–1531 5. Eldar YC (2009) Generalized SURE for exponential familiess: applications to regularization. IEEE Trans Signal Proc 57(2):471–481 6. Flock WL (1983) Propagation effects on satellite systems at frequencies below 10 GHz: a handbook for satellite system design. NASA Reference Publication 1108 7. Kumar P, Foufoula-Georgiou E (1997) Wavelet analysis for geophysical applications. Rev Geophys 35(4):385–412 8. Leroux C, Deuze JL, Goloub P, Sergent C, Fily M (1998) Ground measurements of the polarized bidirectional reflectance of snow in the near-infrared spectral domain: comparisons with model results. J Geophys Res 103(D15):19721–19731 9. Lomb NR (1976) Least-squares frequency analysis of unequally spaced data. Astrophys Space Sci 39:447–462 10. Mallat SG (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 11(7):674–693 11. Nievinski FG, Larson KM (2014) Forward modeling of GPS multipath for near-surface reflectometry and positioning applications. GPS Solutions 18(2):309–322 12. Press WH, George BR (1989) Fast algorithm for spectral analysis of unevenly sampled data. Astrophys J 338:277–280

References

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13. Sardy S (2000) Minimax threshold for denoising complex signals with waveshrink. IEEE Trans Signal Proc 48(4):1023–1028 14. Scargle JD (1982) Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data. Astrophys J 263:835–852 15. Taswell C (2000) The what, how, and why of wavelet shrinkage denoising. Comput Sci Eng 2(3):12–19 16. Zavorotny VU, Larson KM, Braun JJ, Small EE, Gutmann ED, Bilich AL (2010) A physical model for GPS multipath caused by land reflections: toward bare soil moisture retrievals. IEEE J Sel Top Appl Earth Obs Remote Sens 3(1):100–110 17. Zavorotny VU, Voronovich AG (2000) Scattering of GPS signals from the ocean with wind remote sensing application. IEEE Trans Geosci Remote Sens 38(2):951–964

Chapter 5

Sea Surface Altimetry

Altimetry is often used to obtain the accurate topography of land and ocean, which can be utilized for various applications and services. Altimetry can also be used to gain information on the variation in mean sea level, to investigate the global climate change. In addition, sea surface altimetry is useful in the monitoring of ocean disasters such as Tsunami, strong swells and other damaging waves. Altimetry instruments and devices can be carried by satellite, aircraft, or unmanned aerial vehicle, or fixed on ground. There are a range of altimetry technologies which make use of different types of signals including radio signal such as used by radiometer or radar and light signal used by such as laser scanner or Lidar. GNSS-R can also be exploited for altimetry and in fact sea surface altimetry is the first application considered in the early 1990s. Although GNSS-R based altimetry may not achieve the same accuracy of traditional radar and Lidar, it has the advantage of wider coverage in terms of airborne and spaceborne altimetry, so it may be used to complement the traditional technologies. In this chapter GNSS-R based altimetry is studied, with a focus on airborne altimetry.

5.1 Estimating Relative Delay of Reflected Signal Traditional radar or Lidar altimetry simply measures the round-trip time of the radio wave or laser pulse between the device and the surface of interest, so the range between the device and the surface can be determined. The position of the device is measured usually by a precise GNSS receiver. The surface height is equal to the difference between the device vertical position and the range from the device to the surface. Although GNSS-R can be considered as bistatic radar, the traditional altimetry method based on round-trip time cannot be applied directly. GNSS-R altimetry needs to rely on other different approaches. There is one approach which

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_5

89

90

5 Sea Surface Altimetry

uses the estimate of the relative delay of reflected signal to measure the sea surface height (SSH). This section presents the details of one specific method for estimating the relative delay.

5.1.1 Relative Delay The relative delay is estimated by determining the time-of-arrival of the direct signal and that of the reflected signal or the time-difference-of-arrival of the two signal. Basically, two different approaches can be used to estimate the relative delay. One is the delay waveform based approach and the other is the carrier phase based approach. The carrier phase approach may be suitable for scenarios where the sea surface is relatively smooth and the receiver platform is ground-based or on a low altitude aircraft. In the case where the sea surface is rather rough, obtaining the carrier phase of the reflected signal would be a rather challenging problem. It would be useful to conduct more investigations on this approach in order to obtain highly accurate relative delay estimation. There are two methods associated with the delay waveform based method. The first one makes use of the clean code (C/A code), while the second one utilises the interferometry technique. The clean code method deals with the direct signal and the reflected signal separately and only uses the C/A code to generate the delay waveform. In the interferometry technique, on the other hand, both signals are processed together. That is, the two signals are either received simultaneously via the same antenna or cross-correlated when they are received via two different antennas. The interferometry technique is intended to exploit the P(Y) code or military M-code to achieve an accuracy gain at the cost of high-gain and directional antenna, and that the P(Y) code or M-code signals may only be observed at some specific intervals. In addition, a high-bandwidth front-end/receiver is required. Consider the case where the known C/A code (also termed clean code) is employed. The direct and reflected signals are received via two different antennas, for instance, fixed on top and bottom of an aircraft, respectively and then processed separately. The code phase of the direct signal and that of the reflected signal are estimated using correlation delay waveforms, which are then used to estimate the delay of the reflected signal relative to the direct signal. The relative delay (i.e. the time difference of arrival between the reflected and direct signals) is then used to estimate the SSH, which will be discussed in the next section. In the case of airborne experiment or mission, the flight altitude is usually less than 10 km. Thus, the relative delay is smaller than 0.066 ms, which is much less than the code length of 1 ms. However, in the spaceborne case, the relative delay can be a few milliseconds, depending on the altitude of the LEO satellite. Then, an ambiguity would occur, since the distinguishable relative delay by the code phase difference is less than code length which is typically 1 ms. Of course, such an ambiguity can be easily resolved using the known altitude of LEO satellite and the GNSS satellite elevation angle.

5.1 Estimating Relative Delay of Reflected Signal

91

5.1.2 Multipath Interference In the case where the zenith-looking antenna is high above the ground, especially when the receiver is mounted on a satellite or on an aircraft, the code phase of the direct GNSS signal can be readily estimated by determining the location of the peak power of the correlation waveform. It would also be easy to determine the code phase (i.e. the desired code phase) of the reflected signal when the reflection surface is smooth and looks like a mirror. In this case, there is only one reflection path with the signal reflected over the specular point, and the SSH is equal to the mean sea level which is often of the more important interest. On the other hand, it may not be easy to obtain an accurate estimate of the desired code phase of the reflected signal forwarded from a rough sea surface. The main reason is that the location of the peak power of the reflected signal would not be the same as the desired code phase of the reflected signal since the peak power location is shifted due to rich multipath propagation. Clearly, using the peak power location to calculate the relative delay would produce a large bias error. The time shift or offset would depend on a number of factors including the surface roughness and the receiver altitude. The peak power location of the delay waveform derivative can be used as the desired code phase of the reflected signal, but the estimate would also be biased considerably [17]. It is observed that the desired code phase of the reflected signal is somewhere between the peak power location of the delay waveform and that of the waveform derivative. However, the exact location of the code phase is typically unknown. For clarity, it is desirable to explain why the peak power location could shift when a flat sea surface is replaced with a rough sea surface. Figure 5.1 illustrates the idealised correlation diagram (correlogram) of a GNSS signal in the presence of multipath propagation. In the presence of a perfect smooth sea surface, the signal will only be reflected at the specular point and then travels to the receiver. In the presence of a rough sea surface, besides the first reflection path signal with the shortest path length, there will be other multipath signals arriving at the receiver. Suppose that in addition to the first reflection path there are (J − 1) other reflection paths whose Fig. 5.1 Illustration of multipath correlation diagram

1 st

2nd

3rd 4th 1

92

5 Sea Surface Altimetry

lengths relative to the 1st path are less than the GNSS code chip width (τ1 ; i.e. half of the correlogram triangle width). Then, the following result usually holds. The peak correlation power location of the combined multipath signals will shift from the peak correlation power location of the first path signal provided that J 

P j > P1

(5.1)

j=2

where P j is the peak correlation power of the jth path signal. Proof Let C j (t) denote the correlation power of the jth path signal. At time instant τ1 , the combined correlation power of all the paths is equal to

C(τ1 ) =

J 

C j (τ1 )

(5.2)

j=1

At time τ1 + δτ the combined correlation power becomes C(τ1 + δτ ) =

J 

C j (τ1 ) + δC

(5.3)

j=1

where by denoting the slope of the leading edge of the correlogram of the jth path as k j , the correlation power increment δC over the small interval of δτ is given by ⎧ J ⎨

⎧ ⎫ J ⎬ δτ ⎨ δC = k j δτ − k1 δτ = δτ k j − k1 = k j τ1 − k1 τ1 ⎩ ⎭ ⎭ τ1 ⎩ j=2 j=2 j=2 ⎧ ⎫ J ⎬ δτ ⎨ = P j − P1 ⎭ τ1 ⎩ J 

⎫ ⎬

(5.4)

j=2

That is, if (5.1) is valid, then δC is positive and thus the peak correlation power location shifts to the right hand side by at least δτ . Since both the 1st path signal and signals of other paths are reflected signals, the signal power of the 2nd path and a number of following paths can be significant with respect to the 1st path. Thus, intuitively, (5.1) would always be valid with a rough sea surface. It would be interesting to derive the theoretical formulas to describe the location difference and the sea state so that how much shift can be readily determined in the future. Note that the peak correlation power location (τ1 ) of the 1st path signal related to a rough sea surface can be different from that of the single path signal related to a smooth sea surface. However, the difference would be rather small.

5.1 Estimating Relative Delay of Reflected Signal

93

5.1.3 Power Ratio Based Relative Delay Estimation The power-ratio method was proposed in [20] to deal with the challenging issue in the estimation of the desired code phase in the presence of a rough sea surface. A key advantage of the method is that it does not require any a priori knowledge of sea state information or any theoretical model. Thus, it would not be affected by the modelling errors and uncertainties. Figure 5.2 is an illustration of the delay waveform of the reflected signal in the presence of a rough surface. C(τm ) is the peak power of the reflected signal received via the down-looking antenna where τm is the time point at which the peak power occurs, while C(τ1 ) is the correlation power at the time point τ1 where the reflected signal peak power occurs when the surface is perfectly smooth. The time point τ1 may also be considered as the point where the correlation power of the first path signal is the maximum. Since C(τm ) can be measured, the time parameter τm can also be measured. On the other hand, neither C(τ1 ) nor τ1 can be simply measured. The concept of power-ratio is defined as the ratio of the peak power of the reflected signal when the surface is perfectly smooth over the peak power of the reflected signal which is actually received. That is, η=

C(τ1 ) C(τm )

(5.5)

Clearly, in the case of a perfectly smooth sea surface, the power ratio is equal to unity. Otherwise it is less than one. Given a power ratio and the measured peak power, the power at the desired code phase can be calculated using (5.5) and the measured delay waveform. Then, the desired code phase is calculated and then used to calculate the delay of the reflected signal relative to the direct signal. Finally, the SSH estimates are obtained using the method to be studied in the following section. Fig. 5.2 Illustration of code phases/delays and related correlation power of reflected signal

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5 Sea Surface Altimetry

5.2 A Two-Loop Approach for Estimation of Sea Surface Height Suppose that a sequence of delay waveform is produced by processing the logged GNSS data such as digital IF samples. A number (say Nη ) of power-ratio values (are selected to cover the likely values of the actual power-ratio. For instance, since a unit power ratio corresponds to a smooth sea surface, the range of power ratio may set to be between 0.93 and 1.0. It will be seen later that a power ratio of 0.9666 corresponds to a rough sea surface with a significant wave height (SWH) of about 4 m. A power ratio of 0.93 would be associated with a much rougher sea surface. Thus, such a range of power ratio would cover nearly all sea surface conditions. Applying each of the given power-ratios to the sequence of delay waveforms produces a sequence of relative delay estimates and then a sequence of SSH estimates related to each satellite. That is, a given number of power-ratios will yield the same number of sequences of SSH estimates.

5.2.1 Geometrical Relationship As shown in Fig. 5.3, the sea surface height (SSH) is calculated relative to the surface of the theoretical Earth ellipsoid in the WGS84 system, which has zero altitude. The SSH at a specific sea surface point is the distance from the point to the WGS84 Earth ellipsoid surface and the mean SSH is the average of many such distances. The specular point position (SPP) on the WGS84 ellipsoid surface is uniquely determined Fig. 5.3 Geometry of the receiver, WGS84 mean sea level (Earth ellipsoid surface), rough sea surface, direct and reflected signal paths

5.2 A Two-Loop Approach for Estimation of Sea Surface Height

95

when given the position information of the satellite and receiver, since there is only one scattering point, i.e. the SPP, over which the signal propagation path is the shortest. However, the WGS84 ellipsoid model is not exactly the same as the real Earth shape, so the SPP is usually not on the WGS84 ellipsoid, but on a scaled ellipsoid regardless of a calm or rough sea surface. The distance between the WGS84 ellipsoid and the scaled one is equal to the SSH at the SPP.

5.2.2 Algorithm Flowchart One way to determine the SSP is to make use of the time difference of arrival of the direct and reflected signals, in addition to the positions of the GNSS satellite and the receiver as well as the model of Earth ellipsoid. It is also based on the fact that the propagation path through the SPP is the shortest. Figure 5.4 shows the flowchart of the two-loop iterative method for the calculation of the SPP and hence the SSH [6, 20]. The inner loop is for the determination of the SPP, while the outer loop is for the updating of the SSH. Specifically, both the GNSS satellite position (xt , yt , z t ) and the receiver position estimate (xˆr , yˆr , zˆr ) are known, and for a given tentative SSH, the SPP (x˜ S , y˜ S , z˜ S ) on the tentative sea surface, i.e. the scaled WGS84 ellipsoid surface, can be determined. Note that the satellite position is assumed error free, while the receiver position is an estimate.

Fig. 5.4 Flowchart for SSH calculation. δth is a small positive number

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5 Sea Surface Altimetry

5.2.3 Calculating Total Path Length The total path length (TPL) from the satellite through the SPP and to the receiver is given by R˜ t Sr = R˜ t S + R˜ Sr

(5.6)

where (xt − x˜ S )2 + (yt − y˜ S )2 + (z t − z˜ S )2 = (x˜ S − xˆr )2 + ( y˜ S − yˆr )2 + (˜z S − zˆr )2

R˜ t S = R˜ Sr

(5.7)

Minimizing the TPL with respect to the unknown SPP produces the estimated SPP as (xˆ˜ S , yˆ˜ S , zˆ˜ S ) = arg min R˜ t Sr x˜ S , y˜ S ,˜z S

(5.8)

A variety of nonlinear minimization algorithms, either unconstrained or constrained, can be used to implement the minimization [5]. The TPL can also be calculated using the estimated line-of-sight distance ( Rˆ tr ) from the satellite to the receiver and the estimated delay (τˆr d ) of the reflected signal relative to the direct signal. That is, Rˆ t Sr = Rˆ tr + cτˆr d

(5.9)

where c is the speed of light and Rˆ tr =



(xt − xˆr )2 + (yt − yˆr )2 + (z t − zˆr )2

(5.10)

From Fig. 5.4 it can be seen that the SSH estimation accuracy is largely dependent on the performance of the TPL (transmitter-SPP-receiver) measurement, which is determined by (5.9). The measurement error associated with the first term in (5.9) comes from the receiver position estimation error. When the receiver is given, such an error is typically not reducible, although smoothing may improve the accuracy marginally. Thus, it is important to use a GNSS receiver which can achieve satisfactory position estimation accuracy. The measurement error related to the second term in (5.9) results from the estimation of the relative delay of the reflected signal. When an accurate receiver is used, the relative delay estimation error would be dominant. Therefore, it is vital to reduce this error.

5.2 A Two-Loop Approach for Estimation of Sea Surface Height

97

5.2.4 Updating Sea Surface Height Estimate As indicated in Fig. 5.4, the calculated TPL by (5.6) and (5.8) is compared with the measured TPL by (5.9) to determine whether the tentative sea surface height should be increased or decreased. If the TPL calculated with SPP is greater than the one measured by the relative delay, then the tentative SSH is increased by an increment. Otherwise, the tentative SSH is decreased by a decrement. The updated tentative SSH is used to determine the SPP again. The process continues until the difference between the two TPLs is smaller than the pre-defined threshold. To reduce the computational complexity, a simpler technique may be used. For instance, if R˜ t Sr > Rˆ t Sr , the tentative surface height is increased by a relatively larger increment such as 4 m. At the next iteration of the outer loop if R˜ t Sr < Rˆ t Sr , the increment is decreased by half of the previous increment. In this way, the process will quickly converge to the steady state. Note that the specular reflection must satisfy Snell’s Law, i.e. the two angles (θ1 and θ2 in Fig. 5.3) between the incoming wave and the reflected wave, separated by the local surface normal must be equal or the difference is extremely small. Thus the results should be tested to see if this Law is satisfied. From (5.6) the partial derivatives with respect to the coordinates of the tentative specular point can be calculated as u S − xr u S − xt ∂ R˜ t Sr = + , u S ∈ {x˜ S , y˜ S , z˜ S } ˜ ∂u S Rr S R˜ t S

(5.11)

which can be rewritten in a vector form as d S =

S − T S − R + R˜ r S R˜ t S

(5.12)

 R,  and T are the position vectors of the specular point, the receiver and the where S, transmitter, respectively. Equation (5.12) is used to generate an iterative solution to the minimum path length. That is, at time instant n+1 the SPP is updated according to Sn+1 = Sn + κ d S

(5.13)

where κ is a constant which typically should be set as a larger value as the receiver altitude increases. To start the process, an initial estimate of the SPP is required, which can be simply set to be the projection of the receiver position onto the surface. At each iteration, a constraint must be applied to restrain the specular point on the surface which is ˜ m above (being a positive number) or below (being a negative number) the WGS84 ellipsoid surface that has a zero altitude. Thus, the SPP is scaled according to

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5 Sea Surface Altimetry

  ˜ Sn+1 Sn+1 = (r S + ) | Sn+1 |

(5.14)

where the radius of the Earth at the specular point is calculated by

r S = aW G S84

 2 1 − eW zS G S84 , λ = arcsin S 2 2  1 − eW (cos λ ) | S| S G S84

(5.15)

where eW G S84 = 0.08181919084262 and aW G S84 = 6378137 m. Clearly, the altitude of a single specular point cannot be treated as the estimate of the mean sea surface height or mean sea level. However, a reasonable estimate of the mean surface height will be produced through the generation and subsequent processing of the altitude estimates of many specular points over a period of time. Note that in the case where the experimental digital IF samples are stored in a laptop or a hard drive for post-processing, the update rate of the specular point calculation would not be an issue. However, in the case of online processing and large acquisition time, the update rate of the specular point calculation would be affected. Some dedicated GNSS receivers have an FPGA-based signal processing backup to generate the delay-Doppler waveform in real time. As a consequence, the update rate of the specular point calculation can be high enough for a range of applications.

5.2.5 Determining Power Ratio Now the question is how to determine which power-ratio is the best so that the corresponding sequence of SSH estimates has the best performance. To answer this question, two different criteria can be applied. Specifically, in the first criterion the SSH estimation performance is measured by the cost function which is defined as ψ(η j ) =

N 

(m i (η j ) − m total (η j ))2 , j = 1, 2, · · · , Nη

(5.16)

i=1

where data associated with N GNSS satellites are used so that N sequences of SSH estimates are available, m i (η j ) is the mean of SSH estimates related to the ith satellite and jth power ratio, and m total (η j ) is the mean of the SSH estimates associated with all the N satellites and jth power ratio. If only selecting satellites whose elevation angles are greater than say 30°, N would be around five for GPS satellites. Note that N must be greater than one and this requirement is usually satisfied in practice. Note that it is assumed that GNSS signals are captured via a receiver mounted in an aircraft flying at a low altitude, such as a few kilometers above the surface. The

5.2 A Two-Loop Approach for Estimation of Sea Surface Height

99

effective reflection area associated with the signals of the N satellites would be a number of kilometers. Over such a relatively small sea surface area, the mean SSH can be considered the same. On the other hand, in the case of a LEO satellite platform at an altitude of several hundreds of kilometers, the effective reflection area can be hundreds of kilometers. In this case, the mean SSH of one specular point track can be significantly different from that of another track; thus, (5.16) cannot be used. That is, the first criterion is that the power-ratio which minimizes the cost function in (5.6) is selected. Mathematically, this criterion is given by ηˆ = arg min ψ({η j }) [η j }

(5.17)

The selection of such a cost function is based on the observation from processing the data that the best power-ratio associated with one satellite is very similar to that related to another satellite when the elevation angle is greater than 45°. Using the best mean power ratio, the estimated mean SSH related to an individual satellite is very close to the estimated mean SSH associated with all the selected satellites. When the selected mean power ratio is significantly different from the best one, the estimated mean SSHs would be rather different from each other. To verify such an approach, a local reference mean sea level is required, which may be obtained using Lidar measurements. In the second criterion, the cost function is simply defined as ψ(η) = σ (η)

(5.18)

where σ is the standard deviation of all the N sequences of SSH estimates. That is, this criterion is to minimize the cost function in (5.18) to produce the desired power ratio. This criterion selection comes from the consideration that using the desired power ratio would yield estimates that have minimum variations. The performance of the two different cost functions will be evaluated later using the logged experimental data. When the length of the sequence of power ratio evaluated is large, the computational complexity can be high. To reduce the complexity, a technique, similar to finding a minimum of a function using gradient descent method, is described with details given in Sect. 5.2.6. Theoretically, the time difference between the desired code phase corresponding to the time of arrival of the reflected signal and the code phase of the peak correlation power may be determined using the sea state information such as the surface elevation standard deviation. When such a relationship is established, the relative delay of the reflected signal can be readily determined by measuring the code phases of the peak correlation power of the direct and reflected signals. However, it is inevitable that there are some uncertainties associated with such a theoretical model. Therefore, it would be useful to establish a model to describe the relationship between the peak power location shift of the reflected signal and the surface roughness as well as to analyze the effect of the model uncertainties.

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5 Sea Surface Altimetry

5.2.6 Algorithm Complexity Reduction When the length of the sequence of power ratio evaluated is large, the computational complexity can be rather high. To reduce the complexity, a technique, similar to finding a minimum of a function using gradient descent method, may be used, as shown in Fig. 5.5. Initially two tentative power ratios (η1 and η2 where η2 > η1 ) are selected empirically and the difference between these two initial ratios should be such as greater than 0.02. Each of the two ratios is then used to obtain sequences of SSH estimates and calculate the cost functions as described in the preceding subsection. Next, the power-ratio value is updated as illustrated in Fig. 5.6. Specifically, the power-ratio update is dependent on the position and value of the newly updated power ratios and corresponding cost functions. Initially, the ratio update is realized by ⎧ ⎨ η1 − δ2 , ψ2 − ψ1 > a η3 = η + δ2 , ψ2 − ψ1 < −a ⎩ 2 η1 + δ2 /2, |ψ2 − ψ1 | < a

(5.19)

where a is a small positive number that is much smaller than the initial power-ratio increment which can be simply set to be δ2 = η2 − η1

(5.20)

Initial power ratios

SSH calculation (Sat 1)

…

SSH calculation (Sat N)

Cost function calculation

Performance comparison

Power ratio update

Fig. 5.5 Block diagram for determination of power ratio and SSH using a technique similar to the early-late gate method

5.2 A Two-Loop Approach for Estimation of Sea Surface Height

101

Fig. 5.6 Iterative power-ratio updating

Then sequences of the SSH estimates are produced and the corresponding cost function (ψ3 ) is calculated. Also, the three cost function values are ranked as smallest (ψmin ), median, and largest. At the next iteration, the power ratio is updated according to η4 = η(ψmin ) + δ4

(5.21)

The ratio increment (δ4 ) is determined as follows. If the power ratio with the smallest cost function (η(ψmin )) is the median of the three ratios and the power ratio related to the medium cost functions is the maximum, then the increment is updated by δ4 = δ3 /2

(5.22)

Otherwise, if the power ratio related to the medium cost function is the smallest, then δ4 = −δ3 /2

(5.23)

On the other hand, if η(ψmin ) is not the median ratio but the largest one, then δ4 = δ3 Further if η(ψmin ) is the smallest power ratio, then

(5.24)

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5 Sea Surface Altimetry

δ4 = −δ3

(5.25)

This process continues until the power-ratio increment is sufficiently small. Intuitively, such an iterative procedure would quickly approach the steady state and the power-ratio estimate approaches the desired one.

5.2.7 Calibration Since the zenith-looking and nadir-looking antennas, the receiver, and the reference point are not in the same position, it is necessary to calibrate the relative delay measurements to remove the effect of these position differences. Note that the reference point may be set at the centre of the inertial measurement unit (IMU) provided that such a device is used. Figure 5.7 illustrates the configuration of the devices. The two antennas are connected to the receiver via two cables whose lengths are L C R and L C D . The actual measurement of the relative delay of the reflected signal is given by L measur ed = AS + S D + L D R − (AU + L U R ) + ε

(5.26)

where ε is the measurement error. Note that notations in different chapters may have different meanings, for instance, ε denotes the dielectric constant of propagation medium in Chap. 6, dielectric constant of snow in Chap. 11 and that of soil in Chap. 12. On the other hand, using the reference point position, the relative delay is calculated as Fig. 5.7 Geometric relationship between the devices

A

Antenna U

GNSS satellite

Reference point Receiver R

Specular point

S

C D Antenna

V

5.2 A Two-Loop Approach for Estimation of Sea Surface Height

L calculated = AS + SC + L D R − AC

103

(5.27)

where the theoretical and actual specular points are treated to be at the same position since their difference can be neglected. As described earlier, the SSH is estimated by comparing the measured and calculated relative delays. However, typically, there is an offset between the calculated and measured delays even in the absence of measurement error. That is, L o f f set = L measur ed − L calculated = (SC − S D) + (AU − AC) + (L U R − L D R )

(5.28)

where the measurement error is ignored. Therefore, the measured relative delay should be calibrated by subtracting the offset from itself. The lengths of the two cables can be readily measured in advance. In case where a LNA is used to amplify the reflected signal, the path length from D to R or from U to R will be the sum of two cable lengths plus the distance between the two connection points of the LNA. The distance from U to C and that from D to C can be manually determined in advance. The positions of points U and C are estimated by the receiver and frame transformation; thus distances AC and AU can be readily calculated. Calculation of the distances SC and SD requires a knowledge of the SSH, which is unknown in advance. However, initial information about the SSH, or previous SSH estimation results, can be exploited. The uncertainty in the SSH estimate will affect the calculation of both SC and SD in a very similar way, so that a small SSH error will have a negligible impact. That is, distance CV can be estimated and distance SC can be calculated using elevation angle. Calculation of distance SD requires its orientation to determine the angle ∠S DC. In the case where the nadir-looking antenna is fixed directly beneath the reference point, the distance can be simply calculated by SD =



SC 2 + C D 2 − 2 × SC × C D × cos(90 − φ)

(5.29)

where φ is the satellite elevation angle. Once the distances and the cable lengths in (5.28) are known, the relative delay offset can be readily determined. For instance, suppose that points U and D are directly above and below point C respectively; AC = 20000 km; satellite elevation angle is 50°; UC = 0.8 m; CD = 0.4 m; CV = 300 m; LUR = 1 m; and L DR = 0.6 m. Then, the unknown distances are obtained from some simple calculations: AU = 19999.999387 km, SD = 391.32 m, SC = 391.62 m. As a result, the relative relay offset is calculated to be 0.094 m. The SSH estimation error caused by this relative delay offset can be approximated as δ ≈

L o f f set = 6.1cm 2 sin φ

(5.30)

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5 Sea Surface Altimetry

Thus, when the configuration of the devices is arranged properly and the cable lengths are selected based on similar analysis, the SSH error caused by the device configuration will not be large. However, to achieve accurate altimetry, such an error must be compensated for.

5.3 An Airborne Experiment and Results 5.3.1 Experiment Setup A low-altitude airborne experiment was conducted by a UNSW-owned light aircraft off the coast of Sydney between Narrabean Beach and Palm Beach on the 14th of June 2011. The flight height above the sea was between 200 and 500 m. These low flight altitudes were required by the Lidar experiment. The primary payload was for the Lidar experiment whose purpose was to monitor the Sydney coastal areas to provide information for future infrastructure development. The free host payload was for the GNSS signal reception and data logging. Figure 5.8 shows the short flight track segment relative to the coastal area over the duration of about 60 s. The distances between the Waverider buoy and the two points A and B are R AB = 2.98 km, R AW = 8.97 km, and R BW = 9.54 km, respectively. Figure 5.9 shows the basic block diagram of the signal reception and data logging system. All the equipment was secured on stable structures in the aircraft. Figure 5.10

Fig. 5.8 Experimental location off Sydney coast and there was a Waverider buoy monitoring the wave nearby

5.3 An Airborne Experiment and Results Fig. 5.9 Simplified block diagram of the data collection system

105

RHCP Antenna NordNav Software LHCP Antenna

LNA

Laptop

Receiver

Fig. 5.10 Light aircraft used for the experiment

shows the light aircraft used for the experiment, which can accommodate four people. Figure 5.11 shows the GPS software receiver and the Lidar equipment secured in the aircraft. The Lidar device is a Riegl LMS-Q240i laser scanner and the laser wavelength is 905 nm. This device is extremely rugged and thus ideally suited for airborne experiments. The maximum measurement range is around 650 m and ranging accuracy is about 20 mm. The GPS data were logged using the NordNav software receiver, which has four front-ends so that the signals arriving at the LHCP and RHCP antennas were recorded simultaneously via two of the four front-ends. The direct signal was received via the normal zenith RHCP GNSS antenna as shown in the upper left of Fig. 5.12, which was also used for the Lidar experiment. The reflected GNSS signal was captured via the nadir-looking LHCP antenna mounted on the bottom of the aircraft as shown in the upper right of Fig. 5.12. This LHCP antenna is a passive L1/L2 GPS antenna with a 3 dB-beamwidth of 114° in free space. The LNA of a fixed gain 20 dB was connected between the LHCP antenna and the second front-end of the NordNav software receiver as shown in the lower

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5 Sea Surface Altimetry

Fig. 5.11 The software receiver (front left) is secured in the aircraft

Fig. 5.12 The RHCP antenna is secured on the top of the aircraft (upper left); the LHCP antenna (white and round) is secured on the bottom of the aircraft (upper light); the LNA is secured in the aircraft (lower left); the laptop can be secured in the aircraft

left of Fig. 5.12, amplifying the signals arriving at the LHCP antenna. The first frontend of the receiver receives the direct signal that was used to obtain the position and velocity of the satellite and the receiver. The IF and sampling frequency of the software receiver are 4.1304 MHz and 16.3676 MHz respectively. The direct signal was also used to derive the code phase and Doppler frequency information. The raw IF GNSS signals from the output of the receiver were logged to a laptop as shown

5.3 An Airborne Experiment and Results

107

in the lower right of Fig. 5.12. The laptop can also be directly placed on the lap of the operator who needs to operate this laptop alone or most of the time. The wind data was provided by the Australian Bureau of Meteorology (BOM) [1], indicating that the wind speed was about 50 km/h during the experiment. Also, the wave data was provided by Mark Kulmar from the Manly Hydraulics Laboratory (MHL) [15], Sydney, New South Wales, indicating that the sea surface during the experiment was rather rough with maximum wave height greater than 6 m.

5.3.2 Lidar-Based Mean Sea Level and Wave Statistics Measurement

Relative Elevation(m)

Altitude of Points(m)

Figure 5.13 shows the results related to 3983 points on the sea surface from the processing of the Lidar data. The upper plot shows the WGS84 altitudes of the surface points and their mean (dashed straight line), while the lower plot shows the difference when the altitudes are subtracted by a value which is the mean of the altitudes. That is, the lower plot shows the surface elevation variation with respect to the measured MSL which is calculated as 23.44 m. The standard deviation of the MSL estimate is 1.38 m, mainly contributed to the surface roughness. This MSL estimate can be employed as a reference when evaluating the performance of the GNSS-R based altimetry. These samples were taken between 15:27:55 and 15:31:38 for duration of 3 min 43.3 s. 28 26 24 22 20 0

0.5

1

1.5 2 Time (min)

2.5

3

3.5

0

0.5

1

1.5 2 Time (min)

2.5

3

3.5

5

0

-5

Fig. 5.13 WGS84 altitudes of the sea surface points (top) and relative sea surface elevation (bottom) measured by Lidar

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5 Sea Surface Altimetry

Figure 5.14 shows the wave heights derived from the Lidar surface points shown in Fig. 5.13. A wave is defined as the portion of the water between two successive zeroup-crossings relative to the MSL. The wave height is simply calculated as the vertical displacement between the crest and the trough of the wave. Using the Lidar-based wave heights the wave statistics during this period can be calculated, which include the significant wave height (SWH), the root mean square (RMS) wave height, and the maximum wave height. Also, a nearby Waverider buoy continuously measures the wave height statistics and the relevant statistics were obtained from MHL. The wave statistics from Lidar and buoy measurements are listed in Table 5.1. From this table it can be seen that these Lidar-based statistics of the wave height measurements have good agreement with those of the Waverider buoy-based measurements. Note that the distance between the location of the Waverider buoy and the location where the presented data were collected is about 10 km. Due to this location difference some small variations of the wave statistics are expected. Figure 5.15 shows the cumulative distribution function (CDF) of the measured wave heights. It can be seen that the measured wave heights closely follow the 7 6

Wave Heights (m)

5 4 3 2 1 0

50

100

150

200

250

300

350

400

Wave Sample Number

Fig. 5.14 Wave heights derived from the Lidar surface points

Table 5.1 Measured wave statistics

SWH (m)

RMS WH (m)

Max WH (m)

Lidar

3.7

2.6

6.8

Buoy

4.0

2.7

6.4

5.3 An Airborne Experiment and Results

109

1 Measured Theoretical

Cumulative Distribution Function

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3 4 Wave Height (m)

5

6

7

Fig. 5.15 Cumulative distribution of the measured wave heights and the Rayleigh variable

Rayleigh distribution, whose CDF is given by {1−exp(−x 2 /(2σ 2 )} where the distribution parameter σ = 1.81 m is calculated from the measured wave heights. This is in agreement with results reported in the literature [4, 12].

5.3.3 Data Processing and Delay Waveform Generation The collected IF data were processed to generate delay waveforms (correlation power versus C/A code phase). That is, the IF data were cross-correlated with a replica of a C/A code sequence associated with a specific satellite and the Doppler frequency is set to be the central Doppler frequency. Delay waveform is one of the two types of waveforms (the other one is the delay-Doppler waveform) which are the basis for most of the current GNSS-R remote sensing methods especially associated with spaceborne data. In the approach studied in this section, delay waveforms associated with four GPS satellites were generated. The coherent integration time is 1 ms, while the incoherent integration time is 1 s. When the clean code (C/A code) is used and sampling frequency is low, interpolation of delay waveform is necessary to improve resolution of code phase estimation. For instance, Fig. 5.16 shows the top of a measured delay waveform and the interpolated waveform. A software receiver was used to generate digital IF samples with a sampling frequency of 16.3676 MHz. Clearly, the sampling period is equivalent to about 18 m, which would be too large to achieve accurate altimetry. Interpolation can be realized in two steps. Suppose that the original sampling frequency is f 0 and the new desired sampling frequency is L f 0 , where L is a positive integer greater than one. Then, (L −1) zeros are first inserted between each pair of the

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5 Sea Surface Altimetry

Fig. 5.16 Delay waveform interpolation

15.8

Correlation Power

15.6 15.4 15.2 15 14.8 14.6 2.4

original after interpolation 2.45

2.5 2.55 Time (code chip)

2.6

2.65

original samples. The up-sampled data are then passed through a symmetric low-pass filter which allows the original data unchanged, but interpolates the inserted zeros. When using MATLAB software, the library function “Interp” can be directly used; for instance, the dashed line in Fig. 5.12 was generated in this way with L = 50.

5.3.4 Peak Power Based Relative Delay Measurements and Error Statistics The delays of the reflected signal relative to the direct signal associated with four satellites are estimated. First, the relative delay is estimated by determining the location of the peak power of the direct signal and that of the reflected signal. Second, the relative delay is determined under an idealised scenario where the sea surface is smooth and the SSH is equal to the Lidar-based mean SSH measurement. The four selected satellites that have the largest elevation angles are listed in Table 5.2. The selection of the satellites is based on the consideration that the SNR of the reflected signal would increase as the elevation angle increase. Figure 5.17 shows the relative delay estimation results over a period of about 60 s. Clearly, although the shape of the peak power based estimates has a good match with that of the estimates using the Lidar-based mean SSH estimate, there is a relatively large offset between the two curves. As mentioned earlier, this offset is due to the Table 5.2 Elevation and azimuth angles of four satellites Satellite #

22

18

6

21

Elevation (°)

62.67–62.99

58.14–57.41

50.89–50.92

48.57–47.95

Azimuth (°)

238.19–240.29

148.82–149.94

266.29–268.22

78.87–79.93

5.3 An Airborne Experiment and Results

111

Relative Delay (m)

PRN#22 580

540 560 520 540 0

Relative Delay (m)

PRN#18

560

20

40

60

PRN#6

520

500

480

480

460 0

20 40 Time (sec)

60

440

20

40

60

PRN#21

500

500

460

0

0

20 40 Time (sec)

60

Fig. 5.17 Relative delay estimates using known MSL (23.44 m) (dashed black curve); using the peak power code phase (blue dashed curve); and removing mean error (red solid curve)

roughness of the sea surface. When using the Lidar-based results as a reference, the relative delay estimation error of the peak power-based method can be calculated as shown in Fig. 5.18. Table 5.3 shows the mean and STD of the estimation errors associated with individual satellites. It can be seen that the mean error varies significantly and the largest mean error difference is 1.37 m. From this table it is difficult to tell how the estimation error is related to the satellite elevation and azimuth angles. However, if excluding satellite PRN#18, then the error in terms of mean or RMS increases as the elevation angle decreases. The reason why the error mean and STD related to satellite PRN#18 are relatively large may be due to the relationship between satellite azimuth angle (i.e. signal propagation direction) and the wave direction. As shown in Fig. 5.19, during the data collection the wave direction was about 145° (southwest). That is, the reflected signal transmitted from satellite PRN#18 travelled virtually along the wave direction, while the signals from other three satellites basically travelled across the wave direction. It is not clear why the signal travelling along the wave direction produced a larger error than the other signals. It would be necessary to conduct more investigations to determine how the errors are related to the elevation and azimuth angles, wave direction, and other parameters.

5 Sea Surface Altimetry

Relative Delay Error (m)

Relative Delay Error (m)

112

PRN#22

30 25

25

20

20

15

0

20

40

60

PRN#6

30

PRN#18

30

15

0

20

40

60

PRN#21

30

25 25 20 15

0

20 40 Time (sec)

60

20

0

20 40 Time (sec)

60

Fig. 5.18 Relative delay estimation errors of peak power-based method when the dashed black curve in Fig. 5.17 is treated as true relative delay Table 5.3 Mean and standard deviation of measured relative delay errors associated with four satellites Satellite #

22

18

6

Mean (m)

22.29

23.43

STD (m)

1.96

2.16

RMS (m)

22.38

23.53

Fig. 5.19 Wave direction and reflected signal propagation direction

21

All

22.98

23.66

23.09

1.63

1.45

1.89

23.04

23.70

23.18

North From Sat 21

East

From Sat 6

Wave direction From Sat 22 From Sat 18

5.3 An Airborne Experiment and Results

113

5.3.5 Statistics of Ideal Power Ratio As mentioned earlier the receiver position coordinates, including altitude, are required to calculate the SSH. As shown in Fig. 5.20, the aircraft flight height was measured by two different GNSS receivers, NordNav and Novatel, over the duration of about 1 min. Taking the Novatel measurements as a reference, the NordNav measurement error mean is 2.5 m and the error standard deviation is 2 m. Since the Novatel measurements are much more accurate, its measurements of the receiver position including the altitudes were used in the SSH estimation. However, the NordNav receiver has four front-ends which are synchronised, hence the digital IF samples recorded through the NordNav receiver are used. Note that when processing the complex (IF) samples for generating delay waveforms, the coherent integration time is set at 1 ms. The 1 ms delay waveforms are then accumulated over a period of 1 s to generate the final delay waveforms for code phase and SSH estimation. Using the known mean SSH measured by the Lidar device, the power-ratio statistics can be calculated as shown in Fig. 5.21 and the procedure can be described as follows. Given the known mean SSH and using the known satellite position and the measured receiver position, the SPP on the mean sea surface can be determined as described earlier. Accordingly, the TPL can be calculated, and hence the relative delay of the reflected signal can be determined. Then, using the code phase of the direct signal, the desired code phase of the reflected signal can be calculated. As a result, the correlation power of the reflected signal at this code phase can be determined using the delay waveform. Finally, the ratio of this correlation power over 355 NordNav Novatel

350

Flight Height (m)

345 340 335 330 325 320 315

0

10

20

30 40 Time (sec)

50

60

Fig. 5.20 Aircraft altitude measurements by two different GNSS receivers. The measurements are with respect to the WGS84 ellipsoid, not the sea surface

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5 Sea Surface Altimetry

Given mean SSH

Receiver position

SPP calculation

Satellite position

TPL calculation

Relative delay calculation

DS path length

RS code phase calculation

DS code phase

RS power retrieval

RS power ratio calculation

RS peak power

Fig. 5.21 Calculation of power ratio of reflected signal when SSH is given. DS: direct signal, RS: reflected signal

the peak power of the reflected signal can be readily calculated. This is just to evaluate the statistics of the power-ratio calculated over a time series and related to four individual satellites. The procedure of the power-ratio calculation can be repeated for a significant number of delay waveforms obtained over a time interval. For instance, if a delay waveform is produced by accumulating waveforms over a period of 1 s without overlapping, then 1 min measurements would produce 60 delay waveforms associated with a specific satellite, resulting in 60 power-ratio values. If four satellites are considered, 240 power-ratio values would be produced. Then, the mean power ratio (MPR) is calculated. Next, the procedure runs backwards. That is, given the MPR, the correlation power of the reflected signal at the desired code phase can be readily calculated. Then, one locates the point on the delay waveform whose power value equals the calculated power. Note that typically there are two such points on the waveform with one on each side of the peak power point, but only the left-hand-side one is the desired one. The corresponding code phase (i.e. the desired code phase) of this correlation power is then obtained. As a consequence, the relative delay of the reflected signal can be calculated and the SSH is estimated according to the two-loop iterative method described earlier.

5.3 An Airborne Experiment and Results

115

Power Ratio (%)

PRN#22 98

98

97

97

96

96

95

Power Ratio (%)

PRN#18

0

20 40 PRN#6

60

95

98

98

97

97

96

96

95

0

20 40 Time (sec)

60

95

0

20 40 PRN#21

60

0

20 40 Time (sec)

60

Fig. 5.22 Calculated power ratios related to four satellites

Figure 5.22 shows the discrete power-ratios calculated using the above procedure. The mean and standard deviation of the power-ratios associated with the four individual satellites are listed in Table 5.4. The overall MPR is 0.9666 and the overall standard deviation is 0.0049. It can be seen that the four MPRs are very similar and the largest difference is only 0.0017 which is 0.18% of the overall MPR. That is, a single MPR can be used to approximate the MPRs related to the individual satellites for a given sea state. The MPR (0.9666) is then used to calculate the desired code phases and the relative delays of the reflected signal associated with each satellite. Finally, four sequences of SSH estimates are obtained as shown in Fig. 5.23. It can be seen that the mean of the SSH estimates associated with each satellite is very close to the Lidar-derived mean SSH estimate. Table 5.5 shows the error mean and standard STD associated with individual satellites when taking the Lidar measurement (23.44 m) as the error-free mean SSH estimate. The overall error mean and STD are 4.5 cm and 1.08 m, respectively. That is, if the desired MPR can be retrieved, accurate mean SSH estimation can be obtained. Table 5.4 Mean and standard deviation of calculated power-ratios associated with four satellites Satellite #

22

18

6

21

Mean

0.9674

0.9657

0.9664

0.9670

STD

0.0050

0.0063

0.0042

0.0035

5 Sea Surface Altimetry

Height Estimate (m)

Height Estimate (m)

116

PRN#22

28 26

26

24

24

22

22

20

0

20 40 PRN#6

60

20

28

28

26

26

24

24

22

22

20

0

PRN#18

28

20 40 Time (sec)

60

20

0

20 40 PRN#21

60

0

20 40 Time (sec)

60

Fig. 5.23 SSH estimates using a single MPR. Power-ratio-based mean SSH estimate (black dashed), power-ratio-based SSH estimates (red dashed), and Lidar-derived mean SSH measurement (blue solid)

Table 5.5 Power-ratio-based SSH estimation error mean and STD associated with individual satellites Satellite #

22

18

6

21

Mean (m)

0.181

−0.090

−0.023

0.112

STD (m)

0.996

1.409

0.959

0.872

5.3.6 Joint Power-Ratio and SSH Estimation Figure 5.24 shows the effect of varying the MPR on the SSH estimation error mean when all four satellites are considered. Varying the MPR by 2% would result in a mean SSH error of either −5 m or 3.83 m. That is, the performance is rather sensitive to the MPR selection. Figure 5.25 shows how the MPR affects the cost function defined by (5.16) when all four satellites are considered. Clearly, when ignoring the small variations from sample to sample, the overall cost function behaves as a convex function. That is, it is possible to search for the MPR at which the cost function achieves the minimum to obtain the desirable SSH estimate. In this case, the minimum cost function occurs when the MPR is equal to 0.9650. From Fig. 5.23, this MPR produces a SSH error mean of 0.39 m. Figure 5.26 shows the STD of SSH estimation errors associated with all four satellites. In this case the cost function is the STD which reaches the minimum

5.3 An Airborne Experiment and Results

117

10

Mean SSH Error (m)

5

0

-5

-10

-15 0.93

0.94

0.95

0.96 0.97 0.98 Mean Power Ratio

0.99

1

Fig. 5.24 Effect of MPR on mean SSH estimation error 6

1.6

SSH Error STD (m)

Cost Function (m 2 )

5 4 3 2 1 0 0.93

1.5 1.4 1.3 1.2 1.1

0.94

0.95

0.96

0.97

0.98

Mean Power Ratio

0.99

1

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Mean Power Ratio

Fig. 5.25 Cost function defined by (5.16) versus MPR (left panel); Cost function (SSH error STD) versus MPR (right panel). Data related to four satellites are used

when the MPR is equal to 0.9672. Using this MPR produces a mean SSH estimate whose error is −0.08 m. Figure 5.27 show the estimation results when three satellites are considered as well as the two different cost functions are used. There are four different combinations when choosing three from four satellites. Table 5.6 shows the MPRs and the mean SSH errors associated with the four combinations using the two different cost functions. Note that CF1 denotes for cost function defined by (5.16) while CF2 denotes for the one defined by (5.18). Clearly, when the worst combination is selected, the mean SSH error are 0.97 m and 1.51 m for the two different cost functions, respectively. On the other hand, if the best combination in

118

5 Sea Surface Altimetry

PRN#18,6,21

Cost Function

3 2

4

1

2

0

Cost Function

6

0.94

0.96

0.98

PRN#22,6,21

6

1

0

0.94

0.96

0.98

1

PRN#22,18,6

PRN#22,18,21 3

4

2

2 0

1 0.94 0.96 0.98 Mean Power Ratio

1

0

0.94 0.96 0.98 Mean Power Ratio

1

Fig. 5.26 Cost function defined by (5.16) versus MPR. Data related to three satellites are used

SSH Error STD (m)

PRN#18,6,21

PRN#22,6,21 1.6

1.3

1.4

1.2

1.2

1.1

1 0.94

0.96

0.98

1

0.94

SSH Error STD (m)

PRN#22,18,21

0.98

1

PRN#22,18,6

1.6

1.4

1.4

1.3

1.2

1.2 0.94 0.96 0.98 Mean Power Ratio

0.96

1

0.94 0.96 0.98 Mean Power Ratio

1

Fig. 5.27 Cost function (SSH error STD) defined by (5.18) versus MPR. Data related to three satellites are used

5.3 An Airborne Experiment and Results

119

Table 5.6 Estimated MPRs and mean SSH error when combinations of three satellites are considered Satellite # MPR Mean SSH Error (m)

18,6,21

22.6.21

22.18.21

22,18,6

(CF1)

0.9682

0.9648

0.9656

(CF2)

0.9686

0.9596

0.9672

0.9622 0.9672

(CF1)

−0.31

0.44

0.26

0.97

(CF2)

−0.40

1.51

−0.08

−0.08

Table 5.7 Average of estimated MPRs and mean SSH error when using different numbers of satellites Number of Satellites Averaged MPR

2

3

4

(CF1)

0.9659

0.9652

(CF2)

0.9683

0.9657

0.9672

RMS of mean SSH errors (m)

(CF1)

0.78

0.57

0.39

(CF2)

2.87

0.78

0.08

Average of mean SSH errors (m)

(CF1)

0.1841

0.3385

0.3947

(CF2)

−0.6

0.24

0.9650

−0.08

each case is selected, the error would be 0.26 m and −0.08 m, respectively. That is, the performance is also rather sensitive to the satellite selection. Table 5.7 shows the average of the estimated MPRs and RMS of the mean SSH errors when the number of satellites ranges from two to four. For instance, in the case of three satellites the overall error of the four different combinations shown in Table 5.6 is calculated in terms of RMS. As expected, the performance improves as the number of satellites increases. It can also be observed from the table that the first cost function is more suited for the case of two or three satellites, while the second cost function is best used for the case of four satellites. The poor performance associated with the second cost function in the case of two satellites may be due to the limited number of samples. It would be interesting to do more data processing to generate more samples to see if the performance can be improved. In Table 5.7 the average of the mean SSH estimation errors is also listed. In the case of multiple combinations, the performance is improved by averaging. In particular, the mean SSH error is only 0.18 m for the first cost function with two-satellite combinations. Due to the error randomness, averaging would typically cancel the errors with each other to some degree so that an estimation accuracy gain can be achieved as indicated by the results in the table. As for the first cost function, the error with two-satellite combinations is even smaller than that with three- or four-satellite combination. This may be a coincidence since the large errors nearly completely cancel each other.

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5 Sea Surface Altimetry

5.4 Spaceborne Altimetry Spaceborne atlmetry has recently been an active research topic and a lot of results especially related to TDS-1 and CYGNSS have been reported in the literature. In this section, a brief review on a number of selected articles is provided. Clarizia et al. [3] used spaceborne Global Positioing System reflectometry (GPSR) data from TDS-1 satellite to retrieve the ocean sea surface heighet (SSH), which generally agree with the global DTU10 mean surface height after an overall bias is removed. These results represent the first observations of SSH by a spaceborne GPS-R instrument. The South Atlantic and North Pacific regions were selected as the test regions for the altimetric analysis. The leading edge derivative (LED) approach described in [7] was used to estimate the SSH. That is, the first-order derivative is computed for each delay waveform and the delay value at the derivative peak is used to estimate the delay difference between direct and reflected signal. Li et al. [10] performed phase altimetry over sea ice by using data from TDS-1 satellite. It made use of high-precision carrier phase measurements extracted from coherent GNSS reflections at a high elevation angle greater than 57°. The altimetric results show good consistency with a mean sea surface (MSS) model, and the rootmean-square difference is 4.7 cm. The along-track sampling distance is about 140 m and the spatial resolution is about 400 m. The master-slave sampling technique [18] is used for processing the raw samples; i.e. closed-loop tracking of the direct signal and an open-loop tracking of the reflected signal. As a consequence, the power and phase of the reflected signal are extracted. Mashburn et al. [14] investigated the performace of the TDS-1 data for ocean surface altimetry. Compared with the mean sea surface topography, the surface height residuals are found to be 6.4 m. The analysis is performed by considering a range of factors, including the transmitter and receiver orbits, time tag corrections, models for ionospheric and tropospheric delays, zenith to nadir antenna baseline offsets, ocean and solid Earth tides. Thus, an accurate reflection model was established to simulate the true reflection geometry and signal propagation errors as accurately and precisely as possible. Then DDM retracking is performed to precisely determine the desired points such as the peak power point and the 70% power point. Li et al. [11] assessed the ocean altimetry performance of spaceborne GNSS-R by processing the raw data sets collected by CYGNSS constellation. The reflected waveforms of GPS L1, Galileo E1, and BeiDou-3 B1 signals are generated by processing the intermediate frequency signals with a typical software receiver on the ground. The relative delays of the reflected signals are derived from the waveforms generated by retracking algorithms, and the retracking biases are removed with the specular point (SP) delay and power information computed from the corresponding waveform model. The bistatic delay observations are corrected according to the standard procedures and then are used to obtain sea surface height measurements and compared with the mean SSH model. The ranging precision can reach up to 3.9 and 2.5 m with 1-s GPS and Galileo group delay measurement.

5.4 Spaceborne Altimetry

121

Hu et al. [8] evaluated the magnitude of the altimetric error caused by receiver dynamics. The ground-based receiver does not have such an issue and in the case of an airborne platform, the impact of receiver dynamics on the geometric model is negligible because the receiver movement is small. However, the impact of spaceborne receiver dynamics on geometric model is significant. Theoretical studies show that the magnitude of the altimetric error caused by receiver dynamics can be several meters, depending on the trajectory angle of the receiver. The results from processing TDS-1 data also demonstrate that an extra average error of 3.4 m in altimetry is produced if receiver motion is ignored. Xu et al. [19] retrieved global lake levels by using data collected byTDS-1 satellite for the first time. TDS-1 Level 1b data recorded between 2015 and 2017 were used to estimate lake levels of 351 global lakes with area greater than 500 km2 and elevation lower than 3000 m. Although the RMSE in the lake level estimation is large (tens of meters), the spatial and temporal variations in the TDS-1 data based estimation results are consistent with those of other data sources. In the absence of interference from land reflections, lake surface altimetry should produce better results than ocean surface altimetry, since the latter would usually be rougher than the former. Qiu et al. [16] estimated the global mean sea surface height (MSSH) by establishing the relationship between the characteristics of the delay waveform obtained from the CYGNSS DDM data. The relative delay is calculated by using the methods described in [2]. The estimation results were compared with measurements from satellite altimetry; the mean absolute error (MAE) was 1.33 m and the RMSE was 2.26 m. Compared with the sea surface height model DTU10, the MAE was 1.20 m and the RMSE was 2.15 m. Sea surface height estimates obtained from CYGNSS are also consistent with the results by altimetry satellite Jason-2; and the MAE is 2.63 m and RMSE is 3.56 m. By coping with the main challenges in GNSS-R altimetry, including precise delay retracking, correction of ionospheric effects, and spacecraft (LEO satellite) receiver positioning, Mashburn et al. [13] presented a reflection-model-based approach for delay retracking that uses simulated DDMs to estimate the specular delay from measured DDMs. Global ionosphere maps are used to estimate the group delay effect along the direct and reflection paths. The estimate of the LEO satellite position is improved with precise orbit determination techniques and reduction of systematic intersatellite biases. More than 50000 single-point observations recorded over the Indonesia seas and week-long averaged results are used to quantify the noise and systematic characteristics of the retrieved surface heights. The root sum square residual error of the CYGNSS altimetry observations is estimated to be 6-m delay at the average observed CYGNSS SNR using a specific tracking on 1-s observations. Hu et al. [9] studied the validation of the LED in the spaceborne GNSS-R altimetry. Two problems of the weight function were studied, which are related to the assumptions in the LED method that is widely studied in GNSS-R altimetry. The authors theoretically analyzed the monotonicity condition constrained to the weight function and concluded that the weight function needs to be monotonically nonincreasing.

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Also, whether the weight function could meet the monotonicity condition was verified through simulation using the orbits of the COSMIC, GRACE, and GPS satellites, and the antenna gain pattern of the TDS-1.

5.5 Summary Although many results in the literature are not reported in this chapter, it is a fact that lake and sea surface altimetry with ground-based and airborne receiver can achieve an accuracy of sub-decimeter. However, spaceborne GNSS-R altimetry may not be satisfactory for many applications due to the significant estimation error as mentioned in the preceding section. Compared with conventional LEO satellite altimetry, GNSSR altimetry is affected by more factors such as low SNR, transmitter and receiver dynamics, severe multipath interference, and increased complexity in calibration. One significant issue is related to the estimation of the relative delay of the reflected signal. There is still space to reduce the error in the estimation of the delay. It is desirable to develop more advanced signal processing techniques to achieve better delay estimation. Multi-frequency and multi-GNSS would also offer an option to enhance sea surface altimetry.

References 1. BOM (2020, visited) http://www.bom.gov.au/climate/data-services/ 2. Cardellach E, Rius A, Martin-Neira M, Fabra F, Nogues-Correig O, Ribo S, Kainulainen J, Camps A, D’Addio S (2014) Consolidating the precision of interferometric GNSS-R ocean altimetry using airborne experimental data. IEEE Trans Geosci Remote Sens 52(8):4992–5004 3. Clarizia MP, Christopher R, Cipollini P, Zuffada C (2016) First spaceborne observation of sea surface height using GPS-Reflectometry. Geophys Res Lett 43(2):767–774 4. Dean RG, Dalrymple RA (1991) Water wave mechanics for engineers and scientists. World Scientific, Singapore 5. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press, London 6. Gleason S,Gebre-Egziabher D (2009) GNSS Applications and Methods, Artech House 7. Hajj GA, Zuffada C (2003) Theoretical description of a bistatic system for ocean altimetry using the GPS signal. Radio Sci 38(5):1–19 8. Hu C, Benson CR, Rizos C, Qiao L (2019) Impact of receiver dynamics on space-based GNSS-R altimetry. IEEE J Sel Top Appl Earth Obs Remote Sens 12(6):1974–1980 9. Hu C, Benson CR, Qiao L, Rizos C (2020) The validation of the weight function in the leadingedge-derivative path delay estimator for space-based GNSS-R altimetry. IEEE Trans Geosci Remote Sens 58(9):6243–6254 10. Li W, Cardellach E, Fabra F, Rius A, Ribo S, Martin-Neira M (2017) First spaceborne phase altimetry over sea ice using TechDemoSat-1 GNSS-R signals. Geophys Res Lett 44(16):8369– 8376 11. Li W, Cardellach E, Fabra F, Ribo S, Rius A (2019) Assessment of spaceborne GNSS-R ocean altimetry performance using CYGNSS mission raw data. IEEE Trans Geosci Remote Sens 58(1):238–250

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12. Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves. J Mar Res 11(3):245–266 13. Mashburn J, Axelrad P, Zuffada C, Loria E, O’Brien A, Haines B (2020) Improved GNSS-R ocean surface altimetry with CYGNSS in the Seas of Indonesia. IEEE Trans Geoence Remote Sens 58(9):6071–6087 14. Mashburn J, Penina A, Lowe ST, Larson KM (2018) Global ocean altimetry with GNSS reflections from TechDemoSat-1. IEEE Trans Geosci Remote Sens 56(7):4088–4097 15. MHL (2020, visited) https://mhl.nsw.gov.au/Data-Wave 16. Qiu H, Jin S (2020) Global mean sea surface height estimated from spaceborne Cyclone-GNSS reflectometry. Remote Sens 12(3):356 17. Rius A, Cardellach E, Martın-Neira M (2010) Altimetric analysis of the sea-surface GPSreflected signals. IEEE Trans Geosci Remote Sens 48(4):2119–2127 18. Semmling A, Beckheinrich J, Wickert J, Beyerle G, Schön S, Fabra F, Pflug H, He K, Schwabe J, Scheinert M (2014) Sea surface topography retrieved from GNSS reflectometry phase data of the GEOHALO flight mission. Geophys Res Lett 41:954–960 19. Xu L, Wan W, Chen X, Zhu S, Liu B, Hong Y (2019) Spaceborne GNSS-R observation of global lake level: First results from the TechDemoSat-1 mission. Remote Sens 11(12):1438 20. Yu K, Rizos C, Dempster A (2014) GNSS-based model-free sea surface height estimation in unknown sea state scenarios. IEEE J Sel Top Appl Earth Obs Remote Sens 7(5):1424–1435

Chapter 6

Sea Surface Wind Speed Estimation

One of the most useful applications of GNSS-R is the measurement of speed of sea surface wind, especially the tropical cyclone, Typhoon, or Hurricane. Precise measurement and prediction of these disastrous winds is very important to timely warning and thus to avoiding or reducing the loss of economy and life. Different technologies including satellite remote sensing have been used to measure wind speed. GNSS-R is an alternative technology which may play an important role in the monitoring of the intensity of Hurricane or Typhoon such as made by sensors carried by TDS-1, CYGNSS and Bufeng-1 satellites. This chapter presents some basic concepts and principles of GNSS-R based estimation of sea surface wind speed with a focus on the use of airborne data and CYGNSS data.

6.1 Modeling of Sea Wave Spectrum and Received Signal Power Nearly all the existing methods for wind speed retrieval make use of the models of sea wave spectrum and sea surface scattering directly or indirectly. In this section, one model of sea wave spectrum is studied. Also studied are sea surface scattering and the reception of the reflected signal entering the nadir-looking antenna.

6.1.1 Elfouhaily Model Sea surface undulation is a complex process and sea wave heights change randomly in time and space. Sea surface roughness can be described by a number of parameters including significant wave height (SWH) and significant wave period (SWP), mean square slope (MSS) of surface, and root mean square (RMS) of surface elevation. SWH is defined as the average height of the one-third highest waves and SWP is © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_6

125

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6 Sea Surface Wind Speed Estimation

defined as average period of the waves used to calculate the SWH. Alternatively, wave height spectrum and wave direction spectrum can be used to describe the surface roughness. Among the wave height spectral models, the Pierson-Moskowitz model, the JONSWAP model, the Elfouhaily model, and the Phillips models are widely studied [5, 11, 12]. In the Elfouhaily model, the directional spectrum of wind-driven wave height is defined as Ψ (κ, ϕ) =

1 S(κ)Φ(κ, ϕ) κ

(6.1)

where κ = 2π/λ is the wavenumber with λ being the wavelength, ϕ is the angle between the wave direction and the wind direction, S(κ) is the omnidirectional spectrum and Φ(κ, ϕ) is the spreading function. The omnidirectional spectrum is the sum of long wave curvature spectrum B (κ) and the short wave curvature spectrum B (κ): Ψ (κ) = κ −3 (B (κ) + Bh (κ))

(6.2)

B (κ) is defined as B (κ) =

1 cp α p Fp 2 c

(6.3)

here α p is the generalized Phillips-Kitaigorodskii equilibrium range parameter √ α p = 6 × 10−3 Ω

(6.4)

where Ω is the inverse wave age which is equal to 0.84 for a well-developed sea (driven by wind). c is the wave phase speed and c p is the phase speed of the dominant long wave. Note that the parameter c has different meanings in different chapters. The ratio of the two phase speeds can be approximated as cp = c



κ κp

(6.5)

here κ p is the wavenumber of the dominant long wave defined as κp =

gΩc2 2 U10

(6.6)

where U10 is the wind speed at a height of 10 m above the sea. Note that the wind speed at a height of 19.5 m above the sea is related to U10 by U19.5 = 1.026U10 ,

6.1 Modeling of Sea Wave Spectrum and Received Signal Power

127

showing little difference between wind speeds within the vicinity of these heights, and ¯ Ωc = Ω cos(θ)

(6.7)

where θ¯ is the angle between the wind and the dominant waves at the spectral peak, which may be treated as a value close to zero. F p in (6.3) is the long-wave side effect function, given by    κ Ω −1 F p = L P M J p exp − √ κp 10

(6.8)

The functions L P M and J p in (6.7) are defined as   5  κ p 2 L P M = exp − 4 κ Jp = γ Γ

(6.9)

with 

1.7, 0.83 < Ωc < 1 1.7 + 6 log(Ωc ), 1 < Ωc < 5   2 κ 1 −1 Γ = exp − 2 , δ = 0.08(1 + 4Ωc−3 ) 2δ κp γ =

(6.10)

As a result, B (κ) can be modelled as

B (κ) = 3 × 10

−3

    κΩ κ Ω 5  κ p 2 γΓ exp − √ −1 − κp κp 4 κ 10

(6.11)

This is in consistence with the results presented in [19], The short wave curvature spectrum Bh (κ) in the Elfouhaily model is defined as Bh (κ) =

1 cm αm Fm 2 c

(6.12)

here αm is the generalized Phillips-Kitaigorodskii equilibrium range parameter for short waves, defined as,  αm =

10−2 (1 + ln(u ∗ /cm )), u ∗ < cm 10−2 (1 + 3 ln(u ∗ /cm )), u ∗ ≥ cm

(6.13)

128

6 Sea Surface Wind Speed Estimation

where u ∗ is the friction velocity at the water surface, which can be iteratively computed by   −1   10   u = 0.4U10  ln    b(u ∗ ) ∗

b(u ∗ ) = 0.11 × 14 × 10−6

1 0.48(u ∗ )3 Ω + ∗ u gU10

(6.14)

The initial value may be chosen as 10−3 (0.81 + 0.065U10 )U10 . cm is the minimum phase speed at the wavenumber (κm ) related to gravity-capillary peak, cm =



2g/κm = 0.23 m/s

(6.15)

where g is the gravitational acceleration. From (6.15), κm is calculated by κm =

2g = 37.8072 × g 0.232

(6.16)

Similar to (6.5), the ratio of the two phase speeds can be approximated by cm = c



κ κm

(6.17)

Fm is the short-wave side effect function: 

 2 1 κ Fm = exp − −1 4 κm

(6.18)

The spreading function Φ(κ, ϕ) in (6.1) is defined as Φ(κ, ϕ) =

1 (1 + (κ) cos(2ϕ)) 2π

(6.19)

where the upwind-crosswind ratio (κ) is determined by (κ) = tanh{a0 + a p (c/c p )2.5 + am (cm /c)2.5 }

(6.20)

Here tanh(x) is the hyperbolic tangent function and the parameters are: a0 =

ln(2) u∗ , a p = 4, am = 0.13 4 cm

(6.21)

The mean square slopes of the surface in the upwind direction and in the crosswind direction are then respectively calculated by

6.1 Modeling of Sea Wave Spectrum and Received Signal Power

129

κ∗ π mssx =

κ 2 cos2 φ Ψ (κ, ϕ)κ dϕdκ, 0 −π

κ∗ π mss y =

κ 2 sin2 φ Ψ (κ, ϕ)κ dϕdκ

(6.22)

0 −π

where the wavenumber cutoff κ∗ can be calculated according to κ∗ =

2π 3λ

(6.23)

where λ is the wavelength (0.1904 m for the GPS L1 signal). In Garrison et al. [8] the wave number cutoff is modified as κ∗ =

2π sin(θ ) 3λ

(6.24)

where θ is the incidence angle (complementary to the elevation angle of the satellite). Combining the two components in (6.9) produces the total omnidirectional mean square slope as mss = mssx + mss y

κ∗ π = κ 2 Ψ (κ, ϕ)κ dϕdκ

(6.25)

0 −π

6.1.2 Theoretical Waveform of Received Signal Power The signal power received via a down-looking antenna is a function of Doppler frequency and time delay, as described in Chap. 4. The model of the received reflected signal power is given by (4.28). Figure 6.1 shows the simulated decibel delay waveforms of the reflected signal using the theoretical models studied in Chap. 4. For clarity, only two-dimensional delay waveforms (also called delay maps), instead of the three-dimensional delay Doppler waveforms (also called delay Doppler maps), are presented. In fact, the 2D delay waveform is usually used to estimate wind speed and one specific method will be studied in the following section. The six curves correspond to the six different wind speeds (4, 7, 10, 13, 16, and 19 m/s). Four different flight heights were tested: 0.5, 2, 5, and 10 km. As seen from Fig. 6.1, the sea surface roughness caused by wind mainly affects the trailing edge of the waveform. In the case of 0.5 km altitude, the six curves are nearly identical and the difference in the trailing edges is very small. Since the

130

6 Sea Surface Wind Speed Estimation Flight Height = 2 km

Normalized Power (dB)

Normalized Power (dB)

Flight Height = 0.5 km 0

-10

-20 4 m/s 19 m/s -30

0

2

4

6

8

0

-10

-20

0

Flight Height = 5 km

Normalized Power (dB)

19 m/s

4 m/s -30

2

4

6

8

Flight Height = 10 km

0

0

-10

-10

19 m/s

4 m/s

19 m/s

-20

-30

4 m/s

-20

0

2 4 6 A/C Code Phase (chip)

8

-30

0

2 4 6 A/C Code Phase (chip)

8

Fig. 6.1 Normalised correlation powers in decibel of the simulated reflected signals using the Elfouhaily wave elevation model with four different flight heights and five different wind speeds (4, 7, 10, 13, 16, and 19 m/s)

waveforms are insensitive to the wind variation when the flight altitude is less than 0.5 km, it would be inappropriate to use the trailing edge to perform any sea state or wind retrieval. However, as the flight altitude increases, the antenna can capture signals reflected over scattering points which have long distance away from the specular scattering point. As a consequence, the trailing edge becomes wider and the waveforms distinguish from each other better. For a given flight height, the slope of the trailing edge varies with surface wind speed. This is the key feature of the waveform used for inferring the wind speed in the case of airborne data.

6.2 Near Sea Surface Wind Speed Retrieval The basic wind speed estimation method is based on matching the theoretical waveform with the measured one. The matching between the theoretical and measured waveforms can be performed in different ways and the conventional matching method is based on the least-squares fitting. For instance, in the case of delay waveform matching, as illustrated in Fig. 6.2, both the measured and theoretical waveforms are normalised to have the same peak value, and they are truncated to keep their minimum value above a threshold which can be selected based on the shape of the measured waveform. The theoretical waveform is aligned with the measured waveform so that they have the best match in the sense of least-squares fitting. That is,

6.2 Near Sea Surface Wind Speed Retrieval

131

Fig. 6.2 Illustration of measured waveform (solid line) and theoretical waveforms (dotted line and dashed line). The two theoretical waveforms are the same but shifted with respect to each other

the error function Er =

L 

( f i(th) − f i(m) )2

(6.26)

i=1

is minimised through adjusting the relative positions of the two waveforms. Here, f i(th) and f i(m) are the theoretical and measured waveform values, respectively. There are L time points between the first and last samples of the two waveforms. In the interval between the two first samples and that between the last two samples of the two truncated waveforms, one of the two waveforms does not have specified values due to truncation. In these cases, they are assigned the minimum value, i.e. the threshold value, of the power level. For instance, the dashed, instead of the dotted, theoretical waveform has a best match with the measured one. Thus, for each theoretical waveform associated with a wind speed, there is an error function value which is the minimum, referred to as the residual for convenience. Among all the theoretical waveforms corresponding to a range of wind speeds, the theoretical waveform with the smallest residual is selected and the corresponding theoretical wind speed is the estimate of wind speed at the experiment site. Another matching method is simply based on the slope of the trailing edge of the delay waveform, as suggested in [19] and also implied by Fig. 6.1. However, the simple slope matching method may not be suited for scenarios where the slope does not change significantly with wind speed. For instance, as shown in Fig. 6.3, the five theoretical delay waveforms corresponding to five different wind speeds (3, 4, 5, 6, and 7 m/s) are displayed. The measured delay waveform associated with a specific satellite (GPS PRN#13) is also shown. The measured waveform is produced through coherent integration of 1-ms IF signals and then non-coherent integration of 1000 such 1-ms waveforms. In this case, the slopes of the trailing edges of the five theoretical waveforms are very similar, so it is difficult to distinguish them from each other simply based on the slope.

132

6 Sea Surface Wind Speed Estimation 0 Normalised Correlation Power (dB)

Fig. 6.3 Wind speed estimation through matching the waveforms

2.5 m/s 3 m/s 3.5 m/s 4 m/s 4.5 m/s measured

-1 -2 -3 -4 -5 -6 -7 -8

0

0.2

0.4

0.6 0.8 1 1.2 Time (C/A code chip)

1.4

1.6

In cases similar to this one, the area size covered by the delay waveform can be considered [18]. Specifically, given a number of possible wind speeds based on a priori information, a corresponding number of theoretical waveforms are produced. The local weather forecast may be used in the selection of the range of wind speeds. The interval between two neighbouring wind speeds must be considerably smaller than the desired wind speed estimation accuracy. There are four main steps of implementing such an area size matching method: (1) (2) (3)

(4)

interpolating both the measured and theoretical waveforms without changing the original data selecting the cut-off correlation power to retain the waveform above certain power lever so that the slope of the trailing edge does not change abruptly calculating the areas of the interpolated waveforms above the cut-off power and calculating the area difference between the measured waveform and each of the theoretical waveforms selecting the theoretical waveform that produces the minimum area difference and taking the corresponding wind speed as the estimate. Note that when using Matlab the library function ‘INTERP’ can be directly used to perform the interpolation.

Regarding Fig. 6.3, it can be seen that the measured waveform has a good match with the theoretical waveform of wind speed 3.5 m/s, and the theoretical waveform area size of this wind speed has the best match with the measured waveform area size, producing the wind speed estimate of 3.5 m/s.

6.3 An Airborne Experiment and Results

133

6.3 An Airborne Experiment and Results 6.3.1 Experiment Campaign Similar to the airborne experiment that is described in Chap. 5, this airborne experiment was carried out by the same UNSW-owned light aircraft flying off the coast of Sydney near Palm Beach on 4 November 2011. The data were also logged via the NordNav software receiver. Figure 6.4 shows the experiment site and the flight trajectory of the aircraft. There are two nearby coastal weather observation stations where wind speed is measured and recorded, serving as the wind speed reference. Unlike the other experiment with two payloads, this experiment only had one payload which is the GNSS signal reception and data logging system. In total, about 46 Gigabytes of binary raw IF data of the direct and reflected signals were logged over about one and a half hours. The maximum flight height was 3.2 km and the aircraft flew at this height for about 35 min as shown in Fig. 6.5. Such a limited maximum flight height was due to the fact that the aircraft was operated manually and the pilot needed to observe the ground clearly. The aircraft speed, i.e. the receiver platform moving speed, over the duration of the experiment is shown in Fig. 6.6. The speed is basically between 60 and 90 m/s when the aircraft flew above the sea. The sea surface can be treated as well-developed since the wind had blown the surface continuously for a few hours and the variation of the wind speed was not very large. As a result, the corresponding theory can be employed to perform the wind parameter estimation.

Fig. 6.4 Experiment location and aircraft flight trajectory. The picture was generated using GPS Visualizer and Google Earth

134

6 Sea Surface Wind Speed Estimation

Fig. 6.5 Aircraft flight height over the duration of the experiment

3.5

Flight Height (km)

3 2.5 2 1.5 1 0.5 0

0

10

20

30

40

50

60

70

80

90

100

70

80

90

100

Time (min)

Fig. 6.6 Aircraft and receiver velocity during the experiment

120

Aircraft Velocity (m/s)

100

80

60

40

20

0

0

10

20

30

40

50

60

Time (min)

6.3.2 Data Processing In this subsection a problem with the software receiver is first investigated and some data bits of the original data are corrected. Then samples of the two-dimensional delay waveforms and three-dimensional delay-Doppler waveforms are presented.

6.3.2.1

A Problem with the Software Receiver

It was observed that there was a problem with the four front-ends of the NordNav GPS software receiver that has been used for the data logging during the experiment. The problem was associated with the encoding of the GPS IF signals. The 2-bit

6.3 An Airborne Experiment and Results

135 Time domain plot

Amplitude

5

0

-5 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Time (ms)

0

x 10

4

Histogram

Number in bin

6 4 2 0

-8

-6

-4

-2

0 Bin

2

4

6

8

Fig. 6.7 An example of IF data bits statistics when using 4-bit quantisation

quantisation scheme was used so that the output data bits are within {3, 1, −1, −3}. Also, the distribution of the data bits should basically follow some specific pattern such as the one shown in Fig. 6.7 where the 4-bit 16-level quantisation is used. The envelope of the histogram is approximately symmetric with the bins in the middle having the maximal numbers. However, the original data bits collected via this software receiver had an unexpected bit distribution pattern as shown in Fig. 6.8. Clearly, the encoding of the two data bits {−1} and {−3} must have been swapped in the receiver. Such a 2-bit quantisation encoding scheme is actually equivalent to a 1.5-bit quantisation scheme. As a result, certain performance degradation would be incurred although the degradation may be minor in some cases. Figure 6.9 shows the corresponding results of Fig. 6.8 after the {−1} bits are swapped with the {−3} bits. To evaluate the impact of the wrong encoding process in the receiver, the delay waveforms of the reflected signal associated with one specific GPS satellite (PRN#8) were used as shown in Fig. 6.10. When the correlation powers were normalised so that the maximal correlation power is zero dB as seen in Fig. 6.11, it can be clearly observed that the performance improved after swapping the {−1} bits with the {− 3} bits. The noise floor in this case was reduced by about 0.8 dB, approximately 7% of the maximal signal power. From such a decrease in the noise floor, it may be predicted that the decrease in the noise floor or the increase in the signal-to-noise ratio would be more significant when a higher-level quantisation, such as the 3-bit or 4-bit quantisation is used. As a consequence, a performance gain would be achieved.

136

6 Sea Surface Wind Speed Estimation Time domain plot

Amplitude

2

0

-2 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Time (ms)

0

4

Histogram

Number in bin

x 10

4

2

0 -4

-3

-2

-1

0 Bin

1

2

3

4

Fig. 6.8 IF data bits statistics of the NordNav receiver used for the experiment when using 2-bit quantisation Time domain plot

Amplitude

2

0

-2 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Time (ms)

0

4

Histogram

Number in bin

x 10

4

2

0 -4

-3

-2

-1

0 Bin

1

2

3

4

Fig. 6.9 IF data bits statistics of the NordNav receiver after swapping the {−1} with {−3} bits

6.3 An Airborne Experiment and Results Fig. 6.10 Example of impact of swapping the {− 1} and {−3} bits of the IF signals

137

12 original after swapping

Correlation Power (dB)

10 8 6 4 2 0 -2 -4 446

447

448

449

450

451

452

453

454

CA Code Phase (Chip)

0 original after swapping

-2

Correlation Power (dB)

Fig. 6.11 Example of impact of swapping the {− 1} and {−3} bits of the IF signals when the correlation powers are normalised

-4 -6 -8 -10 -12 -14 446

447

448

449

450

451

452

453

454

CA Code Phase (Chip)

Further investigation would be needed if one wants to determine the accurate relationship between the quantisation level and the performance. Also, it can be seen from Fig. 6.11 that the two delay waveforms are nearly identical when the signal power is above −8 dB. That is, in the case where only the waveform with a power greater than −8 dB is used, the swapping of the {−1} bits and {−3} bits will actually not produce any impact on the sea state and wind parameter estimation provided that only the waveform shape is employed. On the other hand, when the whole waveform shape or correlation power is exploited for the parameter estimation, the effect might not be negligible. In the remainder of this section, the results were produced after the original collected data were corrected by swapping the {−1} bits with the {−3} bits.

138

6.3.2.2

6 Sea Surface Wind Speed Estimation

Delay Waveform and Delay-Doppler Waveform

Via the zenith-looking antenna, the receiver recorded the signals transmitted from 12 GPS satellites when the aircraft was above the sea. Figures 6.12 and 6.13 show the delay-Doppler waveforms and the delay waveforms of the direct signals associated with four satellites (PRN#10, PRN#19, PRN#28 and PRN#3), respectively. As expected the delay-Doppler waveforms are symmetric with respect to both the Doppler frequency and the code phase. Also, in the delay waveforms the leading edge and the trailing edge are symmetric with respect to the code phase at the peak of the correlation power when the power is above the noise floor. The coherent integration time is set at 1 ms and the non-coherent integration time is equal to 0.5 s. That is, the results are produced by averaging 500 waveforms of 1 ms. From the delay waveforms it can be seen that the peak powers of the reflected signals can be significantly different from each other. Many factors will affect the reflected signals power, including the configuration of the specific satellite and the receiver and the transmitted signal power, in addition to the surface roughness and the receiver antenna gain. Furthermore, it is observed that over the period of 0.5 s the code phase of the signals can vary significantly up to by 16 samples equivalent to one code chip. Thus, when performing the non-coherent integration, the phase variation must be taken into account. Variation in the Doppler frequency is also observed but it is negligible over a period of a few seconds. The nadir-looking antenna captured the reflected signals associated with ten satellites. Figures 6.14 and 6.15 show the delay-Doppler waveforms of the reflected PRN#19 15 916 10

915 914

5

913 1 Doppler (kHz)

C/A Code Phase (chip)

C/A Code Phase (chip)

PRN#10 917

424

422

15 10

420

5

2

2 3 Doppler (kHz)

8

579

6

578

4 2

577 5 6 Doppler (kHz)

C/A Code Phase (chip)

C/A Code Phase (chip)

PRN#3 10

580

20

421

PRN#28 581

25

423

320

25

319

20

318

15

317

10

316

5

315

-0.5 0 0.5 1 Doppler (kHz)

Fig. 6.12 Delay-Doppler waveforms of direct signals associated with four GPS satellites

6.3 An Airborne Experiment and Results

139 PRN#19 Correlation Power (dB)

Correlation Power (dB)

PRN#10 15 10 5 0 -5

912

15 10 5 0 -5

914 916 918 920 C/A Code Phase (chip)

420 422 424 426 428 C/A Code Phase (chip) PRN#3

Correlation Power (dB)

Correlation Power (dB)

PRN#28 15 10 5 0 -5 576

15 10

578 580 582 584 C/A Code Phase (chip)

5 0 -5 314

316 318 320 322 C/A Code Phase (chip)

Fig. 6.13 Delay waveforms of direct signals associated with four GPS satellites PRN#7

699

8

698

6

697

4

696 2 695

C/A Code Phase (chip)

C/A Code Phase (chip)

PRN#13 259

256

2 3 Doppler (kHz) PRN#23

4

564 2

562 3 4 Doppler (kHz)

C/A Code Phase (chip)

C/A Code Phase (chip)

6

563

2

255

PRN#8

565

4

257

0 0.5 1 1.5 Doppler (kHz)

566

6

258

599 6

598 597

4

596 2

595 -1.5 -1 -0.5 0 Doppler (kHz)

Fig. 6.14 Delay-Doppler waveforms of reflected signals associated with four GPS satellites

140

6 Sea Surface Wind Speed Estimation PRN#19 5

930

4

929

3

928

2

927 926

C/A Code Phase (chip)

C/A Code Phase (chip)

PRN#10 931

1

5

438

4

437

3

436

2

435

1

434 3 2 Doppler (kHz)

2 1 Doppler (kHz)

PRN#3

592

5

591

4

590

3

589

2

C/A Code Phase (chip)

C/A Code Phase (chip)

PRN#28

1

588 5 6 Doppler (kHz)

331

2

330 329

1.5

328

1

327 0.5 326

-0.5 0 0.5 1 Doppler (kHz)

Fig. 6.15 Delay-Doppler waveforms of reflected signals associated with another four GPS satellites

signals associated with eight GPS satellites (PRN#13, PRN#7, PRN#8, PRN#23, PRN#10, PRN#19, PRN#28, and PRN#3). Reflected signals associated with the other two satellites (PRN#5 and PRN#6) were also captured via the nadir-looking antenna. However, the related results are not presented since they are too noisy and distorted due to the rather small elevation angles and the limited antenna beamwidth. It may be difficult to obtain useful information from such noisy and distorted waveforms. The elevation angles and azimuth angles of the eight satellites are shown in Table 6.1, which were used in the generation of the theoretical waveforms. Through comparing the results in Figs. 6.14 and 6.15 with the results in Fig. 6.12, it can be seen that the delay-Doppler waveforms of the reflected signals are not symmetric with respect to a specific code phase that corresponds to the maximal power. Instead, on the top of the delay-Doppler waveforms the signals are spread over two or more code chips, resulting from the roughness of the sea surface. The spreading of the signals over time can be more clearly observed from the delay waveforms as shown in Figs. 6.16 and 6.17. Although the peak power is different with different satellite, the shapes of the trailing edge of all the eight waveforms Table 6.1 Elevation and azimuth angles of the eight GPS satellites Satellite #

13

7

8

23

10

19

28

3

Elevang (°)

65.42

60.51

44.89

35.94

35.35

34.78

26.30

23.55

Azimuth (°)

54.62

186.46

233.02

43.03

284.17

90.35

313.09

126.46

6.3 An Airborne Experiment and Results

141 PRN#7 Correlation Power (dB)

Correlation Power (dB)

PRN#13 10 5 0 -5 694 696 698 700 C/A Code Phase (chip)

10 5 0 -5 254 256 258 260 C/A Code Phase (chip)

702

PRN#23 Correlation Power (dB)

Correlation Power (dB)

PRN#8 10 5 0 -5 560

262

10 5 0 -5 592

562 564 566 568 C/A Code Phase (chip)

594 596 598 600 C/A Code Phase (chip)

Fig. 6.16 Delay waveforms of reflected signals associated with four GPS satellites PRN#19 Correlation Power (dB)

Correlation Power (dB)

PRN#10 10 5 0 -5

10 5 0 -5

926 928 930 932 C/A Code Phase (chip)

432

PRN#3 Correlation Power (dB)

Correlation Power (dB)

PRN#28 10 5 0 -5 590 592 588 594 C/A Code Phase (chip)

434 440 436 438 C/A Code Phase (chip)

10 5 0 -5 324

326 332 328 330 C/A Code Phase (chip)

Fig. 6.17 Delay waveforms of reflected signals associated with another four GPS satellites

142

6 Sea Surface Wind Speed Estimation

Fig. 6.18 Delay waveforms of direct and reflected signals of satellite PRN#19

PRN#19

0

reflected signal direct signal

-2

Correlation Power (dB)

-4 -6 -8 -10 -12 -14 -16 -18 -20

0

1

2

3

4

5

6

7

Time (C/A code chip)

are very similar. That is, in the case of elevation angles of around 20°, the observed waveforms can still be used to estimate wind speed. Figure 6.18 shows the delay waveforms of the direct and reflected signals associated with satellite PRN#19. It can be seen that both the leading edge and the trailing edge of the reflected signal are affected by the sea surface roughness, but the impact on the trailing edge is much more significant. It is this factor that allows the sea surface wind information to be retrieved as described in the preceding subsection. Note that the reflected signal was amplified by a low-noise amplifier with a fixed gain of 20 dB. Without this amplification, the power of the reflected signal will be much smaller.

6.3.3 Wind Speed Estimation Results Using the data associated with a specific satellite, a sequence of wind speed estimates over time can be produced. The data collected when the aircraft flew from Point A to Point B in Fig. 6.4 were employed. During this period of 100 s the elevation angles of eight satellites were greater than 20°. A wind speed estimate is produced using the method described in Sect. 6.2 and data collected over duration of 1 s, so that a sequence of 100 wind speed estimates is produced with respect to each satellite. Figures 6.19 and 6.20 show the estimation results associated with the eight satellites. For comparison, the wind speeds measured by the two closest coastal observation stations (Norah Head and North Head) were also plotted. These wind speed data were provided by the Bureau of Meteorology of Australia and the latest wind speed data over the past three days can be found at the website [1]. As mentioned earlier, since the experiment was conducted near two coastal weather observation stations an accurate independent estimate of the wind speed

Wind Speed (m/s)

Fig. 6.20 Sequences of wind speed estimates associated with satellites PRN# 13, 19, 23, and 28

Wind Speed (m/s)

Wind Speed (m/s)

Fig. 6.19 Sequences of wind speed estimates associated with satellites PRN# 3, 7, 8, and 10. Dashed line and dotted line are for the wind speeds at Norah Head Station and North Head Station respectively

Wind Speed (m/s)

6.3 An Airborne Experiment and Results

6

143

PRN#3, Elevation = 23.6 o

6

5

5

4

4

3

6

0

50

100

PRN#8, Elevation = 44.9 o

3

6

5

5

4

4

3

6

0

50 Time (sec)

100

PRN#13, Elevation = 65.4 o

3

6

5

5

4

4

3

6

0

50

100

PRN#23, Elevation = 35.9

3

o

6

5

5

4

4

3

0

50 Time (sec)

100

3

PRN#7, Elevation = 60.5 o

0

50

100

PRN#10, Elevation = 35.4 o

0

50 Time (sec)

100

PRN#19, Elevation = 34.8 o

0

50

100

PRN#28, Elevation = 26.3 o

0

50 Time (sec)

100

can be used as a reference. Suppose that the wind speed varied from 4.2 m/s at Norah Head Station to 4.7 m/s at North Head Station at a constant rate. Then, the true surface wind speed between Points A and B on the flight route can be approximated as 4.33 m/s. Thus the wind speed estimation errors can be determined. First, each sequence of estimation errors was dealt with independently and the corresponding error statistics were determined. Then the error characteristics were determined using all the estimation errors associated with all the eight satellites. Figures 6.21 and 6.22 show the sequences of wind speed estimation errors associated with the eight satellites. The estimation results are consistent over the duration of 100 s. The mean, standard deviation (STD) and root mean square (RMS) of the estimation errors are listed in Table 6.2. In particular, the RMS error ranges between

144

6 Sea Surface Wind Speed Estimation PRN#7, Elevation = 60.5o

1.5 1

-0.5 0.5 -1

Estimation Error(m/s)

50

0

0

100

PRN#8, Elevation = 44.9

0

o

0.5

0

100

50

PRN#10, Elevation = 35.4 o

0 -0.5 -0.5 -1

0.5

Estimation Error(m/s)

Fig. 6.22 Wind speed estimation errors related to satellites PRN# 13, 19, 23, and 28

PRN#3, Elevation = 23.6o

0

Estimation Error(m/s)

Fig. 6.21 Wind speed estimation errors related to satellites PRN# 3, 7, 8, and 10

0

50 Time (sec)

-1

100

PRN#13, Elevation = 65.4 o

0.5

0

0

-0.5

-0.5

Estimation Error(m/s)

-1

1.5

0

50

-1

100

PRN#23, Elevation = 35.9

o

0.5

1

0

0.5

-0.5

0

0

50 Time (sec)

-1

100

0

50 Time (sec)

100

PRN#19, Elevation = 34.8 o

0

50

100

PRN#28, Elevation = 26.3 o

0

50 Time (sec)

100

Table 6.2 Statistics of wind speed estimation errors PRN#

13

7

8

23

44.9

35.9

10

Eleva (°)

65.4

60.5

35.4

Mean (m/s)

−0.22

0.70

−0.69

0.84

−0.26

STD (m/s)

0.19

0.23

0.13

0.25

0.23

RMS (m/s)

0.29

0.73

0.70

0.86

0.35

19

28

3

34.8

26.3

23.6

−0.30

−0.15

0.56

0.21

0.19

0.18

0.36

0.24

0.59

6.3 An Airborne Experiment and Results

145

0.3 and 0.94 m/s, better than 1 m/s. Except for the relatively large error variation associated with satellite PRN#3 which has the smallest elevation angle; it is difficult to set up a relationship between the error statistics and the elevation angles. Further investigations are needed to investigate how the error statistics are related to some other specific parameters. Figure 6.23 shows the average of wind speed estimates associated with eight satellites when sixteen different wind directions are assumed. One observation is that although the mean wind speed estimate varies with the assumed direction, the variation is limited between 4.25 and 4.78 m/s. Figure 6.24 shows the STD of the wind speed estimation versus the assumed wind directions. The STD ranges from 0.52 to 0.9 m/s. The true wind speed is about 110°, so this STD plot provides some useful information about the wind direction. That is, the wind direction with the 4.9

Mean Wind Speed Estimation (m/s)

Fig. 6.23 Mean wind speed estimate versus assumed wind direction

4.8 4.7 4.6 4.5 4.4 4.3 4.2

0

50

150

100

200

250

300

350

300

350

Wind Direction (deg)

0.9

Wind Speed Estimation STD (m/s)

Fig. 6.24 Wind speed estimation STD versus assumed wind direction

0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

0

50

100

150

200

250

Wind Direction (deg)

146

6 Sea Surface Wind Speed Estimation

smallest STDs may be treated as the actual wind direction. However, there exists an ambiguity of 180°. More accurate wind speed estimation methods can be found such as in [20].

6.4 Spaceborne Wind Speed Estimation As mentioned in Chap. 1, there have been five successful satellite missions related to GNSS-R, which are UK-DMC, TDS-1, CYGNSS, WNISAT-1R, and Bufeng-1. The main focus of GNSS-R application of the five missions is on the monitoring of ocean state especially sea surface wind speed. Figure 6.25 shows the flowchart of the CYGNSS data based wind speed estimation method proposed in [2]. More details of the method are described in the remainder of this section. Although the models and algorithms were developed and validated with CYGNSS data, the basic principles would also be applicable to other satellite data such as TDS-1 and Bufeng-1 data.

6.4.1 DDM Data Preprocessing The left panel of Fig. 6.26 shows the two DDM examples in the presence of large noise (a) and small noise (c). Both DDM data were recorded over the ocean on May 12, 2020. It can be seen that in the presence of large noise (low SNR = 0.5207 dB), significant variation and dispersion occurred in different time-delay waveforms, while in

Fig. 6.25 Flow chart of wind speed estimation algorithm development and validation. The first stage is data preprocessing, the second stage is calculation of observables and the third stage is the model development process and the model performance evaluation

6.4 Spaceborne Wind Speed Estimation

147

Fig. 6.26 DDMs with large noise and small noise and corresponding delay waveforms. a DDM with large noise (SNR = 0.5207 dB), b delay waveforms with large noise, c DDM with small noise (SNR = 2.6943 dB), d delay waveforms with small noise

the presence of small noise (high SNR = 2.6943 dB), the variation in the differential waveforms was much smaller. Large noise may be caused by the space environment of signal propagation, such as atmospheric attenuation, space signal interference, and propagation loss. Data quality is also affected by incident angle, antenna gain, and satellite attitude. The raw DDM data are first denoised by removing the background noise. For instance, the noise power is calculated as the average of the values of a certain number of DDM pixels which are absent of signals. The DDM data are then subtracted by the calculated mean noise power. Next, delay waveforms are extracted from each DDM. The central delay waveform (CDW) is the delay waveform without Doppler frequency shift, and the integrated delay waveform (IDW) is generated by integrating CDW and a number of delay waveforms on both sides of CDW. After normalization, normalized CDW (NCDW) and normalized IDW (NIDW) are produced, and the differential delay waveform (DDW) is defined as the difference between NIDW and NCDW [21]. As shown in the left of Fig. 6.27, the pixels of the DDM marked with black solid dots correspond to the CDW, while the pixels with red hollow circles represent four delay waveforms which are integrated with the CDW to generate the IDW as I DW (τ ) =

8  i=4

D D M(τ, f i )

(6.27)

148

6 Sea Surface Wind Speed Estimation

Fig. 6.27 Denoised and normalized DDM (left panel) and the three delay waveforms (right panel)

Then the CDW and IDW are normalized by 1 × C DW (τi ), i = 1, 2, · · · , 17 C DWmax 1 N I DW (τi ) = × I DW (τi ), i = 1, 2, · · · , 17 I DWmax

N C DW (τi ) =

(6.28)

where C DWmax and I DWmax are the maximum values of {C DW (τi )} and {I DW (τi )}, respectively. The DDW is then simply calculated by D DW (τi ) = N I DW (τi ) − N C DW (τi ), i = 1, 2, · · · , 17

(6.29)

The right panel of Fig. 6.27 shows the calculated three delay waveforms from the DDM on the left panel of the figure. More information on the definition of the delay waveforms can also be found in Chap. 7. After calculating the NCDW and NIDW, followed by DDW, the quality of each DDM data may be evaluated by the RMS of DDW:

R M S D DW

  q 1  = (D DW p )2 q p=1

(6.27)

where D DW p denotes the waveform value at the p-th time delay. It is observed that the RMS of DDW is rather small when the noise is weak. The RMS of DDW would

6.4 Spaceborne Wind Speed Estimation 1

0.8

RMS value of DDW

Fig. 6.28 Root mean square (RMS) statistics of differential delay waveform. Blue line represents the RMS of DDW, red line represents RMS = 0.2, green line represents RMS = 0.15, and cyan line represents RMS = 0.25

149

0.6

0.4

0.2

0

1000

2000

3000

4000

Number of DDM

increase as the noise intensity increases, so that a threshold can be used to judge the DDM quality and select the DDM data of good quality accordingly. Figure 6.28 shows the RMS of DDW for 4767 DDMs and three thresholds for the RMS are set at 0.1, 0.2, and 0.25. After excluding the DDMs with RMS greater than the threshold, manual visual examination is applied to check the DDMs which passed the threshold test. It is observed that about 95.51%, 92.96%, and 88.15% of those DDMs can be finally selected under the three threshold values, respectively. Also, the DDMs which did not pass threshold test are examined visually; only a rather small number of DDMs seem to have good quality. Thus, the RMS based discrimination is basically effective. Nevertheless, a better discrimination method is still desirable. Considering both data utilization and data quality, DDM data with RMS of DDW greater than 0.2 were considered to contain large noise and thus excluded from further processing in this case. It is worth mentioning that data alignment is required in the case where incoherent integration is applied to enhance SNR and thus to improve estimation accuracy. Typically, each DDM generated by onboard GNSS-R receiver comes from the incoherent integration of 1000 one-millisecond DDMs, that is one-second observations, which is equivalent to a distance of 7 km (or 25 km) along the ground track for TDS-1 data (or CYGNSS data). Further incoherent integration of one-second DDMs will decrease the resolution, which may not be desired, especially for CYGNSS data.

6.4.2 Empirical Modeling In many cases, it is often impractical to establish a theoretical model due to the complex theoretical relationship between the parameter to be determined and the observables. As a consequence, an empirical model is preferable due to the simplicity

150

6 Sea Surface Wind Speed Estimation

in model development. In this subsection, empirical models are developed to estimate sea surface wind speed based on two different DDM-based observables.

6.4.2.1

Data Sets and Observables

Two data sets, namely CYGNSS L1 DDM data and European Centre for MediumRange Weather Forecasts (ECMWF) reanalysis data are used for model development and verification. As mentioned in several chapters, in 2016, the CYGNSS was deployed, which was funded by NASA and developed by a group of scientists and engineers led by Prof. Chris Ruff at the University of Michigan. It consists of 8 microsatellites, with an orbital inclination of 35°, a height of 510 km, and a spatial resolution of 25 × 25 km [10]. Delay Doppler mapping instrument (DDMI) [16] is equipped onboard each CYGNSS satellite, which is specially designed for GNSS reflectometry. The CYGNSS products mainly include four levels of products, namely, level 1, level 2, level 3, and level 4. The first three levels of products are open to the public, and now available in NASA’s Physical Oceanography Distributed Active Archive Center (PODDAC) [13, 14], with the data being in NetCDF format. Here, level-1 V2.1 product is employed, which can be downloaded from the website [4]. In particular, the observation data from 12 May 2020 to 12 August 2020, with dayof-year ranging from 133 to 225, are used. The DDM data of CYGNSS level 1 is composed of 17 time-delay rows and 11 Doppler columns. The ECMWF data can be downloaded from Copernicus climate change service (C3S) climate database [3]. The database contains hourly-sampled data on the atmosphere, land surface, and ocean waves from 1979 to the present. The spatial resolution of wind speed products is 0.5° × 0.5°, and there are two data formats: GRIB and NetCDF. For long-term large-scale observations of sea surface wind speed, the current wind speed product of ECMWF can be used as the ground-truth data in CYGNSS wind speed retrieval. ECMWF assimilates meteorological data from different sources and obtains an ERA-5 reanalysis data set containing wind speed of 10 m above the sea surface. Therefore, the observables derived from CYGNSS DDM data are matched with wind speed of ECMWF reanalysis data, respectively, to develop empirical models and validate the models as well. The first observable is the normalized bistatic radar cross section (NBRCS) which is defined by σ0 =

  π ||2 q 4 q⊥ P − qz4 qz

(6.28)

which is the same as (4.32) in Chap. 4 and the parameters were already defined there. Each of the DDM pixels corresponds to a NBRCS value, and a calibrated NBRCS value needs to be determined based on the 17 × 11 NBRCS values for CYGNSS data (or 128 × 20 NBRCS values for TDS-1 data). Since NBRCS has better accuracy near specular reflection points, it would be better to use up to such as 0.75 delay

6.4 Spaceborne Wind Speed Estimation

151

chips and 2.5 kHz Doppler shifts to calculate NBRCS. In the case of CYGNSS data, such calibrated NBRCS is provided, but when using TDS-1 data, NBRCS may not be provided and thus needs to be calculated. The second observable is the leading edge slope (LES) of the IDW. Trailing edge slope (TES) has also been used as an observable for wind speed estimation as mentioned in the use of airborne data. Whether using LES or TES would depend on the receiver platform and wind speed as well. It is worth conducting more investigations on this issue and it is of an option to combine LES and TES to achieve a performance gain. The LES can be calculated through least squares fitting: n 

L E Sn =

τi wi − n τ¯ w¯

i=1 n  i=1

(6.29) τi2 − n τ¯ 2

where n (n ≥ 2) denotes the number of selected points used in the slope fitting; τi represents the abscissa value of each point, namely the time delay values; wi is the leading edge waveform value at τi ; τ¯ and w¯ are the means of {τi } and {wi }, respectively. The calculated observables and the ground-truth wind speed data are then used to establish their mathematical relationship. The DDM data are randomly divided into training data set and test data set according to the proportion of 70% and 30%, respectively. Such a proportion is just a choice and the important thing is that a large amount of data should be used for model construction so that the model is suited for many different scenarios.

6.4.2.2

NBRCS Based Modeling

Based on the observation of the distribution of wind speed versus the NBRCS observable as well as the results reported in the literature such as those presented in [6, 9], three different models may be considered as follows: Model-1: f (σ0 ) = Aσ0B , 0 < σ0 ≤ 200

(6.30a)

f (σ0 ) = A exp(Bσ0 ) + C, 0 < σ0 ≤ 200

(6.30b)

Model-2:

Model-3:  f (σ0 ) =

Aσ0B + C, if 0 δT

where the double summation is with respect to the delay and Doppler shift, respectively. The weighted area computed by (7.4) is approximately equal to the volume of the section of DDM with pixel values above the threshold. Similarly, if divided by δτ · δ f D , (7.4) becomes PS =



DDM(τ, f D )

(7.5)

DDM(τ, f D )>δT

Then, the observable is just the summation of DDM powers which are above the threshold. (3)

Center-of-Mass (CM) Distance:

This CM distance observable and the following two observables were proposed by Rodriguez-Alvarez et al. [5]. Denote the coordinates of DDM center of mass as CM = (CMτ , CM f D )

(7.6)

which are defined as 

CMτ = I0−1

τ · DDM(τ, f D ) · δτ · δ f D

DDM(τ, f D )>δT

CM f D =

I0−1



(7.7)

f D · DDM(τ, f D ) · δτ · δ f D

DDM(τ, f D )>δT

where I0 has the same expression of the weighted area given by (7.4). Here, the coordinates and position are referred to the delay coordinate and the Doppler coordinate. The distance observable is defined as the distance from the center of mass to the position of the peak DDM:  d1 = δτ δ f D

MAXτ − CMτ δτ

2

 +

MAX f D − CM f D δ fD

2 (7.8)

where (MAXτ , MAX f D ) are the coordinates of the peak power (i.e. peak DDM, peak pixel).

168

(4)

7 Sea Ice Detection

Geometrical Center

The geometrical center observable is defined as the distance from the position of the maximum DDM, (MAXτ , MAX f D ), to the position of the geometric center, (GCτ , GC f D ):  d2 = δτ δ f D (5)

MAXτ − GCτ δτ

2

 +

MAX f D − GC f D δ fD

2 (7.9)

Taxicab Center of the Mass

The taxicab center observable is defined as the taxicab distance (Manhattan distance) from the position of the center of mass to the position of the maximum DDM:      MAXτ − CMτ   MAX f D − CM f D  −  d1 = δτ δ f D     δ δ τ

(6)

(7.10)

fD

DDMA

The DDMA observable and the following two ones were proposed by Alonso-Arroyo et al. [1]. DDMA is the average value of the normalized DDM around its peak. Three different regions on the DDM were selected: (a) (b) (c)

3 × 3: 3 Doppler bin cells × 3 delay bin cells. 3 × 5: 3 Doppler bin cells × 5 delay bin cells. 3 × 7: 3 Doppler bin cells × 7 delay bin cells The region may be further enlarged to see if better performance can be achieved.

(7)

DIW TES

Due to the fact that the trailing edge slope (TES) of delay waveform decreases with the increased roughness, TES can be used as an observable [1]. To improve SNR, Doppler integrated waveform (DIW) can be used, which is generated by adding 28 delay waveforms of different Doppler shifts within the normalized DDM in the case of TDS-1. The slope observable is computed using a number of the largest values of the trailing edge of the normalized DIW. For instance, three different versions related to three different numbers of bins (normalized DIW samples) may be considered: (a) (b) (c)

Three-Bin: Approximately 750 ns after the peak power. Six-Bin: Approximately 1.5 µs after the peak power. Nine-Bin: Approximately 2.25 µs after the peak power. Through least squares fitting, the TES can be simply computed by

7.2 DDM Observables

169 n 

TES =

τi · DIWi − n · τ · DIW

i=1

n  i=1

(7.11) τi2 − n · τ 2

where n DIW bins are used for calculating the slope, τ and DIW are the means of time delay and DIW values of the n DIW points applied for fitting. The most suited number of bins may be determined based on experimental results. (8)

MF

This matched filter (MF) observable is defined as the similarity (correlation) between the unitary energy DIW waveform and the Doppler cut of the unitary energy Woodward ambiguity function (WAF) for the same pseudorandom noise code or coherent scattering model. (9)

Weighted Area of dDDM Pixels with Values Above Threshold

This observable and the following observable [10] are associated with the differential DDM (dDDM) generated by subtraction between two neighboring DDMs: dDDM(τ, f D ) = DDMi (τ, f D ) − DDMi+1 (τ, f D )

(7.12)

The observable is defined as the weighted area of the dDDM pixels with values above a predefined threshold: 

WAdDDM =

dDDM(τ, f D ) · dτ · d f D

(7.13)

|dDDM(τ, f D )|>DDMT where the testing uses the absolute value of the dDDM pixel value since it can be negative. This observable is similar to that one defined by (7.4), and if dτ · d f D is dropped by normalization, (7.12) becomes 

PSdDDM =

dDDM(τ, f D )

(7.14)

|dDDM(τ, f D )|>DDMT which is similar to (7.5). (10)

Abstract Number of dDDM Pixels with Absolute Values Above Threshold

This observable is defined as AN =

 |dDDM(τ, f D )|>δT

dDDM(τ, f D )   ·δ ·δ dDDM(τ, f D ) τ f D

(7.15)

170

7 Sea Ice Detection

If dτ · d f D is dropped by normalization, AN is the abstract number of the dDDM pixels which have absolute values greater than a predefined threshold, and AN is a positive or negative integer. (11)

DDW-TES

This observable and the next one are proposed in [8]. The differential delay waveform (DDW) is defined as the difference between the normalized integrated delay waveform (NIDW) and the normalized central delay waveform (NCDW). Note that the integrated delay waveform (IDW) and the DIW mentioned earlier are the same. Also, given DDM(τ, f D ), NIDW and IDW would be the same. Figure 7.4 shows an example of three delay waveforms computed from the TDS-1 DDM for both sea ice and seawater. It can be seen that The three delay waveforms of sea ice are dramatically different from those of seawater. The NIDW and NCDW of sea ice are very similar, which is in consistence with the fact that the specular reflection or coherent scattering is dominant for sea ice surface. But for seawater, the diffuse reflection or incoherent scattering is rather significant. This results in the much wider spreading of the trailing edge. The DDW-TES observable is defined as the TES of the DDW and Zhu et al. [8] selected three different numbers of delay bins: (a) (b) (c)

Three-Bin: 0.75 chips, the length of time delay is about 750 ns; Five-Bin: 1.25 chips, the length of time delay is about 1.25 µs; Seven-Bin: 1.75 chips, the length of time delay is about 1.75 µs;

Fig. 7.4 TDS-1 DDM(τ, f D ) and three delay waveforms of seawater (left) and seawater (right)

7.2 DDM Observables

171

It is necessary to choose the suited number of delay bins using experimental results, if impractical theoretically. Figure 7.5 shows the probability density functions (PDFs) of three different TESs calculated with DDM data collected over the two polar regions. NIDW TES is the same as DIW TES mentioned earlier, and NCDW TES can be considered as a special case of DIW TES. Five delay bins are used, corresponding to 1.25 µs of length of time delay, and over 20,000 DDMs were used to compute the slopes. As expected, sea ice produces much higher TES than seawater. The threshold of the TES observables may be set to be the value at which the two PDF curves intersect. The reason for such a selection of threshold is to obtain a high probability of detection (PD) and a low probability of false alarm (PFA) for both sea ice and seawater. Next section will present more discussions on the selection of the threshold. Thus, if TES is greater than the threshold, the relevant sea surface footprint is covered by sea ice; Otherwise, it is covered by seawater. The ground-truth seawater and sea ice states of the surface were downloaded from the OSISAF web site [4]. (12)

DDW-TEWS

This DDW-TEWS observable is defined as the summation of the pixel values of the trailing edge of DDW. In [8], three different lengths of delay bins are selected for computing the observable: (a) (b) (c)

Seven-Bin: 1.75 chips, the length of time delay is about 1.75 µs; Nine-Bin: 2.25 chips, the length of time delay is about 2.25 µs; Eleven-Bin: 2.75 chips, the length of time delay is about 2.75 µs;

Similar to Figs. 7.5 and 7.6 shows the probability density functions of three different TEWSs when seven bins are used. The TEWS threshold can also be set at the value when the two PDF curves intersect. Although the thresholds are quite different among different observables, the two thresholds of the same observable for the two regions are almost the same, showing the geographical versatility. However, this may not always be true. It can be seen from Fig. 7.4b and e that there is a big difference between the two thresholds. It is not clear whether or not an undeliberate error was introduced when processing the data. If the results are correct, the threshold should not always be the same when detecting the sea ice in the two polar regions. (13)

Effective Zone Distance

In [7] 13 observables are listed, including the effective zone distance (EZD) observable and the following four observables. The EZD is the distance between two pixels with a value below the average of the pixel values, which are immediately before and after the peak power point of a waveform such as CDW or DIW. These two points basically confine the boundary of the “effective zone” of the waveform. (14)

Effective Zone STD

The effective zone STD observable is defined as the standard deviation of pixel values over the effective zone.

172

7 Sea Ice Detection

Fig. 7.5 Probability density function of three different TESs computed with five delay bins using DDMs collected over the Northern Hemisphere (NH, Arctic region) and Southern Hemisphere (SH, Antarctic region) and the OSISAF SIE data were used as the ground truth

7.2 DDM Observables

173

Fig. 7.6 Probability density function of three different TEWSs computed with seven delay bins using DDMs collected over the Arctic region (NH) and Antarctic region (SH) and the OSISAF SIE data were used as the ground truth

174

7 Sea Ice Detection

Fig. 7.7 Average probability of false alarm (POF) of sea ice and seawater detection using six different observables

(15)

Offset Centre of Gravity (OCOG)

This OCOG observable is defined as the distance between the CM point and peak power point of a waveform. (16)

Cut-off Power Position Offset (CPPO)

This CPPO observable is defined as the coordinate differences between the position of peak power and the position where the power is 85% of the maximum. (17)

Kurtosis

The Kurtosis observable is defined as the kurtosis of the power distribution in the DDM. All the above 17 observables can be readily calculated using the preprocessed DDM DDM(τ, f D ). It is necessary to evaluate the sea ice detection performance of the different observables, so that suitable ones can be selected. Of course, most of them have already been investigated and the results already available in the literature can be used as a reference. Certainly, new observables can be defined and they can be applied to sea ice detection if they perform well.

7.3 Sea Ice and Seawater Detection 7.3.1 Performance Index In the statistical signal processing, hypothesis testing is often used to decide whether a hypothesis is correct or not. The surface area covered by the ground footprint or effective reflection/scattering zone can be divided into three different cases: sea ice,

7.3 Sea Ice and Seawater Detection

175

seawater, mixture of both. For simplicity, only the first two cases are considered and in fact the probability of the third case would be much smaller than those of the first two. That is, one hypothesis is that the reflection surface is sea ice and the other one is that the reflection surface is seawater. The probability that the hypothesis of sea ice is accepted when the surface is sea ice is defined as the probability of detection of sea ice, denoted as PDi. Similarly, the probability that the hypothesis of seawater is accepted when the surface is seawater is defined as the probability of detection of seawater, denoted as PDw. On the other hand, the probability that the hypothesis of sea ice is accepted when the surface is seawater is defined as the probability of missing of seawater, denoted as PMw; or the probability of false alarm of sea ice, denoted by PFAi. Also, the probability that the hypothesis of seawater is accepted when the surface is sea ice is defined as the probability of missing of sea ice, denoted by PMi; or the probability of false alarm of seawater, denoted by PFAw. Take Fig. 7.5f as an example to have a better understanding of these probabilities. (i) (w) (pink solid curve) and pTEWS (blue dashed curve) be the PDF of TEWS Let pTEWS of sea ice and that of seawater, respectively. Then, the probabilities are calculated by δT PDi =

(i) pTEWS (u i )du i

0

∞ PDw =

(w) pTEWS (u w )du w

δT

PMw = PFAi δT =

(7.16)

(w) pTEWS (u w )du w

0

PMi = PFAw ∞ (i) = pTEWS (u i )du i δT

where δT is the threshold which is about 0.365 in Fig. 7.5f. Also, the following two equations always hold: PDi + PFAw = 1 PDw + PFAi = 1

(7.17)

In hypothesis testing, if the purpose is just to determine whether or not an event occurs, the usual strategy is to maximize the PD when giving a rather small PFA. In general, for a given testing method, it is impractical to increase PD and decrease PFA simultaneously; PFA will increase with increased PD. When treating the presence of

176

7 Sea Ice Detection

sea ice and seawater as two individual events, the performance index may be defined as the average PFA of sea ice and seawater (APFA), which is denoted as POF in [8]: APFA =

1 (PFAi + PFAw) 2

(7.18)

Alternatively, the performance index may be defined as the average of PDi and PDw: APD =

1 (PDi + PDw) 2

(7.19)

It can be shown than APD and APFA satisfy APD + APFA = 1

(7.20)

That is, decreasing APFA would result in a decreased APD at the same time. However, by choosing the most suited threshold, the APD would be maximized and the APFA would be minimized. Such an optimal threshold may not be the same as the value at the intersection of the two density functions. In the presence of PDF formulas, the optimal threshold could be derived theoretically. Otherwise, a numerical method may be used to obtain the optimal threshold.

7.3.2 Detection Performance Figure 7.6 shows examples of APFA (POF) for six different observables under three different lengths of delay bins. Data collected over both polar regions were used to detect sea ice and seawater and the OSISAF data were used as the ground truth. The data are the same as those used in the preceding section, consisting of over 20,000 DDMs [8]. For both polar regions, the POFs of detection based on NIDW TEWS and DDW TEWS are the smallest. In particular, the DDW TEWS achieves the best performance. Also, NIDW performs better than NCDW. This is in agreement with the results shown in Fig. 7.4. The integration and differentiation produce a detection performance gain. Table 7.1 shows the detection performance of eight different methods in terms of PD, PFA, and POF. The overall performance may be evaluated with POF, as mentioned earlier [8]. Then, DDW TEWS achieves the lowest POF, i.e. the highest APD for both polar regions. NIDW TEWS, MF, and PN-D have similar performance, while NCDW TES and NCDW TEWS have the worst performance. PN-D denotes for the detection method based on the 10th observable defined in the preceding section, which is the number of dDDM pixels with absolute values above threshold. Also, the performance in the Arctic region is significantly better than the Antarctic region.

7.3 Sea Ice and Seawater Detection

177

Table 7.1 Sea ice and seawater detection performance NH

SH

PDi (%)

PDw (%)

PFAi (%)

PFAw (%)

NCDW TES

98.55

88.83

11.17

1.45

POF (%) 6.31

NIDW TES

98.25

92.17

7.83

1.75

4.79

DDW TES

95.66

92.06

7.94

4.34

6.14

NCDW TEWS

97.07

88.96

11.04

2.93

6.98

NIDW TEWS

98.74

93.78

6.22

1.26

3.74

DDW TEWS

97.56

98.87

1.13

2.44

1.78

MF

97.17

96.42

3.58

2.83

3.21

PN-D

97.12

95.33

4.67

2.88

3.78

NCDW TES

95.97

82.02

17.98

4.03

11.00

NIDW TES

94.19

88.24

11.76

5.81

8.78

DDW TES

93.15

91.09

8.91

6.85

7.88

NCDW TEWS

94.44

84.91

15.09

5.56

10.33

NIDW TEWS

93.84

94.55

5.45

6.16

5.80

DDW TEWS

95.22

98.08

1.92

4.78

3.35

MF

95.01

96.11

3.89

4.99

4.44

PN-D

95.18

95.14

4.86

4.82

4.84

Thus, the four methods (NIDW TEWS, DDW TEWS, MF, PN-D) are preferable, and DDW TEWS would be the best choice. Figure 7.8 shows the detection performance in the Antarctic region for four days in 2015 and two days in 2016. The ice coverage from May to November is large, while the ice coverage from January to March is rather small, corresponding to the winter and summer seasons, respectively. The DDW TEWS method is used to obtain the results. It can be seen that the detection results have a good match with the ground-truth data. Similarly, Fig. 7.9 shows the detection results for the Arctic region. Contrarily, the ice coverage in the arctic region is largest in January and February, while it is smallest in August. The ice is rather sparsely distributed excepted for that in the central area. On the other hand, the ice is mainly distributed on the boundary of land area in the Antarctic region. This may be the main reason that the detection performance in the Antarctic region is much inferior to that in the Arctic region. The 9th and 10th observables defined in Sect. 7.1.1 are associated with dDDM [10]. It is useful to take a look at the detection performance using these two observables. dDDM provides a different way to use the DDM data, which may be particularly useful in detecting the transition from one type of surface to another. Since one dDDM involves two DDMs associated with two effective reflection zones on the surface, four different cases of transition would be encountered, which are. (a) (b)

Seawater → Seawater Sea ice → Sea ice

178

7 Sea Ice Detection

Fig. 7.8 GNSS-R sea ice detection results compared with the OSISAF SIE map of six different days for the Antarctic region; SH denotes Southern Hemisphere

7.3 Sea Ice and Seawater Detection

179

Fig. 7.9 GNSS-R sea ice detection results compared with the OSISAF SIE map of six different days for the Arctic region; NH denotes Northern Hemisphere

180

(c) (d)

7 Sea Ice Detection

Seawater → Sea Ice Sea ice → Seawater.

In DDM based methods, either seawater hypothesis or sea ice hypothesis is accepted based on the observable value and the threshold. However, regarding dDDM based methods, one of the four different hypothesis is accepted, involving two surface states of the same or different. Figure 7.10 shows four dDDMs associated with the four different transitions. It can be seen that the dDDMs under different transitions are significantly different from each other. Among the four different transitions, the ice-water and water–ice transitions can be distinguished from each other more easily due to the negative and positive powers. Also, the transition between the same type of surface can be readily distinguished from the transition between the different types of surface. Distinguishing between the two transitions of the same type of surface would be more affected by noise. Figure 7.11 shows a sequence of 30 consecutive TDS-1 dDDMs. From the 45th to the 58th dDDM, the transition is from seawater to seawater. From the 65th dDDM to 74th dDDM, the surface is sea ice. At the 72nd dDDM, the first DDM related

Fig. 7.10 Normalized dDDMs for four different observation situations: a water-water transition, b water–ice transition, c ice-ice transition and d ice-water transition

7.3 Sea Ice and Seawater Detection

181

Fig. 7.11 Example of a series of dDDMs related to sea surface areas of sea water, mixture of seawater and sea ice, and sea ice

surface is mostly seawater and the second DDM related surface is mostly sea ice. The surface areas related to 59th–61st dDDMs would have a small percentage of sea ice, while those related to 63th and 64th dDDMs could have a small percentage of seawater. That is, sea ice concentration is a more important parameter to describe the property of a sea surface area, which will be studied in the following section. Using GPS signals collected by TDS-1 in March 2015, June 2015, October 2015, January 2016 and March 2016, a large number of DDMs were employed to validate the dDDM based sea ice detection method [10]. Only DDMs collected over sea ice and seawater are used and DDMs over land are excluded based on the position of specular point and geographic information. Five methods are evaluated, which are the power summation method (PS-N) [5], pixel number method PN-N [2] and matching filter (MF) method [1], PN-D, and PS-D which denotes the detection based on the 9th observable defined in the preceding section. Thresholds are selected individually based on the results of detection, that is, the thresholds that produce the best detection results are used for each method. Table 7.2 shows the probability of false alarm of sea ice detection for the five methods. The sea ice edge data provided by OSI SAF are used as the ground-truth data to verify whether or not the decision is correct. In this case, the two dDDM based methods produces the lowest PFAi, and they also achieve the highest PDi as shown in Fig. 7.12. Figure 7.13 shows the sea ice detection results for the Arctic region based on the PS-D method using data collected over six periods (24 March 2015, 04 June 2015, 23 August 2015, 26 October 2015, 21 January 2016 and 26 March 2016). It can be seen

182

7 Sea Ice Detection

Table 7.2 Probability of false alarm of sea ice detection for five observables using GNSS-R differential DDM Date of dataset

Number of dataset Number of tested PS-D PN-D PS-N PN-N MF DDM

24 March 2015

22

2815

0.28

0.36

3.20

3.48

1.88

4 June 2015

25

2906

0.34

0.41

3.99

3.92

1.72

23 August 2015

20

2668

0.22

0.30

3.67

3.37

1.95

26 October 2015 22

3411

0.29

0.23

3.28

3.34

1.91

21 January 2016

21

2764

0.22

0.29

4.12

3.98

1.77

26 March 2016

20

2782

0.29

0.29

3.52

3.24

1.83

Total

130

17,346

0.28

0.31

3.62

3.55

1.84

Fig. 7.12 Sea ice detection probability of five different detection methods

that the sea ice and seawater detection results have a good match with the groundtruth data provided by OSI SAF. It is worth noting that the sea ice coverage changes as the season changes, however, the change mainly occurs in the areas marked by blur rectangles (A, B and C) in Fig. 7.13a. Sea ice in these areas can be recognized as young ice or first-year ice which includes both closed ice and open ice, while sea ice in area D marked with magenta rectangle can be regarded as multi-year ice (only closed ice).

7.4 Sec Ice Concentration Estimation The preceding sections focus on the detection of sea ice and seawater. However, as mentioned earlier, in some cases especially in the surface areas near the boundary between sea ice and seawater, the scattering area of the GNSS signal associated with one DDM would contain both sea ice and seawater. Thus, it is useful to determine the sea ice and seawater concentration.

7.4 Sec Ice Concentration Estimation

183

Fig. 7.13 Sea ice and seawater detection results with the PS-D method using differential and normalized DDM in six different days: Lines in green and red represent detected sea ice and seawater respectively. The white area depicts the ground-truth sea ice coverage (sea ice edge product of the EUMETSAT Ocean and Sea Ice Satellite Application Facility, OSI SAF, www.osi-saf.org)

184

7 Sea Ice Detection

7.4.1 Estimation Method Unlike sea ice detection which decides to accept or reject the hypothesis of the presence of sea ice, sea ice concentration (SIC) is a parameter which needs to be estimated. In the case where the surface area is completely covered by sea ice, SIC is equal to one. If there is no sea ice over the surface area, SIC is equal to zero. If both seawater and sea ice occur in the surface area, SIC is greater than zero and smaller than one. Figure 7.14 shows three TDS-1 DDMs and corresponding three delay waveforms under three surface conditions with SIC being 0, 0.5, and 1, respectively [9]. The three types of delay waveforms (CDW, IDW, DDW) are already mentioned in the preceding section. It can be seen that each of the three delay waveforms changes as SIC varies, and a larger SIC variation would result in a more different delay waveform. Similarly, an observable can be defined or selected from one of the observables defined in Sect. 7.2. Consider that the delay waveform DDW is used and the three DDWs in Fig. 7.14 are put together as shown in Fig. 7.15. The observable DDW TEWS can be directly used, but it should be noted that the concepts of leading edge and trailing edge may not be suited for DDW; in fact, they are differential leading and trailing edges. For this reason, “TEWS” is replaced with “REWS” (right edge waveform summation) in [9], Also, DDW TEWS, denoted by rDT , is the power (strictly differential power) summation, starting from the zero delay for a specified number of delay bins: rDT =

n 

DDWi

(7.21)

i=1

where DDWi is the ith value of DDW, starting from the zero delay bin, and n is the predefined number of delay bins such as 7. To obtain an estimate of the unknown SIC, a mathematical relationship (i.e. model) must be established to link the observable and SIC. The model can be theoretical, empirical, or semiempirical. Clearly, in this case, it may be rather difficult, if not impossible, to develop a theoretical model, so an empirical model would be more practicable. Zhu et al. [9] developed an empirical model using TDS-1 DDM data and the ground-truth SIC data downloaded from [3]. About 71,000 DDMs of Arctic region and 69,000 DDMs of Antarctic region are randomly selected from twice the number of DDMs, respectively. The least-squares fitting is then applied to obtain the empirical model: SIC = Aε B(rDT −μ) + C

(7.22)

where {A, B, C, ε, μ} are model parameters which are listed in Table 7.3. Figure 7.16 shows the scatterplot of the calculated DDW TEWS by (7.21) and the ground-truth SIC. Although the distribution of the SIC with respect to the TEWS is

7.4 Sec Ice Concentration Estimation

185

Fig. 7.14 Three types of delay waveforms related to three sea ice concentrations

rather dispersed, the relationship can still be well described by the model in (7.22). It would be useful to significantly decrease the dispersion such as by taking other relevant factors into account.

186

7 Sea Ice Detection

Fig. 7.15 The differential delay waveform (DDW) with different SIC. The waveform is divided into left edge and right edge by the dotted line in red

Table 7.3 Model parameters obtained through least squares fitting for Arctic and Antarctic regions A

B

C

ε

μ

Arctic

0.8145

0.8215

0.0083

31.8

0.15

Antarctic

0.7504

0.8507

0.0133

51.6

0.26

Fig. 7.16 Scatterplot of ground-truth SIC versus the observable DDW TEWS (REWS) calculated by (7.21) over the Arctic (left) and Antarctic (right); the curve fitting based model is also shown by the dashed black line

7.4.2 Estimation Performance The models developed above are then used to estimate the SIC with the other 71,000 and 69,000 DDMs collected over Arctic region and Antarctic region, respectively; these DDMs were not used for the model development. The DDW TEWS is calculated

7.4 Sec Ice Concentration Estimation

187

Fig. 7.17 Scatterplot of the estimated SIC versus the ground-truth SIC

Table 7.4 SIC estimation performance for Arctic and Antarctic regions

Error mean (%)

RMSE (%)

Corrcoef

Arctic

1.67

11.7

0

Antarctic

1.94

12.1

0

for each DDM and then (7.22) is used to calculate the SIC. Figure 7.17 shows the scatterplot of the estimated SIC and the ground-truth SIC for both regions. It can be seen that although the dispersion is significant, the SIC estimates are largely close to the true SIC values. Table 7.4 shows the estimation performance in terms of the bias (mean error) and the RMSE. Although the bias is minor, the RMSE is considerable for both polar regions. Figure 7.18 shows the distribution of sea ice on the map for both polar regions, one day in winter and another day in summer. The line segments denote the surface specular scattering tracks of TDS-1 GPS signal, and the line colors represent the SIC values as specified by the colormap. To avoid confusion, the ground-truth SIC and the estimated SIC use two different colormaps, cool and parula. In general, the estimated SIC is in agreement with the ground-truth data.

7.5 Summary Coherent scattering of GNSS signal over sea ice is dominant, while diffuse scattering over seawater is usually significant. As a result, the sea ice related DDM pattern is significantly different from the seawater related DDM, enabling relatively easy to distinguish sea ice from seawater in general, as demonstrated in this chapter. The detection performance would degrade as the roughness of seawater surface becomes mild. Many observables have been proposed in the literature and some of them were summarized. A number of observables were used for detection of sea ice and seawater and their performance was evaluated. More useful information on ice coverage such

188

7 Sea Ice Detection

Fig. 7.18 Visualization of SIC estimation results based on the TDS-1 DDM and the SIC model for both polar regions. ASI SIC data are the ground-truth values measured by the left colormap, while the right colormap is used as a ruler to measure the estimated SIC values

as in the two polar regions will be obtained if more satellite signals are used for ice detection. Accurate estimation of SIC will provide useful information on the dimensions and locations of the boundary between sea ice and seawater. It is also useful to perform classification of ice types, which may include young ice, first-year ice and multi-year ice [6].

References 1. Alonso-Arroyo A, Zavorotny VU, Camps A (2017) Sea ice detection using U.K. TDS-1 GNSSR data. IEEE Trans Geosci Remote Sens 55(9):4989–5001 2. Marchan-Hernandez JF, Rodríguez-Álvarez N, Camps A, Bosch-Lluis X, Ramos-Pérez I, Valencia E (2008) Correction of the sea state impact in the L-band brightness temperature by means of delay-Doppler maps of global navigation satellite system signals reflected over the sea surface. IEEE Trans Geosci Remote Sens 46(10):2914–2923

References

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3. Meereisportal (visited, 2020) https://www.meereisportal.de/en/ 4. OSISAF (visited, 2020) https://osisaf.met.no/p/ice_extent_graphs.php 5. Rodriguez-Alvarez N, Dennis MA, Zavorotny VU, Smith JA, Camps A, Fairall CW (2013) Airborne GNSS-R wind retrievals using delay-Doppler maps. IEEE Trans Geosci Remote Sens 51(1):626–641 6. Rodriguez-Alvarez N, Holt B, Jaruwatanadilok S, Podest E, Cavanaugh KC (2019) An Arctic sea ice multi-step classification based on GNSS-R data from the TDS-1 mission. Remote Sens Environ 230:111202 7. Yan Q, Huang W (2019) Sea ice remote sensing using GNSS-R: a review. Remote Sensing 11(2565):1–26 8. Zhu Y, Tao T, Yu K, Li Z, Qu X, Ye Z, Geng J, Semmling M, Wickert J (2020) Sensing sea ice based on Doppler spread analysis of spaceborne GNSS-R data. IEEE J Selec Topics Appl Earth Observ Remote Sens 13:217–226 9. Zhu Y, Tao T, Zou J, Yu K, Wickert J, Semmling M (2020b) Spaceborne GNSS reflectometry for retrieving sea ice concentration using TDS-1 data. IEEE Geoscience and Remote Sensing Letter 10. Zhu Y, Yu K, Zou J, Wickert J (2017) Sea ice detection based on differential delay-Doppler maps from UK TechDemoSat-1. Sensors 17(7):1–18

Chapter 8

Snow Depth and Snow Water Equivalent Estimation

Variation in regional and global snowfall significantly affects the ecological and climate systems, which is usually used for policy-making in water resource management and disaster prevention. The amount of snowfall is measured by two different parameters, snow depth and snow water equivalent (SWE). SWE is defined as the product of snow depth and snow density, which is also equal to the depth of water after the snow completely melts without evaporation, penetration and run-off. In this chapter, the focus is on the use of GNSS-R for estimating snow depth and SWE. Four different methods of snow depth estimation are studied with details, while only one SWE estimation method is described, indicating more investigations are needed to enhance SWE estimation.

8.1 SNR-Based Snow Depth Estimation GNSS receivers usually provide the measurements of SNR, which is defined as the carrier power to noise power ratio (CNR). As a fact, there is some minor difference between SNR defined in communications and signal processing and CNR defined in GNSS, although SNR and CNR are often treated as the same. SNR was first exploited for snow depth estimation by Larson et al. [11]. It was motivated by the fact that there are a few thousand continuous operating reference stations (CORS) deployed in the cold northern atmosphere. The SNR data recorded by those CORS receivers can be used to measure snow depth without the need of any new infrastructure. At the same time, the CORS data will increase the temporal and spatial resolution of snow depth measurement considerably.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_8

191

192

8 Snow Depth and Snow Water Equivalent Estimation

8.1.1 Basic Method As studied in Chap. 4, the composite direct and reflected signal captured by a face-up geodetic antenna may be modelled as ˜ s(t) = A(t) sin ψ(t)

(8.1)

˜ where A(t) and ψ(t) are the composite amplitude and the composite carrier phase. After some mathematical manipulations and the detrending operation as performed in Chap. 4, the SNR becomes S N Rd (t) =

2 Ad (t)Am (t) cos δφ(t) PN

(8.2)

where the definitions of the parameters are already given in Chap. 4 and the interferometric phase is rewritten here for convenience: δφ(t) =

4π h sin θ (t) λ

(8.3)

which results from the time difference of arrival of the direct and reflected signals. Figure 8.1a shows the SNR time series associated with the L2 signal of a GPS satellite recorded by a Trimble NetR9 receiver with antenna TRM55971.00 at Shangli, a village in Mudanjiang, China, on the 10th of January 2018. The third term in (8.2) contains the parameter of antenna height (relative to the reflection surface), which is the term of interest and the vibration in Fig. 8.1 reflects this term. The slowly increase trend is produced by the first two terms in (8.2). Thus, to extract the information of the antenna height, it is necessary to get rid of the first two terms. For instance, the two terms can be modeled as a low-order polynomial, or can be extracted through a low-pass filter. Then the recorded SNR is subtracted by the polynomial or by the output of the low-pass filter. Figure 8.1b shows the detrended SNR time series after removing the first two terms. Since both Ad (t) and Am (t) vary with time, the detrended SNR is a quasisinusoidal signal with respect to time variable sin θ (t), which is in accordance with the SNR curve observed in Fig. 8.1b. The original SNR and the detrended SNR contains noise components, which include receiver thermal noise and multipath interference. Note that, (8.1) through (8.3) are derived under the condition that only one reflection path is considered, which is the specular reflection path. In reality, there are multiple reflection paths especially when the surface is rough. As studied in Chap. 4, the detrended SNR is a quasi-sinusoidal signal with a main frequency given by f =

2h λ

(8.4a)

8.1 SNR-Based Snow Depth Estimation

193

Fig. 8.1 Example of SNR data recorded by a GNSS receiver, original SNR (a) and detrended SNR (b)

which can be written as h=

1 fλ 2

(8.4b)

Thus, the relative antenna height can be determined by (8.4b) if the spectral peak frequency of the detrended SNR time series is known. Figure 8.2 shows the power spectral density of the detrended SNR time series given in Fig. 8.1a. Note that although the SNR data are evenly sampled temporally such as at a rate of 1 Hz, θ (t) and sin θ (t) are not uniformly distributed. Therefore, as

194

8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.2 Example of power spectral density of detrended SNR data

mentioned in Chap. 4, the spectrum analysis cannot be performed using the conventional methods such as fast Fourier transform. Instead, a special method such as the Lomb-Scargle spectral analysis [8] has to be employed. The Lomb-Scargle spectral analysis was used to generate the results in Fig. 8.2. Although it does not have the ideal pulse of a single sinusoidal signal, a sharp spike does occur. Thus, the spectral peak frequency can be readily determined. Here, since sin θ (t) is the time variable which is not the typical time variable. the frequency is dimensionless and sometimes expressed as m · m−1 . Thus, the antenna height equals the scaled signal wavelength with the scaling coefficient being of half the main frequency. With the known snow-free antenna height, the snow depth estimate is generated. In the event that the snow-free antenna height is not available or the snow-free ground surface elevation is changed, (8.4b) can be used to estimate the actual snow-free antenna height using the average of multiple estimations if available. An estimate of snow-free antenna height should be obtained for the rising period and falling period of each satellite, respectively, so that the effect of topography and estimation bias can be mitigated.

8.1.2 Data Fusion Based Improvement As mentioned earlier, the snow depth estimation error mainly comes from the noise including the modeling error and the receiver thermal noise. The noise would not only decrease the amplitude of the power spectral density, but it also causes the deviation of signal main frequency, and thus produces a bias in snow depth estimation. Figure 8.3 shows an example of simulated multipath SNR series of GPS L1 signals and the results of Lomb-Scargle spectral analysis. The antenna height is set at 2.5 m

8.1 SNR-Based Snow Depth Estimation

195

Fig. 8.3 Simulated multipath SNR series and their spectrograms with three different noise levels

and the radiation pattern of GNSS antenna TRM55971.00 is used for the calculation of amplitudes of SNR time series. For convenience, three Gaussian noise components are respectively added to the noise-free SNR time series to produce three series with different noise levels. The SNR (S/N0 ) of those three simulated series are 15 dB, 10 dB, and 5 dB respectively. It can be observed that the presence of noise results in variation in the main frequency; stronger noise causes a larger main frequency shift. The main frequency error is about 0.4 when the SNR of simulated detrended SNR series is 5 dB. The frequency error would lead to a bias of about 4 cm in snow depth estimation when using GPS L1 signals. Thus, a quantitative indicator of noise level is useful as the noise level indicates the precision of main frequency estimation and hence the snow depth estimation accuracy. It can also be seen from Fig. 8.3 that the peak of power spectral density (PSD) is inversely proportional to the noise level. This implies that, the peak of PSD obtained by spectral analysis on the detrended SNR series could be used as an indicator to weight the main frequency estimation results when data fusion is applied. To evaluate the impact of noise, 20 different Gaussian noise levels are selected, coresponding to S/N0 from 1 to 20 dB with an interval of 1 dB. For a given noise

196

8 Snow Depth and Snow Water Equivalent Estimation

level, 1000 noise sequences are generated and each of them is added to the noisefree detrended SNR series to produce a noise-corrupted detrended SNR series. Then, the Lomb-Scargle spectrum analysis is applied to the SNR series to obtain the peak PSD and the corresponding main frequency. Subtracting the main frequency by the theoretical one in the noise-free case produces the noise-induced frequency error. Repeating the procedure for all other 999 noise-corrupted multipath-induced SNR series under the same noise level, a sequence of 1000 main frequency errors and 1000 peak PSD values are obtained. After repeating the procedure for all other 19 noise levels, 20,000 samples are obtained for main frequency error, corresponding to 20,000 samples of peak PSD. All the peak PSD samples are arranged into bins centered at 8 different peak PSD values (0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55). and the width of each bin is 0.05. Each of the central peak PSD values represents all the peak PSD samples within the bin. RMS of main frequency error within each bin is calculated. Thus, a distribution of the RMS of main frequency error versus peak PSD is obtained, as shown in Fig. 8.4, where three different antenna heights are tested. Obviously, the RMS of main frequency error decreases with the increased peak PSD or the decreased noise level. RMS of main frequency error is marginally different under a different antenna height. However, the variation pattern of RMS of main frequency error versus peak PSD is almost the same. Using the exponential fitting produces the exponential fitting function: R M S = ae−bp

(8.5)

Fig. 8.4 RMS of the main frequency estimation error as a function of the peak spectral density under three different antenna heights (2, 3, and 4 m)

8.1 SNR-Based Snow Depth Estimation Table 8.1 Fitting parameters and correlation coefficients for different antenna heights

197

Antenna height (m)

a

b

R

1.0

2.12

−5.79

0.9886

1.5

2.06

−5.43

0.9840

2.0

2.14

−5.94

0.9903

2.5

1.91

−5.05

0.9807

3.0

2.03

−5.80

0.9907

3.5

1.99

−5.34

0.9864

4.0

2.08

−5.67

0.9877

4.5

2.16

−5.67

0.9887

5.0

2.06

−5.47

0.9849

where p is the peak PSD and {a, b} are the fitting coefficients, as listed in Table 8.1. The fitting curves under three different antenna heights are also shown in Fig. 8.4. The correlation coefficients between the simulated and the fitted RMS of main frequency error are also listed in Table 8.1, which are all greater than 0.98 for three different antenna heights, indicating a strong correction and valid fitting. Accordingly, the accuracy of SNR-based antenna height estimation and hence snow depth estimation is closely related to the peak PSD. The fitting parameter a ranges from 1.9 to 2.2, while the fitting parameter b varies between −6 and −5. As those two fitting parameters are not sensitive to the antenna height, it is reasonable to take the average of each parameter’s values as the final value to simplify the model. Then the exponential fitting function becomes: R M S = 2.06e−5.57 p

(8.5b)

Based on the exponential relationship between the accuracy of antenna height estimation and the peak PSD, a fusion model is developed to combine multi-satellites snow depth estimations as: h=

1 N sat 

e5.57· pi

·

Nsat 

e5.57· pi h i

(8.6)

i=1

i=1

where superscript (i) denotes the ith satellite and Nsat snow depth estimates {h i } are produced using data of Nsat satellites over a certain observation period. In addition to SNR, GNSS receivers also provide carrier phase observation which can also be used for snow depth estimation especially when SNR data is not available or the quality of SNR data is poor.

198

8 Snow Depth and Snow Water Equivalent Estimation

8.2 Daul-Frequency Carrier Phase Combination Based Method Ozeki and Heki [18] proposed the carrier phase based snow depth estimation method which is termed L4 method. It makes use of the combination of the carrier phases of dual-frequency signals of the same satellite, defined as M1,2 (t) = λ1 ψ˜ 1 (t) − λ2 ψ˜ 2 (t)

(8.7)

where the noise term is droped for simplicity, λi and ψ˜ i (t) are the wavelength and composite carrier phase of the ith frequency band signal, respectively. The carrier phase is recorded as the number of cycles, which is usually a non-integer number. Substituting the second equation in (4.6) (see Chap. 4) into (8.7) yields M1,2 (t) = (λ1 ψ1 (t) − λ2 ψ2 (t)) + (λ1 β1 (t) − λ2 β2 (t))

(8.8)

The navigation satellite signals travel through the ionosphere which slows down the signal propagation and produces frequency-dependent ionospheric delays (δi (t)). Thus, the individual weighted phase becomes λi ψi (t) = ds,r (t) + δi (t) + (t), i = 1, 2

(8.9)

where ds,r (t) is the Euclidian distance between the satellite and the receiver and (t) accounts for clock errors, tropospheric delays, and all other carrier-independent effects. Substituting (8.9) into (8.8) yields M1,2 (t) = (λ1 δ1 (t) − λ2 δ2 (t)) + (λ1 β1 (t) − λ2 β2 (t))

(8.10)

Therefore, the combination does not depend on geometric parameters such as coordinates of receiver or satellites and thus this is called geometry-free dualfrequency carrier phase combination. The combination is a function of the carrier wavelengths, the ionospheric delays and the interferometric phases which depend on antenna height, satellite elevation angle, antenna gain pattern, and surface reflectivity. As shown in Fig. 8.5a, the combination is overwhelmed by the ionospheric delay and the desired multipath-induced signal can hardly be seen. The effect of such ionospheric delay must be removed from the combination in (8.10) such as by low-pass filtering or high-pass filtering. When using low-pass filtering, the original combination data is filtered by a low-pass filter and the result is subtracted from the original data to obtain the detrended data. The results shown in Fig. 8.5b are produced in this way. Alternatively, the original combination data can be directly filtered using a high-pass filter to generate the detrended data without involving the subtraction. From Fig. 8.5b, it can be seen that the residual trend caused by ionospheric delay still can be observed. As a consequence, although the main spectral spike can be

8.2 Daul-Frequency Carrier Phase Combination Based Method

199

Fig. 8.5 Example of results of dual-frequency carrier phase combination: a original combination result, b detrended combination result, and c power spectral density of detrended combination result

200

8 Snow Depth and Snow Water Equivalent Estimation

clearly noticed from Fig. 8.5c, there are strong side lobes. It is desirable to design better filters to remove the effect of ionospheric delay in a more effective way. After removing the ionospheric delays and ignoring the residual ionospheric delays, (8.10) becomes M1,2 (t) = λ1 β1 (t) − λ2 β2 (t)

(8.11)

From (4.13) in Chap. 4, the composite excess phase β1 (t) can be approximated as a quasi-sinusoidal signal. Thus, the detrended combination would produce two spectral spikes. However, it would produce a large error to calculate the antenna height by the spectral peak frequencies obtained through spectral analysis in the similar way to the SNR method described in the preceding section. Thus, it is preferable to establish the theoretical relationship between the spectral peak frequency and the antenna height. More details about the development of such a model will be provided in the following subsection

8.3 Triple-Frequency Carrier Phase Combination Based Method To remove the effect of geometry and ionosphere simultaneously, the carrier phase combination of triple-frequency signals can be used [28]. It consists of two main steps, prior theoretical modeling and real-time model-based estimation. The triplefrequency phase combination is used in both steps, so it is studied first as follows.

8.3.1 Triple-Frequency Carrier Phase Combination The triple-frequency phase combination is defined as the linear combination of three carrier phase differences: M1,2,5 (t) =(λ25 (λ1 ψ˜ 1 (t) − λ2 ψ˜ 2 (t)) + λ21 (λ2 ψ˜ 2 (t) − λ5 ψ˜ 5 (t)) + λ22 (λ5 ψ˜ 5 (t) − λ1 ψ˜ 1 (t)))/(1 m2 )

(8.12)

where the composite carrier phase ψ˜ i (t) as defined by (4.6) in Chap. 4 is in cycles and the wavelength λi is in meters so that the combined phase M1,2,5 (t) is in meters due to the division by 1 m2 ; For simplicity, such a division is dropped from now on; Taking the GPS L1, L2 and L5 signals as an example, the three wavelengths are λ1 = 0.1902937 m, λ2 = 0.2442102 m and λ5 = 0.2548280 m. Assume that the antenna gain is exactly the same for the three different frequencies, although

8.3 Triple-Frequency Carrier Phase Combination Based Method

201

each signal of different frequency is received with slightly different antenna gain, depending on the antenna’s frequency response. Considering that the carrier phases are recorded by GNSS receivers as number of cycle and integer ambiguities are involved, (8.12) can be written as 





M1,2,5 (t) = κ1 ψ 1 (t) + κ2 ψ 2 (t) + κ5 ψ 5 (t) + u 

(8.13)

βi (t) 2π κ1 = λ1 η1 ; κ2 = λ2 η2 ; κ5 = λ5 η5

ψ i (t) = φi (t) +

η1 = (λ25 − λ22 ); η2 = (λ21 − λ25 ); η5 = (λ22 − λ21 ) u = κ1 N 1 + κ2 N 2 + κ5 N 5

(8.14)

where  Note that {ηi } are unitless and {ψ i } are the composite phase (unit in cycle) after the integer ambiguity is removed. Provided that there is no cycle slip in the raw phase observation data, u is a constant which is the weighted sum of the integer ambiguities {N1 , N2 , N5 } of the three carriers. Otherwise, cycle slips need to be detected and compensated so that u remains a constant over the observation period. By dropping this unknown constant u, (8.13) becomes 





M˜ 1,2,5 (t) = κ1 ψ 1 (t) + κ2 ψ 2 (t) + κ5 ψ 5 (t)

(8.15)



Using the raw phase observations {ψ i (t)} at a sequence of time instants, a combined phase time series can be readily produced by (8.15).

8.3.2 Property of Triple-Frequency Phase Combination Using (4.6), (8.12) can be written as (d) (r ) + M1,2,5 M1,2,5 = M1,2,5

(8.16)

where (d) =λ25 (λ1 ψ1 (t) − λ2 ψ2 (t)) + λ21 (λ2 ψ2 (t) − λ5 ψ5 (t)) M1,2,5

+ λ22 (λ5 ψ5 (t) − λ1 ψ1 (t)) (r ) =λ25 (λ1 β1 (t) − λ2 β2 (t)) + λ21 (λ2 β2 (t) − λ5 β5 (t)) M1,2,5

+ λ22 (λ5 β5 (t) − λ1 β1 (t))

(8.17)

202

8 Snow Depth and Snow Water Equivalent Estimation

(r ) It can be seen that the second term in (8.16), M1,2,5 , is only a function of the three carrier wavelengths and three composite excess phases which involve antenna height, satellite elevation angle, antenna gain pattern, and surface reflectivity. It is thus independent of both geometry and ionospheric delays. Meanwhile, in the presence of ionospheric delays {δi }, the individual weighted phase is modelled as

λi ψi (t) = ds,r + δi + , i = 1, 2, 5

(8.18)

where ds,r is the Euclidian distance between the satellite and the receiver and accounts for clock errors, tropospheric delays, and all other carrier-independent (d) can be effects. Both δi and have the unit of distance. Accordingly, the term M1,2,5 written as (d) = λ25 (δ1 − δ2 ) + λ21 (δ2 − δ5 ) + λ22 (δ5 − δ1 ) M1,2,5

(8.19)

where the Euclidean distance and have been removed due to subtraction. The ionospheric delay is produced when the radio signals travel through the ionosphere, which is described by δi =

40.7 T EC f i2

(8.20)

where TEC is the total electronic density of the ionosphere of the propagation path, which is the same for all the carrier signals transmitted from the same satellite and captured by the same receiver and f i is the carrier frequency of the signal. That is, the delay is inversely proportional to the squared carrier frequency. Substituting (8.20) into (8.19) and making use of λi = c/ f i where c is the speed of light produce (d) =0 M1,2,5

(8.21)

As a consequence, the combined composite phase becomes (r ) (t) = κ1 β1 (t) + κ2 β2 (t) + κ5 β5 (t) M1,2,5 (t) = M1,2,5

(8.22)

Thus, the triple frequency phase combination is independent of geometry and ionospheric delays. Note that the formula (8.20) for calculating the ionospheric delay is not perfect, so that residual ionospheric delay will still exist in the combined phase, but it would be insignificant.

8.3 Triple-Frequency Carrier Phase Combination Based Method

203

8.3.3 An Example of Triple-Frequency Phase Combination Analytically, as indicated by (8.22), the combined composite carrier phase is actually equal to the combined composite excess phase or error phase. Figure 8.6 shows examples of the combined phase pattern with respect to sin θ (t), calculated by the third equality (error phase) in (4.6), (8.19) and (8.22). The error phases are calculated using typical snow permittivity and the antenna gain patterns of TRM59800.00; and the antenna height range is between 1.47 and 6 m. For a given antenna height, the combined phase oscillates with a nearly constant period, but the amplitude decreases significantly with time (a damping process). The phase series pattern at a given antenna height can be very different from that at a different antenna height. Note that the real measurements of the combined phase for two antenna heights (1.47 and 1.56 m) are also shown in Fig. 8.6. Basically, there is a good match between the analytical and measured combined phase.

Phase error(mm)

1

1

Observed data

0

-0.5

-0.5 0.2

0.4

0.8

0.6

1

Phase error(mm)

1.56 m

0.5

0

-1

Observed data

1.47 m

0.5

-1

0.2

0.4

3.00 m

0.5

0

0

-0.5

-0.5 -1 0.2

0.4

0.6

0.8

Phase error(mm)

0.2

0.4

0.8

0.6

2

2

6.00 m

5.00 m

1

1

0

0

-1

0.8

4.00 m

0.5

-1

0.6

1

0.2

0.4

0.6

sin θ (elevation angle)

0.8

-1

0.2

0.4

0.6

0.8

sin θ (elevation angle)

Fig. 8.6 Combined carrier phase series under six different antenna heights (1.47 m through 6 m)

204

8 Snow Depth and Snow Water Equivalent Estimation

To compare the magnitude of the oscillations of the combined phase to the expected noise level, the magnitude of the noise level can be estimated via propagation of uncertainties. Assuming the phase observation errors of the three carriers have the same standard deviation of 1 mm, σ0 = σ1 = σ2 = σ5= 1 mm, the standard deviation of the combined phase error becomes σ1,2,5 = σ0 κ12 + κ22 + κ52 = 37.4 μm. This value is shown as two horizontal dotted gray lines in Fig. 8.6. In this case, since the oscillation amplitude is within the one standard deviation (σ1,2,5 ) when the elevation angle goes to 30°, the cutoff elevation should be set below 30°.

8.3.4 Theoretical Modeling From (8.22) and (4.22), it can be seen that the result of the triple-frequency carrier phase combination is approximately equal to the weighted sum of three quasi-sinusoids: M1,2,5 (t) ≈ κ1 α1 (t) sin(δφ1 (t)) + κ2 α2 (t) sin(δφ2 (t)) + κ5 α5 (t) sin(δφ5 (t)) (8.23) As mentioned, treating sin θ (t) as the time variable, the main frequencies of the three quasi-sinusoids are fi =

2h , i = 1, 2, 5 λi

(8.24)

Figure 8.7 shows a typical example of the periodogram of the combined phase time series M1,2,5 in (8.22) when antenna height is 2.5 m and sin θ (t) is evenly sampled for the elevation angle range between 10 and 30°. The spectrum exhibits a well-defined spectral peak frequency, despite there being three error phase terms 1400 2 -1 Power Spectral Density (rad /(m m )

Fig. 8.7 Example of periodogram of the combined error phase when antenna height is 2.5 m and elevation range is 10–30°

1200 1000 800 600 400 200 0 5

10

15 20 25 Frequency (m m-1)

30

35

8.3 Triple-Frequency Carrier Phase Combination Based Method

205

associated with three wavelengths. This is probably because the difference between L2 and L5 wavelength is very small and the scaling factor for the L1 phase term is much smaller than those for the L2 and L5 phase terms. In this case, the spectral peak frequency is 18.8 (m · m−1 ), close to the value of the 20 (m · m−1 ) calculated by (8.24) for L2 and L5 signals. The spectral peak frequency of the combined phase is a function of the antenna height h, so it is feasible to develop a model to describe the relationship between the antenna height and the spectral peak frequency. Given the amplitude attenuation factor and the three wavelengths, the spectral peak frequency is independent of satellite selection and elevation range selection as long as the error phase measurements over the same elevation range are employed to estimate antenna height. As observed from Fig. 8.6, the oscillations are actually heavily damped, and data at high elevation angles is basically useless since the combined phase magnitude is rather small. Therefore, observations associated with satellite elevation between 10 and 30°, corresponding to the sin θ (t) range between 0.174 and 0.5, are considered. Note that in some cases, the elevation angle range may be set to be between 5 and 25° (or 30°), depending on the signal quality. Calculating the combined error phase with a number of different antenna heights produces a sequence of spectral peak frequencies. Figure 8.8 shows the results for the relationship between antenna height and spectral peak frequency under the specific elevation range. It can be seen that the antenna height can be well modelled as a linear function of the spectral peak frequency (through least squares fitting) given by h = 0.1318 × f + 0.0163

(8.25)

where h (with unit of meter) is the height of antenna, f is the peak frequency of combined phase time series, which is dimensionless or with the unit of meters-permeter. Accordingly, the snow depth estimate is calculated by 7 6 5

Antenna Height (m)

Fig. 8.8 Linear relationship between antenna height and spectral peak frequency obtained using least squares fitting

4 3 2 1 10°~30° (simulated) 10°~30° (linear model)

0 -1

0

10

20 40 30 Spectral Peak Frequency (m⋅m-1)

50

206

8 Snow Depth and Snow Water Equivalent Estimation

dsnow = h 0 − (0.1318 × f + 0.0163)

(8.26)

where h 0 is the snow-free antenna height, which can also be estimated using (8.25) when the ground surface is not covered by snow. Note that the model is general-purpose as long as the surface is flat and the variation pattern of AAF is very similar. In the case where the AAF is significantly different due to the difference in antenna gain pattern or snow permittivity, or antenna height is out of the range tested, it is necessary to establish a new model. As antenna height increases, the useful range of sin θ (t) decreases. That is, one needs to select an appropriate elevation range based on the phase error series pattern when antenna height is assumed to be between the original snow-free antenna height and the relative height above possible maximum snow depth. The ground surface, even if planar, is often slightly tilted around an antenna, so that the reflector height is not comparable at different azimuths. If the tilted angle is known, its effect can be included in the analysis. On the other hand, in the presence of unknown surface slope, one needs to make use of observations related to multiple ranges of azimuths such as from multiple satellites or associated with the rising and setting arcs of the same satellite, in order to mitigate the surface slope effect. In the following section, experimental data is applied to evaluate the performance of the model (8.26).

8.3.5 A Practical Example The data were collected by the CORS STK2 in Japan with latitude and longitude coordinates (+43°31 43.12 , +141°50 41.35 ). STK2 is equipped with a Trimble NetR9 GPS receiver and a geodetic antenna (TRM59800.00), which can receive and process the GPS triple-frequency signals. STK2 does not have daily snow depth observations, but there are four Automatic Meteorological Data Acquisition System ultrasonic snow depth sensors located at Takigawa, Atsuta, Bibai, and Tags around STK2 as shown in Fig. 8.9a and Fig. 8.9b. Their snow depth observations can be downloaded from the website [9] which is operated by Japan Meteorological Agency (JMA). In order to obtain the ground-truth snow depth at STK2, one way is to interpolate the daily snow depth observations at the four stations. However, although STK2 does not have daily snow depth records, there are eight in situ measurements of snow depth after new snowfall over each of the two seasons. If using the corresponding sixteen observations at each of the four stations as the ground truth at STK2, then the mean error and the mean absolute error of snow depth are shown in Table 8.2. Also shown in Table 8.2 are the distances between STK2 and the four other stations. It can be seen that there is no specific relationship between distance and snow depth error, so it would be inappropriate to use distance-based interpolation to estimate the ground truth. However, the measurements at Bibai have the best match with those at STK2, probably because both snowfall and environmental conditions are very similar at

8.3 Triple-Frequency Carrier Phase Combination Based Method

207

Fig. 8.9 a Locations (latitude and longitude) of Japan’s STK2 station and four stations providing ultrasonic sensor based snow depth observations. b Elevated terrain around the five stations Table 8.2 Distance and snow depth difference between STK2 and other four stations Takigawa

Babai

Atsuta

Distance to STK2 (km)

14.28

31.26

59.89

Tags

Mean Error (mm)

−5.8

0.19

16.8

−15.2

Mean Absolute Error (mm)

12.6

4.1

20.4

16.8

8.20

208

8 Snow Depth and Snow Water Equivalent Estimation

these two stations. For this reason, the daily snow depth observations at Bibai are directly used as the ground-truth snow depth at STK2. Because the distance from STK2 to Bibai is 31.26 km, the snow depth difference between the two stations can be greater for other days in the two winter seasons. However, the difference would not be too much since the sixteen known snow depths are approximately evenly distributed over the two seasons. The SNR and phase observations at STK2 were recorded from 13 July 2012, while snowfall in this region in 2012 began in mid-November. Observation data over two winter seasons (2012–2013, 2013–2014) were employed for this study as shown in Fig. 8.10. The sixteen in situ snow depth measurements at STK2 are also shown in the figure. Phase measurement time series of the triple-frequency signals are extracted for a period equivalent to the range of the elevation angle or sin θ (t). The triple-frequency carrier phase combination is then performed by using (8.15). As mentioned earlier 200 180 160

Snow Depth (cm)

140

T akigawa

(a)

Bibai Atsuta T ags Observations of ST K2

120 100 80 60 40 20

0 01 Nov 2012

01 Dec 2012

01 Jan 2013

01 Feb 2013

01 Mar 2013

01 Apr 2013

01 May 2013

180 160 140

Snow Depth (cm)

120

T akigawa

(b)

Bibai Atsuta T ags Observations of ST K2

100 80 60 40 20

0 01 Nov 2013

01 Dec 2013

01 Jan 2014

01 Feb 2014

01 Mar 2014

01 Apr 2014

01 May 2014

Time (day)

Fig. 8.10 Daily snow depth observations at four nearby stations (Takigawa, Atsuta, Bibai, and Tags) and eight snow depth observations at STK2 in the 2012–2013 and 2013–2014 winter seasons

8.3 Triple-Frequency Carrier Phase Combination Based Method

209

in this chapter and Chap. 4, Lomb-Scargle spectral analysis is applied to obtain the periodogram of the combined carrier phase series, since the specified time variable sin θ (t) is unevenly sampled. From the periodogram, the spectral peak frequency is obtained, and finally the snow depth estimate is generated by (8.26). Figure 8.11 shows the snow depth estimation errors in the 2012–2013 and 2013– 2014 snowfall seasons using three different methods (SNR, dual-frequency combination and triple-frequency combination). The antenna height relative to the bareground reflector was estimated during the summer and a sequence of the estimates were produced and averaged to generate the final estimate of bare-ground antenna height. One apparent observation from Fig. 8.11 is that all three methods underestimate the ground-truth snow depth. Table 8.3 shows the mean error, error standard deviation (STD) and RMSE of the three methods over the two snowfall seasons. The results indicate that the estimation accuracy of the first season is very similar to that of the second season for all three methods, showing good consistency. It can be seen that a better match occurs at the beginning and ending of snowfall process (early November and late April), while the estimation error is relatively

Snow Depth Error (mm)

20 0 -20 -40 -60

SNR L4 Proposed

-80 1 Nov 2012 1 Dec 2012 10

(a) 1 Jan 2013

1 Apr 2013 1 May 2013

SNR L4 Proposed

0 Snow Depth Error (mm)

1 Feb 2013 1 Mar 2013

-10 -20 -30 -40

(b)

-50 -60 1 Nov 2013 1 Dec 2013

1 Jan 2014

1 Feb 2014 1 Mar 2014 Time (day)

1 Apr 2014 1 May 2014

Fig. 8.11 Snow depth estimation errors of the SNR, dual-frequency and triple-frequency combination methods based on data collected in the 2012–2013 and 2013–2014 winter seasons

210

8 Snow Depth and Snow Water Equivalent Estimation

Table 8.3 Mean, STD and RMS of errors in snow depth estimation for the SNR, dual-frequency and triple-frequency phase combination methods Method

Seasons

STD (mm)

RMSE (mm)

SNR method

2012–2013 2013–2014

−8.84

6.15

10.76

−8.07

5.83

Dual-frequency

9.95

2012–2013

−14.86

8.83

17.28

Triple-frequency

2013–2014

−14.80

8.28

16.94

2012–2013

−7.33

5.10

2013–2014

8.92

−7.05

5.34

8.83

Table 8.4 Mean and STD of the snow depth underestimation percentage of the three algorithms

Mean (mm)

Method

Mean (%)

STD (%)

SNR method

19.03

15.44

L4 method

29.77

17.15

Proposed method

17.34

14.44

larger over the middle period of the snowfall season. The main reason would be that the snow depth is virtually zero in early November and late April, so the ground truth is in fact error free. On the other hand, non-trivial errors in the assumed ground truth would be produced when snow depth varied significantly, which could be one of the main estimation error sources in this case. Table 8.4 shows the mean and STD of the underestimation percentage of the three algorithms, which is defined as the ratio of snow depth estimation error over the snow depth ground truth. The underestimation is more significant, probably because when the snow depth is small, the underestimation percentage can be large. Besides, the reason behind the underestimation of the snow depth shown in Fig. 8.11, and the results in Table 8.3 and Table 8.4 could be the penetration of the L-band signal through the media. It will be useful to evaluate and compare the performance of the different methods using in situ snow depth measurements at the GNSS station as ground truth. When the snow depth suddenly increases or decreases a lot, the estimation error may become large due to significant changes in the snow surface characteristics such as roughness or surface slope, which may be caused by wind. Both the detrended signal SNR and the error phase change as the surface slope changes. The snow depth was recorded once a day at a specific time point, while the snow depth estimation was performed based on signal observations over a specific period of about half an hour. Such a difference may also contribute to the estimation error.

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

211

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation There are two different schemes related to the combination of pseudorange and carrier phase, single-frequency combination and dual-frequency combination, depending on the availability of data. Both of them can be used for snow depth estimation [12, 29], which are studied in this section.

8.4.1 Single-Frequency Combination Based Method 8.4.1.1

Combination of Single-Frequency Pseudorange and Carrier Phase

Denote the pseudorange and carrier phase observations as ρ(t) ˜ and ϕ(t), ˜ which can be written as [8], ⎧ ˜ = ds,r (t) + δ I (t) + (t) + (t) ⎪ ⎨ ρ(t) λϕ(t) ˜ = λ(ϕ(t) ˜ + N ) − λN (8.27) ⎪ ⎩ = ds,r (t) − δ I (t) + β(t) + (t) − λN where ds,r (t) is the Euclidian distance between the satellite and the receiver; δ I (t) is the ionospheric delay; and (t) accounts for all other effects including clock errors and tropospheric delays; N is the integer ambiguity of carrier phase observations; (t) and β(t) are the psedorange multipath error and carrier phase multipath error in meters, which can be written as [1, 18]: 2h sin θ (t) · α(t) · cos δφ (t) 1 + α(t) cos δφ (t)

α(t) sin δφ (t) λ · tan−1 β(t) = 2π 1 + α(t) cos δφ (t) (t) =

(8.28)

where h is the antenna height above the snow surface; θ (t) is the GNSS satellite elevation angle; λ is the GNSS signal wavelength; α(t) is the amplitude attenuation factor (AAF), which is relevant to the elevation angle, complex permittivity of snow and GNSS antenna gain [17]; and δφ (t) = (4π h/λ) sin θ (t). Note that the third equation in (4.6) and the second equation in (8.28) are the same, but the unit for the former is radians and the unit for the latter is the same for the wavelength (i.e. meters). The single-frequency pseudorange and carrier phase combination (SF-PCC) is defined as:

212

8 Snow Depth and Snow Water Equivalent Estimation

M1 (t) = ρ(t) ˜ − λϕ(t) ˜

(8.29)

Substituting (8.27) into (8.29), the Euclidean or physical distance d s,r and are removed, yielding: M1 (t) = (t) − β(t) + 2δ I (t) + λN

(8.30)

In the absence of cycle slip in the raw carrier phase observation data, the last term in (8.30) related to integer ambiguity is a constant, so it does not affect the frequency of the combined signal. Compared with the combined signal of pseudorange multipath error and carrier phase multipath error, the ionospheric delay has a much lower frequency, and can be well recovered by low-pass filtering or ionospheric delay modeling and then removed by subtraction. By dropping those parameters, (8.30) becomes, M1 (t) = (t) − β(t)

(8.31)

where the residual error is not considered, which is mainly from the incomplete removal of the ionospheric delay and other un-modelled errors and noise. Thus, the combination of raw pseudorange and carrier phase observation is equal to the combination of pseudorange multipath error and carrier phase multipath error. Figure 8.12 shows examples of the SF-PCC series with respect to sin θ (t) under five antenna heights. The typical complex dielectric constant of snow surface and the radiation pattern of GNSS antenna of TRM55971.00 are used to calculate the combined multipath error. As observed, the period of the time series is basically constant for a given antenna height. It is a damping oscillation with the amplitude gradually attenuating with increased sin θ (t) in general mainly due to the antenna radiation pattern and the RHCP component of reflected signal being reduced with increased elevation angle. The series generated from the combination of real observations of pseudorange and carrier phase of GPS L1 signal is also shown, when the antenna height above the snow surface is 2.98 m. Clearly, the results show good agreement between the theoretical time series calculated by (8.31) and the combination of the real pseudorange and carrier phase observations. Figure 8.13 shows an example of the RMS of the peak frequency estimation error with respect to the peak power spectral density of the combined series for GPS L1 signal when the antenna height is 3.5 m. It can be seen that the RMS of the peak frequency estimation error significantly decreases with the increase of the peak power spectral density and similar results were mentioned earlier. The correlation coefficient between the peak power spectral density and the RMS of the peak spectral frequency estimation error is 0.9739, indicating a strong linear correction between them. This observation is helpful, because it is possible to estimate the accuracy of snow depth estimation roughly based on the peak power spectral density in a way similar to the SNR based method studied earlier.

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

213

Fig. 8.12 Time series of combined pseudorange and carrier phase errors for GPS L1 signal

Fig. 8.13 RMS of the peak spectral frequency estimation error with respect to the peak power spectral density

214

8.4.1.2

8 Snow Depth and Snow Water Equivalent Estimation

Model Development

Similar to the dual-frequency and triple-frequency carrier phase combination methods, a theoretical model needs to be established to describe the relationship between the spectral peak frequency (i.e. main frequency) and the antenna height above the reflection surface. Considering the use of CORS data for snow depth measurement, the height of a CORS antenna would usually range from 3.0 to 6.0 m, so the relative antenna height above the snow surface can set to be between 2 and 6 m. The radiation pattern of antenna TRM55971.00 is considered, the snow permittivity is set to be 2.025 − 0.0005j, and the elevation angle series ranges from 5 to 30°. The combined multipath error series are then calculated theoretically for a number of relative antenna heights. The Lomb-Scargle spectrum analysis is carried out over the combined error series under each antenna height to obtain the main frequency. Figure 8.14 shows the antenna height with respect to the main frequency of the combination of GPS L1 pseudorange and carrier phase errors. Clearly, the relationship between antenna height and main frequency can be well described by a linear function. The least squares fitting was used to obtain the linear function as: h i = a × fi + b

(8.32)

where f i is the main frequency of the combined series associated with the GNSS signal of specific frequency, and a and b are the fitting parameters. Table 8.5 shows the fitting parameters for different GNSS signal frequency and the fitting errors. Note that, although those models were developed based on a special antenna gain pattern (TRM55971.00) and reflection surface (dry snow), the antenna gain pattern and properties of reflection surface of snow (i.e. complex dielectric constant of snow) have marginal effect on the main frequency of the combined series, as shown in

Fig. 8.14 Linear relationship between antenna height and peak frequency

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

215

Table 8.5 Fitting parameters for GPS, BDS, and Galileo Satellite constellations

GNSS signal band

a (m)

b (m)

RMSE (mm)

GPS

L1

0.0951

0.0016

0.5

L2

0.1221

0.0026

0.7

BDS

B1

0.0960

0.0005

0.3

B2

0.1241

0.0031

0.7

E1

0.0951

0.0016

0.5

E5

0.1257

0.0037

0.9

Galileo

Table 8.6 Fitting parameters for GPS L1 band signal with different complex dielectric constant of snow surface and antenna gain pattern Antenna type

Complex dielectric constant of snow surface a (m)

TRM41249.00 2.025 − 0.0005j

b (m)

RMSE (mm)

0.0951 0.0016 0.5

9.50 − 1.80j

0.0951 0.0018 0.3

TRM55971.00 2.025 − 0.0005j

0.0951 0.0015 0.8

9.50 − 1.80j

0.0951 0.0016 0.5

Table 8.6. The fitting parameters of the linear model for GPS L1 signal are very similar under two significantly different antenna gain patterns and snow surface dielectric constants (dry snow and wet snow). The error of antenna height estimations caused by the difference in antenna gain pattern and dielectric constant is smaller than 1 mm. As shown in Fig. 8.13, the RMS of the main frequency estimation error is inversely proportional to the peak power spectral density. Accordingly, the accuracy of GNSSbased antenna height and hence snow depth estimation is proportional to the peak power spectral density. Thus, in the presence of observations related to multiple satellites, an improved snow depth estimate can be obtained as the weighted sum of the individual estimates: h = H −

1 n 

· pi

n 

( pi h i )

(8.33)

i=1

i=1

where pi is the peak power spectral density of the ith satellite; H is the antenna height when the ground is snow-free. The observation periods of the satellite signals are not required to be exactly the same, but the snow depth variation over space and observation duration should be negligible. Otherwise, the estimated snow depth would be the mean snow depth over the space and duration. Such a weighting scheme is simpler than that studied in the section of SNR based method, and a better scheme may be considered if a considerable accuracy gain can be achieved.

216

8.4.1.3

8 Snow Depth and Snow Water Equivalent Estimation

Removal of Ionospheric Delay

The combination performed by (8.29) only removes the effect of geometry, so further processing is required to remove the effect of ionosphere. The frequency of the ionospheric delays is lower than 0.1 MHz, whereas the spectrum of combined multipath error is significant in the range of 1 MHz to several tens of mHz [19]. This means that the ionospheric delays can be estimated by a low-pass filter such as the moving average method. Consider the combined raw observations (see 8.30) with k samples from sample t − k + 1 to sample t, the moving average produces t 1 1  M1 (i) = k i=t−k+1 k



t 

((i) − β(i)) +

i=t+k−1

t 

(2δ I (i) + λN )

(8.34a)

i=t−k+1

where the average of the k multipath error samples would be a small number close to zero and the k samples of the ionospheric delay would be very similar. Thus, (8.34a) can be written as 2δ I (t) + λN =

t 1  M1 (i), t ≥ k k i=t−k+1

(8.34b)

Then, the term (2δ I (t) + λN ) in (8.30) is removed by subtracting the calculated results of (8.34b) from the combined error series, producing the combination of free-geometry and free-ionospheric delay: (t) − β(t) = M1 (t) −

t 1  M1 (i), t ≥ k k i=t−k+1

(8.35)

Figure 8.15 shows an example of the time series of the raw combinations (8.29) and the ionoshperic delay related term estimated by (8.34b) with respect to time t (seconds). The sampling rate of GNSS observations is 1 Hz and the length of moving window k is 310 epochs. As a descending satellite, the ionospheric delays increase with the increasing of the total electron content (TEC) on the propagation path. It can be seen that the presence of ionospheric delays results in an ascendant trend for the raw combinations. The estimated ionospheric delay fluctuates with a linear increasing trend as the elevation angle decreases. The fluctuation may be caused by the effect of the combined multipath error in addition to the variation in the TEC. Figure 8.16 shows the combined multipath error series estimated from real data by (8.35), and the simulated one with the real antenna height of 2.98 m. It can be observed that the estimated and simulated results have a good match in terms of oscillating trend and frequency. Note that, the raw multipath error combinations with elevation angle differences smaller than 0.1° could be treated as one combination with a given elevation angle. Then, mean filtering method could be used to improve signal-to-noise ratio of raw

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

217

Fig. 8.15 Example of the time series of raw combinations for BDS B1 signal and the estimated time series of 2δ I (t) + λN

Fig. 8.16 Time series of raw combination of multipath error (red), mean filter result (green), simulated result (black), and the spectrogram for those series

218

8 Snow Depth and Snow Water Equivalent Estimation

multipath error combinations. It can be seen from Fig. 8.16 that the mean filtering generates a higher peak spectral density, and the spectral peak frequency is closer to the simulated one. This is in accordance with the theoretical analysis related to Fig. 8.13. Note that the violent fluctuation of ionosphere, which is often caused by solar flares, earthquake, or hurricane, might lead to the failure of effective ionospheric removal.

8.4.1.4

Experimental Results

As mentioned in Sect. 8.1, an experiment was conducted in northeast China, from December 26, 2017 to January 17, 2018. The Trimble NetR9 GNSS receiver equipped with GNSS antenna TRM55971.00 was used for data collection. As shown in Fig. 8.17, the GNSS antenna was fixed on a pole which was installed in a flat and open area. The height of the GNSS antenna is 3.43 m when the surface is snow-free. GNSS data of 14 satellites with an elevation angle range of 5–30° are used to evaluate the performance of the snow depth estimation method based on SF-PCC. Also shown are the footprints of specular reflection for those satellites when the antenna height is 3.43 m. A wooden ruler of one-meter length was used to measure the ground-truth snow depth three times a day. Since the difference between the three individual rulerbased daily snow depth measurements is smaller than 1.5 cm for most of the days, the average is treated as the daily in situ snow depth. Figure 8.18 shows the in situ snow depth observations and the daily average snow depth estimations obtained by the SNR method and the SF-PCC method for BDS, GPS, and Galileo signals.

Fig. 8.17 Local environment around the GNSS receiver. The dash and solid line represent the specular reflection footprint of descending satellite and ascending satellite respectively

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

219

Fig. 8.18 GNSS-R based daily average snow depth estimations and in situ snow depth observations

Table 8.7 Mean, STD and RMS of BDS-based snow depth estimation error Method

Band

Mean(cm)

STD(cm)

RMS(cm)

SNR

B1

−4.45

1.23

4.61

B2

−4.06

1.34

4.28

SF-PCC (NA)

B1

−3.09

2.22

3.81

B2

−1.90

2.01

2.77

SF-PCC (WA)

B1

−2.60

2.40

3.54

B2

−1.61

1.75

2.38

Table 8.7 shows the mean, STD and RMS of BDS-based snow depth estimation errors of SNR method and the SF-PCC method with two different schemes, normal average (NA) and weighted average (WA). It can be observed that for both methods, there exists an obvious negative mean error. The reason behind this underestimation is the interference of reflected GNSS signals coming from underneath of snow surface [29]. The SF-PCC method has a smaller mean error than that of the SNR method, probably because the GNSS signals reflected from underneath the snow surface has a smaller effect on the pseudorange and carrier phase multipath errors [13]. Certainly, it is useful to make a correction for the systematic error to improve the accuracy of snow depth estimation. However, the systematic error is related to the characterization of snow (e.g., density, depth, and temperature) and GNSS signals (e.g., wavelength or frequency, and signal strength), indicating a complex mechanism. It is thus desirable to establish the relationship between the systematic error and the different factors. The WA scheme of the SF-PCC method performs better than the NA scheme for both frequency signals, although the improvement is not very significant. In addition,

220

8 Snow Depth and Snow Water Equivalent Estimation

the accuracy with B2 signal is considerably better than that with B1 signal for the SF-PCC method. The main reason would be that the bandwidth of the B2 signal is 20.46 MHz, much larger than the 4.092 MHz of B1 signal. A higher signal bandwidth will result in a larger SNR. Table 8.8 shows the error statistics of GPS-based snow depth estimation with L1 and L2 signals. It can be seen that the snow depth estimation error statistics (mean, STD, and RMS) of the GPS L2 signal are significantly larger than those of the L1 signal for both methods. The main reason for this is that the strength of legacy GPS signal L2P(Y) is considerably lower than that of the L1 signal strength, as shown in Fig. 8.19. Therefore, the accuracy of the SNR method and the SF-PCC method with L1 signal is better than that with L2P(Y) signal. Because the strength of modernized GPS signal L2C is much higher than that of L2P(Y) signal [24], the performance of the former would be better for both methods. Table 8.9 shows the mean, STD, and RMS of snow depth estimation error when using Galileo E1 and E5 signals. The use of E5 signal produces better performance than using the E1 signal for the SF-PCC method. The performance gain comes from Table 8.8 Mean, STD and RMS of GPS-based snow depth estimation error Method

Band

Mean (cm)

STD (cm)

RMS (cm)

SNR

L1

−2.54

1.88

3.16

L2

−6.01

3.48

6.95

L1

−2.48

1.83

3.08

L2

−1.45

3.78

4.05

L1

−2.06

1.71

2.68

L2

−1.47

2.93

3.28

PM-NA PM-WA

Fig. 8.19 SNR observations for an ascending GPS satellite (#25)

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

221

Table 8.9 Mean, STD and RMS of Galileo-based snow depth estimation error Method

Band

Mean (cm)

STD (cm)

RMS (cm)

SNR

E1

−3.36

1.77

3.80

E5

−2.29

2.76

3.59

PM-NA

E1

−3.46

2.19

4.09

E5

−1.58

1.96

2.52

E1

−2.78

2.58

3.79

E5

−1.61

1.85

2.45

PM-WA

the smaller pseudorange measurement error with the E5 signal. As studied by Braasch et al. [3], the accuracy of pseudorange measurement is inversely proportional to the pseudorandom noise code period, which is 244.4 ns for C1X of the Galileo E1 signal and 97.8 ns for C8X of the Galileo E5 signal. The smaller code period produces smaller measurement noise and more accurate snow depth estimations.

8.4.2 Dual-Frequency Combination Based Method 8.4.2.1

Dual-Frequency Pseudorange and Carrier-Phase Combination

Dual-frequency pseudorang and carrier phase combination (DF-PCC) of GNSS signals is defined as [27], M1,2 (t) = ρ˜1 (t) + κ1 λ1 ϕ˜1 (t) + κ2 λ2 ϕ˜ 2 (t)

(8.36)

where ρ˜1 (t) is the measured pseudorange of first frequency signal (e.g. BDS B1) in meters; ϕ˜1 (t) and ϕ˜ 2 (t) are the measured carrier-phase of the dual-frequencysignals (e.g. BDS B1 and B2) in cycle respectively; λ1 and λ2 are the wavelengths of the dual-frequency signals in meters; and κ1 =

λ21 + λ22 −2λ2 , κ2 = 2 1 2 2 2 λ1 − λ2 λ1 − λ2

(8.37)

The pseudorange and carrier-phase observations in (8.36) can be written as ⎧ ρ˜1 (t) = ds,r (t) + δ I 1 (t) + (t) + (t) ⎪ ⎪ ⎪ ⎪ ⎪ λ ϕ ⎪ ⎨ 1 ˜1 (t) = λ1 (ϕ˜1 (t) + N1 ) − λ1 N1 = ds,r (t) − δ I 1 (t) + β1 (t) + (t) − λ1 N1 ⎪ ⎪ ⎪ λ2 ϕ˜2 (t) = λ2 (ϕ˜ 2 (t) + N2 ) − λ2 N2 ⎪ ⎪ ⎪ ⎩ = ds,r (t) − δ I 2 (t) + β2 (t) + (t) − λ2 N2

(8.38)

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8 Snow Depth and Snow Water Equivalent Estimation

where the parameters have the same definitions as defined for (8.27) except that subscript is used to specify the frequency band. As studied in [8] and mentioned earlier, the ionospheric delay is inversely proportional to the squared carrier frequency, thus, proportional to the squared carrier wavelength: λ2 δ I1 (t) = 12 δ I2 (t) λ2

(8.39)

By substituting (8.38) and (8.39) into (8.36), the Euclidean distance, ionospheric delays and (t) have been removed, yielding: 

M1,2 (t) = 1 (t) + κ1 β1 (t) + κ2 β2 (t) − u u = κ1 λ 1 N 1 + κ 2 λ 2 N 2

(8.40)

Therefore, the DF-PCC results are also related to the multipath-induced pseudorange error and carrier-phase error. Provided that the cycle slips are compensated, the parameter u is a constant so it does not affect the frequency of the combined signal. By dropping this parameter, the DF-PCC can be rewritten as M1,2 (t) = 1 (t) + κ1 β1 (t) + κ2 β2 (t)

(8.41)

Thus, the DF-PCC is actually equal to the combination of multipath-induced single-frequency pseudorange error and dual-frequency carrier-phase error. Figure 8.20 shows an example of the time series of DF-PCC with respect to sin θ (t), calculated by (8.41). The typical dry snow permittivity and the antenna gain pattern of TRM55971.00 are used for the calculation. For a given antenna height, the DF-PCC error signal oscillates with a nearly constant period, but due to the effect of the AAF (see Chap. 4), the amplitude gradually decreases with sin θ (t) in general. The oscillation patterns of the error signal under different antenna heights can be very different. Notice that the real measurements of the DF-PCC data are also shown in Fig. 8.20, when the antenna height is 2.89 m. The results clearly show that the theoretical formula given by (8.41) is a good model for describing the actual DF-PCC. Figure 8.21 shows an example of the spectrogram of the simulated DF-PCC time series added with white Gaussian noise when antenna height is 3.5 m. It can be seen that the maximum of power spectral density reduces significantly with the increase of noise level, and the curves of power spectral density are flattened to some degree. The spectral peak frequencies of the DF-PCC with noise deviate from the spectral peak frequency without noise, indicating the influence of the noise; the deviation is inversely proportional to the maximum power spectral density in general. This observation is useful, because we could roughly estimate the errors of snow depth estimation of the proposed method based on the maximum power spectral density as mentioned in the study of SNR method earlier. Thus, the accuracy of snow depth estimation can be improved by weighting each snow depth estimate differently. In

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

223

Fig. 8.20 Time series of DF-PCC error signal using the combination of BDS B2 pseudorange, B2 and B1 carrier phases when the antenna height is 2.50 m, 2.89 m, 3.50 m, 4.00 m and 4.50 m

addition, the maximum power spectral density shows the quality of the error signal, so that selecting the error signal larger power spectral density will improve the accuracy of GNSS-based snow depth estimation.

8.4.2.2

Theoretical Modeling

As studied in [28], the triple-frequency carrier phase combination generates a quasisinusoidal signal with respect to sin θ (t) and has a spectral peak frequency proportional to antenna height. It can be derived that the DF-PCC also yields a quasisinusoidal signal with a similar spectral peak frequency. Figure 8.22 shows an example of the spectrogram of the DF-PCC time series displayed in Fig. 8.21. Clearly, a single spike occurs in the spectrum of each error time series and the spectral peak frequency can be readily determined. Notice that, because of the existence of observation noise, the power spectral density of observed data is reduced to some degree; however, the spectral peak frequency of observed data is very close to the simulated one.

224

8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.21 Spectrogram of DF-PCC error series added with white Gaussian noise using BDS B2 pseudorange, B2 and B1 carrier phases when antenna height is 3.5 m

Fig. 8.22 Spectrogram of DF-PCC error series when using BDS B2 pseudorange, B2 and B1 carrier phases under five different antenna heights, and elevation angle range is 5–30°

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

225

Fig. 8.23 Linear relationship between antenna height and the spectral peak frequency of DF-PCC using least squares fitting for the combination of BDS B2 pseudorange, B2 and B1 carrier phases

Although the spectral peak frequency can be clearly observed, it is a challenging problem to derive a mathematical formula to calculate it. However, since a different antenna height corresponds to a different spectral peak frequency as observed in Fig. 8.22, it is possible to develop a model to describe the relationship between antenna height and peak spectral frequency similar to the methods of triple-frequency carrier phase combination and the SF-PCC. As shown in Fig. 8.23, the antenna height can be well modeled as a linear function of the spectral peak frequency. By using least squares fitting, the relationship between antenna height and spectral peak frequency of the DF-PCC error signal can be established as, h =a× f +b

(8.42)

where f is the spectral peak frequency, a and b are the fitting coefficients. Then, the relationship between snow depth and the spectral peak frequency can be written as h = H − (a × f + b)

(8.43)

Table 8.10 shows the fitting coefficients for three different satellite constellations and the fitting errors in terms of root-mean-square error (RMSE) of the linear model. After obtaining the spectral peak frequency of the DF-PCC data by exploiting LombScargle spectral analysis [26], the snow depth can be readily calculated by (8.43).

226

8 Snow Depth and Snow Water Equivalent Estimation

Table 8.10 The fitting coefficients for GPS, BDS and Galileo GNSS system

GNSS observation

GPS

L1 L2

L2

BDS

B1

B1

B2

B2

E1 E5

Galileo

8.4.2.3

L1

a (m) L2

b (m)

RMSE (mm)

0.0951

0.0041

1.1

L1

0.1222

−0.0066

1.6

B2

0.0959

0.0068

1.5

B1

0.1242

−0.0055

1.5

E1

E5

0.0951

0.0024

0.9

E5

E1

0.1258

−0.0046

1.2

Experimental Results

Two field data sets collected in two different environments are used to evaluate the DF-PCC method. The first data was collected in an experimental campaign conducted in a village close to Mudanjiang, Heilongjiang Province, China, over 23 days in the winter, which was already used for evaluating the SF-PCC method. The second data set was collected from IGS (International GNSS service) station STK2 in Takigawa, Hokkaido, Japan, over a longer period (up to 7 months), which was also used for evaluating the triple-frequency carrier phase combination method. Table 8.11 shows the GNSS satellite signals, the pseudoranges and/or carrier phases which are used to estimate snow depth. Figure 8.24 shows the daily snow depth estimation error over a period of 23 days obtained by the DF-PCC method with observation data of BDS B1 and B2 pseudorange and carrier phase of four different satellites. Two different estimates were produced for each satellite since either B1 or B2 pseudorange can be used in the combination of pseudorange and carrier phase. Table 8.12 shows the error statistics of the snow depth estimation with BDS signals. Note that PR and CP denote the pseudorange and carrier-phase respectively. It can be seen from Fig. 8.24 and Table 8.12 that there exists a significant negative mean error for all satellites in two different combinations, similar to what observed in triple-frequency combination method and the SF-PCC method. The use of B1 pseudorange produces significantly worse mean error than the use of B2 pseudorange. This is probably because the penetration depth Table 8.11 The GNSS observation codes for GPS, BDS and Galileo

GNSS

Band

Signals Pseudorange

Carrier phase

GPS

L1

C1C

L1C

L2

C2W

L2W

BDS

B1

C2I

L2I

B2

C7I

L7I

E1

C1X

L1X

E5

C8X

L8X

Galileo

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

227

Fig. 8.24 Snow depth estimation errors obtained by using the combination of BDS B1 pseudorange, B1 and B2 carrier phases (red), and the combination of B2 pseudorange, B2 and B1 carrier phases (blue), respectively

Table 8.12 Mean, STD and RMS of errors in snow depth estimation with BDS signals Satellites

Combinations (PR, CP, CP)

Mean (cm)

STD (cm)

RMSE (cm)

C07

B1, B1, B2

−4.16

1.86

4.55

B2, B2, B1

−0.99

2.35

2.55

B1, B1, B2

−2.67

2.84

3.89

B2, B2, B1

−1.71

2.69

3.19

C10

B1, B1, B2

−6.29

2.49

6.76

B2, B2, B1

−3.67

2.46

4.41

C13

B1, B1, B2

−2.60

3.29

4.19

B2, B2, B1

−2.42

3.02

3.88

C08

is inversely proportional to the wavelength [2] and the wavelength of B2 signal is longer. This may also be due to that the SNR of B2 signal is a bit higher than that of B1 signal. Table 8.13 shows the mean error, error STD and RMSE of the snow depth estimates of DF-PCC method using GPS signals. Clearly, the error of GPS signals is significantly larger than that of BDS signals, probably because the error in GPS pseudorange measurement is larger. The accuracy of pseudorange measurement is proportional to the SNR, which is about 38 dB-Hz, 40 dB-Hz and 43 dB-Hz for GPS L1, BDS B2 and Galileo E5 band signals in the elevation angle range of 5–30°; and

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8 Snow Depth and Snow Water Equivalent Estimation

Table 8.13 Mean, STD and RMS of errors in GPS snow depth estimation Satellites

Combinations (PR, CP, CP)

Mean (cm)

STD (cm)

RMSE (cm)

G02

L1, L1, L2

−2.85

5.58

6.26

L2, L2, L1

−4.13

8.51

9.46

G18

L1, L1, L2

−5.96

3.72

7.03

L2, L2, L1

3.35

3.57

4.89

L1, L1, L2

−4.37

3.45

5.56

L2, L2, L1

−1.23

5.89

6.02

G24

L1, L1, L2

−5.52

3.49

6.53

L2, L2, L1

0.48

4.53

4.55

G25

L1, L1, L2

−3.03

3.68

4.76

L2, L2, L1

3.21

6.67

7.40

G20

is inversely proportional to the PRN (pseudorandom noise) code chipping period [3], which is 977.6 ns for GPS L1 signal, 488.8 ns for BDS B1 and B2 signals, and 97.8 ns for Galileo E5 signal. The higher SNR and smaller code chipping period produce smaller measurement noise and a cleaner DF-PCC error time series. In addition, the GNSS receiver may use different signal processing algorithm for signals of different constellation, which may contribute to the performance difference. However, as the commercial GNSS receivers do not provide the specific signal processing technique, so more field experiments with different GNSS receivers are needed to verify the assumption. Figure 8.25 shows three typical time series of DF-PCC and their spectrograms. The RMS errors of the observed time series for GPS, BDS, and Galileo in the elevation angle range of 5–30° are 0.38 m, 0.17 m, and 0.10 m, respectively. As the GPSbased time series has larger measurement noise than that of BDS-based one, the GPS-based peak power spectral density is smaller than BDS-based one, degrading peak frequency and snow depth estimation performance. This is in accordance with the simulated example shown earlier. Note that, unlike the SNR and carrier-phase of reflected signals, the multipath-induced pseudorange error is influenced by both AAF and additional path length of reflected signal. The AAF decreases with the increase of elevation angle for typical RHCP antenna, decreasing the amplitude of combined multipath error, while the additional path length increases with the increase of elevation angle, increasing the amplitude of combined multipath error. The conflicting effect of the AAF and additional path length on the attenuating pattern of combined multipath error is complex and non-linear. In general, the amplitude of combined multipath error is attenuating with the increase of elevation angle, although the attenuation rate can be different. Table 8.14 shows the mean, STD and RMS of the snow depth estimation error based on DF-PCC when using observations of E1 and E5 pseudoranges and carrier phases of five different Galileo satellites. Clearly, the estimation errors are considerably smaller than the results shown in Table 8.11 and 8.12. Notice that because of the

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

229

Fig. 8.25 Time series of DF-PCC error and their spectrograms for elevation angle in the range of 5–30°. The pseudorange observations used are C1C in GPS L1 band, C7I in BDS B2 band, and C8X in Galileo E5 band, respectively

Table 8.14 Mean, STD and RMS of errors in Galileo snow depth estimation Satellites

Combinations (PR, CP, CP)

Mean (cm)

STD (cm)

RMSE (cm)

E01

E1, E1, E5

−2.48

1.76

3.05

E5, E5, E1

−1.14

1.96

2.27

E02

E1, E1, E5

−2.05

1.17

2.36

E5,E5,E1

−1.35

1.07

1.73

E12

E1, E1, E5

−2.02

1.48

2.50

E5, E5, E1

−0.53

1.18

1.29

E1, E1, E5

−2.01

2.36

3.10

E5, E5, E1

−0.86

1.24

1.51

E1, E1, E5

−0.89

2.60

2.75

E5, E5, E1

−0.25

1.82

1.84

E24 E26

orbital revolution period of Galileo satellites is about 14 h, and the daily observation time of the experimental campaign was between 9 AM and 16 PM, suitable elevation range may not be available some days for the same satellite. As mentioned before, the higher SNR and shorter PRN code chipping period produces more accurate pseudorange measurements, so the pseudorange measurement and snow depth estimation with C8X (E5) could be more accurate.

230

8 Snow Depth and Snow Water Equivalent Estimation

Figure 8.26 shows the daily averages of the snow depth estimates using the DFPCC method with BDS, GPS, and Galileo signals, respectively. For comparison, the ruler-based ground-truth snow depths are also shown. In general, there is good agreement between the ground truth and the estimates for all three GNSS constellations. As the pseudorange measurement of Galileo signals is more accurate, the snow depth estimation accuracy with Galileo signals is the best. Table 8.15 shows the averaged mean, STD and RMS of the snow depth estimation errors. It is observed that the daily averages of snow depth estimates are closer to the ground truth than most of the single satellite based estimates. Averaging the individual satellite based estimates is a useful method for improving the accuracy of snow depth estimation, as it is usually unknown which satellite produces the best performance in advance. Basically, the deviation of daily average snow depth estimations increases with increased snow depth. This could be seen clearly in Fig. 8.27 which shows the scatterplot of the DF-PCC snow depth estimation errors versus in situ snow depth. This is in accordance with the fact that the DF-PCC error signal amplitude is proportional to antenna height and hence inversely proportional to snow depth. When the true snow depth is between 27.5 and 36 cm, the deviation of the estimates is a bit small, probably because of the relatively small number of the estimated snow depth. The second data set was collected from IGS (International GNSS service) station STK2 in Japan, which does not have daily snow depth observations, but only have a small number of in situ snow depth observations made after new snowfall at each winter season as mentioned earlier. However, as indicated in [28], there is an Automatic Meteorological Data Acquisition System (AMeDAS) ultrasonic snow depth sensor at Bibai, about 30 km south of STK2, and the snow depth measurements at Bibai have a good match with the in situ snow depth measurements at STK2. Thus, the daily averages of snow depth observations at Bibai are used as the ground truth of snow depth at STK2, as did for the evaluation of the triple-frequency carrier phase combination method earlier. As shown in Fig. 8.28, the reflection environment around STK2 is a bit complicated. Since the picture was taken during a spring season, it gives more details about the environment. Clearly, there are some trees and buildings near the station, which might produce multipath interference. Data of ten GPS satellites ascending from south of STK2 are used to evaluate the performance of the DF-PCC method, as there is less interference from trees and buildings in the south of the station, and the ground surface is flat enough in absence of snowpack. Because STK2 stores GPS observations only and the combination of GPS L1 pseudorange, L1 and L2 carrier phases performs slightly better than the combination of L2 pseudorange, L1 and L2 carrier phases in terms of the single satellite based estimates, the snow depth estimation results based on the former combination are presented. Figure 8.29 shows the individual and averaged daily snow depth estimates by the DF-PCC method. For comparison, the ground-truth data produced by the AMeDAS sensor are also displayed. It can be seen that the sharp increase and decrease in snow depth are well captured by the proposed GPS based estimates. The daily standard deviations of the estimates significantly increase with the increase of snow depth. The main reason could be that the GPS signal sampling intervals are rather irregular and do not occur during the same period in a day, whereas the variation of

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

231

Fig. 8.26 Daily averages of the snow depth observations for (a) BDS, (b) GPS and (c) Galileo, respectively. The error bars show the daily standard deviations of GNSS-based estimations

232

8 Snow Depth and Snow Water Equivalent Estimation

Table 8.15 Mean, STD and RMS of eerrors of averages of DF-PCC snow depth estimation GNSS

Combination

Mean (cm)

STD (cm)

RMS (cm)

BDS

B1, B1, B2

−3.93

1.69

4.28

B2, B2, B1

−2.20

1.41

2.61

GPS

L1, L1, L2

−4.38

2.43

5.02

L2, L2, L1

0.37

3.00

3.02

E1, E1, E5

−1.95

1.46

2.43

E5, E5, E1

−0.90

1.25

1.54

Galileo

Fig. 8.27 Scatterplot of DF-PCC snow depth estimation error versus in situ snow depth. The two purple dashed lines show the three times standard deviation of the estimation errors

snow depth may be dramatic in a complex manner in a day, as evidenced in Fig. 8.30. In addition, there are only 10 GPS based estimates, while there are 24 AMeDAS based measurements. Moreover, the snow surface characteristics such as roughness or surface slope, which may be caused by wind or trees, may also contribute to the deviations. In the early and mid-November, the second half of April and the whole May, there was no snow on the ground. However, the DF-PCC method generates nonzero snow depth estimates. This could be caused by the scattering medium which is grass-covered soil in the absence of snow; the height of grass was treated as the snow depth. In general, the comparison in Fig. 8.29 shows good agreement between DF-PCC snow depth estimates and the ground truth. The averaged mean, STD and RMS of the DF-PCC snow depth estimation errors are 1.62 cm, 8.71 cm, and 8.86 cm, respectively. Obviously, the accuracy of the

8.4 Carrier Phase and Pseudorange Combination Based Snow Depth Estimation

233

Fig. 8.28 Local environment around STK2 in the spring season; footprints of specular reflection point of GPS satellites under antenna height of 5.075 m, and elevation angle is 5–30°

DF-PCC method at STK2 is considerably inferior to that with the data collected during the experimental campaign in northeastern China, as shown in Table 8.12. The main reason would be that the signals reflected from the trees or buildings around STK2 interfere with the signals reflected from the snow surface, and hence distort the spectral peak frequency of the combined multipath signals reflected from the snow surface. In addition, the snow depth was recorded once a day at some specific time points, whereas the GPS based snow depth estimates were produced using signal observations over irregular periods, as shown in Fig. 8.30. Such a temporal mismatch may also contribute to the estimation error especially in the presence of significant increase or decrease in snow depth.

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8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.29 Daily averages of the snow depth observations by AMeDAS at Bibai, and individual and averaged daily snow depth estimates by DF-PCC method; The error bars show the daily standard deviations of DF-PCC estimations

Fig. 8.30 The time instants of snow depth measurements for AMeDAS at Bibai (red dot), and the GPS signal sampling period (solid line); snow depth observations by AMeDAS (green dashed lines), and DF-PCC snow depth estimations (green dotted lines)

8.5 Dual Receiver System Based Snow Depth Estimation

235

8.5 Dual Receiver System Based Snow Depth Estimation Considering the case where a typical ground GNSS receiver is used for positioning at a fixed station such as a continuous operating reference station (CORS) in an outdoor environment. Although various techniques are developed to mitigate multipath interference, the ground reflected signal will pass through the antenna even though with amplitude greatly reduced. When the direct signal and the reflected signal are superimposed at the receiving antenna and the receiver front end, they are simultaneously down-converted and then correlated with a replica of the pseudorandom noise code of the satellite produced by the receiver baseband digital signal processing function module, which results in multipath-induced pseudorange and carrier phase errors in GNSS measurements as mentioned in the preceding sections. Basically, there are two different types of reflection [7]. One is the specular reflection, producing a reflected signal scattered at a single angle; the other is the diffuse reflection, resulting in multiple reflected signals scattered at many different angles, the sum of which is often modeled as an additional noise term. The specular reflection has the shortest path for the signal to reach the receiver in the ground-based observation platform when the ground surface is flat, and has a power significantly greater than that of the diffuse reflection signal in the case of a flat reflection surface. Because the surface of interest is the outdoor snow surface, only the specular reflection signal is considered for the theoretical modeling and analysis for simplicity, as did earlier, which is simply called the reflected signal. Due to power loss caused by penetration, absorption and propagation over a longer distance, the power of reflected signal is significantly lower than the direct signal. Figure 8.31 illustrates the dual receiver system in the presence of snow on the ground [30]. A ground-based synchronous observation is conducted with two individual receivers separated not far away from each other. The selection of the distance between the two receivers mainly depends on the ionospheric delay. The satellite signals propagating to the two receivers should have approximately the same propagation path. It would be useful to conduct theoretical study to determine the effect of distance between the two receiver on snow depth estimation accuracy. In the experiment to be discussed later, the distance between a pair of receivers is about 10 m. Two antennas of the same type are respectively used to capture the GNSS signal for the two receivers and there is a considerable height difference between the two antennas to form a dual receiver system. is the height of the taller antenna relative to the snow-free ground, and h is the taller antenna height relative to the snow surface. Snow depth D is the difference between H0 and h. h is the antenna height difference and θ is the satellite elevation angle.

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8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.31 The dual receiver system for snow depth estimation

8.5.1 Combination Methods This section studies two dual-receiver based combination methods. The first one makes use of observations of two single-frequency receivers, while the second one utilizes observations of two dual-frequency receivers. For convenience, they are termed single-frequency combination (SFC) method and dual-frequency combination (DFC) method, respectively. These two methods use the similar combination procedure. That is, the combination between the observations from a single receiver is performed first, and then the combination results from the individual receivers are combined again. It is assumed that the snow depths around the two antennas are the same or the snow depth difference due to the location difference can be neglected. The snow depth measurement system consists of GNSS satellites, two GNSS-R receivers, and two antennas. For each of the two receivers, the observation equation of the carrier phase measurement is [1] λϕ˜ = ρ0 − cVt R + cVt S − λN +λβ − Vion − Vtr op

(8.44)

where ϕ˜ (unit in cycle) is the carrier phase observation multiplied by carrier wavelength λ; ρ0 is the geometric distance between the receiver and the satellite; c is the speed of light; β is the multipath-induced carrier phase error; Vt R is the receiver clock error; Vt S is satellite clock error; N is the integer ambiguity; Vion (unit in meters) is the ionospheric delay; and Vtr op (unit in meters) is the tropospheric delay. Note that except for c and λ, all other parameters in (8.44) are time varying. Because the

8.5 Dual Receiver System Based Snow Depth Estimation

237

ionospheric delay of pseudorange observation and that of carrier phase are equal in value, opposite in sign, the observation equation of the pseudorange measurement is given by ρ˜ = ρ0 − cVt R + cVt S + l + Vion − Vtr op

(8.45)

where ρ˜ is the pseudorange observation, and l is the multipath-induced pseudorange error. Other error sources (e.g. satellite ephemeris errors, and measurement noise) are not modeled for simplicity, but they will incur an additional noise term to the final solution, if they are not removed by differential operation. The two receivers in the dual receiver system make synchronous observation and are close to each other, so that the signal propagation paths from the same satellite to the two receivers at the same time are actually the same. The ionospheric delay is a function of the signal frequency and the Total Electron Content (TEC) on the signal propagation path, so that the ionospheric delay of the same frequency is the same for both receivers under this circumstance. That is (Vion )(1) = (Vion )(2)

(8.46)

In the SFC method, the carrier phase observations and pseudorange observations of each receiver are first combined according to ρ˜ (i) − λϕ˜ (i) = λN (i) − λβ (i) + l (i) + 2(Vion )(i)

(8.47)

Clearly, the subtraction eliminates the geometric effect. The satellite clock error and receiver clock error are also removed by the subtraction since the pseudorange and carrier phase data are received by the same receiver and transmitted by the same satellite. The tropospheric delay is also eliminated since the propagation path associated with pseudorange and that related to carrier phase are exactly the same. Then, performing subtraction between two receivers produces [ρ˜ (1) − λϕ˜ (1) ] − [ρ˜ (2) − λϕ˜ (2) ] = [l (1) − λβ (1) ] − [l (2) − λβ (2) ] + N S FC

(8.48)

Here N S FC = λN (1) − λN (2) is a constant during the observation period in the absence of cycle slip. The combination is repeatedly carried out over a consecutive number of carrier phase and pseudorange observations to generate a combination time series. Note that the combination or subtraction between the two time-series in (8.48) is with respect to the sine of the elevation angle, i.e. sin θ (t). However, the receivers at different locations usually have different elevation angles of the same satellite. To solve this problem, the observation of one receiver with an elevation angle is combined with that of the other receiver with the most similar elevation angle. In reality, the difference between the sines of the two elevation angles would be very small. For instance, the difference between the two elevation angles considered in

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8 Snow Depth and Snow Water Equivalent Estimation

the processing of the data to be discussed later is less than 0.05°. Accordingly, the difference between the sines of the two elevation angles is less than 8.73 × 10−4 . The average magnitude of the difference would be close to zero. In addition, the maximum time difference between the two time-series of (8.48) in experimental data processing is 0.5 s. Thus, the effect of the time lag on the removal of errors by differential operations would be negligible. The DFC method uses dual-frequency carrier phase observations. In the first combination, the carrier phase observations of two different frequencies of each receiver are combined as (i) λ1 ϕ˜1(i) − λ2 ϕ˜2(i) = λ2 N2(i) − λ1 N1(i) + λ1 β1(i) − λ2 β2(i) + (Vion )(i) (2) − (Vion )(1) (8.49)

Clearly, the geometric parameters are completely excluded due to the subtraction of two ranges. In the second combination, the first combination result of the first receiver is subtracted by that of the second receiver, yielding [λ1 ϕ˜1(1) − λ2 ϕ˜2(1) ] − [λ1 ϕ˜ 1(2) − λ2 ϕ˜2(2) ] =[λ1 β1(1) − λ2 β2(1) ] − [λ1 β1(2) − λ2 β2(2) ] + N D FC

(8.50)

Here N D FC = [λ2 N2(1) − λ1 N1(1) ] − [λ2 N2(2) − λ1 N1(2) ] is a constant during the observation period. The combinations established by the two methods are the linear combination of the multipath-induced pseudorange errors and carrier phase errors. Both multipath-induced errors are a function of carrier wavelength, relative antenna height, amplitude ratio (or AAF) and the sine of satellite elevation angle. The sine of satellite elevation angle is considered as an independent time variable, and the amplitude ratio is empirically selected for modeling and generating theoretical results, so that different combined multipath error sequences can be obtained at different antenna height differences.

8.5.2 Spectral Peak Frequency Analysis Considering that the AAF α(t) is much smaller than one due to reflection and antenna radiation pattern, the multipath-induced pseudorange error and carrier phase error in (8.28) can be approximated as (t) ≈ 2h sin θ (t) · α(t) · cos δφ (t) λ α(t) sin δφ (t) β(t) ≈ 2π

(8.51)

As the elevation angle varies with time and the AAF is also changing with time or with elevation angle especially due to the GNSS antenna radiation pattern which

8.5 Dual Receiver System Based Snow Depth Estimation

239

is designed to mitigate the multipath effect. As a consequence, the pseudorange and carrier phase errors are quasi-sinusoidal signals with many components of different frequencies. That is, in the frequency domain, instead of a spike at a single frequency point, a pulse-like spike with certain bandwidth would occur. Examples of the pulselike spike are displayed in Fig. 8.32. Fourier spectral analysis is commonly used to convert the time domain sequences distributed continuously and uniformly into frequency spectral sequences. Although the GNSS observation data are sampled uniformly in the time domain such as once per second, the distribution of the sine of the satellite elevation angle is a nonlinear function of time. That is, the so-called time variable, the sine of the elevation angle, is not uniformly distributed. Therefore, Fourier transform cannot be employed for spectral analysis on the combined error signals. Instead, the Lomb-Scargle method

Fig. 8.32 The variation of combined sequence with the sine of the satellite elevation angle and spectrograms under different antenna heights

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8 Snow Depth and Snow Water Equivalent Estimation

is used to perform spectrum analysis for an unevenly distributed time series [14, 20] as mentioned before. Since two receivers have two different antenna heights, the SFC method will produce signal components with two different spectral peaks. For the DFC method, there would be four different spectral peaks. Signal components of spectral peaks at different frequencies will be superimposed under certain conditions. Therefore, one or two peaks will appear in the power spectrum of the SFC method, while the DFC method will have one to four peaks, depending on the two antenna heights. Figure 8.32 shows a few examples of the time series and power spectrum of the two combination methods.

8.5.3 Theoretical Model The L-band carrier commonly used in GNSS satellites is the right-hand circularly polarized (RHCP) plane wave [8]. Due to the irregularity of the terrain and the different reflection characteristics of the surface medium, the energy of the weak signal transmitted from the satellite to the earth will be attenuated again and some signal components will be changed from the RHCP signal to the left-hand circularly polarized (LHCP) signal after being reflected by the surface. That is, the received reflected signal will consist of both RHCP and LHCP components. As the satellite elevation angle increases, the RHCP component of the reflected signal decreases and the LHCP component increases as studied in Chap. 4. When the ground or snow surface is flat and reflection coefficient is a constant, the individual spectral peak frequencies of the combined multipath-induced error are independent of the selection of satellite and satellite azimuth angle range and are very little affected by the selection of the elevation angle range of interest. Due to such zenith-looking placement of the CORS antennas and the antenna radiation pattern, the reflected signal is stronger when the satellite elevation angle is smaller, as observed in a number of figures earlier. In general, when the satellite elevation angle is in the range of 5 to 30°, better multipath information can be obtained. The combined multipath-induced error sequence (i.e. time series) is supposed to be generated over a suitable range of satellite elevation angles, and its periodogram can be obtained by Lomb-Scargle spectrum analysis. The spectral peak of the combined error time series will change from single peak to multiple peaks or vice versa as the two antenna heights change. The inevitable ambiguity of the number of peaks brings trouble in modeling and subsequent processing of measured data. In the case of a single spectral peak, it is trivial to select the spectral peak frequency for offline modeling and online snow depth estimation. In the case of multiple spectral peaks, it is better to choose the frequency with the possible highest spectral power in the offline modeling, to reduce the effect of noise. As shown in Fig. 8.32, for instance, the spectrograms have one to three peaks. In the case of the second spectrogram, the second peak has the largest power spectral density. The corresponding peak frequency is selected for modeling. The modeling should be

8.5 Dual Receiver System Based Snow Depth Estimation

241

carried out with respect to a specific antenna and a specific signal wavelength in the case of DFC. Accordingly, in the online snow depth estimation, the frequency with the highest peak may be simply selected. With Lomb-Scargle spectrum analysis, the relationship between antenna height and the spectral peak frequencies of the combined error time series can be obtained by least squares fitting. Figure 8.33 shows the modeling case where the antenna height difference is 0.32 m. The height of the low antenna above the reflection surface is assumed to be from 0.8 to 1.5 m. The selection of the antenna height difference is based on the antenna heights measured in the field experiment. Clearly, the simulation based estimation results with both SFC and DFC are in good agreement with the linear model which is in the same form of (8.42), i.e. h = a × f + b. Table 8.16 shows the fitting coefficients and fitting errors in terms of RMSE for the two different satellite constellations (GPS and BDS). The DFC method has the fitting accuracy better than the SFC method, mainly because the former produces up to four peaks superimposed and perhaps blurred in the spectrum analysis and the latter produces up to only two spectral peaks. The DFC method with BDS generates larger fitting error. This is because when the height of the lower antenna equals or close to 0.8 m, the points significantly deviate from the linear model. This indicates that the antenna height should be adequate to minimize the fitting error. Assuming that the ground surface is flat and the antenna gain mode remains unchanged, the same model can be applied as long as the elevation angle range and the two antenna heights remain the same. In the measurement of snow depth,

Fig. 8.33 Linear relationship between the lower antenna height and spectral peak frequency by least squares fitting using GPS observation, when the antenna height difference is 0.32 m

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8 Snow Depth and Snow Water Equivalent Estimation

Table 8.16 Fitting coefficients and fitting errors of SFC and DFC models for GPS and BDS signals Method

GNSS

Antenna

a (m−1 )

b (m)

RMSE (cm)

DFC

GPS

h1

0.0991

−0.0021

0.70

h2

0.0991

0.3179

BDS

h1

0.0961

0.0363

h2

0.0961

0.3563

h1

0.0932

−0.3030

h2

0.0932

0.0170

SFC

GPS

1.16 0.47

the pseudorange and carrier phase data are collected from the receiver, the data are combined according SFC or DFC to produce the error time series, the Lomb-Scargle spectrum analysis is performed to obtain the spectral peak frequency of the time series, the antenna height relative to the snow surface is directly calculated by (8.42), and finally the snow depth estimate is generated by (8.43).

8.5.4 Experimental Results On 31 December 2016, a team from the School of Geodesy and Geomatics at Wuhan University started the experiment in a village close to Mudanjiang, Heilongjiang Province, China. The location of the experiment is shown in the upper panel of Fig. 8.34. Three receivers marked with 1, 2, and 3 were used to collect GNSS observations. The latitude and longitude of these receivers are (44.5934°, 128.3986°), (44.5936°, 128.3987°), and (44.5935°, 128.3985°), respectively. The length and width of the experiment area are about 55 m by 55 m. As a typical climatology of northeast China, snowfall occurs usually from November to March in the experimental area. The ground was covered by snow throughout the observation period. The snow depth was measured three times a day at 8:00 AM, 11:00 AM, and 15:00 PM, respectively. The average snow depth in the location was between 28 and 60 cm during the experiment period. The increase in the snow depth is mainly caused by snowfall. As the period of the experiment is not very long, the minor decrease in the snow depth is mainly caused by wind. The snow depth changes caused by metamorphism of the snowpack are rather marginal. Considering the factors such as traffic and topography, a flat field adjacent to the village was selected for the experiment. The lower panel of Fig. 8.34 shows the field when the ground was covered by snow. The Trimble Zephyr2 geodetic antennas were used to capture GNSS signals, adopting an advanced technology to generate a desired antenna radiation pattern to mitigate multipath effect. Figure 8.35 shows the typical antenna radiation pattern of a geodetic antenna. When installing, the antennas were facing up in a way similar to CORS setup. The collected GNSS signals were processed by Trimble R9 GNSS receivers, which can process signals transmitted from satellites of all four constellations (GPS, GLONASS, BDS and Galileo).

8.5 Dual Receiver System Based Snow Depth Estimation

243

Fig. 8.34 (upper) Location of the experiment; (lower) Three receivers for GNSS data collection are not far away from each other

The ground truth data need to be collected and compared with the estimation result based on the GNSS-R method. An optical plummet was used to search for the snow surface point which is on the plummet line starting from the antenna phase center point. A sampling tube of length 1 m was used to dig a hole through the snow from the surface point. A steel-made tape was then used to measure the distance from a marked point on the antenna edge to the bottom of the hole, which is on the ground surface. Since the tape is not rigid but flexible, an error in the antenna height measurement would occur. However, it has been shown that the error is typically smaller than 1 cm based on many tests. A wooden ruler was used to measure the snow depth during another experiment as mentioned in the preceding sections. In order to obtain as many available observation data as possible, three receivers about 10 meters away from each other were set for simultaneous observation, as shown in Fig. 8.34. Table 8.17 shows the GNSS satellite signals of pseudorange and carrier phase which are used to estimate snow depth. The antenna height is from 0.8 to 1.6 m, and the antenna height difference ranges from 9.43 to 40.56 cm. The experimental campaign was conducted over 12 consecutive days. The GNSS

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8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.35 Example of radiation patterns of a typical geodetic GNSS antenna

Table 8.17 GNSS observation codes for GPS and BDS

GNSS

Signal band

Pseudorange

Carrier phase

GPS

L1

C1C

L1C

L2

–

L2W

B1

C2I

L2I

B2

–

L7I

BDS

antennas remained fixed, while the receiver batteries were taken away for recharging after 8-h observation each day. Figure 8.36 shows the skyplots of GPS and BDS satellites observed at one of the receivers over an interval of about 8 h on 2 January 2017. Clearly, the number of visible GPS satellites is much larger than that of BDS satellites. Note that China has launched a good number of BDS-3 satellites over the three years between 2017 and 2019, so the number of visible BDS satellites has been significantly increased as mentioned in Chap. 2. Figure 8.37 shows a typical example of a multipath-induced time series produced by DFC method and its power spectral density. Results from both modeling and experiment are presented. Clearly, the time series generated from experimental data shows a clear oscillating pattern, which has a good match with the modeled time series. Although the power spectrum patterns of the modelled time series and the observed one are very similar in terms of the two spikes, the spectral power of the observed data is significantly lower than the modelled one due to noise. Also, the

8.5 Dual Receiver System Based Snow Depth Estimation

245

Fig. 8.36 The skyplots of different satellite constellations on 2 January 2017. (left) The skyplot of GPS satellites. (right) the skyplot of BDS satellites

Fig. 8.37 Time series and power spectral density produced by DFC method with simulated data (dashed red line) and observed data (solid blue line) of the GPS G18 when the heights of the two antennas are 1.486 m and 1.689 m, respectively

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8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.38 Observed (ground-truth) antenna heights versus estimated antenna heights obtained by the two methods with GPS and BDS observation data

two spectral peak frequencies of the observed data slightly deviate from those of modelled data also due to noise corruption. The observation data collected by two receivers with fixed antenna height difference in the same satellite system are processed to obtain the observation antenna heights corresponding to the estimated antenna heights under the two estimation methods. Figure 8.38 shows the comparison between the estimated antenna heights of the two combination methods with different satellite systems and the true antenna heights. The straight line represents the zero-error line. Figure 8.39 shows the daily snow depth observed (i.e. ground truth) and estimated results associated with three receivers. In general, there is good agreement between the ground truth and all estimates using the two combination methods under different satellite systems. It can be seen that the estimation results with GPS signals are generally better than the estimation results with BDS signals, probably because the BDS pseudorange observation data contains stronger noise under this circumstance. The SFC method of the L1 signal is superior to the DFC method with the same satellite system, whereas the SFC method of the L2 signal has different characteristics, which shows the worst estimation result. The main reason could be that the SFC method is considerably less affected by the superposition of multiple spectral peaks, and the theoretical modeling of DFC method is less accurate than the SFC method. However, the SFC method is more sensitive to data quality, and pseudorange observations of the L1 signal contain larger noise than that of the L2 signal. Table 8.18 also confirms this conclusion from a statistical point of view, which shows the mean, STD and RMS of the snow depth estimation error for three methods

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247

Fig. 8.39 The three scatter plots show the daily snow depth observed and estimated results for GPS and BDS by SFC and DFC methods

Table 8.18 Mean, STD, and RMS of snow depth estimation error of different methods Method

GNSS

Band

SNR

GPS

L1 L2 L1, L2

Mean(cm)

STD(cm)

RMS(cm)

0.84

4.23

4.31

0.92

4.36

4.46

−0.21

5.04

5.04

DFC

GPS BDS

L1, L2

0.02

6.26

6.26

SFC

GPS

L1

−1.28

3.96

4.16

L2

−1.22

7.54

7.64

(i.e. SNR method [11], DFC method, and SFC method) using GPS and BDS signals. SFC method of the L1 signal slightly outperforms SNR method. Both SFC and DFC with GPS observations tend to underestimate the actual antenna height, resulting in overestimation of the snow depth. On the other hand, BDS observations produce the results which underestimates the snow depth. The penetration of the signal and inner reflection beneath the snow surface can also bring estimation errors. The observation area is not perfectly flat, and there were some dried grasses on the floor mixed with the snow, producing additional errors. In addition, when the snow depth suddenly increases or decreases such as caused by wind, additional estimation error will be produced. Such a wind-caused variation would dependent on the wind speed and

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8 Snow Depth and Snow Water Equivalent Estimation

direction as well as its duration. Furthermore, strict simultaneous observations are not guaranteed between the two receivers, making the effects of geometry and ionospheric delays incompletely eliminated. These factors may also contribute to the snow depth estimation error.

8.6 Snow Water Equivalent Estimation Estimating SWE by GNSS signals is a much more difficult problem than estimating snow depth, partly because the complex effect of snow components (e.g. ice particles, air and liquid water) on the dielectric constant of snowpack [25]; Stein [22]. Henkel et al. [5] and Steiner et al. [23] make use of the relation between GNSS carrier phase and SNR observations and the dielectric constant of snow to estimate SWE, but the method belongs to GNSS refraction technique and the data collected by CORS receivers cannot be used. Since GNSS-R based snow depth estimation has been investigated extensively [11, 12, 18, 28, 29], it is a convenient way to estimate SWE by making use of GNSS-R snow depth estimates. Based on the observations made by traditional SWE measurement instrument, a number of empirical models have been developed to convert snow depth to snow density or convert snow depth to SWE [6, 10, 15, 16]. Those models would be accurate, since they use meteorological parameters for model development. However, GNSS stations may not be equipped with weather instrumentation to obtain all the required meteorological parameters in general, so that the mentioned models would not be applicable in some circumstances. It is useful to develop empirical models for converting snow depth estimations to SWE estimations without involving any meteorological parameter, significantly simplifying the SWE estimation [31].

8.6.1 Data Preprocessing The data set used to develop the empirical model was collected at the sites of snow telemetry (SNOTEL) network which is operated by Natural Resource Conservation Service (NRCS), U.S. [4, 21]. Although the SNOTEL network is distributed over 800 sites where all snow depth and SWE data are collected automatically, about 200 sites do not have snowfall or only have rather small snowfall for a short period each year. Snow depth and SWE data collected at 612 SNOTEL sites from August 1, 2010 to August 1, 2015 are used to develop the empirical model. The data in the five years are then examined to exclude those with snow cover duration less than 90 days during a winter season. Such selection will make sure the maximum snow depth is greater than 50 cm and get rid of the data associated with multiple cycles of snowfall and snow melting process in a single winter season. As shown in Fig. 8.40, the elevations of the selected sites range from ~800 to 3500 m although they are all distributed in the western of U.S.

8.6 Snow Water Equivalent Estimation

249

Fig. 8.40 Distribution of SNOTEL stations, measurements at which are used for modeling

Physically implausible snow depth and SWE data measured by ultrasonic sensor and snow pillow are removed through examining snow density which is defined as the ratio of SWE over snow depth. The SWE and snow depth data with a normal snow density between 0.03 and 0.7 g/cm3 are retained, while the data out of the range are excluded. Figure 8.41 shows a typical series of snow depth and SWE measurements used for developing the empirical model. The data is obtained from SNOTEL site of #350, and the daily observations are produced based on the average of hourly observations each day. It can be observed that there is a good match between snow depth and SWE in terms of the variation pattern. Because of the significant

Fig. 8.41 The variation of snow depth and SWE recorded at SNOTEL site 350

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8 Snow Depth and Snow Water Equivalent Estimation

fluctuation of daily snow depth observations, which are caused by noise, interference, or environmental factors, a moving average filtering method with the filtering window length of 7 days is used to process the raw daily snow depth observations, that is, j=i+k 

h(i) =

˜ j) h(

j=i−k

2k + 1

, i >k

(8.52)

˜ j)} are the raw daily snow depth observations, and 2k + 1 is the filtering where {h( window length. Clearly, the filtered snow depth curve is much smoother than that of raw daily snow depth especially during the snowfall period. The filtered daily snow depths and raw SWE observations are used to establish the empirical model.

8.6.2 Empirical Model The snow depth usually increases until reaching its maximum during the snowfall period, and then it sharply decreases during the snow melting period until there is no snow on the ground. Therefore, as shown in Fig. 8.42, the variation of SWE versus snow depth presents a hysteresis loop and produces multivalued SWE for each snow depth in a snow season. It can be observed in Fig. 8.42 that, before the snow depth reaches its maximum within the snow accumulation period (Paccum),

Fig. 8.42 Typical variation of SWE versus snow depth over a snow season

8.6 Snow Water Equivalent Estimation

251

the SWE slowly increases with the increase of snow depth. After SWE reaches maximum and thus the snow melting period (Pmelt) starts, as snow depth decrease rate is much larger than snow density increase rate, SWE decreases continuously with the decrease of snow depth. Although the increase rate and decrease rate of SWE are considerably different, SWE is proportional to snow depth in both of those two periods. In the period between the two days when the snow depth and SWE reach the maximum separately (i.e. snow transition period, Ptran), the SWE increases with the decrease of snow depth basically. Because of the significant difference of variation pattern of SWE versus snow depth in the three periods, it is necessary to develop three separate regression models to convert snow depth to SWE for Paccum, Pmelt and Ptran, respectively. Taking the snow depth as independent variable and SWE as dependent variable, a scatterplot of SWE versus snow depth for Paccum and Pmelt are shown in Fig. 8.43. It can be observed that SWE increases with the increase of snow depth for both phases in general. Fitting those discrete points by first-, second-, and third-order polynomial produces three fitting equations for three individual schemes. The fitting coefficients and fitting errors for the Paccum and Pmelt phases are listed in Table 8.19 and Table 8.20, respectively. It can be seen that the fitting performance in terms of both correlation coefficient and RMSE improves with higher order polynomials. However, the improvement is very marginal, for example, the correlation coefficient

Fig. 8.43 Scatterplot of SWE versus snow depth and the polynomial fitting results in Paccum (upper) and Pmelt (lower)

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8 Snow Depth and Snow Water Equivalent Estimation

Table 8.19 Fitting polynomial coefficients and correlation coefficients for Paccum and Pmelt Phase Paccum

Pmelt

Polynomials

Fitting polynomial coefficients

R

3rd

2nd

1st

c

1st

–

–

0.3484

−5.8202

0.9696

2nd

–

0.0004

0.2417

−1.1102

0.9775

3rd

−0.92·10−6

8.7·10−4

0.1761

0.7331

0.9784

1st

–

–

0.4748

−4.9221

0.9679

2nd

–

0.0002

0.4301

−1.478

0.9695

3rd

−1.78·10−6

1.2·10−4

0.2719

2.6027

0.9709

Table 8.20 Mean, STD and RMS of model fitting errors for Paccum and Pmelt Phase

Polynomials

Fitting error Mean (cm)

Paccum

Pmelt

STD (cm)

RMS (cm)

1st

0.00

5.93

5.93

2nd

0.27

5.12

5.13

3rd

−0.01

5.01

5.01

1st

0.00

8.39

8.39

2nd

−0.64

8.24

8.27

3rd

0.00

7.99

7.99

difference between the 2nd and 3rd order polynomials is only 0.0009 and the RMSE difference is just 0.12 cm for Paccum. It would be a better option to choose the simpler model with a lower order polynomial at the cost of negligible performance degradation. Therefore, the second-order polynomials are selected as the regression models for Paccum and Pmelt:

SW E Paccum = 0.0004 · h 2 + 0.2417 · h − 1.1, h > 4.6 cm (8.53) SW E Pmelt = 0.0002 · h 2 + 0.4301 · h − 1.5, h > 3.4 cm Note that, the constraints on snow depth are applied to guarantee positive SWE estimations in the presence of estimation error. As most of the snow depth observations used for developing the models are smaller than 500 cm, those two models may not be suited for scenarios where the snow depth is significantly greater than 500 cm. Figure 8.44 shows the scatterplot of SWE versus the snow depth in the transition period Ptran. For clarity, only eight results related to six SNOTEL sites are presented. Contrary to Paccum and Pmelt, the SWE basically increases with the decrease of snow depth in Ptran. The statistical results show that there is a significantly linear correlation between snow depth and SWE in the period, and the relationship at each site can be modeled with a linear regression equation:

8.6 Snow Water Equivalent Estimation

253

Fig. 8.44 Scatterplot of SWE versus snow depth and the polynomial fitting results in Ptran

SW E Ptran = a · h + b

(8.54)

Analysis of all of the stations revealed that there is no significant relevance between the slope parameter a and snow depth, or climatological parameters (e.g. air temperature, and precipitation). Although the slope basically ranges between −3.5 and 1, most of the values are clustered around the range from −1.5 to 0.5, which are account for 98% of the total number as shown in Fig. 8.45. Therefore, one takes the average

Fig. 8.45 Distribution of parameter a

254

8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.46 Scatterplot of snow depth maximums versus parameters b, and the fitting line obtained by using least squares method

of the parameters between −1.5 and 0.5 as the value of parameter a, which is given by: a = −0.3515

(8.55)

The intercept parameter b varies greatly for different sites and snow seasons, which also could be seen clearly in Fig. 8.46. It can be readily derived that the least-squares solution to the intercept is given by: b=

n 1 (yi − axi ) n i=1

(8.56)

where xi and yi are snow depth and SWE observations in Ptran for a given site and snow season, respectively; n is the number of snow depth or SWE observations in Ptran. It would be desired to generate a simple model to describe parameter b without using all the snow depth and SWE observations in Ptran for each site. Intuitively, the parameter b is relevant to the maximum of snow depth for each snow season. This can be seen clearly in Fig. 8.47, which shows a scatterplot of parameter b and corresponding maximum of snow depth of the snow season. Based on the linear distribution pattern shown in Fig. 8.47, the relationship between the maximum snow depth and parameter b can be obtained by using the least-squares fitting as:

8.6 Snow Water Equivalent Estimation

255

Fig. 8.47 Scatterplot of maximum snow depth versus h tm and the linear fitting result

b = 0.7745 · h max − 17.03

(8.57)

where h max is the maximum snow depth in a given snow season. Substituting (8.57) and (8.55) into (8.54) produces: SW E Ptran = −0.3515 · h + 0.7745 · h max − 17.0, h max > 40.3 cm

(8.58)

where the constraint on maximum snow depth is to guarantee a positive SWE. When the maximum snow depth is smaller than 40.3 cm, the Ptran would be very short, so the whole snow season could be split into just two periods (Paccum and Pmelt) at the point of maximum snow depth, and (8.53) is used to convert the snow depth into SWE. In addition, the fitting results of SNOTEL station 1000 obtained by (8.58) is also shown in Fig. 8.44. It can be seen that, there is a good accordance between the in situ SWE observations and the fitted ones basically. The mean, STD and RMS of fitting error for Ptran model are −2.43 cm, 10.9 cm, and 11.17 cm, respectively, which are considerably larger than those for Paccum and Pmelt models. However, as the duration of Ptran is much shorter than Paccum and Pmelt, the modeling error would only have a marginal effect on the SWE estimation over the whole snow season. By using (8.53) and (8.58), snow depth observations can be converted into SWE in the corresponding period. In the presence of three periods, Paccum and Ptran are separated by the day when the maximum of snow depth occurs, which can be easily obtained from series of SNR-based snow depth estimations. On the other hand, it is

256

8 Snow Depth and Snow Water Equivalent Estimation

non-trivial to determine the day when Ptran ends on or Pmelt starts from; because that day when the maximum SWE occurs in a snow season is unknown in advance. However, the day separating the two periods Ptran and Pmelt can be determined by using the fact that the modeled SWEs of Ptran and Pmelt are equal on the separating day: SW E Ptran (h tm ) = SW E Pmelt (h tm )

(8.59)

where h tm is the snow depth on the intersection point of the two models, corresponding to the maximum SWE. Substituting (8.58) and the second equation of (8.53) into (8.59), the snow depth h tm can be determined by: 0.0002h 2tm + 0.7815h tm − 0.7745h max + 15.552 = 0

(8.60)

When the maximum snow depth is greater than 40.3 cm and smaller than 500 cm, (8.60) has two real solutions. The positive solution is retained, while the negative one is removed. The scatterplot of maximum snow depth versus h tm is shown in Fig. 8.47. Using the least-squares fitting produces: h tm = 0.8893h max − 9.6

(8.61)

In summary, the complete SWE models are described by: If h max ≤ 40.3 cm is satisfied, then

SW E(h) =

SW E Paccum (h), 4.6cm < h < h max SW E Pmelt (h), h max ≥ h > 3.4cm

(8.62a)

If 40.3cm < h max < 500cm holds, then ⎧ ⎪ SW E Paccum (h) , 4.6cm < h < h max ⎪ ⎨ SW E(h) =

⎪ ⎪ ⎩

SW E Ptran (h) , h max ≥ h > h tm

(8.62b)

SW E Pmelt (h) , h tm ≥ h > 3.4cm

Figure 8.48 shows the distribution of SWE residuals versus snow depth (left panel) and the residual distributions (right panel) for the three periods. Basically, the fitting error of SWE increases with the increase of snow depth for all three models. The mean SWE fitting residuals of the three periods are 0.27, −2.43 and −0.64 cm. That is, there is a bias in all three periods and the one in the transition period is significant. It can be seen from the left panel of the figure that the bias mainly appears when the SD is larger than 350 cm. The main reason is the relatively small number of the SD and SWE observations over this range, so the modeled function cannot well describe the in situ relationship between SD and SWE. Although the model is established

8.6 Snow Water Equivalent Estimation

257

Fig. 8.48 Heat map of modeled SWE residuals versus snow depth (left) and the residuals probability density (right) for three periods

with snow depth observations ranging from 0 to 500 cm, the model is more suited for snow depth less than 350 cm.

8.6.3 Algorithm of SNR-Based SWE Estimation Making use of the SNR observations recorded by GNSS receiver and SWE conversion models established above, the SNR-based SWE estimation algorithm can be developed. Figure 8.49 shows the flowchart of the SWE estimation algorithm. Inputs of the algorithm are GNSS observation files and navigation files; both files could be in RINEX format. The coordinates of GNSS station and navigation file can be used to calculate the satellite elevation angle in each observation epoch. Combined with the SNR observations which are contained in the GNSS observation file, the sequence of SNR versus sine of elevation angle in the range of 5 to 25° can be extracted. Due to the effect of trees, buildings or other ground objects around the GNSS station, some SNR sequences may not show a clear periodical oscillation pattern, so those sequences should be abandoned. In addition, each selected SNR sequence with respect to sine

258

8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.49 Flowchart of SNR-based SWE estimation algorithm. H 0 and H are the antenna height when the ground is snow free and in the presence of snow cover, respectively; hmax is the maximum snow depth in a snow season; htm is the snow depth corresponding to the maximum SWE; Paccum, Ptran and Pmelt denote the snowfall period, snow transition period and snowmelt period, respectively

8.6 Snow Water Equivalent Estimation

259

of elevation angle should have a PSD peak larger than a threshold such as 0.1 in this case to obtain reliable snow depth estimation. After the suited sequences of SNR and related elevation angles are selected, sinθ (t) is calculated and Lomb-Scargle spectral analysis is performed to obtain the spectral peak frequencies of each SNR sequence in snow-covered case and in snow-free case. Antenna heights are then calculated with (8.4b), followed by snow depth calculation. Assume that the surface topography around the GNSS station in the snow-free case is the same as or very similar to that of snow-covered case, the antenna height H 0 in the snow-free case can be estimated in the same way using (8.4b). In order to avoid the effect of vegetation height around the station, the data used to estimate the antenna height in the snow-free case may be collected over 7 days prior to the snow-covered season and over another 7 days following the snow-cover season. The sequence of SNR and the related sequence of sinθ (t) for each satellite and each day are processed to generate one antenna height estimate. Then the multiple antenna height estimates associated with multiple satellites are combined to generate a weighted average by (8.6) where snow depth is replaced with antenna height. Then the 14 individual daily antenna height estimates are averaged to generate the final antenna height estimation in the snow-free case. Note that, because the GNSS satellite rotates around the earth about twice a day, there might be multi-reflection tracks in a day and thus multisequences of SNR and sine of elevation angle for each GNSS satellite, as shown in Fig. 8.50 where four ground reflection tracks (blue solid lines) occur within the range of 5–25°. In the case where the topography changes significantly over azimuth

Fig. 8.50 Ground reflection tracks for GPS satellite of #12 and the green text indicates the GPS time of the day. The data was collected in Wuhan, China on June 21, 2019

260

8 Snow Depth and Snow Water Equivalent Estimation

angle, the antenna height in the snow-free case and the snow-covered case for each track should be estimated separately. Once the SNR-based snow depth estimations for each satellite are obtained, (8.6) is used to fuse the multi-satellite snow depth estimations for each day, producing the weighted average of daily snow depth estimations. When estimating the snow depth and SWE for the past snow season, the complete sequence of snow depth estimations can be obtained first. If the maximum of daily snow depth in a snow season is larger than 40.3 cm, (8.61) is used to calculate the snow depth htm corresponding to the maximum SWE in the snow season. Then, the snow depth sequence in the snow season is divided into three sequences (Paccum, Ptran, and Pmelt) by the maximum snow depth and the snow depth htm . Finally, (8.62b) is used to convert those three sequences of snow depth into SWE sequence. If the maximum snow depth in a snow season is smaller than 40.3 cm, the snow depth sequence is divided into two sequences (Paccum and Pmelt), and (8.62a) is used to convert those two sequences of snow depth into SWE sequence. In the case where real-time daily snow depth and SWE estimation is required, the maximum snow depth is unknown in advance, so the procedure is modified slightly as follows. Suppose that the snow depth and SWE estimation is performed immediately after snowfall starts. Then, after getting the snow depth estimates, the SWE is calculated daily with the Paccum model if the snow depth estimate is in an increase trend. If the daily snow depth estimate decreases considerably a couple of times compared to the previous ones, then Paccum is likely over and this may be further validated by historical snow depth data. If the maximum snow depth estimate is smaller than 40.3 cm, then the Pmelt model is used to calculate the SWE; otherwise, the Ptran model is employed to calculate the SWE. In the Ptran case, when the snow depth estimate drops to htm a few times, the Pmelt model is used to calculate the SWE.

8.6.4 Model Verification 8.6.4.1

Verification of Snow Depth Based SWE Conversion Model

The data used for verifying the SWE empirical model are snow depth and SWE observations collected at SNOTEL sites from August 2, 2015 to August 1, 2020. Section 8.6.1 already provided information about SNOTEL and the main difference is that the data used here were collected over the past five years, while the data used in Sect. 8.6.1 were recorded from 2010 to 2015. Scatterplots of the model based SWE estimates versus SNOTEL SWE observations (i.e. in situ data) for the three periods (accumulation period, transition period and melting period) are shown in Fig. 8.51. RMS of the model based SWE estimation errors are 6.03 cm, 9.97 cm and 8.85 cm for the three periods respectively. This can be also seen in Table 4, which shows the mean, STD and RMS of errors of the empirical model based SWE estimations for the three periods. It can be observed from Fig. 8.51 and Table 8.21 that, the empirical model based SWE estimates agree

8.6 Snow Water Equivalent Estimation

261

Fig. 8.51 Scatterplot of snow depth based SWE versus in situ SWE for three different periods

Table 8.21 Mean, STD and RMS of errors for empirical model based SWE estimation

Period

Mean (cm)

STD (cm)

RMS (cm)

Paccum

−0.20

5.76

5.77

Ptran

−1.27

9.40

9.48

Pmelt

−0.07

8.12

8.13

well with the SNOTEL SWE measurements in general. This indicates that empirical models established in Sect. 8.6.2 are valid and effective. Nevertheless, the RMS of the errors is relatively large, so it is still desirable to develop new techniques to reduce the SWE estimation error.

8.6.4.2

Verification of SNR-Based SWE Estimation Method

The data set used for this verification was collected in an experimental campaign conducted in a village between Mudanjiang and Harbin, Heilongjiang Province, China, from the 26th of December, 2017 to the 17th of January, 2018, as described in Sect. 8.4.2.3. Four BDS satellites with B1 and B2 signal SNR observations and five GPS satellites with L1 and L2 signal SNR observations are used to test the method. According to the snow depth records over the years in the area, the snow depth hit its maximum in February basically. Therefore, the empirical model of Paccum was used to convert the SNR-based snow depth into SWE. Figure 8.52 shows the scatterplot of in situ SWE versus SNR-based SWE estimates, and the

262

8 Snow Depth and Snow Water Equivalent Estimation

Fig. 8.52 Scatterplot of in situ SWE versus SNR-based SWE estimates (left), and scatterplot in situ snow depth versus SNR-based snow depth estimates (right)

scatterplot of in situ snow depth versus SNR-based snow depth estimates. Note that, SNR-based snow depth estimations produced by two different average techniques are presented in the figure; NA stands for results by normal average and WA stands for results by weighted average which is obtained by (8.6). Therefore, two different SWE estimations, calculated by those two different snow depths respectively, are also produced and presented in the figure. Clearly, the SNR-based SWE estimations contain a significant negative bias, as observed earlier. This can be also seen in Table 8.22 which shows the mean, STD and RMS of errors in SNR-based SWE estimation. The estimation error of SNR-based SWE mainly comes from two error sources: SNR-based snow depth estimation error and SWE empirical model error. The main reason for the negative bias of SNR-based SWE estimations shown in Fig. 8.52 is the underestimation of snow depth by the Table 8.22 Mean, STD and RMS of errors in SNR-based SWE estimation Band

Method

Mean (cm)

STD (cm)

RMS (cm)

–

Ruler

−0.82

0.98

1.28

BDS B1

NA

−2.03

1.02

2.28

WA

−1.76

1.05

2.05

BDS B2

NA

−1.93

0.95

2.15

WA

−1.77

0.99

2.03

NA

−1.52

1.05

1.85

WA

−1.41

1.01

1.73

NA

−2.47

1.30

2.79

WA

−2.07

1.24

2.42

GPS L1 GPS L2

8.6 Snow Water Equivalent Estimation

263

SNR-based method. By making use of in situ (or, ruler measured) snow depth and the Paccum SWE empirical model, in situ snow depth based SWE estimations can be calculated, as shown in the left panel of Fig. 8.52 and Table 8.22. The other source of the negative bias of SNR based SWE estimation is the SWE empirical model error. This can be seen from Table 8.22 that the mean error of in situ snow depth based SWE estimations is −0.82 cm. Compared with the use of normal average of snow depth, the use of weighted average by (8.6) can reduce the SWE estimation errors by 9.7, 5.8, 6.3 and 13.5% for BDS B1, BDS B2, GPS L1, and GPS L2 signals, respectively. This is in accordance with the fact that a performance gain will be achieved by the use of a weighting scheme when there exists significant difference in measurement errors between different measurement devices. In addition, the use of GPS L2 signal for SWE estimation produces relatively large error. This is mainly caused by the considerably lower strength of legacy GPS L2P(Y) code used for the evaluation; a lower strength of GNSS signal would produce a larger error in snow depth estimation, degrading SWE estimation performance.

8.7 Summary Data recorded by typical ground-based GNSS receivers can be used to measure snow depth and SWE, which include SNR, carrier phase and pseudorange data of such as CORS receivers, as demonstrated in this chapter. The accuracy is still a major issue when precise snow depth and SWE measurement is required. More studies may be needed to better understand the mechanism of internal snow reflection and buried ground surface reflection which causes the underestimation of the snow depth and SWE. Noise and interference mitigation is always useful to improve the observation data quality and thus enhance the estimation accuracy. The horizontal and vertical snow density distribution are usually uneven and the characteristics of the snow components may also vary especially vertically. It is thus a challenging issue to establish accurate models to describe the relationship between SWE and relevant observations such as snow depth and density. The results presented are all related to the use of data collected by ground-based receivers. It seems that there are no reports on snow depth and SWE estimation by exploiting airborne and spaceborne data collected by GNSS receiver. Thus, it is interesting and useful to do more investigations on this issue

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References 1. Axelrad P, Larson KM, Jones B (2005) Use of the correct satellite repeat period to characterize and reduce site-specific multipath errors. In: Proceedings of ION GNSS, The Institute of Navigation, Long Beach, CA, September 2005, pp 2638–2648 2. Babic DI, Corzine SW (1992) Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors. IEEE J Quantum Electron 28(2):514–524 3. Braasch MS, van Dierendonck AJ (1999) GPS receiver architectures and measurements. Proc IEEE 87(1):48–64 4. Dressler KA, Fassnacht SR, Bales RC (2006) A comparison of snow telemetry and snow course measurements in the colorado river basin. J Hydrometeorol 7:705–712 5. Henkel P, Koch F, Appel F, Bach H, Prasch M, Schmid L, Schweizer J, Mauser W (2018) Snow water equivalent of dry snow derived from GNSS carrier phases. IEEE Trans Geosci Remote Sens 56:3561–3572 6. Hill DF, Burakowski EA, Crumley RL, Keon J, Hu JM, Arendt AA, Wikstrom J, Wolken GJ (2019) Converting snow depth to snow water equivalent using climatological variables. Cryosphere Discuss 13:1767–1784 7. Hinrikus H (2006) Electromagnetic waves. Wiley Encyclopedia of Biomedical Engineering 8. Hofmann-Wellenhof B, Lichtenegger H, Wasle E (2008) GNSS—Global navigation satellite systems: GPS, GLONASS, Galileo and More, Springer Wien New York 9. JMA (2020, visited) http://www.jma.go.jp/jma/index.html 10. Jonas T, Marty C, Magnusson J (2009) Estimating the snow water equivalent from snow depth measurements in the Swiss Alps. J Hydrol 378(1–2):161–167 11. Larson KM, Gutmann ED, Zavorotny VU, Braun JJ, Nievinski FG (2009) Can we measure snow depth with gps receivers? Geophys Res Lett 36(17):L17502 12. Li Y, Chang X, Yu K, Wang S, Li J (2019) Estimation of snow depth using pseudorange and carrier phase observations of GNSS single-frequency signal. GPS Solut 23(4):1–13 13. Liu L, Amin MG (2007) Comparison of average performance of GPS discriminators in multipath. In: Proceedings of IEEE international conference on acoustics, speech and signal processing (ICASSP), Honolulu, HI, USA, 15–20 April 2007, pp 1285–1288 14. Lomb NR (1976) Least-squares frequency analysis of unequally spaced data. Astrophys Space Sci 39:447–462 15. McCreight JL, Small EE, Larson KM (2015) Snow depth, density, and SWE estimates derived from GPS reflection data: Validation in the western U.S. Water Resour Res 50(8):6892–6909 16. Mizukami N, Perica S (2008) Spatiotemporal characteristics of snowpack density in the mountainous regions of the western United States. Journal of Hydrometeorology 9(6):1416–1426 17. Nievinski FG, Larson KM (2014) Inverse modeling of GPS multipath for snow depth estimation—Part II: Application and validation. IEEE Trans Geosci Remote Sens 52(10):6564–6573 18. Ozeki M, Heki K (2012) GPS snow depth meter with geometry-free linear combinations of carrier phases. J Geodesy 86(3):209–219 19. Pugliano G, Robustelli U, Rossi F, Santamaria R (2016) A new method for specular and diffuse pseudorange multipath error extraction using wavelet analysis. GPS Solutions 20(3):499–508 20. Scargle JD (1982) Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data. Astrophys J 263:835–852 21. Serreze MC, Clark MP, Armstrong RL, Mcginnis DL, Pulwarty RS (1999) Characteristics of the western United States snowpack from snowpack telemetry (SNOTEL) data. Water Resour Res 35(7):2145–2160 22. Stein J, Laberge G, Lévesque D (1997) Monitoring the dry density and the liquid water content of snow using time domain reflectometry (tdr). Cold Reg Sci Technol 25(2):123–136 23. Steiner L, Meindl M, Geiger A (2019) Characteristics and limitations of GPS L1 observations from submerged antennas. J Geodesy 93:267–280 24. Tabibi S, Nievinski FG, Dam TV (2017) Statistical comparison and combination of GPS, GLONASS, and multi-GNSS multipath reflectometry applied to snow depth retrieval. IEEE Trans Geosci Remote Sens 55(7):3773–3785

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25. Tiuri ME, Sihvola A, Nyfors E, Hallikaiken M (1984) The complex dielectric constant of snow at microwave frequencies. IEEE J Oceanic Eng 9(5):377–382 26. Vander Plas JT, Connolly AJ, Ivezic Z, Gray A (2012) Introduction to astroML: Machine learning for astrophysics. In: Proceedings of Conference on Intelligent Data Understanding (CIDU), Boulder, Colorado, USA, 24–26 October 2012, pp 47–54 27. Vázquez GE, Bennett R, Spinler J (2013) Assessment of pseudorange multipath at continuous GPS stations in Mexico. Positioning 4(3):253–265 28. Yu K, Ban W, Zhang X, Yu X (2015) Snow depth estimation based on multipath phase combination of GPS triple-frequency signals. IEEE Trans Geosci Remote Sens 53(9):5100–5109 29. Yu K, Li Y, Chang X (2018) Snow depth estimation based on combination of pseudorange and carrier phase of GNSS dual-frequency signals. IEEE Trans Geosci Remote Sens 57(3):1817– 1828 30. Yu K, Wang S, Li Y, Chang X, Li J (2019) Snow depth estimation with GNSS-R dual receiver observation. Remote Sensing 11(17):2056 31. Yu K, Li Y, Jin T, Chang X, Wang Q, Li J (2020) GNSS-R-based snow water equivalent estimation with empirical modeling and enhanced SNR-based snow depth estimation. Remote Sensing 12(23):3905

Chapter 9

Soil Moisture Measurement

Knowledge of soil moisture content is critical for drought and irrigation management, so as to increase crop yields and to gain a better understanding of natural processes linked to the water, energy and carbon cycles. Soil moisture can be measured by analyzing surface scattered signals transmitted and received by active radar sensors or natural surface emission detected by microwave radiometers. GNSS reflectometry is an alternative technique for soil moisture measurement. This is because the GNSS L-band frequencies (such as 1.5 and 1.2 GHz) are among the frequencies which have the highest sensitivity to soil moisture. That is, GNSS signals can be used to effectively perform soil moisture measurement. In this chapter, a number of GNSS-R based soil moisture estimation methods are studied, which include those based on signal power, SNR, and data related to geostationary satellites.

9.1 A Classic Soil Moisture Estimation Method Soil moisture measurement based on microwave remote sensing has a history of over 4 decades. Early efforts focused on developing the relationship between soil dielectric constant and soil moisture content and different empirical models and semi-empirical models were established. It was observed from the early efforts that there are two distinct features related to the relationship between the two variables. First, for all types of soil the dielectric constant increases slowly with moisture content initially. After reaching a transition moisture value, the dielectric constant increases steeply with moisture content. Second, the transition moisture is found to vary with soil type or texture. In fact, the transition moisture is observed to be strongly correlated with the wilting point of soils and one empirical model is given by S Mt = 0.49 × W P + 0.165

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_9

(9.1)

267

268

9 Soil Moisture Measurement

where WP is the wilting point (WP) which is related to soil texture. WP can be described as a function of the contents of key soil constituents such as clay and sand and in terms of volumetric water content (cm3 /cm3 ) as W P = 0.06774 − 0.00064 × S AN D + 0.00478 × C L AY

(9.2)

where CLAY and SAND are the clay and sand contents in percent of dry weight of a soil. Depending on whether the soil moisture is greater or less than the transition moisture, a specific formula can be used to calculate the soil dielectric constant (ε). This model employs the mixing of the dielectric constants of ice (εi ), water (εw ), rock (εr ), and air (εa ), and treats the transition moisture value as an adjustable parameter [20]. Specifically, the empirical formulas for calculating soil moisture content can be written as:  ε−S P εa −(1−S P )εr , S M ≤ S Mt a (9.3) S M = ε+S Mt εw −S Mεt εx x−ε −S P εa −(1−S P )εr , S M > S Mt εw −εa Here S P is the porosity of the dry soil, defined as Sp = 1 −

ρs ρr

(9.4)

where ρs is the density of the dry soil and ρr is the density of the rock. At the frequency of 1.4 GHz, ρs lies in the range of 1.1–1.7 g/cm3 , while ρr varies between 2.6 and 2.75 g/cm3 . These ranges of values would also be suited for the GNSS signals which have frequencies around 1.2 and 1.5 GHz. In the case where it is difficult to calculate the porosity of the dry soil, the parameter S P may set at 0.5. Parameter εx stands for the dielectric constant of the initially absorbed water, defined as  εx =

εi + (εw − εi ) SSMMt × γ , S M ≤ S Mt εi + (εw − εi ) × γ S M > S Mt

(9.5)

where γ is a parameter which can be chosen to best fit (9.3) to the experimental data to compensate the modeling error. Thus, the model is quite general since it was developed by considering the typical constituents of soils and the use of the transition moisture. Equation (9.3) establishes the relationship between soil moisture and soil dielectric constant. However, the soil dielectric constant is usually unknown and may not be easy to determine. It is a fact that the relationship between soil dielectric constant and reflectivity has well been established. The Fresnel reflection coefficient for vertical polarization (ΓV V ) and that for horizontal polarization (Γ H H ) are functions of the surface soil dielectric constant (ε), given by:

9.1 A Classic Soil Moisture Estimation Method

269

 ε − sin2 θinc  ΓV V = cos θinc + ε − sin2 θinc  ε cos θinc − ε − sin2 θinc  ΓH H = ε cos θinc + ε − sin2 θinc cos θinc −

(9.6)

where θinc is the incidence angle which is complementary to the elevation angle of the visible satellite of interest. In the case of circular polarization such as for a GNSS signal which is transmitted as right hand circularly polarized (RHCP), there are two different Fresnel reflection coefficients due to the fact that some components of the scattered signals are left hand circularly polarized (LHCP), while other components are RHCP. The reflection coefficient for the scattered RHCP (Γ R R ) and scattered LHCP (Γ R L ) are given by: 1 (Γ H H + ΓV V ) 2 1 = (Γ H H − ΓV V ) 2

ΓR R = ΓRL

(9.7)

It can be seen that given the soil moisture content and the type of soil, the dielectric constant and hence the reflectivity of the soil can be calculated. On the other hand, the dielectric constant of the soil can be estimated by measuring the surface reflectivity which is measured from the observed signal powers. Figure 9.1 shows the reflection coefficient of surface soil with respect to elevation angle under three different soil wetness conditions. Clearly, the reflectivity presents a good sensitivity 0.7 0.6

Reflectivity

0.5 0.4 0.3 0.2 Dry SMC Moist SMC

0.1 0

Wet SMC

0

10

20

30

40

50

60

70

80

90

Elevation Angle (degree)

Fig. 9.1 Simulated soil reflectivity versus elevation angle under three different soil moisture contents. The corresponding complex permittivity is set to be 0.035 cm3 /cm3 , 0.18 cm3 /cm3 and 0.36 cm3 /cm3 , respectively

270

9 Soil Moisture Measurement

to soil moisture variation virtually over the whole range of elevation angle. In the following sections, a number of different approaches are studied, which make use of the outputs of GNSS receiver or different statistics derived from the receiver outputs.

9.2 Signal Power Based Soil Moisture Estimation Using GPS signals for soil moisture measurement was originally investigated by a number of researchers [15, 24]. In these initial investigations the basic observation was that the reflected signal power varies as surface soil moisture changes. The soil moisture experiment 2002 (SMEX02) is the first experiment in which both GPS reflected signals from land and in situ data were collected throughout the state of Iowa, USA in June-July 2002. The U.S. National Snow and Ice Data Center (NSIDC) web site provides the complete data acquired during the SMEX02 [16]. The SMEX02 data have been used by researchers to investigate soil moisture retrieval based on GNSS reflectometry. For instance, in [12] calibration of the reflected GPS signals was investigated for soil reflection and dielectric constant estimation. Later on, researchers have conducted many ground-based and airborne experiments as well as exploited satellite data to investigate GNSS-R based soil moisture estimation. For instance, the interferometric method makes use of the power associated with the coherent sum of the direct and reflected GNSS signals [17]. When two antennas (or a composite antenna) with respective horizontal and vertical polarization are installed in a bare soil field with the antenna(s) facing horizontally, an interesting phenomenon would be observed. A notch would occur in the received GNSS signal power after continuously measuring the signal power for a few hours with respect to satellite elevation angle. The occurrence of the notch is only for the vertical polarization and the location of the notch is related to the soil moisture. Theoretical modeling and simulation results also show clear notches in the received signal power variation pattern. That is, by establishing a model to describe the relationship between the location of the notch and soil moisture, it is possible to measure the soil moisture based on the notch observation and the established model. The horizontal-polarization and vertical-polarization Fresnel reflection coefficients related to two layers (indexed by i and i + 1) can be described by 

rvi,i+1

 εri+1 rh i,i+1 =   εri − εri · sin2 (θinc ) + εri+1   εri+1 εri − εri · sin2 (θinc ) − εri εri+1  =  εri+1 εri − εri · sin2 (θinc ) + εri εri+1 εri − εri · sin2 (θinc ) −

− εri · sin2 (θinc ) − εri · sin2 (θinc ) − εri · sin2 (θinc )

(9.8)

− εri · sin2 (θinc )

where εri is the dielectric constant (also called relative permittivity) of the ith layer and θinc is the incidence angle. As indicated in [17], in the case of three layers (air

9.2 Signal Power Based Soil Moisture Estimation

271

and two soil layers), the surface reflectivity coefficient may be described by     4π σs 2 ri,i+1 + ri+1,i+2 exp(s + j2ψ) R = exp − 2 λ 1 + ri+1,i+2 exp(s + j2ψ)

(9.9)

where σs is the soil surface roughness, λ is the signal wavelength (for GPS L1 signal, λ = 19.05 cm), ri,i+1 and ri+1,i+2 are the Fresnel coefficients of the first and second soil layers, and s is the surface roughness correction factor, given by  2  π σ s = −8 εri+1 − εri · sin2 (θinc ) λ

(9.10)

where σ is the standard deviation of the roughness of the interface between layers, assumed to be small such as less than 2 cm. The phase term ψ is given by ψ=

 2π δi+1 εri+1 − εri · sin2 (θinc ) λ

(9.11)

where δi+1 is the thickness of (i + 1)th layer. The total power received thus may be described as P = ηFn (θ )|1 + R · exp( jδφ )|2

(9.12)

where η is a scaling factor, Fn (θ ) is the antenna pattern/gain, θ is the satellite elevation angle, R is the reflectivity coefficient, δφ is the phase difference between the incident and reflected signals. The scaling factor η may be simply set to be unity only when using the power notch position for soil moisture estimation, while it can be adjusted when using the pattern or shape of the total power for soil moisture estimation. Fn (θ ) may be approximated by  Fn (θ ) = −12

θ θant

2 dB

(9.13)

where θant is the 3 dB or half-power beamwidth of the antenna. The phase difference δφ , which was studied in Chaps. 4 and 6, is given by δφ =

4π h ant sin θ λ

(9.14)

where h ant is the antenna height. The power notch occurs at the Brewster’s angle [17], which is computed by  θB = arctan

n2 n1

 (9.15)

272

9 Soil Moisture Measurement

where n 1 is the refractive index of the first medium (i.e. the air) and n 2 is the refractive index of the second medium (i.e. the soil). Making use of ni =

√

εi μi

(9.16)

where μi is the relative permeability, (9.8) becomes

ε2 μ2 ε1 μ1 √ = arctan ε2 μ2

θB = arctan

(9.17)

Here the relative permeability μ1 and relative dielectric constant ε1 of the air are both set to be one. The soil relative dielectric constant ε2 and relative permeability μ2 vary greatly, depending on soil components and soil moisture. Thus, given the observed Brewster’s angle, the soil moisture cannot be determined by the above formulas, indicating the need of further modeling such as those to be discussed in the following subsections. It is an advantage to use existing GPS receivers, installed primarily for geophysical and geodetic applications, to estimate soil moisture [23]. The zenith-looking antenna at a geodetic reference receiver station captures both direct and reflected GPS signals, although the reflected signals are received at a negative elevation angle where the antenna radiation gain is non-zero. The amplitude (or signal-to-noise ratio) and phase (or frequency) of the multipath signal can be used to infer soil moisture information. These receivers, if exploited effectively, could provide a global network for services on soil moisture, vegetation and snow cover [13, 21].

9.3 SNR Based Method There are a range of models proposed to simulate the emission characteristics of a rough surface including the geometric optics model, optical physical model, small disturbance model, integration model, and advanced integration model [6, 11]. That is, the forward scattering coefficient can be calculated when the input parameters are given, which include the surface roughness and the signal incidence angle. As indicated in [14], the simulated forward scattering coefficient varies significantly with both the incidence angle and the surface roughness. When giving a specific incidence angle and surface roughness, a linear relationship between forward scattering coefficient and soil moisture can be established. Another way to modelling is to establish an empirical model based on field measurements. For instance, SNR data recorded by a GNSS receiver were used in [5] to develop an empirical model for soil moisture estimation. The details of such an empirical model based method are described below.

9.3 SNR Based Method

273

9.3.1 Fundamental Theory As studied in Chap. 4, the SNR data recorded by a ground-based GNSS receiver can be modelled by a concise theoretical formula when only considering one reflection path. The signal captured by an antenna of zenith-looking or horizon-looking is combination of direct and reflected signals, so that the SNR consists of three components, the direct signal SNR component, the reflected signal SNR component, and the multipath-induced component. As shown in lower panel of Fig. 9.2, the original raw SNR series exhibits a trend of increasing with sin θ (t) where θ (t) is the satellite elevation angle. This trend can be well modelled by a low-order polynomial such as S N Rd = −12.94 sin2 θ (t) + 22.94 sin θ (t) + 32.78

(9.18)

which is almost equal to the direct signal SNR. After removing the trend by subtracting the SNR term in (9.18) from the original SNR data, a detrended SNR series is generated as shown in the upper panel of Fig. 9.2. Theoretically, the detrended SNR series can be approximated by S N Rm (t) =

2α(t)A2d (t) cos δφ (t) PN

(9.19)

where α(t) is the amplitude attenuation factor (AAF) and δφ (t) is the reflection induced excess phase (see Chap. 4). To reduce the effect of the noise power and the amplitude of the direct signal, which is a function of transmission power, propagation

Fig. 9.2 An example of SNR series recorded by a GNSS receiver and the detrended SNR series

274

9 Soil Moisture Measurement

loss, and antenna radiation pattern, the detrended SNR is normalized by: S N Rmn (t) =

S N Rm (t) A2d (t) PN

= 2α(t) cos δφ (t)

(9.20)

where the divisor is in fact the SNR of direct signal, which can be well approximated by the trend model such as (9.18). Thus, the normalization can be readily implemented in reality. The AAF is a function of a number of parameters like Woodward ambiguity function and antenna gain of both direct and reflected signals. Since the radiation pattern of the antenna is usually known in advance, (9.20) may be further normalized by: S N Rmn2 (t) =

S N Rmn (t) 2α(t) = cos δφ (t) ρG (t) ρG (t)

(9.21)

where ρG (t) is the square root of the ratio of antenna gain of direct signal (G d (t)) over that of reflected signal (G r (t)):

ρG (t) =

G r (t) G d (t)

(9.22)

9.3.2 Definition of a Statistic As seen from Fig. 9.2, the SNR data is noisy although a geodetic receiver and antenna were used to obtain the data. A variety of filters can be applied to smooth the data and reduce the noise. For instance, the simple median filter can be used, which is easy to implement and even more effective than the average algorithm. A 0.1-degree window of elevation angle is used, and the median of the SNR data within the window is treated as the output SNR value. The window is moving from the start to the end of the selected SNR series without overlapping. Thus, this filtering technique may be termed sliding window moving median (SWMM). The selection of such sliding window mainly depends on the sampling rate of the SNR data and the elevation angular velocity of GNSS satellite. Except for the geostationary satellites, the orbital periods of all GNSS satellites are about 12 h. That is, about 6 h cover 180 degrees of elevation angle, resulting in an average angular velocity of 0.0083 degrees per second. Thus, a 0.1-degree window of elevation angle covers about 12 angle samples when the sampling rate of SNR data is 1 Hz. The blue dashed curve shown in Fig. 9.3 is the SNR series smoothed by such a median filter. Compared with the original detrended SNR series in Fig. 9.2, the noise has been greatly reduced. The smoothed SNR data are still a bit noisy, so it is preferable to do the data processing one more time. For

9.3 SNR Based Method

275

Fig. 9.3 An example of smoothed SNR series by a median filter and fitted SNR series by sinusoidal fitting

instance, the parabolic fitting is used in [5] for fitting each semi-cycle of positive or negative waveform with the fitting function given by: F(θ ) = a2 sin2 θ + a1 sin θ + a0

(9.23)

where {ai } are the fitting coefficients of parabolic function,which would be different for different semi-cycle waveform. It can be readily derived that the local maximum or minimum function values at the crest or trough points are equal to FM =

4a2 a0 − a12 4a2

(9.24)

In order to perform modelling, it is required to choose or design at least one statistic which is sensitive to soil moisture. From processing experimental data, it was noticed that those maximum and minimum values are sensitive to the variation in soil moisture. Thus, the minimums and maximums can be used to design a statistic. For instance, it may be defined as the mean of the absolute value of the extremum, which is termed mean peak in [5]: q 1 |Fi | F¯ = q i=1

(9.25)

276

9 Soil Moisture Measurement

where q maximum and minimum values are used to calculate the statistic. It is selected by visual inspection such that the extremum is significantly greater than noise level and can be clearly observed. The use of mean or average is intended to reduce the effect of noise and interference. The observation variable defined in (9.25) will be used for the establishment of empirical models as follows.

9.3.3 Definition of Observation Variable To develop an empirical model, a large data set is usually used, which is collected from field experiments. Some empirical models can be widely used since they are developed using data obtained from a variety of different fields and a few model parameters are adjustable depending on the specific environment. Some other models may only be suited to similar environments because they are developed with data collected in a particular environment or several similar environments and the model parameters are fixed. The statistic defined in (9.25) is mainly affected by the scattering environment and the soil composition, both of which can be rather different in different geographical locations. Thus, it is necessary to establish an independent empirical model to describe the relationship between the soil moisture and the statistic. Here the data from PBO H2O project was used for modeling, which was funded by the NSF GEO directorate, specifically the Atmospheric Sciences, Hydrology, and EarthScope programs. The data can be downloaded from [19]. The PBO H2O project makes use of the data from the Plate Boundary Observatory (PBO) to study the GNSS-R based measurement of water-cycle parameters, including snow depth, snow water equivalent, soil moisture and vegetation. The data recorded at the MFLE station was processed to generate the time series ¯ The selection of the station is based on the fact that the surface is roughly flat, of F. the vegetation is rather sparse and the vegetation variation over the four seasons is insignificant. As a consequence, the effect of vegetation can be ignored. Nevertheless, vegetation effect has to be considered for soil moisture measurement in scenarios where vegetation affects the signal scattering significantly. Otherwise, large errors in soil moisture estimation would occur. Currently, it is still a challenging problem to handle vegetation effect since there are so many different kinds of vegetation and the coverage and density also vary a lot with time and space. Thus, more efforts are needed to find solutions to mitigate vegetation effect in the future. As mentioned earlier, except for the geostationary satellites, all GNSS satellites have an orbital period of about 12 h. Thus, each satellite has two times of ascending and two times of descending, so that four F¯ values can be generated each day for each satellite. That is, there are four possible F¯ sequences over the year for each satellite. In total, 62 complete statistic sequences were produced, associated with 28 GPS satellites. Figure 9.4 shows only three such sequences of the statistic F¯ as well as the sequence of the in situ soil moisture values measured by the probe as ground-truth data. When the soil moisture increases, the electrical conductivity of soil increases, more power of signal is absorbed by soil and hence less power is reflected to the

9.3 SNR Based Method

277

Fig. 9.4 Three examples of F¯ series and soil moisture series over one year. “G28.dscen.2” denotes the results obtained by using GPS signal observed over the second descending periods of satellite G28

receiver. This is in accordance with the results shown in Fig. 9.4 where the F¯ time series show a negative correlation with the in situ soil moisture. Based on this fact, the reciprocal of each mean peak F¯ is used, that is, the observation variable is defined as (q) Fr eci pr ocal

1 1 = k j=1 F¯ j(q) k

(9.26)

where q is the index of days, there are k mean peaks of all satellites in a day, and (q) F¯ j is the jth mean peak on the qth day. The daily time series of Freciprocal is then used in regression analysis for modeling.

9.3.4 Development and Verification of Empirical Models To develop an empirical model, a large amount of data is collected by conducting experiments in one or more geographical locations over an interval of time period. An empirical model can only be applied to the same situation or situations very similar to that where the data were collected for the model development.

278

9.3.4.1

9 Soil Moisture Measurement

Using PBO H2 O Data

In this case 6-month GPS data collected over a single GPS station is used to establish an empirical model. Thus, the model to be developed may not be suited to other GNSS stations and it is desired to develop different models for different stations, although some models may be very similar. The ground-truth daily soil moisture data were generated by an in situ probe installed at 2.5 cm depth within 250 m of the GPS antenna. Through regression analysis by using the Freciprocal time series and the daily average of in situ probe based soil moisture measurements, it is observed that a parabola function is the best regression function in terms of minimum fitting error, which is given by S M = 0.0651Fr2eci pr ocal − 0.1397Fr eci pr ocal + 0.0423

(9.27)

where SM denotes soil moisture content. Figure 9.5 shows the model curve produced by the parabolic fitting. The curve basically represents the data well, although the deviation is also considerable. To verify the developed model, a different data set was used, which was also collected at the same GPS station but over a time interval of another nine months. That is, a set of six-month data was used for modelling, while a different set of ninemonth data was used for verification. Figure 9.6 shows the scatterplot of soil moisture estimation of the PBO-H2O method [7, 8] and the method based on the model given by ((9.20). The root-mean-square error of the former is 0.0558 cm3 cm−3 , while that of the latter is 0.0345 cm3 cm−3 , respectively. The in situ soil moisture ranges between about 0.04 and 0.29 cm3 cm−3 . The performance difference between the two different methods would mainly come from the difference in the modeling error.

Fig. 9.5 Parabolic modeling based on regression analysis of MFLE station

9.3 SNR Based Method

279

Fig. 9.6 Scatterplot of soil moisture estimation of two different methods. “Proposed method” denotes results using (9.20)

9.3.4.2

Using Data Collected in Wuhan

The second data set was obtained by conducting a field experiment for a period of two weeks in June 2019 in Wuhan, China. A Trimble R9 GNSS receiver and a zenithlooking geodetic antenna were used to collect the direct and reflected GNSS signals. The vegetation coverage is relatively sparse, so the ground surface can be treated as bare soil. The ground truth soil moisture was measured by a sensor installed 5 cm beneath the ground surface with a distance less than 3 m from the antenna, as shown in Fig. 9.7. The sensor generates soil moisture measurement with a sampling period of 10 min. Two rainfalls occurred during the experiment, so that the soil moisture changed significantly over the observation period. The recorded SNR series were processed according to the procedure described earlier to generate the statistic F¯ and the observation variable Freciprocal . Due to the Fig. 9.7 A soil moisture sensor probe was inserted into the soil horizontally about 5 cm beneath the surface

280

9 Soil Moisture Measurement

Fig. 9.8 Parabolic modeling based on regression analysis based on data collected over two weeks

limited observation period of two weeks, all the available data were divided into two parts based on satellites instead of days. One part of data is used for modeling and the other is used for verification. Figure 9.8 shows the calculated observation variable under a range of soil moisture values between 0.14 and 0.45 cm3 cm−3 . In this case, the parabolic fitting also generates the minimum fitting error, which is given by S M = −0.0214Fr2eci pr ocal + 0.1998Fr eci pr ocal − 0.0951

(9.28)

Although both (9.27) and (9.28) indicate that the models are second-order polynomial (or parabola), the model parameters are quite different, mainly due to the difference in soil composition, surface roughness and vegetation. This reminds that independent modeling should be carried out for different scenarios. Then, the established empirical model is used to measure the same soil moisture values used for modeling, but the second part of GPS data was used to generate the sequence of observation variable Freciprocal . The calculated observation variable values and (9.21) are used to calculate the soil moisture values and the results are shown in Fig. 9.9. It can be seen that there is a good match between the in situ measurements and the estimates by the empirical model. In several short periods when the soil moisture values are high due to heavy rain, the empirical model considerably underestimates the ground-truth soil moisture. One possible reason is that the surface is basically covered by a thin layer of rain water, or the soil is almost in a saturated state. As a consequence, the observation variable may not change with the further increase in the soil moisture.

9.4 GEO Satellite Based Method

281

Fig. 9.9 Ground-truth soil moisture sequence (black solid line) and empirical model based soil moisture estimates (red dots)

9.4 GEO Satellite Based Method As mentioned earlier, the orbital periods of typical MEO GNSS satellites are about 12 h, so that GNSS signals recorded over the two ascending intervals and two descending intervals of the satellite can be used to retrieve geophysical parameters when using a zenith-looking antenna for data collection. It is a fact that China’s BDS-2 contains five geostationary (GEO) satellites and China’s BDS-3 contains three GEO satellites. Figure 9.10 shows the projected locations of the five BDS-2 GEO satellites on the Earth surface. Also shown are the coverages of the satellites. In addition, a number of satellite based augmentation systems (SBAS) have been established for enhancing regional navigation capability by EU, U.S., Russia, Japan and India. All SBAS satellites are GEO satellites and the footprint of the network

Fig. 9.10 Projected positions and coverage of five BDS-2 GEO satellites

282

9 Soil Moisture Measurement

of the GEO satellites covers most of the Earth surface, enabling GEO satellite based reflectometry to be a real-time and continuous global remote sensing technique. This section presents the GEO satellite based soil moisture estimation method reported in [2].

9.4.1 GEO-IR Based Soil Moisture Estimation Although all GEO satellites remain static with the Earth, it does not mean that the satellite elevation angle remains exactly the same. Instead, it is observed that the elevation angle of a GEO satellite typically varies between 0.5° and 1° around a mean value, which is a quasi-sinusoidal signal with a period of about 24 h. On the other hand, the elevation angle of a MEO satellite can gradually change from zero to a peak value such as 70° and then gradually decreases to zero. That is, the GEO satellites are still moving relative to the Earth, but the movement is very slow so that the Doppler frequency can be ignored. As a consequence, the variation in the SNR amplitude and excess phase signals of a GEO satellite is much smaller and slower than those of a MEO satellite. Figure 9.11 shows two examples of the simulated SNR and the combined carrier phase (see Chaps. 6 and 8 for more details) with two 46

SNR (dB)

45 44 43

Excess Phase (mm)

42 1 0 -1 -2 -3

0

1

2

3

4 5 Time (s)

6

7

8 x 10

4

Fig. 9.11 Simulated SNR and excess phase time series of a GEO satellite signal over a period of a full day with a different elevation angle and antenna height respectively. Solid blue line: antenna height 1 m and elevation angle 29.1°–29.9°. Dashed red line: antenna height 3 m and elevation angle 8°–9.9°

9.4 GEO Satellite Based Method

283

different antenna heights (1 and 3 m) and two different elevation angle range. The SNR expression is given by (4.17) and the combined excess phase is given by (8.22), but they are repeated here for convenience: S N Rd (t) =

2 Ad (t)Am (t) cos δφ(t) PN

(r ) M1,2,5 (t) = M1,2,5 (t) = κ1 β1 (t) + κ2 β2 (t) + κ5 β5 (t)

(9.29) (9.30)

It can be seen that both SNR and combined excess phase are significantly affected by antenna height and elevation angle. The SNR and excess phase of a MEO satellite signal are also affected by antenna height and elevation angle as studied in Chaps. 4 and 8, but their variation patterns are rather different. The main reason is that the range of elevation angle is different as mentioned earlier and that the gain pattern of a geodetic antenna varies with elevation angle significantly.

9.4.1.1

Development of Empirical Models

Similar to the GNSS-R based snow depth estimation studied in Chap. 8, the SNR and carrier phase, two of the ordinary observations of GNSS receivers, can be used to estimate soil moisture. As studied in Chaps. 4 and 8, the amplitudes of both the detrended SNR and combined carrier phase increase with the increased amplitude of the reflected signal. As the soil moisture increases, the soil permittivity or soil dielectric constant increases and hence the amplitude of reflected signal decreases. That is, the amplitudes of SNR and combined carried phase are a function of soil moisture. It is impractical to establish a model to describe the relationship between the soil moisture and the reflected signal without involving the soil type and texture as mentioned in Sect. 9.1. Nevertheless, for a give type of soil with similar compositions, a specific empirical model can be developed. The typical range of soil moisture is between 0.10 and 0.40 cm3 cm-3 , so a range of different soil moisture contents within the range can be selected. The elevation angles and positions of GEO satellite C04 of BDS observed in Hubei CORS were employed to simulate the SNR signal and combined excess phase for a full day. Figure 9.12 shows how the soil moisture affect the SNR and excess phase. For clarity, only four curves related to four soil moisture contents are displayed for SNR and combined excess phase. It can be seen that over a full day, except for two small periods for the SNR signal, the amplitudes of both SNR and combined excess phase are quite sensitive to soil moisture variation. Two time periods are selected and marked by two empty column bars, which correspond to the highest and lowest elevation angle ranges, respectively. Over each of the periods, the variation of the signal related to a soil moisture content is quite small, which needs a special receiver with significantly high SNR resolution. However, when the station is in higher latitude and/or the antenna height is higher, the difference in SNRs related to different soil moisture

284

9 Soil Moisture Measurement 45.7

SNR (dB)

(a) 45.65

0.1 0.2 0.3 0.4

45.6

Excess Phase(mm)

45.55

-10

-15

-20

-25

0

1

2

3

4

5 Time (s)

6

7

8

9 x 10

4

41

(b) 0.1 0.2 0.3 0.4

SNR (dB)

40

39

38

Excess Phase(mm)

37 40 30 20 10 0 -10

0

1

2

3

4

5 Time (s)

6

7

8

9 x 10

4

Fig. 9.12 Simulated GEO SNR and combined excess phase signals time series with four different soil moisture contents and two different antenna heights. a The station latitude is 30.5° and antenna height is one meter; b The station latitude is 66.3° and antenna height is two meters

contents become larger as shown in Fig. 9.12b and thus traditional geodesy receiver with SNR resolution of 0.01 dB can be used. Table 9.1 shows the SNR and combined excess phase values for seven given soil moisture contents. “L” and “R” denote respectively for the data of left and right column bars and the time duration is 3000 s. Each of the amplitude values in the table is equal to the average of the amplitudes over the time duration. Note that in the case of CORS, the antenna height is usually in the range of 3–5 m and thus the SNR

9.4 GEO Satellite Based Method

285

Table 9.1 SNR and excess phase amplitudes versus soil moisture contents with station latitude 66.3° and antenna height two meters SM (cm3 cm−3 )

0.10

0.15

0.20

0.25

0.30

0.35

0.40

SNR (dB)

L

37.15

37.21

37.30

37.41

37.52

37.62

37.76

R

40.22

40.16

40.08

39.99

39.90

39.81

39.69

Phase (mm)

L

6.648

5.479

3.931

2.280

0.663

−0.856

−2.811

R

39.94

39.49

38.68

37.67

36.60

35.52

34.04

difference between the neighboring soil moisture contents will increase, enabling better estimation performance. Using least squares (LS) curve fitting, empirical models can be developed based on the results shown in Table 9.1. The linear and second-order models are given as follows: Linear model a1 ϑ + a0 , (9.31) SM = b2 ϑ 2 + b1 ϑ + b0 , Second - order model where SM denotes the soil moisture content with unit cm3 cm−3 , ϑ is the SNR magnitude with unit dB or the combined excess phase magnitude with unit mm, and {ai } and {bi } are the coefficients. The values of these coefficients are displayed in Table 9.2. Figure 9.13 shows the original simulated soil moisture contents versus SNR and combined excess phase as well as the linear and second-order fitting curves. It can be seen that both the linear fitting and parabolic fitting are appropriate, although the latter is slightly better. It is worth noting that these models are produced using elevation angle of a specific GEO satellite, a specific receiver antenna gain pattern, and under the assumption of a type of bare soil. When a different GEO satellite is used, or the antenna gain pattern or the soil type is significantly different, it is necessary to establish new models. Table 9.2 Coefficients in the linear and parabolic models and fitting errors SNR Excess error

SNR Excess error

a1

a0

Fitting error

L

0.4832

−17.83

0.0115

R

−0.5627

22.75

0.0063

L

−0.0315

0.319

0.0065

R

−0.0495

2.103

0.0166

b2

b1

b0

Fitting error

L

−0.2361

18.16

−348.9

0.0063

R

−0.2957

23.07

−449.4

0.0038

L

−0.0005

−0.0293

0.3223

0.0038

R

−0.0374

0.2276

−3.019

0.0080

286

9 Soil Moisture Measurement

-3

Soil Moisture (cm cm )

0.4 3

0.35 0.3 left linear model

0.25

right linear model simulated SM

0.2

left second-order model simulated SM

0.15 0.1 37

right second-order model

37.5

38

38.5 39 SNR(dB)

39.5

40

40.5

-3

Soil Moisture (cm cm )

0.4

3

0.35 0.3 0.25 0.2 0.15 0.1 -10

0

10

20

30

40

Excess Phase (mm) Fig. 9.13 The linear and parabolic fitting results for SNR based soil moisture estimation (left panel), and those for combined excess phase based soil moisture estimation (right panel)

9.4.1.2

GEO-IR Experimental Results

A data set was collected from the beginning to the end of November 2015 with a geodetic GNSS receiver and a geodetic antenna in HBHN station in Hubei, China. The surface around the antenna is flat enough and the soil around the antenna was bare without vegetation. During the data recording, there were intermittent rains. The collected data are only the observations of signals transmitted by GPS and BDS satellites, but unfortunately the real in situ soil moisture data are not available. Thus, only a comparison study is performed to evaluate the feasibility and accuracy of the GEO-IR method studied in the preceding subsection. Since only dual-frequency signals were recorded, the combined carrier phase based method is not evaluated. The satellite signals were captured via the zenith-looking antenna, the GPS-IR method

9.4 GEO Satellite Based Method

287

proposed by Zavorotny et al. [23] was also evaluated. Thus, this is really a preliminary study on the GEO-IR method due to the rather limited experimental data. Figure 9.14 shows the soil moisture estimation results using the two SNR methods (GPS-IR and GEO-IR) over the whole month of November 2015. SNR observations associated with two GEO satellites (C04, C05) and one GPS satellite (PRN 09) are used to estimate soil moisture. A number of observations can be made from the results as follows. The results related to the two GEO satellites are very similar, showing a good consistency between results produced by the use of signals from different GEO satellites. The results of the two different methods have a very similar pattern especially with peaks occurring almost at the same time points, indicating that the two methods would produce similar results and achieve similar accuracy. However, although the results are similar, the estimated soil moisture by GEO-IR is considerably greater than that by GPS-IR on average. The difference may be contributed by a number of different factors. One factor is that the received signal strength of a BDS GEO satellite is different from that of a GPS satellite. Another factor is that the time interval of GEO signal observation can be quite different from that of GPS signal observation, so that the two methods may estimate the soil moisture at two different time periods. Due to lack of groundtruth soil moisture data, it is difficult to tell whether the GPS-IR, GEO-IR, or both underestimates or overestimates the soil moisture. Nevertheless, it is expected that the true soil moisture would have a similar pattern and be close to the results by the two methods. It is worth noting that the SNR based GEO-IR method is suited for scenarios where latitude is high and thus the elevation angle is low as well as antenna height is high such as greater than 2 m. On the other hand, the phase combination method does not have such limitation. 0.35

Soil Moisture (cm 3cm-3 )

0.3 0.25 0.2 0.15 0.1

GEO C05 GEO C04 GPS PRN09

0.05 0 0

5

10

15

20

25

30

November (day)

Fig. 9.14 Results of soil moisture estimation by SNR based GEO method and GPS method [23]

288

9 Soil Moisture Measurement

9.4.2 GEO-R Based Soil Moisture Estimation 9.4.2.1

A Few Basic Concepts

GEO-R is a special case of GNSS-R, which utilizes signals from geostationary satellites. GNSS-R makes use of two or more individual antennas to receive the direct and the reflected signal separately. Note that GNSS-R is often refereed as a general concept for all the techniques which use reflected GNSS signals to infer the information of reflection or scattering surface, including GNSS-IR (or GNSS-MR) and GNSS-R. Regarding the so-called GNSS-R, an up-looking RHCP antenna is used for collecting the direct signal, while one or more down-looking LHCP antennas (and RHCP antenna) are used for capturing the reflected signal. The use of a LHCP antenna is due to the fact that the polarization of the signal has largely changed from RHCP to LHCP when the signal is reflected by a surface and the elevation angle is not small. Various studies indicate that the peak power of the reflected signal is associated with the soil moisture content [3, 7]. However, it may be difficult to precisely locate the peak power point corresponding to a specific Doppler frequency and code delay. This is because there is the resolution issue, that is, a relatively large frequency interval and code delay interval are typically used to generate the DDM onboard an instrument to save memory space and reduce computational complexity. For instance, Fig. 9.15 shows an example of the DDM of the reflected GPS signal over a land surface, received by a static GNSS-R receiver with frequency interval of 25 Hz and code delay interval of one-eighth code chip. The approximate location instead of the precise location of the peak power can be identified and the ambiguity increases as the frequency and code delay interval increase. This difficulty can be considerably reduced when employing GEO-R method and a ground static receiver. Since there is no Doppler frequency, the three-dimensional DDM is reduced to two-dimensional delay waveform termed delay map (DM). Although the code delay interval or resolution is still an issue, simpler one-dimensional interpolation can be used to smooth the delay waveform as shown in Fig. 9.16, while more complex two-dimensional interpolation is required to smooth the DDM. Thus, in addition to the complexity reduction, more accurate peak power location and value may be achievable when using one-dimensional interpolation.

9.4.2.2

GEO-R Empirical Modeling

As mentioned earlier, land surface reflection coefficients are complicated functions of satellite elevation angle and dielectric permittivity of surface soil. In addition to the models described in Sect. 9.4.1, a variety of empirical or semi-empirical models were developed for soil moisture retrieval in the past four decades such as the one studied in Sect. 9.1 [20] which is still often referred to. However, these models require the knowledge of the percentages of textural components (e.g. sand and clay) of soil

9.4 GEO Satellite Based Method

289 dB

0.5

78

1

76 74

1.5

72 Code Delay(chip)

2 70 2.5

68

3

66 64

3.5

62 4 60 4.5

58

5 -200

-150

-100

-50

0

50

100

150

200

Doppler Delay(Hz)

Fig. 9.15 An example of DDM of received GPS satellite signal reflected over a land surface 78 Observed DM Fitting line

76 74

Power (dB)

72 70 68 66 64 62 60 58 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Code Delay (chip)

Fig. 9.16 Original delay waveform (red dotted line) of reflected GEO satellite signal and the smoothed delay waveform (solid line) based on interpolation

290

9 Soil Moisture Measurement

and one or more other parameters such as the transition moisture parameter. Such knowledge of those parameters may not be available in practice, so it is difficult to apply these models. Thus, it is desirable to develop models which do not rely on the knowledge of soil components, such as those studied in Sect. 9.3. Here the GEO-R method for soil moisture retrieval is described as follows. Signal Peak Power Based Modeling The basic idea is to develop a polynomial or a linear model to describe the relationship between the peak power and the soil moisture. Figure 9.17 (left panel) shows the delay map of the reflected GEO satellite signals which were recorded by the Integrated GNSS Remote Sensing Instrument manufactured by Chinese Academy of Science. The instrument antennas were fixed above a ground surface of bare soil. Data were collected over a period of two weeks, during which a rainfall occurred, so that the GEO signals were recorded under a range of twelve soil moisture contents. For clarity, only four DM curves, corresponding to four different soil moisture contents, are plotted. As mentioned earlier, the original data were smoothed for better locating the peak power point as shown in the right panel of Fig. 9.17. After obtaining the peak power value for each soil moisture content from the smoothed DM, a simple regression analysis is applied to establish the mathematical relationship between the signal peak power and the soil moisture content through least-squares based curve fitting. Both linear model and second-order model are considered, which are yielded as SM =

0.02088 ∗ p − 1.321, Linear model 0.00207 ∗ p − 0.2892 ∗ p + 10.26, Second - order model

(9.32)

where SM is soil moisture content with unit cm3 cm−3 and p is peak power of DM of the reflected signal. Figure 9.18 shows the soil moisture contents versus the original 80

80 35.5% 24.0% 18.0% 15.3%

Power

70 65 60 55

75 Power (dB)

75

35.5% 24.0% 18.0% 15.3%

70 65 60

0

0.5 1

1.5 2 2.5 3 3.5 4 Code Delay(chip)

4.5 5

55

0

0.5 1

1.5 2 2.5 3 3.5 4 Code Delay(chip)

4.5 5

Fig. 9.17 Original (left) and smoothed (right) reflected signal power versus code delay under four different soil moisture contents

9.4 GEO Satellite Based Method

291

0.5 0.45

observed SM linear model second-order polynomial

Soil Moisture (cm 3cm -3)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 68

70

72

74

76

78

80

82

Peak Power(dB)

Fig. 9.18 Linear and second-order models generated through least-squares fitting

peak power values and the two fitting curves. The fitting error in terms of RMSE is 0.0381 and 0.01001 cm3 cm−3 for the linear and second-order model, respectively. For the range of soil moisture values tested, the second-order model is preferable due to the much smaller fitting error. However, this second-order model is not suitable for dry soil with soil moisture below 0.15 cm3 cm−3 as shown in Fig. 9.18. Therefore, it is necessary to select the model based on the peak power magnitude or some prior information or initial estimation of the soil moisture. For instance, in this case, the second-order model is selected when the peak power is greater than 68 dB, while the linear model is used for peak power less than 68 dB. In addition to the station site where the data were collected for the model development, the model can also be used to estimate soil moisture in scenarios where the surface and soil have a similar type (e.g. bare soil) and the antenna gain is similar. In the case where there is a significant difference in the soil type or in the antenna gain, it is necessary to determine the new model parameters. Some GNSS receivers provide power observations while more GNSS receivers including traditional ones provide SNR observations or both power and SNR observations. Therefore, it is useful to make use of SNR observations to infer soil moisture information, as already studied in Sect. 9.3. SNR Based Modeling In Sect. 9.3, the detrended SNR data are normalized to remove the effect of the SNR component of the direct signal. Here, the SNR of the received reflected signal is directly normalized through dividing it by SNR of the received direct signal as

292

9 Soil Moisture Measurement

γS N R =

S N Rr S N Rd

(9.33)

which is also the ratio of the reflected signal SNR over the direct signal SNR. Note that in the case of interferometric reception studied in Chap. 8 and Sect. 9.3, the direct and reflected signal SNRs can only be estimated instead of direct observation. Regarding the GEO-R considered here, both direct and reflected signal SNRs are recorded directly by receiver. Equation (9.26) can be written as γ S N R = αn · ηs

(9.34)

where αn and ηs are the signal power ratio and noise power ratio, respectively, defined as αn =

Pn,d Ps,r ; ηs = Pn,r Ps,d

(9.35)

Here the subscript “n” is for noise, “s” for signal, “r” for reflected signal, and “d” for direct signal. Thus, the SNR ratio defined by (9.34) is the scaled surface reflection coefficient which is directly associated with soil composition, soil moisture and surface roughness. In the case of a flat ground surface, the effect of roughness may be ignored. Since the noise power in the received reflected signal would be usually larger than that in the received direct signal, the scaling factor (αn ) is less than unity. This scaling factor would be receiver-dependent, but the same type of receiver would have very similar scaling factor. A field experiment was conducted in Baoxie Town, Wuhan, China, from 9 November 2014 to 7 December 2014, using two ground-based BDS receivers. A zenith-looking RHCP antenna was used to receive the satellite direct signal and a nadir-looking LHCP antenna was used to capture the reflected signals. The two antennas were mounted on top of two individual pillars separated by seven meters, and the antennas are two meters above the ground. The ground-truth soil moisture data were obtained using two FY-H2 soil hygrometers manufactured by Fortune Flyco Electronic Technology Co., Ltd. The penetration depth of the soil hygrometers was 5 cm which is the typical depth of interest. The SNR observations of BDS satellite C04 were employed to generate the SNR ratio. A specific sequence of ground-truth soil moisture values and the corresponding calculated SNR ratios are used for modeling. Since the soil moisture varies with time, seven different time periods were selected. Over each time period of 20 min, the variation in soil moisture is usually small and the average of the soil moisture is used. In the same way, the values of SNR ratio over the same period are averaged. The selected soil moisture contents should not very close to each other, but should be roughly evenly distributed, covering the range of soil moisture contents of interest. Thus, seven suitable soil moisture contents are selected, corresponding to seven

9.4 GEO Satellite Based Method

293

0.7 observed SM linear model second-order polynomial

0.5

3

-3

Soil Moisture (cm cm )

0.6

0.4

0.3

0.2

0.1

0 0.6

0.65

0.7

0.75

0.8

0.85

0.9

SNR Ratio

Fig. 9.19 Linear and second-order polynomial relationship between SNR ratio and soil moisture using LS fitting

values of SNR ratio. The selected seven soil moisture contents and seven SNR ratios are used to generate the model through regression analysis. Similarly, both linear fitting and second-order polynomial fitting are employed, based on LS technique. The linear and second-order models are generated as SM =

Linear model 1.249 ∗ γ S N R − 0.7046, 4.443 ∗ γ S2N R − 5.328 ∗ γ S N R + 1.699, Second - order model

(9.36)

The fitting error in terms of RMSE is 0.0284 and 0.0087 for the linear and secondorder model, respectively. Figure 9.19 shows the original soil moisture measurements (dotted line) and the linear and second-order fitting curves. Clearly, the second-order polynomial model has a better match with the ground-truth data and thus is preferable.

9.4.2.3

GEO-R Model Verification

The in situ soil moisture data and the SNR data collected between 9 November 2014 and 7 December 2014 were used for model building and verification. However, the data used for modeling are different from the data used for verification; they were collected at different time periods. In fact, the data for verification are much larger than the data for modeling. After calculating the values of SNR ratio, they are used to calculate the soil moisture contents by (9.29). Figure 9.20 shows the in situ soil moisture data and the GEO-R based estimation results using the linear model and the second-order model, respectively.

294

9 Soil Moisture Measurement 0.3 real data linear model second-order model

Soil Moisture(cm3cm-3)

0.25

0.2

0.15

0.1

0.05 2014/11/09

2014/11/16

2014/11/23

2014/11/30

2014/12/07

Time

Fig. 9.20 Soil moisture estimation results by GEO-R and in situ soil moisture measurement data

It can be seen that the GEO-R estimation results basically have a good match with the ground-truth data, indicating the feasibility of the GEO-R method. The estimates of soil moisture fluctuate around the ground-truth soil moisture and the RMSE of linear and second-order polynomial fit are computed to be 0.0231 cm3 cm-3 and 0.0201 cm3 cm-3 , respectively. The second-order polynomial fit has a better match with the ground-truth data when the soil moisture is low; however, the linear model is better when the soil moisture is high. The fluctuation and error may be caused by the variation of the quality of SNR observations as well as by the difference between the length of the soil hygrometer in the soil and penetration depth of BDS satellite signals in the soil. Although the GEO-R method can quickly response to the raining event, there is a rather big gap between the in situ soil moisture measurement and the estimated one during the period between November 23 and 26. It might be possible that there existed an error in the measurement by the soil hygrometer due to the big rain. It is also likely that the in situ measurement provided by the soil hygrometer is correct. Then, the GEO-R method cannot be able to follow the sudden and great change in the soil moisture.

9.5 An Airborne Experiment for Soil Moisture Measurement This section focuses on the campaign of an airborne experiment and preliminary data processing results. The purpose of the experiment was to collect GNSS data over land areas to investigate soil moisture estimation using an airborne GNSS-R receiver. The aircraft, the receiver and antenna used for the experiment were the same

9.5 An Airborne Experiment for Soil Moisture Measurement

295

as those already introduced for the other two experiments for investigation on sea surface altimetry in Chap. 5 and wind speed estimation in Chap. 6.

9.5.1 Selection of Experimental Site As mentioned earlier, soil moisture is very important to a range of applications and sciences, including agricultural application since it is one of the key factors limiting crop production. Therefore, it is desirable to conduct the airborne experiment in agricultural fields. Since UNSW owned a small aircraft which had been used for a number of other experiments, the experimental site should not be too far away from Bankstown Airport, west Sydney, where the aircraft was parked. In addition, to evaluate the performance of the GNSS-based technique, an independent validation technique is required to provide the ground-truth reference. Clearly, the Yanco region in NSW Australia is a good choice for the experimental site, which is related to the Soil Moisture Active Passive (SMAP) mission. SMAP is a mission recommended by the U.S. National Research Council Committee on Earth Science and Applications from Space in 2007. SMAP satellite was successfully launched in January 2015 and started operation in April 2015. The prime mission was completed in 2018 and it has been in extended operation phase since then [18]. A radiometer and a synthetic aperture radar (SAR) are the main SMAP instruments to measure surface emission and backscatter to sense soil conditions. SMAP has provided measurements of soil moisture and its freeze/thaw state globally, which have both high science value and high application value. SMAP has a range of application areas, including weather and climate forecasting, drought, floods and landslides, agricultural productivity, and human health. The Soil Moisture Active Passive Experiments (SMAPEx) are the pre-launch SMAP validation campaigns in Australia, which consist of a series of three aircraft and field experiments specifically designed to contribute to the development of soil moisture retrieval algorithms from radar and radiometer for the SMAP mission. Prof. Jeff Walker from the Department of Civil Engineering, Monash University led a research group to work on the SMAPEx. The three experiments were conducted on 5–10 July 2010, 4–8 December 2010 and 5–23 September 2011. The experiment site is located in the Yanco region, a semi-arid agricultural area in the Murrumbidgee Catchment, about 530 km west of Sydney, as shown in Fig. 9.21. Concurrently with each flight, supporting ground data on soil moistures, soil temperature, and surface roughness were collected at intensive monitoring sites. In the ground SMAPEx network there were 24 surface monitoring stations (0–5 cm) and profile monitoring stations (0–90 cm) which are unevenly distributed over an area of 36 km x 38 km as shown in Fig. 9.22. Note that only the soil moisture data measured by surface monitoring stations can be used as the ground-truth data, since the GNSS signals can only penetrate soil up to a number of centimetres. These monitoring stations/sites were established in 2009 and denoted by small squares. Also, between 2001 and 2005, thirteen stations were established in the Yanco region, denoted by

296

9 Soil Moisture Measurement

Fig. 9.21 SMAPEx Yanco study area (provided by Dr. Alessandra Monerris-Belda at Monash University)

Fig. 9.22 Distribution of soil moisture monitoring stations in Yanco region. The picture was generated using GPS Visualizer and Google Earth

stars. The WGS84 positions of these SMAPEx stations are listed in Table 9.3. Listed are also the station ID and the year when the station was established. These data were provided by Dr. Alessandra Monerris and Prof. Jeff Walker from Monash University. Note that the 24 monitoring stations in Yanco region are distributed in two areas, 13

9.5 An Airborne Experiment for Soil Moisture Measurement

297

Table 9.3 Positions of 13 monitoring stations: “Y” for Yanco Region Station ID

Established

Latitude

Longitude

Elevation (m)

YA1

2009

−34.6889

146.0855

130

YA3

2009

−34.677153

146.139695

132

YA4a

2009

−34.706005

146.079365

131

YA4b

2009

−34.703062

146.105287

132

YA4c

2009

−34.714213

146.094253

130

YA4d

2009

−34.714202

146.075058

130

YA4e

2009

−34.721393

146.102972

132

YA5

2009

−34.712858

146.127712

132

YA7a

2009

−34.7326

146.0802

132

YA7b

2009

−34.737835

146.098668

130

YA7d

2009

−34.7544

146.077773

129

YA7e

2009

−34.750728

146.094928

132

YA9

2009

−34.741377

146.153637

133

in “YA” area and 11 in “YB” area. To save space, only data related to “YA” area is presented.

9.5.2 Design of Flight Trajectory and Actual Ground Tracks Figure 9.23 shows the enlarged “YA” area in the Yanco region and the distances between every pair of stations are listed in Table 9.4. The shortest distance between a pair of stations is 1.13 km, while the longest distance is 10.28 km. The dimensions of the whole area shown in Fig. 9.23 are about 8.96 km (width) × 11.13 km (height). To cover most of the stations in the Yanco “YA” study area, the aircraft would fly over the areas in a zigzag pattern as shown by the dashed blue lines in Fig. 9.23, which was the flight trajectory designed prior to the flight. Specifically, the stations are covered in the order: YA3→YA1→YA4a→YA4d→YA7a→YA7d→ YA7e →A7b→YA4e→YA4c→ YA4b→A5→YA9 in Yanco YA area. The aircraft would not exactly follow these trajectories, but along the suggested directions, and the aircraft would fly on smooth trajectories. The flight direction selection is made by considering ease of flying of the aircraft as well as minimisation of flight time, whilst ensuring aircraft safety. The airborne experiment was conducted on the 9th of May 2013 when the weather is sunny and calm. Professor Jason Middleton, Head of Department of Aviation, UNSW, was the pilot for this experiment. Mr Greg Nippard, the engineer, did the assembling and disassembling of the equipment for this experiment. Mr. Scott O’Brien, a Ph.D. student from ACSER, UNSW, operated the laptop to log data in the aircraft.

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9 Soil Moisture Measurement

Fig. 9.23 Enlarged “YA” area in Yanco study region. The picture was generated using GPS Visualizer and Google Earth Table 9.4 Distance (km) between each pair of stations in YA area YA3 YA4a YA4b YA4c YA4d YA4e YA5 YA7a YA7b YA7d YA7e YA9 YA1 YA3 YA4a YA4b YA4c YA4d YA4e YA5 YA7a YA7b YA7d YA7e

5.1

2.0

2.4

3.0

3.0

3.9

4.7

4.9

5.6

7.3

6.9

8.5

6.4

4.3

5.9

7.2

6.0

4.1

8.2

7.7

10.3

9.1

7.2

2.4

1.6

1.0

2.8

4.5

3.0

4.0

5.4

5.2

7.9

1.6

3.0

2.0

2.3

4.0

3.9

6.2

5.4

6.1

1.8

1.1

3.1

2.4

2.7

4.7

4.1

6.2

2.7

4.8

2.1

3.4

4.5

4.4

7.8

2.5

2.4

1.9

4.3

3.3

5.1

4.9

3.8

6.5

5.2

4.0

1.8

2.4

2.4

6.8

2.7

1.5

5.1

1.6

7.1 5.5

9.5 An Airborne Experiment for Soil Moisture Measurement

299

Fig. 9.24 Flight ground track in Yanco YA area. The picture was generated using GPS Visualizer and Google Earth

Figures 9.24 shows the specular reflection tracks in YA area. There are two trajectories since the aircraft flew over the Yanco YA stations twice. Since the flight height relative to the ground is not high, either around 200 m or about 400, the specular reflection tracks will not greatly deviate from the aircraft ground tracks which are thus not presented. At the closest points, the reflection tracks are close to the in situ stations. The shortest distance from the stations to reflection tracks are listed in Table 9.5 for GPS satellite PRN#29. The mean shortest distance in Table 9.5 is 96 m and 51 m for the two trajectories, respectively. Interestingly, there are three stations which have a shortest distance of less than 10 m to one of the reflection tracks. Table 9.6 lists the elevation and azimuth angle of six satellites in YA area. During Table 9.5 Shortest distance (metre) from each station to the two aircraft ground tracks in YA area YA1 YA3 YA4a YA4b YA4c YA4d YA4e YA5 YA7a YA7b YA7d YA7e YA9 Track 1

5

6

88

29

30

129

11

16

54

15

15

43

29

Track 56 2

14

24

19

49

103

61

7

56

45

22

82

31

300

9 Soil Moisture Measurement

Table 9.6 Satellite elevation and azimuth angles (degree) in YA area Satellite

29

1st Elevation 58.9–63.1 track (deg) Azimuth (deg)

5

12

21

63.6–57.4

48.8–45.8

36.0–29.8 26.3–30.5

2 15.8–11.8

205.5–197.1 337.3–342.6 111.7–118.8 14.6–13.7 286.7–281.9 123.6–120.6

2nd Elevation 63.1–68.7 track (deg) Azimuth (deg)

25

57.4–48.3

45.8–40.2

29.8–21.2 30.5–36.5

11.8–6.1

197.1–181.0 342.6–347.4 118.8–126.8 13.7–12.9 281.9–274.1 120.6–116.0

the flight over the individual tracks, the elevation angle variation can be greater than eight degrees. Such a variation may have a non-trivial effect on the ground parameter (e.g. soil moisture) retrieval.

9.5.2.1

Initial Data Processing Results

Using the shortest distance points on the ground reflection tracks, a short reflection track segment is selected associated with each station and flight. That is, there are two track segments related to an individual station. The data length for each of the track segments is 30 s, equivalent to about 1.8 km, and the shortest distance point is around the middle of the track segment. The logged data are processed to generate a delay waveform by coherently integrating 1 ms IF samples. These delay waveforms are then non-coherently accumulated over 250 ms and the peak power is obtained. The noise magnitude is calculated by averaging the delay waveform data of one chip duration (16 samples) which is nine chips away from the peak power location on the left-hand side and another one on the right-hand side. Over these regions on the waveform, there would be no signals but noise. The preliminary results generated are the SNR of the reflected signal collected when flying over the track segments close to four YA stations (YA4b, YA5, YA7a, and YA7d) and four YB stations (YB1, YB2, YB5e, and YB7e) as shown in Fig. 9.25. The SNR magnitude variation over time/location attribute to the surface characteristics. Figure 9.26 shows the ground surfaces where the reflection tracks are located. Also shown are the reflection tracks colourised by the SNR magnitude. The SNR variation can be clearly observed when the track goes from one field to another in three areas shown in Fig. 9.26. The variation could be contributed to by the different crops in these fields as well as the difference in soil moisture and surface roughness. Table 9.7 shows the start time instants in local time for the period of 30 s related to each of the eight 30 s track segments. Table 9.8 shows the in situ soil moisture measurements observed at the eight stations on the flight day. These stations continuously measure the soil moisture every 20 min and the data were downloaded from the stations in the field manually or they were forwarded wirelessly so that the data can directly be downloaded in the laboratory at Monash University. From the tables it

9.5 An Airborne Experiment for Soil Moisture Measurement Measurements over tracks close to YA4b

10 5 0

0

5

10

15

20

25

8 6 4 2

30

10 5 0

10

15 20 Time (sec)

25

5 0

10

0

5

10

15

20

25

30

15 20 Time (sec)

25

30

0

5

10

15

20

25

4 2

Measurements over tracks close to YA7d

15 10 5 0

30

0

5

10

0

5

10

15

20

25

30

15 20 Time (sec)

25

30

15 SNR (dB)

10 SNR (dB)

5

6

0

30

SNR (dB)

SNR (dB)

5

Measurements over tracks close to YA7a

10

5 0

0

8 SNR (dB)

SNR (dB)

15

0

Measurements over tracks close to YA5

10 SNR (dB)

SNR (dB)

15

301

0

5

10

15 20 Time (sec)

25

30

10 5 0

Fig. 9.25 SNR measurements of reflected signals over reflection track segments closest to four stations in YA area

can be seen that the soil moisture measurement at each station is nearly constant over the period. In fact, the soil moisture typically does not change significantly during a day provided that there are no significant weather changes, such as from sunny to raining or vice versa. However, the soil moisture can vary dramatically from one location to another. These in situ measurements can be used as a reference when the GNSS measurements are employed to estimate the soil moisture. However, one needs to be cautious when using these data since the reflection tracks do not exactly cross the station locations. The minimum distance from the station to the reflection track is between 7 and 213 m. The in situ measurements can be used as an approximate reference only when the surface conditions at the station location and the reflection track area are quite similar. GNSS-based soil moisture estimation is complicated by a number of issues including surface roughness, vegetation canopy, and variation in percentage of individual soil components. To achieve reliable and accurate soil moisture estimation, these issues must be taken into account through processes such as modelling and compensation.

302

9 Soil Moisture Measurement

12

10

8

6

4

2

Fig. 9.26 Reflection tracks colourised by SNR around the in situ stations in YA area. The picture was generated using GPS Visualizer and Google Earth

Table 9.7 Start time instants for each of the eight 30-second data collected on 9 May 2013 YA4b 1st flight

YA5

YA7a

YA7d

YB1

YB3

YB5e

YB7e

11:43:11 11:46:12 11:38:04 11:38:48 12:15:07 12:16:43 12:28:21 12:23:49

2nd flight 12:01:32 12:04:43 11:56:41 11:57:26 12:43:09 12:44:51 12:54:13 12:51:25

Table 9.8 In-situ soil moisture measurements (%vol) versus local time (hour: minute) at YA stations on 9 May 2013 (data provided by Dr. Alessandra Monerris-Belda at Monash University) Time-> 10:00 10:20 10:40 11:00 11:20 11:40 12:00 12:20 12:40 13:00 13:20 YA4a

6.65

YA4b

24.26 25.76 26.97 27.99 29.25

YA5

5.81 7.1

YA7a

4.08

YA7d

7.15

YA7e

6.42

6.66

5.97

6.32

6.98

6.49

30.89 31.43

7.37

7.25

7.67

7.39

7.67

7.54

4.40

3.96

4.27

4.09

4.36

4.15

4.16

6.58

6.85

6.93

6.52

6.79

7.15

5.80

6.08

6.10

5.70

5.79

6.28

6.61 32.46 8.11

7.81

7.17

7.19

7.28

6.13

5.92

4.45

9.6 Summary

303

9.6 Summary This chapter studied soil moisture estimation using data recorded by ground-based GNSS receivers. Both signals of MEO and GEO satellites have been used to retrieve soil moisture through establishing empirical models with different observables. Although an airborne experiment was introduced, but no soil moisture estimation results were presented. More information on the use of airborne data for soil moisture estimation can be found in the literature such as [10]. The use of spceborne data for soil moisture estimation is also an active research and a lot of results have been reported [1, 3, 4, 9, 22]. It is worth noting that the received signals after reflected over soil is affected by a range of different factors, including soil structure and components, surface roughness, and vegetation, making soil moisture estimation a challenging issue. More efforts are needed to deal with the problem, especially by accommodating the various effect in the model development.

References 1. Al-Khaldi MM, Johnson JT, O’Brien AJ, Balenzano A, Mattia F (2019) Time-series retrieval of soil moisture using CYGNSS. IEEE Trans Geosci Remote Sens 57(7):4322–4331 2. Ban W, Yu K, Zhang X (2018) GEO-satellite-based reflectometry for soil moisture estimation: signal modeling and algorithm development. IEEE Trans Geosci Remote Sens 56(3):1829– 1838 3. Camps A, Park H, Pablos M, Foti G, Gommenginger CP, Liu PW, Judge J (2016) Sensitivity of GNSS-R spaceborne observations to soil moisture and vegetation. IEEE J Sel Top Appl Earth Obs Remote Sens 9(10):4730–4742 4. Carreno-Luengo H, Luzi G, Crosetto M (2019) Sensitivity of CyGNSS bistatic reflectivity and SMAP microwave radiometry brightness temperature to geophysical parameters over land surfaces. IEEE J Sel Top Appl Earth Obs Remote Sens 12(1):107–122 5. Chang X, Jin T, Yu K, Li Y, Li J, Zhang Q (2019) Soil moisture estimation by GNSS multipath signal. Remote Sens 11(21):1–16 6. Chen KS, Wu TD, Tsang L, Li Q, Shi J, Fung AK (2003) Emission of rough surfaces calculated by the integral equation method with comparison to three-dimensional moment method simulations. IEEE Trans Geosci Remote Sens 41(1):90–101 7. Chew CC, Small EE, Larson KM (2016) An algorithm for soil moisture estimation using GPS-interferometric reflectometry for bare and vegetated soil. GPS Solution 20:525–537 8. Chew CC, Shah R, Zuffada C, Hajj G, Masters D, Mannucci AJ (2016) Demonstrating soil moisture remote sensing with observations from the UK TechDemoSat-1 satellite mission. Geophys Res Lett 43(7):3317–3324 9. Clarizia MP, Pierdicca N, Costantini F, Floury N (2019) Analysis of CYGNSS data for soil moisture retrieval. IEEE J Sel Top Appl Earth Obs Remote Sens 12(7):2227–2235 10. Egido A, Paloscia S, Motte E, Guerriero L, Pierdicca N, Caparrini M, Santi E, Fontanelli G, Floury N (2014) Airborne GNSS-R polarimetric measurements for soil moisture and aboveground biomass estimation. IEEE J Sel Top Appl Earth Obs Remote Sens 7(5):1522–1532 11. Fung AK (1994) Microwave scattering and emission models and their applications. Artech House Inc 12. Katzberg SJ, Torres O, Grant MS, Masters D (2006) Utilizing calibrated GPS reflected signals to estimate soil reflectivity and dielectric constant: results from SMEX02. Remote Sens Environ 100(1):17–28

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13. Larson KM, Small EE (2013) Using GPS to study the terrestrial water cycle. EOS 94(52):505– 512 14. Mao K, Zhang M, Wang J, Tang H, Zhou Q (2008) The study of soil moisture retrieval algorithm from GNSS-R. In: Proceedings of international workshop on education technology and training & 2008 international workshop on geoscience and remote sensing, pp 438–442 15. Masters D, Zavorotny VU, Katzberg S, Emery W (2000) GPS signal scattering from land for moisture content determination. In: Proceedings of the IEEE international geoscience and remote sensing symposium (IGARSS), pp 3090–3092 16. NSIDC (2020, visited) http://nsidc.org/data/amsr_validation/soil_moisture/smex02/ 17. Rodriguez-Alvarez N, Bosch-Lluis X, Camps A, Vall-Llossera M, Valencia E, MarchanHernandez JF, Ramos-Perez I (2009) Soil moisture retrieval using GNSS-R techniques: Experimental results over a bare soil field. IEEE Trans Geosci Remote Sens 47(11):3616–3624 18. SMAP (2020, visited) https://smap.jpl.nasa.gov/mission/description/ 19. UNAVCO (2020, visited) https://www.unavco.org/data/gps-gnss/data-access-methods/dai2/ app/dai2.html) 20. Wang JR, Schmugge TJ (1980) An empirical model for the complex dielectric permittivity of soils as a function of water content. IEEE Trans Geosci Remote Sens GE-18(4):288–295 21. XENON (2000, visited) http://xenon.colorado.edu/portal/ 22. Yan Q, Gong S, Jin S, Huang W, Zhang C (2020) Near real-time soil moisture in China retrieved from CyGNSS reflectivity. IEEE Geosci Remote Sens Lett. https://doi.org/10.1109/ LGRS.2020.3039519 23. Zavorotny VU, Larson KM, Braun JJ, Small EE, Gutmann ED, Bilich AL (2010) A physical model for GPS multipath caused by land reflection: toward bare soil moisture retrievals. IEEE J Sel Top Appl Earth Obs Remote Sens 3(1):100–110 24. Zavorotny VU, Voronovich AG (2000) Bistatic GPS signal reflections at various polarizations from rough land surface with moisture content. In: Proceedings of the IEEE international geoscience and remote sensing symposium (IGARSS), pp 2852–2854

Chapter 10

Tsunami Detection and Parameter Estimation

A Tsunami can be really disastrous, causing tremendous damage and much loss of life, such as that triggered by the 2004 Indian Ocean earthquake and the 2011 Japan’s Tohoku earthquake. Although a Tsunami cannot be stopped, it is very helpful to obtain the knowledge of whether a Tsunami will occur at a specific region and how strong it will be. A number of Tsunami warning centers have been established over the world to provide warning information so that the impact can be greatly reduced. These warning systems basically make use of a network of sensors such as pressure gauges moored on the ocean bottom to derive the earthquake and Tsunami parameters. A Tsunami can also be monitored in real time by a network of buoymounted sensors, measuring the wave height, wavelength and wave propagation speed. These buoyed sensors only cover a very small part of the ocean surface and the deployment and maintenance of such sensors are usually very expensive. Another direct way to monitor a Tsunami is to use a satellite altimeter such as the Jason-1 radar altimeter which provided the relative sea level measurement two hours after the Indian Ocean earthquake happened. Although the accuracy of a satellite altimeter can be as high as a few centimeters, the chance that a Tsunami is captured by a satellite altimeter is rather small due to the fact that the coverage is limited to the narrow swath beneath the satellite. This chapter studies the potential of GNSS-R for Tsunami detection and parameter estimation based on surface height estimation.

10.1 Tsunami Modeling A Tsunami wave is a specific type of sea wave which may result from one of a number of different causes including the sudden sea floor displacement due to an earthquake, underwater landslide and volcano eruption. In this section, examples of typical Tsunami waveforms are first presented, which were observed by surface buoys. Then, modeling of two-dimensional Tsunami waveform with a single triangle

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Yu, Theory and Practice of GNSS Reflectometry, Navigation: Science and Technology 9, https://doi.org/10.1007/978-981-16-0411-9_10

305

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10 Tsunami Detection and Parameter Estimation

is studied, followed by modeling with two triangles. The model parameters are determined by least-squares fitting.

10.1.1 Examples of Tsunami Waveforms In its simplest form a Tsunami comprises a single solitary wave. In the ocean, this single wave (or a more complex waveform, depending on the disturbance that generates it) can evolve into a leading crest ‘the lead’, and a trailing wave train of (ideally) progressively shorter and lower crests. A Tsunami lead wave typically consists of a complete cycle of waveform which is rather different from a typical sinusoidal wave. The first half wave usually has larger magnitudes than the second half one which has negative magnitudes, followed by a number of waves with irregular patterns. Although the patterns of lead wave can be significantly different from each other, the first half waves have some specific patterns. Figure 10.1 shows the real SSH measurements including four Tsunami lead waves and the starting point of the horizontal axis is arbitrary. These real Tsunami waveforms were generated by Tsunami data measured by buoy-based sensors when powerful Tsunamis were triggered by the 2011 Japan’s earthquake. The data were obtained from the web site [4], where a download link is provided and a map is given to show the locations and identification numbers of observation stations as well as the location of the center of the earthquake. The observed wave amplitude is a function of time and the Tsunami propagation speed is not provided since it would be infeasible to measure the speed simply based on SSH measurements of individual buoys. However, knowledge of the propagation speed and wavelength of the lead wave is useful especially when considering SSH measurement by a Low Earth Orbit (LEO) satellite. Thus, to facilitate the analysis, the Tsunami propagation speed is approximated using the shallow water gravity wave theory as v≈



gh w

(10.1)

where g is the acceleration of gravity and h w is the water depth. This approximation is appropriate for shallow-water waves such as Tsunamis, whose wavelength is more than twenty times the water depth. Using this approximation, the Tsunami propagation speed at the four stations can be estimated to be between 731 and 778 km/h. The identification numbers and locations in terms of latitude and longitude of the four stations, the time delay of the wave arriving at the stations, the approximate wave propagation speed, and the water depth at these stations are listed in Table 10.1. For simplicity, Fig. 10.1 is generated under the assumption that the Tsunami travels at the same speed over the sea surface area around the four stations, which is 760 km/h. Note that there exists a SSH measurement offset in the first three plots of Fig. 10.1 probably due to the residual error in the removing of the solar and lunar tides.

10.1 Tsunami Modeling

307

Station ID 46409

4

Station ID 46403 5

2

SSH (cm)

0

0

-2 -5 -4 -10

-6 0

SSH (cm)

10

500

1000

1500

2000

0

500

1000

1500

Distance (km)

Distance (km)

Station ID 46402

Station ID 21415

5

20

0

10

-5

0

-10

-10

2000

-15 0

1500

1000

500

2000

0

500

1500

1000

2000

Distance (km)

Distance (km)

Fig. 10.1 Tsunamis triggered by 2011 Japan earthquake and measured by buoy sensors at four observation stations

Table 10.1 Locations of the stations (for logging of Tsunami data), time delays of the wave arriving at the stations, water depth, and approximate wave speed at the stations Station ID

Latitude (degree)

Longitude (degree)

Time delay (hour)

Water depth (km)

Approx speed (km/h)

46,409

55.300

−148.492

6.20

4210

731

46,403

52.649

−156.921

5.45

4510

757

46,402

51.068

−164.005

4.70

4760

778

21,415

50.176

171.849

2.90

4681

771

The amplitude of a Tsunami lead wave can be less than a decimeter over a deep ocean surface such as the first three waves displayed in Fig. 10.1, while it can be higher than ten meters when approaching a coast where the water depth is shallow. For a specific Tsunami event triggered by an earthquake or another natural phenomenon, the wave parameters would change as the wave propagates, depending on the location of the measurement station and the arrival time of the wave at the station.

308

10 Tsunami Detection and Parameter Estimation

10.1.2 A Single Triangle Based Modeling The main purpose of the modeling is to simplify theoretical analysis to facilitate the discussion of parameter selection for Tsunami detection in the following section as well as for waveform parameter estimation. The question is how to choose a function to model the Tsunami waveform or more specifically the lead wave. Tsunami lead waves would have different shapes, but many of them typically have a spike shape which is similar to a number of well-known mathematical functions such as parabola, Laplace function, and triangular function. Examples include the lead waves shown in Fig. 10.1. Other examples include the lead waves of the Tsunami that occurred in Solomon Islands on 6 February 2013, the one that occurred off the Southern Coast, Chile on 27 February 2010, the one that occurred off the Coast of Central Peru on 15 August 2007, and the one that occurred in Kuril Islands, Russia on 15 November 2006. However, it is observed that the triangular function may be the best choice for describing Tsunami lead waves, so a triangle function is used to model the lead wave shape. Note that the lead waves of some Tsunamis may not have a triangular shape, in which case the use of a triangular function for the shape modeling might be inappropriate. In the Cartesian coordinate system, the horizontal axis is the distance, the vertical axis is the sea surface height (SSH), the zero-distance point is arbitrary, and it is assumed that there exists a SSH measurement bias error, Then, using a simple isosceles triangle as illustrated in Fig. 10.2, the leading edge and trailing edge of the lead wave can be described by  h=

b − k|d − d0 |, d0 − L ≤ d ≤ d0 + L b − kL, elsewher e

(10.2)

where h (cm) is the SSH, b (cm) is the sea surface height corresponding to the triangular peak which occurs at the distance point d0 (km), k (cm/km) is the slope of leading edge, and L (km) is half the width of the triangle base. Suppose that there are n 1 wave magnitude samples on the leading edge, while there are m 1 samples on the trailing edge and typically the two numbers of samples would be exactly the same. Fig. 10.2 Geometry of single triangle based modeling of Tsunami lead wave

h b

d0

L

d L

10.1 Tsunami Modeling

309

Station ID 46403

Station ID 46409

SSH (cm)

2

5

0 0 -2

-4

200

0

600

400

-5

200

0

600

400

Distance (km)

Distance (km)

Station ID 46402

Station ID 21415

30

10 20

SSH (cm)

5

10

0

0

-5 -10

-10

0

100

200

300

400

500

-20

Distance (km)

0

100

200

300

400

500

Distance (km)

Fig. 10.3 Measured (red solid curve) and modeled (black dashed line) Tsunami lead waves triggered by the 2011 Japan earthquake

Substituting each pair of wave magnitude and distance point samples {h i , di } into (10.2) generates a group of n 1 + m 1 simultaneous nonlinear equations h i = b + k(di − d0 ), i = n 0 , n 0 + 1, · · · , n 0 + n 1 − 1 h j = b − k(d j − d0 ), j = m 0 , m 0 + 1, · · · , m 0 + m 1 − 1

(10.3)

where the unknown parameters are b, k and d0 . Since it involves the product of two unknown parameters, the equations are nonlinear. The number of equations is greater than the number of unknown parameters, so the problem is technically overdetermined. Note that this is in fact the nonlinear least squares regression problem. By introducing an intermediate variable, η = kd0

(10.4)

the nonlinear equations become linear and can be written in a compact form as

310

10 Tsunami Detection and Parameter Estimation

Aθ = h

(10.5)

where ⎡

⎤T 1 ··· 1 1 ··· 1 A = ⎣ dn 0 · · · dn 0 +n 1 −1 −dm 0 · · · −dm 0 +m 1 −1 ⎦ ∈ R (n 1 +m 1 )×3 −1 · · · −1 1 ··· 1

(10.6)

θ = [ b k η ]T ∈ R 3×1 h = [ h n 0 , · · · , h n 0 +n 1 −1 , h m 0 , · · · , h m 0 +m 1 −1 ]T ∈ R (n 1 +m 1 )×1 The least-squares solution is then given by θˆ = (AT A)−1 AT h

(10.7)

bˆ = θˆ1,1 , kˆ = θˆ2,1 , dˆ0 = θˆ3,1 /kˆ

Table 10.2 shows the three model parameters calculated by (10.7) for the four Tsunami lead waves shown in Fig. 10.1. Note that the parameter L is calculated using the estimates of b, k, and d0 and the approximate sea level before the lead wave based on the actual measurements. Tsunami height H is simply calculate by H = k × L. In the case of the four lead waves considered, as the wave height increases, the slope increases nearly proportionally. Both the modeled (denoted by a dashed curve) and the observed (denoted by a solid curve) Tsunami lead waves are shown in Fig. 10.3. It can be seen that there is a good match between each pair of the observed real Tsunami lead wave and the modeled one. The second half of the lead wave (the trough, not the crest) is not modeled for two reasons. The first one is that the amplitude of the second half wave is much smaller and has an irregular pattern especially when the wave height is rather low such as the first two in Figs. 10.1 and 10.3. The second one is that the first half lead wave will be employed in the detection method to be studied in Sect. 10.2. Here and in Sect. 10.2, Tsunami wave, or lead wave actually means the first half lead wave for simplicity and the wave length is the triangle base length (i.e. the length of the first half lead wave). The wave length is less than (and probably not exactly equal to half of) the wavelength of the complete lead wave. Table 10.2 Model (triangular function) parameters determined by least-squares method. “H” denotes the triangle height Station ID

k (cm/km)

d0 (km)

H (cm)

L (km)

46,409

b (cm) 2.67

0.0288

293.75

5.42

195.75

46,403

6.29

0.0629

285.75

10.11

160.75

46,402

9.76

0.1232

267.02

15.28

124.02

21,415

28.48

0.2722

235.81

30.16

110.81

10.1 Tsunami Modeling

311

Since the introduced parameter (η) in (10.4) is a function of the other two parameters (k and d0 ), the solution given by (10.7) is not optimal. The estimation accuracy may be improved using iterative optimization. Specifically, a cost function is defined to account for the difference between the observed real lead wave and the modeled one. For instance, the cost function can be defined as the sum of the square of the difference of their magnitudes at each sampling point. That is, (b, k, d0 ) =

n 0 +n 1 −1

(h i − b − k(di − d0 )) + 2

i=n 0

m 0 +m 1 −1

(h j − b + k(d j − d0 ))2

j=m 0

(10.8) Note that this is also the standard least squares minimization function and the minimization can be realized using such as the Matlab optimization toolbox. Note that Matlab is the registered trademark of The Mathworks Inc. Minimizing the cost function with respect to the three unknown parameters produces their estimates: ˆ k, ˆ dˆ0 ]T = arg min (b, k, d0 ) [ b, b,k,d0

(10.9)

To start the minimization process, an initial guess of the unknown parameters is required and the initial parameter values can be simply set to be those produced by the linear least-squares solution given by (10.7). However, it is observed that the model parameters obtained via iterative optimization are virtually the same as those determined by (10.7). The reason for such a result would be due to the fact that a large number of simultaneous equations are employed to obtain an estimate of the three parameters by the solution of (10.7). Thus, the solution provided by (10.7) would be usually satisfactory to produce an accurate model.

10.1.3 Two Triangles Based Modeling The single triangle based modeling is simple and convenient for theoretical analysis in Tsunami detection. However, the second half of the lead wave may also be significant in some cases, so two triangles can be used to generate a more accurate model to describe Tsunami lead wave. Figure 10.4 shows another two Tsunami waveforms (denoted by red circle) triggered by 2011 Japan’s Tohoku earthquake. The waves were recorded by ocean buoy-based sensors at the station numbered 21,413 (latitude 30.528° and longitude 152.123°) and another station numbered 21,401 (latitude 42.617° and longitude 152.583°). These two waveforms are modeled with three line segments to form two triangles for the two half waves. The triangle for the first half wave is isosceles for simplicity, while the other may not be. The parameters of the modeled two half waves are displayed in Fig. 10.5. The coordinate system origin is

312

10 Tsunami Detection and Parameter Estimation

Station 21401 Original

60

Modeled

SSH (cm)

40 20 0 -20 200

400

600

800

1000

1200

1400

Distance (km)

Station 21413

80

Original

SSH (cm)

60

Modeled

40 20 0 -20 -40 400

600

800

1000

1200

1400

1600

Distance (km)

Fig. 10.4 Original and modeled Tsunami waveforms triggered by 2011 Japan’s earthquake using least squares curve fitting Fig. 10.5 Two triangles based linear modeling for Tsunami waveform

Line 2

Line 1

Line 3

10.1 Tsunami Modeling

313

simply set at the instant point of the first SSH sample in the selected sequence of SSH measurements. Model Structure. The SSH measurements are described as 

u i = μi + εi , i = 1, 2, · · ·, N

(10.10)



where {εi } are the noise samples and {μi } are the samples of the waveform which is modeled by ⎧ a1 d + b1 , ds ≤ d < d p ⎪ ⎪ ⎨ −a1 d + b2 , d p ≤ d < dv  μ(d) = ⎪ a d + b3 , dv ≤ d ≤ de ⎪ ⎩ 3 0, elsewher e

(10.11)

It is clear that the waveform is completely determined by the three blue solid line segments (Line 1, Line 2, and Line 3). The three line segments correspond to three linear equations with slopes of {a1 , −a1 , a3 } and y-axis intercepts of {b1 , b2 , b3 } as indicated by (10.11). ds and de are the distances to the points where the modeled Tsunami starts and ends, respectively, while d p and dv are respectively the distances to the peak and valley points of the modeled wave. Given the three linear equations, the four distance quantities {ds , d p , dv , de } can be described as ds = −

b1 b2 − b1 b2 − b3 b3 , de = − , dp = , dv = a1 2a1 (a1 + a3 ) a3

(10.12)

Since the height and length of the first half wave can be significantly different from those of the second half wave, it is useful to take all the four parameters (h1 , h2 , L 1 and L 2 ) into consideration, which are given by 1 b2 + b1 (b2 + b1 ), L 1 = 2 a1 a1 b3 + a3 b2 a1 b3 + a3 b2 h2 = − , L2 = − a1 + a3 a1 a3 h1 =

(10.13)

Thus the height and length parameters are directly linked to the wave linear shape parameters. The wave height (h = h 1 + h 2 ) and wavelength (L = L 1 + L 2 ) are then associated with the shape parameters by h=

a1 (b1 + b2 − b3 ) + a3 b1 a3 b1 − a1 b3 ,L= 2(a1 + a3 ) a1 a3

(10.14)

314

10.1.3.1

10 Tsunami Detection and Parameter Estimation

Linear Least Squares Fitting for Parameter Determination

Using the samples on the leading edge (line 1) produces u i = a1 di + b1 , i = q1 , q1 + 1, ..., N1

(10.15)

where di is the horizontal distance between the position of the ith SSH sample and that of the first SSH sample (i.e. the coordinate origin) in the sample sequence. Similarly, using the samples on the trailing edge (line 2) yields u i = −a1 di + b2 , i = q2 , q2 + 1, ..., N2

(10.16)

where the sampling instant q2 can be either equal to N1 + 1 or an integer around N1 . Combining (10.15) and (10.16) produces a compact form as u = Aρ

(10.17)

where u = [ u q1 ... u N1 u q2 ⎡ dq1 ... d N1 −dq2 A = ⎣ 1 ... 1 0 0 ... 0 1

... u N2 ]T

⎤T ... −d N2 ... 0 ⎦ ... 1

(10.18)

ρ = [ a1 b1 b2 ]T The least squares solution is simply given by ρˆ = (AT A)−1 AT u

(10.19)

Parameters a3 and b3 can be determined in the same way, so we do not repeat the procedure to save space. Table 10.3 lists the resulting parameter values of the models of the two waveforms shown in Fig. 10.4. It can be seen that the modeled waveform and the original one have a good match. The difference between the modeled and the original SSH data in terms of root mean square error (RMSE) is 4.9 cm and 5.8 cm for station 21,401 and station 21,413, respectively. The RMSE associated with station 21,413 is about 20% larger mainly due to the significant variation in the observed wave samples around line 3.

10.1 Tsunami Modeling Table 10.3 Model parameters for the two waves observed by two stations

315 Station # a1 (cm/km)

21,413

0.604

0.836

b1 (cm)

−365.0

−728.2

b2 (cm)

493.5

886.6

a3 (cm/km) b3 (cm)

10.1.3.2

21,401

0.202 −196.7

0.112 −154.6

h 1 (cm)

63.8

79.3

L 1 (km)

211.3

189.7

h 2 (cm)

23.5

31.8

L 2 (km)

154.9

322.0

Tsunami Waveforms Observed by Altimetry Satellites

The above studied two waveforms observed by ocean surface buoy-based sensors can be treated as the real Tsunami waveforms. When observed by a satellite-carried sensor, the observed wave shape can be quite different, depending on the satellite moving direction and the sensor SSH measurement accuracy. For instance, if the satellite is moving exactly in the same (or opposite to the) direction of the Tsunami propagation, the length of the observed wave is slightly longer (or shorter) than that of the real one. Suppose that the Tsunami wavelength is 400 km, Tsunami propagation speed is 700 km/h, and the LEO satellite ground speed is 7 km/s.

700 the  observed

 Then, × = 411 km wavelength can be approximately calculated by: L ≈ 400+ 400 4 3600 when the satellite and Tsunami move in the same direction and the satellite ground track is initially thewave. On the other hand, the observed wavelength is about  700

behind × = 389 km when they move exactly towards each other. L ≈ 400 − 400 4 3600 In some cases, the observed wave length by a satellite sensor can be significantly longer. Figure 10.6 shows the two waveforms of the Tsunami (triggered by 2011 Japan’s earthquake) observed by Jason-1 altimetry satellite (pass 146, cycle 338) and Envisat satellite (pass 449, cycle 100). The distance is calculated using the sampling time intervals and the satellite ground velocity which is about 5.8 km/s for Jason-1 and 6.86 km/s for Envisat. The start distance point on the horizontal axis is arbitrary. There are a few sampling time intervals in which SSH measurements are not available so that they are simply left empty. The data were downloaded from the web site [5] which is managed by the Technical University of Delft, the Netherlands. The initially sensed SSH data were affected by a range of factors including troposphere, ionosphere, barometer, dynamic atmosphere, solid earth tide, ocean tide, load tide, pole tide and sea state bias. The downloaded data are the processed data after those effects are corrected. Thus, the resultant SSH (or sea level anomaly) could only consist of components contributed by the Tsunami, the residual error from the various corrections, and the sensor measurement error. The measurement accuracy of the Envisat and Jason-1 altimeters is about a few centimeters, while the residual error

316

10 Tsunami Detection and Parameter Estimation

Envisat Satelite

SSH (cm)

100 50 0 -50

0

1000

2000

3000

4000

5000

6000

4000

5000

6000

Distance (km)

Jason-1 satellite

SSH (cm)

100 50 0 -50

0

1000

2000

3000

Distance (km)

Fig. 10.6 Tsunami waveforms triggered by 2011 Japan’s earthquake and observed by Envisat and Jason-1 altimetry satellites

might also be a number of centimeters due to the increasingly enhanced performance in the modeling of the various factors. The sea surface locations of the two waves are listed in Table 10.4, which are quite far away from the two buoy stations. Nevertheless, the waves observed by the two satellites and those observed by buoy sensors have a similar shape. It is worth noting that Envisat presents a significant negative triangle prior to the positive triangle, while Jason-1 and the buoy-based observations do not have. This negative triangle does not look like an artifact and it should be real signal of the data, although its presence is a bit abnormal. It can be seen that Envisat observed the Tsunami leading edge first, indicating that the satellite and the Tsunami roughly move towards each other. On the other hand, Jason-1 observed the Tsunami trailing edge first, meaning that the satellite ground track initially was behind the wave, but quickly caught up with the wave trailing edge, went through the whole wave, and finally moved out of the leading edge. In the same way, the waveform can be linearly modeled as two triangles as shown in Fig. 10.7 using LS fitting. The leading edge of the waveform observed by Jason-1 Table 10.4 Locations of two Tsunami waveforms observed by two altimetry satellites Satellite

Latitude (deg)

Longitude (deg)

Start point

End point

Start point

End point

Jason-1

−38.2351

−41.9297

40.4309

43.0984

Envisat

−36.4802

−30.3041

156.4616

154.6962

10.1 Tsunami Modeling

317

Envisat

100

SSH (cm)

Original Modeled

50 0 -50

0

1000

500

1500

Distance (km)

Jason-1

100

SSH (cm)

Original Modeled

50 0 -50

0

200

400

600

800

1000

1200

Distance (km)

Fig. 10.7 Original and modeled Tsunami waveforms observed by Envisat and Jason-1

Table 10.5 Model parameters for the two waves observed by two altimetry satellites

Satellite a1 (cm/km)

Envsat 0.563

Jason-1 0.585

b1 (cm)

−244.1

−206.6

b2 (cm)

438.3

385.3

a3 (cm/km)

0.301

0.497

b3 (cm)

−309.0

h 1 (cm)

96.9

−414.7 89.3

L 1 (km)

344.8

361.1

h 2 (cm)

48.6

47.4

L 2 (km)

247.1

207.7

has been placed on the left-hand side. The model parameters are listed in Table 10.5. Basically, there is good agreement between the observed and modeled waveforms.

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10.2 Average Bin Based Tsunami Detection 10.2.1 Noise Corrupted SSH Measurement As mentioned earlier, the height of a Tsunami lead wave can be as small as a few decimeters or even sub-decimeter. It would be a challenging problem to detect a Tsunami of such a small height especially when using noisy GNSS-R based SSH measurements. The accuracy of a satellite altimeter such as that carried by the Jason2 satellite can be a few centimeters, while the accuracy of GNSS-R and satellitebased SSH measurements would be significantly lower. As indicated in [1], when considering the instrument parameters (e.g. the transmitted signal power, the antenna directivity, the incoherent integration time, the receivers’ noise temperature, and the bandwidth) and the sea state as well as given the elevation angle of 55 degrees, the height precision (STD of the SSH estimation error) of the interferometric GNSS-R would range from 30.5 to 60.5 cm for GPS L1 and L5 signals and from 13.7 to 27.7 cm for Galileo E1 and E5 signals. The SSH performance can be improved with larger directivity antennas and less noisy receivers.

Actual SSH (cm)

Scaled Buoy Based SSH Measurements 50

0

-50 0

500

1000

1500

2000

Modeled SSH Measurement(cm)

Distance (km)

Modeled GNSS-R Based SSH Measurements

300 200 100 0 -100 -200

0

500

1000

1500

2000

Distance (km)

Fig. 10.8 Scaled Actual SSH measurements observed at station 21,415 and modeled GNSS-R based SSH measurements by adding the actual SSH measurements with a Gaussian noise variable of zero mean and STD 76 cm

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319

Figure 10.8 shows the modeled GNSS-R based SSH measurement which is the sum of the scaled actual SSH measurement at Station 21,415 (fourth plot in Fig. 10.1) and a Gaussian random variable with a zero mean and a STD of 76 cm. Clearly, the Tsunami is significantly corrupted by the measurement noise and is difficult to see whether or not a Tsunami is present even for this case where the Tsunami height is relatively large. The situation would be worse if the wave height is smaller such as the one observed at Station 46,409 or the error STD is even larger. Although there are few reports on the modeling of the GNSS-R SSH measurement noise, it may be appropriate to model the noise as a Gaussian random variable and it will be useful to do more investigations to confirm such an assumption.

10.2.2 Method Description A bin average (BA) method may be used to detect the weak Tsunami lead wave [10]. That is, the SSH measurement samples over a bin distance interval are taken and averaged to produce a BA output. Then, the bin moves one bin distance and covers a completely new sequence of samples which are averaged to produce another BA output. It is worth noting that this BA technique is based on a simple boxcar filer function for simplicity. The resulting filtered samples are independent only when they have a distance apart corresponding to the cutoff wavenumber of the filter (for a boxcar filter function, this is approximately the inverse of the bin width; for a raised cosine filter it is twice that width). The boxcar filter function also has poor sidelobe suppression, so alternative filter functions such as cosine or Lanczos functions could be considered to avoid sidelobe contamination when the filter is applied in the time domain at the cost of a longer filter bin. This is especially important when data being filtered are subject to large impulses. Alternatively, filtering could be performed in the wavenumber domain, in which case, a spectral window might need to be applied. Depending on the size and shape of the underwater disturbance, Tsunami wavelength can range from 10 to 500 km. In the event that the bin size is set to be 200 km, there are basically four different cases regarding the geometry between the bins and the modelled wave as shown in Fig. 10.9. The first case is that the modeled lead wave is completely covered by a single bin and this occurs especially when the bin size is much larger than the lead wave length which is equal to the triangle base length. The second case is that the wave is covered by two neighboring bins and this can occur even if the bin size is much larger than the wave length. The third case can occur when the wave length is greater than the bin size but not greater than three times the bin size. The Fourth case is that four neighboring bins cover the complete modeled wave, which can occur when the wave length is greater than twice but not greater than three times the bin size. Note that when the wavelength is longer or the bin size is smaller, there will be more than four cases. As the bin moves along the horizontal distance axis, it will go through the SSH measurements taken in the presence of a Tsunami. This procedure continues to

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Fig. 10.9 Four cases of geometrical relation between the modeled Tsunami and the bins

SSH Case 1

Distance SSH Case 2

Distance SSH

Case 3

Distance SSH Case 4

Distance

generate BA outputs one by one. In the event that one or a few neighboring BA outputs are significantly greater than others, it is likely that a Tsunami is detected. In order to detect the Tsunami, it is necessary to have the knowledge of the mean sea level of the locations of interest in the absence of a Tsunami event. Except for the Tsunami’s impact, sea level is affected by a range of other factors or parameters including tides caused by Sun and Moon, small-scale wind-driven waves, large-scale and low frequency wind-driven ocean circulation, and the small long-term sea level variation due to climate change, which can be ignored. That is, the GNSS-R based initial SSH measurements may be modeled as S = ST sunami + Stide + SW W + SOC + ζ

(10.20)

where ST sunami , Stide , SW W , and SOC are the SSH components contributed by Tsunami, tide, small-scale wind-driven wave, and large-scale wind-driven ocean circulation, respectively, and ζ is the noise. The tides can be predicted so that their effects can be removed from the GNSS-R measurements such as using tidal datum and other relevant information. The large-scale wind-driven ocean circulations are

10.2 Average Bin Based Tsunami Detection

321

produced by Ekman wind-forcing and associated quasi-geostropic dynamics. They can be predicted from global ocean circulation models so that their effects can also be removed from GNSS-R measurements. As mentioned earlier, tides also have long wavelengths, but they would not be confused with Tsunamis since they are basically removed by tidal datum correction. Also, one may design a filter to get rid of the tides whose propagation speed is much slower than Tsunamis’ speed. Thus Tsunami detection is mainly affected by SSH measurement error and wind-driven waves induced component and thus (10.20) can be rewritten as S = ST sunami + SW W + ζ

(10.21)

The wind-driven waves induced sea surface elevation variation may be simply modeled as a Gaussian random variable, although the more complex Tayfun distribution may be more accurate. Thus in the absence of a Tsunami the SSH measurements can be modeled as the sum of wind induced surface variation component and noise component. Wind waves have a rather short wavelength of typically a few meters to tens of meters and the surface elevation variation caused by wind is a random variable with a zero mean, while the GNSS-R based SSH measurements are usually produced using delay waveforms based on coherent integration of one to a few milliseconds, which are accumulated incoherently over an interval which can be greater than a number of seconds. The incoherent accumulation would alleviate the sea state impact to some degree, but residual sea state effect would contribute a noise-like variable to the SSH measurement. As a consequence, in the absence of a Tsunami √ the BA would produce another sequence of noise samples with the STD reduced by N times, where N is the number of samples covered by the bin. When the bin covers a significant part of the Tsunami, it is likely that the Tsunami can be detected provided that the Tsunami component of the BA output is large enough to exceed the reduced noise STD. Figure 10.10 shows an example of the BA outputs when both the modeled and the scaled real Tsunami data are used and the noise/error STD is set to be 76 cm. The real Tsunami data were observed at Station 21,415 (the fourth plot in Fig. 10.1) with a scaled lead wave height about 80 cm. The bin size is set to be 120.4 km equivalent to 34 samples with a sampling distance 7.08 km. The difference between the modeled and real bin output is illustrated in Fig. 10.10 for the lead waves. Also, the modeled wave data beyond the lead wave on both sides is set to be constant, while the real data is typically not constant. Note that the BA output peaks typically are not located at the peak of the lead wave because the bin moves at a distance of the bin size (no overlapping between neighboring bins) each time and the BA output corresponds to the distance of the mid-bin point. That is, it is unlikely that the midpoint of the bin with the peak output is exactly located at the peak of the lead wave due to the randomness of the bin’s initial location.

10 Tsunami Detection and Parameter Estimation

Relative Sea Level (cm)

322

Scaled Buoy Based SSH Measurements 50 0 -50 0

500

1000

2000

1500

Distance (km) BA Output (cm)

50 Model based 0

-50

0

500

1000

1500

2000

Distance (km) BA Output (cm)

50 Real data based 0

-50

0

500

1000

1500

2000

Distance (km)

Fig. 10.10 Example of BA outputs when wave height is about 80 cm and the noise STD is 76 cm

10.2.3 Hypothesis Testing Although the abnormal high BA outputs can be manually examined, it is useful to notify the outcome automatically. To achieve the goal, statistical hypothesis testing theory is exploited. There are only two hypotheses for the problem considered, the presence or absence of a Tsunami. The BA outputs are examined one by one against a predefined threshold. In the one stage scheme, if a BA output is greater than the threshold, then the hypothesis of presence of a Tsunami is accepted. Otherwise, the hypothesis of absence of a Tsunami is accepted and the bin is moved by a bin length to cover a completely new sequence of SSH samples along the specular reflection track direction, generating a new BA output and performing another test. Depending on the geometry between the wave propagation direction and the direction of motion of the specular reflection of the GNSS signal, the bin may first cover part of the leading edge of the lead wave and later the trailing tail or part of the long trailing tail first and later the trailing edge and finally the leading edge of the lead wave. In the two stage scheme, if the threshold is crossed by a BA output, the hypothesis of presence of a Tsunami is first tentatively assumed. The tentative assumption is then verified by examining the next BA output to see whether the output is greater than a

10.2 Average Bin Based Tsunami Detection

323

second threshold. If the second threshold is crossed, then the assumption is accepted. Otherwise, the assumption of presence of a Tsunami is rejected.

10.2.3.1

Basic Formulas

For convenience, as shown in Fig. 10.9, the triangle apex is set on the vertical axis and the base on the horizontal axis. Assume that the total number of Tsunami samples is M, and that the bin can cover N SSH samples. Also, for simplicity, let us assume that there are N1 = (N − 1)/2 samples on each of the two edges and one on the apex. Regardless of the Tsunami, the bin always covers N noise samples whose average is sn =

N 1  εi N i=1

(10.22)

where εi is the ith noise sample. Note that the signal is originated only from a Tsunami, while effects of all other phenomena have been removed from the SSH measurement and only produce noise which is added to the original SSH measurement noise, resulting in {εi }. The Tsunami component of the BA output can be derived under specific conditions arising in one or more of the four cases illustrated in Fig. 10.9. The BA output under other conditions can be derived in the same way. The four conditions are: (1) (2)

(3) (4)

The lead wave is completely covered by a single bin as shown in Case 1. Part or complete leading or trailing edge is covered by a bin such as the first bin in Case 2, the first bin and the third bin in Case 3, and the first and the fourth bin in Case 4. The bin covers the complete leading (or trailing) edge and part of the trailing (or leading) edge such as the second bin in Case 2. The bin covers part of the leading edge and part of the trailing edge such as the second bin in Case 3 and second bin in Case 4. Then, under all the four conditions the BA output can be described by s = sn + stw

(10.23)

where sn is the noise component given by (10.22) and stw is the component contributed by the wave samples. Under the first condition, stw can be easily derived to be ⎞ ⎛ N1 1⎝  stw = k j + b⎠ 2 N j=1 (10.24) 1 = (N1 (N1 + 1)k + b) N

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where k is the slope of the leading edge and is the distance interval. Under the second condition, stw is given by stw =

q 1  k j N j=1

(10.25)

1 = q(q + 1)k , 1 ≤ q ≤ N1 + 1 2N Under the third condition, we obtain ⎞ ⎛ q N1 +1  1 ⎝ stw = k j + (b − ki )⎠ N j=1 i=1

(10.26)

1 qb = ((N1 + 1)(N1 + 2) − q(q + 1))k + , 1 ≤ q < N1 2N N Under the fourth condition, stw becomes

stw =b−

⎞ ⎛ q N −q−1   1 = ⎝ (b − k j ) + (b − ki )⎠ N j=0 i=1

(10.27)

1 ((N − q − 1)(N − q) + q(q + 1))k , 1 < q ≤ N1 2N

It is worth noting that besides the BA approach, an alternative approach for Tsunami detection is the sliding window moving average (SWMA) approach which moves one sample each time. Although SWMA is more time consuming, it would produce more accurate detection performance. This is because that the window will completely cover the lead wave or cover the central part of the wave. At this time instant, the SNR would be maximized.

10.2.3.2

Hypothesis Testing Analyses

Suppose that the GNSS-R based SSH measurement noise samples are mutually independent Gaussian random variables and hence the average of the noise samples covered by the bin as given by (10.22) is also a Gaussian random variable with a probability density function (PDF) given by   N ε2 1 exp − 2 pn (ε) = √ 2σ 2π/N σ

(10.28)

When the Tsunami samples are covered by the bin, the average of the N SSH measurements of signal plus noise is still a Gaussian random variable with a PDF

10.2 Average Bin Based Tsunami Detection

325

given by   N (ε − stw )2 1 exp − pwn (ε, stw ) = √ 2σ 2 2π/N σ

(10.29)

where stw is determined by (10.24, 10.25, 10.26 or 10.27), depending on the location of the bin relative to the modeled or real Tsunami. Note that the specific location of the bin relative to the Tsunami is not required for the detection and in fact it is not known in advance. The cases considered are simply for convenience in theoretical analysis. Figure 10.11 shows two examples of the PDFs of the BA outputs, calculated by (10.28) and (10.29), respectively. As mentioned earlier the hypothesis of presence of a Tsunami is directly accepted once the BA output is greater than a threshold in the one stage scheme. According to the statistical hypothesis testing theory [3], the threshold (γ ) can be determined by giving the probability of false alarm (PFA) denoted by α which is the probability that the hypothesis of presence of a Tsunami is accepted, but in fact the wave is not present. The PFA is given by

Noise Wave+Noise

0.12

0.1

PDF

0.08

0.06

0.04

0.02

0 -15

-10

-5

0

5

10

15

20

25

30

Modeled SSH Measurement (cm)

Fig. 10.11 Two probability density functions of the BA output variable in the absence and in the presence of a modeled Tsunami respectively

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10 Tsunami Detection and Parameter Estimation

∞ α(γ ) = γ

  ∞ 1 N ε2 pn (ε)dε = √ exp − 2 dε 2σ 2π/N σ γ    1 Nγ = 1 - erf 2 2σ

(10.30)

where erf(x) is the error function. Since α is given but γ is unknown, (10.30) can be written as  2 (10.31) σ er f −1 (1 − 2α) γ = N where er f −1 (x) is the inverse error function which can be directly calculated by the Matlab library function “erfinv” in the case where simulation is performed by Matlab. The PFA should be as small as possible to avoid delivering unnecessary incorrect warning information. As shown in Fig. 10.11, the red solid area is the PFA, while the blue-shaded area is the probability of missed detection (PMD) which is the probability that the hypothesis of absence of a wave is accepted but a wave is present, given by γ m(γ , stw ) = −∞

  γ 1 N (ε − stw )2 pwn (ε, stw )dε = √ exp − dε 2σ 2 2π/N σ −∞ (10.32)    N γ − stw 1 = 1 + erf 2 2 σ

The probability of detection (PD) is the probability that the hypothesis of presence of wave is accepted when the wave is present, given by ∞ β(γ , stw ) = γ

  ∞ 1 N (ε − stw )2 dε pwn (ε, stw )dε = √ exp − 2σ 2 2π/N σ γ (10.33)    1 N γ − stw = 1 − erf 2 2 σ

In fact, PMD and PD are related by β(γ , stw ) + m(γ , stw ) = 1

(10.34)

The overlapping portion between the two density functions will increase as the STD of the measurement noise increases, making the discrimination more difficult.

10.2 Average Bin Based Tsunami Detection

327

When the parameters of a Tsunami and the noise STD are given, reducing the PFA or increasing the threshold will increase the PMD, and vice versa, as shown in Fig. 10.11. Thus, it is impossible to reduce the PFA and the PMD at the same time. Equations (10.32) and (10.33) are only suited for calculating the probabilities for a BA output associated with a specific bin location. Let us consider a general case where the Tsunami samples corrupted by noise are completely covered by κ neighboring bins, producing κ BA outputs. In the one stage scheme, the Tsunami is detected if any one of the κ outputs is greater than the threshold. The PD is thus given by (1) )+ P D = β(γ , stw

 κ   j−1 ( j) (i)  m(γ , stw ) β(γ , stw ) i=1

j=2 κ

=1− 

i=1

(10.35)

(i) m(γ , stw )

(i) where stw is the Tsunami component (stw ) of the ith BA output, calculated by (10.24, 10.25, 10.26 or 10.27), depending on which Tsunami samples are covered by the bin. The PFA of the one stage scheme can be derived as

⎛

⎞  κ   j−1 (i) P F A = ⎝1 +  m(γ , stw ) ⎠α(γ ) j=2

(10.36)

i=1

In hypothesis testing for signal detection a two-stage approach is typically employed to improve detection performance. It is thus desirable to consider both the one-stage approach and the two-stage one in the case of Tsunami detection. As for the two stage scheme it is a bit tedious to do a thorough analysis. More details about the PD and PFA of the two stage scheme can be found in [10].

10.2.4 Signal-To-Noise Ratio and Bin Size As mentioned earlier the number of bins covering the lead wave will depend on the bin length, the wave length, and their relative positions, which may range from one to more than ten. For instance, if the lead wave length is 300 km and the bin size is 30 km, then ten or eleven bins are needed to cover the wave. On the other hand, if the wave length is 100 km and the bin size is 120 km, then only one or two bins are needed to cover the wave. Thus, a significant issue is the selection of the bin size or the number of samples covered by the bin and a criterion is required to determine the most suitable number of samples. Since the lead wave length is unknown, it is infeasible to choose the best bin size in advance. However, it is useful to determine the best bin size when given the wave length to provide a guideline in the selection

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of bin size. Due to the fact that the lead wave has a typical length of several hundred kilometers, the bin length may be simply set to be a few hundred kilometers in the absence of any wave dimensional information. Let us study the SNR of a BA output using the formulae derived in Sect. 10.2.3. Since the purpose is to distinguish the case of the presence of a wave from the case of the absence of a wave, it is natural to maximize the ratio of the BA output power in presence of a wave over that in absence of a wave. As mentioned earlier, SSH is affected by several factors including tides and wind-driven ocean circulations. It is assumed that these effects have been corrected using models and thus they only produce noise which increases SSH measurement noise level, while the signal is originated only from a Tsunami. If εi is a Gaussian random variable that has a zero mean and a variance equal to σ 2 , then sn in (10.22) is also a Gaussian random variable of zero mean and variance σ 2 /N . In the presence of a Tsunami, the output is still a Gaussian random variable, but with a mean of stw . It is obvious that the BA output would be maximal when the bin covers the central part of the wave. Then, the average of the lead wave amplitude samples covered by the bin is given by

stw

⎞ ⎛ N1  1⎝ = (b − k j )⎠ b+2 N j=1

(10.37)

k 2 (N − 1) =b− 4N which can also be obtained by substituting q = N1 into (10.27). Since N 2 is typically much larger than one, N 2 − 1 can be approximated by N 2 , resulting in stw ≈ b −

k N 4

(10.38)

The power ratio of BA output in presence of a Tsunami (s2 = stw + sn ) over BA output in absence of a Tsunami (s1 = sn ) can then be obtained as σ 2 + N (b − k N /4)2 E[s22 ] = σ2 E[s12 ]

(10.39)

To maximize the ratio in (10.39) is equivalent to maximize  2 1 (N ) = N b − k N 4

(10.40)

Differentiating the above with respect to N produces   k 2 2 ∂(N ) =3 N − bk N + b2 ∂N 4

(10.41)

10.2 Average Bin Based Tsunami Detection

Letting

∂(N ) ∂N

329

= 0 and then solving the equation produce two solutions N1 =

4b 4b , N2 = 3k k

(10.42)

To determine which solution is the desired one, second-order differentiation is needed and is given by   ∂2 (N ) k 2 = 6 N − bk ∂N2 4

(10.43)

It can readily be seen that  8 b ∂2 (N )  0 = ∂ N 2  N =N2 3 k

(10.44)

Thus,  is maximal when N satisfies N=

4b 3k

(10.45)

The corresponding bin length (km) is given by L win =

4b 3k

(10.46)

Since the triangle base length is equal to 2b/k, the bin length is two thirds of the triangle base length. Note that the number of samples given by (10.45) or the bin length given by (10.46) also maximizes the SNR of the BA output s2 . The best number of samples is only associated with the model parameters and is independent of the measurement noise statistics for the case where a single bin covers the wave samples. Substituting (10.45) into (10.39) produces 16 b3 E[s22 ] =1+ 2 27 k σ 2 E[s1 ]

(10.47)

For a given Tsunami, the model parameters (k, b, ) are constant. If the noise variance is given, then the power ratio is nearly inversely proportional to the distance sampling interval. That is, increasing the SSH sampling frequency will increase the power ratio. However, increasing sampling frequency may result in performance degradation in SSH measurement. As indicated in [2], SSH measurement precision decreases and noise variance increases as the sampling interval or incoherent integration time decreases. Therefore, the impact of the parameters on both Tsunami

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10 Tsunami Detection and Parameter Estimation

detection and SSH measurement should be taken into account when selecting the parameters. As a Tsunami propagates from deep water to water of moderate depth the propagation speed as well as the wavelength would decrease significantly. It would be useful to develop an adaptive technique to anticipate topographic changes in the propagation region so as to optimize the detection procedure to improve detection performance. This can be considered as an open research issue. Over deep water Tsunamis propagate extremely fast such as at a speed of 800 km/h; such a speed may considerably affect Doppler shift in the received reflected GNSS signal depending on the orbiting transmitter velocity vector. This issue also needs further investigation in the future. Another prominent characteristic of Tsunamis is the long-crested form of the waves. If signals from multiple GNSS satellites are reflected from the long-crested lead wave and captured by the same receiver, then the approach studied above can be employed to process multiple sequences of SSH measurements to increase detection probability such as initially through processing separately and probably later through a joint processing. If a preliminary analysis reveals a wave in several locations, and these observations can be connected to provide the lead wave orientation, then additional sets of observations could perhaps be averaged along-crest to improve the SNR of the observations, and produce a more reliable estimate of the wave direction and height.

10.2.5 Simulation Results Simulation is conducted by using both the modeled triangle-shaped Tsunami and the real Tsunami data observed at Station 46,402 with the lead wave length about 248 km (69 samples). The data were originally sampled at a period of 15 s, or a distance interval of 3.54 km. This is equivalent to a sampling period of about half a second for a receiver logging data on a LEO satellite. The number of wave samples covered by the first bin is randomly generated and one or more bins will cover all the remaining wave samples. The number of bins required to cover all the wave samples depends on the wave length, bin length, as well as the number of wave samples in the first bin, as mentioned earlier. In addition, it is reasonable to assume that the statistics (mean and STD) of GNSS-R based SSH measurements are known in advance such as through processing data collected in the absence of any Tsunami, so that the error bias can be corrected or removed from the measurements for analytical simplicity. For simplicity, the SSH measurement consists of the Tsunami component, the wind wave effect and the noise, and the wind effect only contributes to the increase in the measurement noise STD. Nevertheless, it would be more realistic to consider the variation of the sea state along the Tsunami’s path. That is, over the wave length of 248 km, the wind induced surface elevation is modeled as a Gaussian variable of zero mean but a location-dependent STD, which is a significant issue, limiting the Tsunami detection performance. As mentioned earlier, the GNSS-R based SSH

10.2 Average Bin Based Tsunami Detection

331

measurement precision depends on a number of factors, including the incoherent integration time equivalent to the along-track averaging distance, delay detection technique, and whether the clean-replica method or the interferometric method is used. The clean-replica method makes use of the GNSS codes such as C/A code known to civilians, while the interferometric method exploits the full transmitted signal bandwidth including that of the encrypted M code and P(Y) code. When the peak of the first derivative of the delay waveform is used for the delay estimation and the interferometric GNSS-R is exploited, the altimetric precision would range from 0.15 to 0.76 m as the along-track averaging distance decreases from 130 to 7 km as listed in Table IV in [2]. Therefore, it is a great challenge to employ GNSS-R to perform weak Tsunami detection when the lead wave length is less than about 100 km, since very few Tsunami samples are available to perform detection. Assume that the along-track averaging distance is 7 km which is equivalent to incoherent integration time of one second. Accordingly, the standard deviation of the SSH measurement noise would be 0.76 m. The scaled waveform of Station 46,402 is used throughout the simulation and the wave height is set to be 45 cm if not mentioned elsewhere. Figure 10.12 shows the PFA with respect to the first threshold using the theoretical formulae given by (10.36) for the one-stage scheme and (34) in [10] for the two-stage scheme, as well as the corresponding simulated results. It can be seen that the analytical PFAs and the simulated ones have a good match and the two-stage scheme produces smaller PFAs than the one stage scheme. The corresponding PDs are shown in Fig. 10.13. The analytical results are produced based 0.9

theory-one stage simulated-one stage theory-two stages simulated-two stages

0.8 0.7

PFA

0.6 0.5 0.4 0.3 0.2 0.1 0 -20

-10

0

10

20

30

40

50

60

Threshold (cm)

Fig. 10.12 Probability of false alarm versus the threshold for the one-stage case and that for the first stage in the two-stage scheme where the second stage threshold is smaller than the first stage one by 30 cm

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10 Tsunami Detection and Parameter Estimation 1

0.8

PD

0.6

0.4

theory-one stage model-one stage real data-one stage theory-two stages model-two stages real data-two stages

0.2

0 -20

-10

0

10

20

30

40

50

60

Threshold (cm)

Fig. 10.13 Probability of detection versus the threshold for the one stage case and that for the first stage in the two stage scheme where the second stage threshold is smaller than the first stage one by 30 cm

on (10.35) and (33) [10]. One main observation is that the theoretical, model based and real data based PDs are in good agreement with each other. As expected, the PD of the two stage scheme is smaller than that of the one stage scheme. However, this does not mean that the two-stage scheme performs poorly. The key criterion for the statistical detection performance evaluation is the PD when giving a specific PFA as shown in Fig. 10.14. It can be seen that the two-stage scheme only slightly outperforms the one-stage scheme at very low PFA for the simulation setup described as follows. The SSH measurement noise STD is assumed to be 76 cm, while the model triangle height and the scaled real Tsunami height are 45 cm. The bin length is equivalent to 19 samples and the second stage threshold is less than the first one by 30 cm. On the other hand, when the bin covers 30 samples, the one-stage scheme outperforms the two-stage scheme as shown in Fig. 10.15. It is observed that when a suitable bin length is selected for each of the two schemes, they produce very similar results. This may be mainly because in the one stage scheme the threshold and the bin size can be set at larger values to improve detection performance, while the performance of the two-stage scheme may degrade if increasing the bin size. The suitable bin length of the one-stage scheme is significantly larger than that of the two-stage scheme. In the case where the threshold of the second stage is set very low, the performance of the two schemes is virtually the same. Since the two-stage scheme needs two thresholds to be defined, the one-stage scheme is preferable. From now on, only the results related to the one-stage scheme are presented.

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Fig. 10.16 Effect of bin length on the PD and PFA

Figure 10.16 shows how the bin length affects the performance of the one-stage scheme under four different bin lengths with the same SSH error STD and wave height as mentioned above. For a given PFA of 10%, the PD with a bin length of 106 km (14 samples) is about 60%, increased by about 4% compared to that with a bin length of 212 km (28 samples). In this case the best choice of bin coverage is around 14 samples since the performance with 10 and 22 samples degrades slightly and more significantly when covering more than 28 samples or less than 10 samples. That is, it seems that the most suitable bin length is around half the wave length that is 34 samples with an integration time of one second, which is smaller than the bin length that maximizes the SNR. In the simulation, as mentioned, the first bin always covers part of the leading or trailing edge of the wave depending on the propagation directions of the wave, and the measurement sensor ground track. Even for the one-stage scheme, multiple adjacent bins covering wave samples would be examined instead of a single one especially when the threshold is high to reduce the PFA. Nevertheless, the formulae in (10.45) and (10.46) may be used as a reference for bin length selection. Figure 10.17 shows the detection performance of the studied method with respect to the SSH data sampling rate. As mentioned earlier, there are 69 samples of real Tsunami lead wave data collected at Station 46,402, corresponding to a sampling period of about half a second in the case of a satellite-borne receiver. Three different sampling periods (one second, one and a half second, and two seconds) are examined. Three different bin lengths (15, 10 and 5 samples) are selected for the three sampling periods respectively for possible best performance. As expected, under the same SSH error STD of 76 cm a higher sampling rate can successfully detect the wave with

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Fig. 10.17 Effect of SSH measurement sampling rate on the PD and PFA under the same SSH error STD of 76 cm

a higher probability for a given PFA. However, when considering the integration time dependent error STD (76, 64, and 52 cm for integration time of 1, 1.5, and 2 s respectively), the performance is almost the same for the three sampling rates, with the sampling interval of 2 s achieving slightly better performance as shown in Fig. 10.18. Also, it is useful to evaluate the cases where the precision is improved (or decreased) due to using larger (or smaller) directivity antennas and less (or more) noisy receivers. Figure 10.19 shows the detection performance with respect to five SSH measurement error STDs (56, 66, 76, 86, and 96 cm), while the integration time is one second. For a given PFA of 5%, the PD is about 70% when the error STD is 56 cm, while the PD is reduced to about 30% when the error STD is increased to 96 cm. That is, when the error STD is reduced by 40%, the PD is improved by 130%. Finally, let us evaluate how the lead wave height affects the detection performance when the error STD is 76 cm and the wave height ranges from 15 to 55 cm as show in Fig. 10.20. If the wave height is significant such as 55 cm, the PD is about 60% for a given PFA of 5%. On the other hand, if the wave height is rather low such as 15 cm, then the PD is only about 10% for the same given PFA. Therefore, it is a challenging problem to reliably detect the lead wave when the error STD is rather high while the wave height is rather low. As mentioned in Sect. 10.2.3, the SWMA method would be more accurate than the BA method. Here, simulation is conducted to compare the performance of the two methods. Figure 10.21 shows the waveforms of the Tsunami triggered by the same 2011 Japan’s earthquake but observed by Jason-1 satellite altimeter. The second waveform was already shown in Fig. 10.6. Since the accuracy of Jason-1 altimeter is

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Fig. 10.18 Effect of SSH measurement sampling rate on the PD and PFA when the SSH error STD depends on integration time, which are 76, 64, and 52 cm, respectively 1 0.9 0.8

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Fig. 10.21 Observed waveforms of the Tsunami (triggered by the 2011 Japan’s earthquake), recorded by Jason-1 satellite altimeter and corrected for a number of impacts

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Fig. 10.22 Modeled GNSS-R SSH measurements in the presence of a Tsunami

a few centimeters, the observed waveforms would have the same pattern of the real Tsunami waveform except for the measurement error and the residual errors after the imperfect model-based corrections. When a Tsunami is sensed by a satellite-carried GNSS-R sensor, the SSH measurement would be much noisier. The measurement noise standard deviation may be around 70 cm if the sampling interval is one second. Figure 10.22 shows the modeled GNSS-R based SSH measurement when the measurement noise standard deviation is 60 cm. In this case, the Tsunami waveforms are overwhelmed by the noise so that it is difficult to see whether or not a Tsunami waveform is present. Figure 10.23 shows the detection performance of the BA approach and the SWMA approach when the bin length and sliding window length are 30 and the STD of the SSH measurement error is set to be two different values (70 cm and 100 cm), corresponding to rather poor SSH measurement. When the PFA equals 3% and the STD of the SSH measurement error is 70 cm, the PD is about 60.3% and 67.7% for the BA approach and the SWMA approach, respectively [11]. Comparatively, the PD of the SWMA approach is increased by about 7.4%. A similar increase in PD is obtained for the case where the SSH error STD is 100 cm.

10.3 Tsunami Reconstruction In Chap. 6, the basic principle of wavelet theory is studied. This section focuses on the use of wavelet theory to reconstruct Tsunami waveform. Real Tsunami data

10.3 Tsunami Reconstruction

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Fig. 10.23 PD versus PFA for the BA and SWMA approaches

Fig. 10.24 Block diagram of the wavelet based Tsunami measurement noise reduction and waveform reconstruction approach

were used to simulate GNSS-R based SSH observations. Performance evaluation is performed when Tsunami is present and absent respectively.

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10.3.1 Wavelet Based Noise Reduction for Tsunami Reconstruction The functional diagram of the wavelet based noise mitigation and Tsunami waveform reconstruction method is shown in Fig. 10.24. Suppose that a sequence of SSH measurements is collected by a satellite-carried GNSS sensor. Since Tsunami wavelength can be more than 500 km, the length of the data should cover the whole Tsunami. The SSH data go through the decomposition and reconstruction process (see Fig. 4.14 in Chap. 4). That is, the SSH samples are first simultaneously high-pass and low-pass filtered and then down-sampled. The low-pass filtered and down-sampled data sequence is again high-pass and low-pass filtered and then down-sampled. The procedure repeats for the predefined number of levels. The downsampled output from each high-pass filter is then processed to mitigate the noise such as through thresholding. Then, signal reconstruction is performed to reconstruct the waveform using the high-pass and low-pass filters, up-sampling, the data sequence from the low-pass filter of the final stage (or level), and the data sequences resulting from processing data sequences generated by the high-pass filters during signal decomposition and noise mitigation. The reconstructed waveform is then modeled mathematically so that the wave heights and wave lengths can readily be determined. The parameters of the low-pass and high-pass filters depend on which wavelet is used. As mentioned in Chap. 4, there are a number of popular wavelets. But it is not clear about which wavelet is more suited for noise reduction for Tsunami reconstruction. One way to select the wavelet is to use experimental data, if available, to test which one achieves the best performance. Another parameter is the number of decomposition stages or levels, which is constrained by the number of samples in the SSH sample sequence. The number of levels can be up to int(log2 N ) where int(x) is the operation of taking the integer part of x and N is the length of SSH sample sequence. As will be discussed in the next section, the number of levels should be much smaller than int(log2 N ).

10.3.2 Simulation Results 10.3.2.1

Simulation Results in Presence of Tsunami

In this section modeled GNSS-R based SSH measurements using real Tsunami data observed by satellite Envisat as shown in the upper panel of Figs. 10.6 and 10.7 are used to evaluate the Tsunami reconstruction and parameter estimation approach. The SSH measurements are corrupted by Gaussian noise with a zero mean and STD of 50 cm as shown in Fig. 10.25. The distance interval is 6.86 km equivalent to sampling period or incoherent integration time of one second. Over 10,000 simulation runs, the RMSE is 49.93 cm which is virtually the same as the given STD of the noise (i.e. 50 cm). This is in accordance with the fact that the RMSE and the STD are

10.3 Tsunami Reconstruction

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Fig. 10.25 Modeled SSH measurements by adding noise of STD of 50 cm to Envisat SSH data

equal in the case of zero mean noise. The addition of such a large noise is based on the studies by [2] that the SSH measurements observed by a satellite-carried GNSS sensor (e.g. CYGNSS sensor) could be rather noisy especially when the incoherent integration time is not long. Matlab wavelet toolbox was used for simulation and data processing. The Matlab library function ‘wden’ is directly used for noise mitigation. Figure 10.26 shows the examples of the reconstruction results using four different wavelets. The results show that when wavelet parameters are properly chosen, the four wavelets produce very similar reconstruction results. From now on, only results using the Daubechies wavelet are presented to evaluate the effect of different parameters. Five parameters need to be selected to call the function. If not specified elsewhere, they are respectively set to be: ‘heursure’ for threshold selection based on heuristic variant of Stein’s unbiased risk, ‘s’ for soft thresholding, ‘mln’ for rescaling done using level dependent estimation of level noise, ‘4 for decomposition level used, and ‘db20 for Daubechies wavelet with 20 coefficients. Figure 10.27 shows the results when using four different orders of highest vanishing moment. The results indicate that the reconstruction performance is rather insensitive to the order of highest vanishing moment. Note that the order of highest vanishing moment limits the highest order of a polynomial which the wavelet can be used to describe the signal. A higher vanishing moment order means that a more complex signal can be represented. Because Tsunami waveforms are usually not too complex, very similar results would be produced using a range of different orders of highest vanishing moments.

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Fig. 10.26 Noise free SSH measurements (red solid line) and four wavelets based reconstruction results (black dashed line) from noise corrupted SSH measurements as shown in Fig. 10.25

Figure 10.28 shows the impact of the number of decomposition and reconstruction level. The best results for this case are produced when using four levels of decomposition and reconstruction. When two levels are used, a significant part of the noise still remains in the reconstructed signal. On the other hand, when using five levels, the length is much overestimated while the height is greatly underestimated. The results indicate that more decomposition and reconstruction levels may not produce better results. The suitable number of levels could be determined by the number of wave samples instead of the number of the SSH samples of the whole sequence. The determination of the best suitable level theoretically is an interesting future research issue. Figure 10.29 shows the impact of noise STD when four different STD values are evaluated. The number of decomposition level is set to be four, and the order of highest vanishing moment is set to be twenty. As expected, the performance degrades as the noise STD increases. However, the impact is mainly on the wave height, while the wavelength is less affected. When noise STD is 90 cm, useful information about the wave height and length can still be obtained. That is, this technique is suited for Tsunami reconstruction and parameter estimation with noisy measurements such

10.3 Tsunami Reconstruction

3rd Vanishing Moment

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Fig. 10.27 Effect of order of highest vanishing moment on lead wave reconstruction

as made by satellite missions not dedicated for surface altimetry or wave height measurement such as CYGNSS, Bufeng-1 and the potential GEROS-ISS mission. The above results are obtained when giving a specific sequence of noise samples. To have a better evaluation of the wavelet based method, it is necessary to use a large number of noise sample sequences and use curve fitting to obtain the estimate of the shape parameters and the wave height and length from each sequence of data. Then the statistics of the wave parameter estimation errors can be determined. To achieve the goal, 2000 sequences of noise samples are generated and a modeled waveform is obtained from the denoised data with each sequence of the noise samples. Figure 10.30 shows four typical examples of filtered waveform and linear LS curve fitting results. The number of decomposition and reconstruction levels is four, the order of highest vanishing moment is 20, and the noise STD is 50 cm. To avoid overestimating the wavelength, a threshold is used to constrain the slope a3 to be greater than 0.15. It can be seen that the wave model parameters related to one sequence of noise samples can be quite different from those related to another sequence of noise samples, but all the reconstructed waveforms can be well modeled by triangular functions. Table 10.6 shows the statistics of the wave model parameter estimation errors, which are the mean, STD and root mean square of the errors. For comparison, the

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Fig. 10.28 Effect of filtering levels on the performance of Daubechies wavelet based lead wave reconstruction

calculated root square CRLB values are also listed. It can be seen that the estimation accuracy of all the parameters is close to the CRLB, but the estimation accuracy in terms of RMSE of several parameters is better than the CRLB. This could be mainly due to the fact that the CRLB is calculated with the modeled wave, while the simulation uses the real wave data. Also, the estimator is not unbiased, so the CRLB can only be used as a reference. Table 10.7 shows the performance of the wave height (h = h 1 + h 2 ) and wavelength (L = L 1 + L 2 ) estimation. Both STD and RMSE are close to the square root of CRLB values.

10.3.2.2

Simulation Results in Absence of Tsunami

The above results were obtained under the assumption of presence of a Tsunami. It is useful to examine what will be produced when using the reconstruction method in the absence of a Tsunami. Figure 10.31 shows four typical examples of the reconstruction results when the SSH measurement sequence consists of only simulated noise samples with a STD of 50 cm. It was observed that the reconstructed noise basically varies between−30 and 30 cm. One can readily decide that a significant

10.3 Tsunami Reconstruction

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Fig. 10.29 Effect of noise STD on wave reconstruction. Red solid line is for Envisat observed SSH and black dashed line is for reconstruction results

Tsunami with amplitude greater than 50 cm is not present most of the time. However, it is difficult to determine if a small Tsunami occurs due to the strong noise. Although the four waveform patterns look like noise ones, it is not sure whether a Tsunami is sensed. Also, occasionally such as in the third subplot of Fig. 10.31 where the maximum magnitude is over 50 cm, it is difficult to judge whether or not a significant Tsunami with amplitude greater than 50 cm occurs. Therefore, it is necessary to combine with results provided by one or more other Tsunami detection methods to make a more reliable and accurate decision on whether a Tsunami is present. To better approximate the realistic scenario, satellite altimetry results in absence of a Tsunami are used. Figure 10.32 shows four typical SSH sequences produced by Envisat (pass 101, cycle 100). The variations would result from the residual errors in the model-based correction of different effects. Adding the same four sequences of noise samples used for generating Fig. 10.31 to the SSH sequence in the topleft panel of Fig. 10.32 and performing reconstruction produces the reconstruction results shown in Fig. 10.33. Compared with Fig. 10.31, some of the SSH peak values in Fig. 10.33 are increased by around 15% due to the significant amplitude of the SSH residual error of about 20 cm. As a consequence, it is more likely to accept the presence of a small Tsunami mistakenly only based on the reconstruction

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Fig. 10.30 Four examples of wavelet based wave reconstruction and linear LS fitting results. Red solid line is for Envisat observed SSH and black dashed line is for reconstructed waveform

Table 10.6 Model parameter estimation performance Model parameters

True value

Error mean

Error STD

RMSE

CRLB 0.091

a1 (cm/km)

0.563

−0.005

0.084

0.084

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193.7

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0.301

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0.12

0.22 37.7

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−74.4

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22.9

23.0

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96.9

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12.9

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30.6

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48.5

0.4

11.7

11.7

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L 2 (km)

247.1

29.6

72.4

78.2

69.9

results, indicating the importance of Tsunami detection based on multiple source data. Figure 10.34 shows the correspondingly reconstruction results when the SSH sequence in bottom-right panel of Fig. 10.32 is used. In this case the mean of the SSH samples is about 10 cm, so the measurements are biased considerably. As a result, it can be readily seen that all the four reconstructed SSH sequences are also

10.3 Tsunami Reconstruction Table 10.7 Wavelet based wave height and wavelength estimation error statistics

347 Wave Height (h = 145.5 cm) Mean (cm)

STD (cm)

RMSE (cm)

SR-CRLB (cm)

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19.8

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18.6

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RMSE (cm)

SR-CRLB (cm)

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86.1

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Fig. 10.31 Reconstruction results with simulated zero-mean SSH noise data in absence of a Tsunami

biased. If the bias in the altimetry data is removed, then the amplitude is only about 10 cm and the impact of such residual SSH errors would be considerably smaller, compared with that shown in Fig. 10.33.

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Fig. 10.32 Envisat altimetry results in absence of a Tsunami

10.3.3 Cramer-Rao Lower Bound In this subsection, theoretical formulas of the CRLB are derived for estimation of the modeled Tsunami wavelength and wave height as well as shape parameters [12].

10.3.3.1

Model Parameters

Define the SSH measurement vector as u = [ u 1 u 2 ... u N ]T

(10.48)

where u i is defined by (10.10). To simplify the analysis, the origin of the coordinate system shown in Fig. 10.5 is shifted to the crossing point between line 2 and the horizontal axis as shown in Fig. 10.35. As a consequence, the y-axis intercept b2 is equal to zero and the magnitudes of the Tsunami samples are described by:

10.3 Tsunami Reconstruction

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Fig. 10.33 Reconstruction results with simulated zero-mean noise data plus altimetry data shown in top-left panel of Fig. 10.32

⎧ a1 d + b˜1 , ⎪ ⎪ ⎨ −a1 d, μ(d) ˜ = ⎪ a3 d + b˜3 , ⎪ ⎩ 0,

d˜s ≤ d < d˜ p d˜ p ≤ d < d˜v d˜v ≤ d ≤ d˜e elsewher e

(10.49)

where the two new y-axis intercepts b˜1 and b˜3 are related to the previous ones by: a3 b2 b˜1 = b1 + b2 ; b˜3 = b3 + a1

(10.50)

These two y-axis intercepts are calculated as given in Table 10.8, while the other parameters remain the same as those listed in Tables 10.3 and 10.5. Also, due to the coordinate transformation, the four distances are changed to be: b˜1 b˜1 ˜ b˜3 b˜3 , d˜e = − d˜s = − , d˜ p = − , dv = − a1 2a1 (a1 + a3 ) a3 Now, the parameter vector becomes a four-component vector:

(10.51)

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Fig. 10.34 Reconstruction results with simulated zero-mean noise data plus altimetry data shown in bottom-right panel of Fig. 10.32

Fig. 10.35 Linear Tsunami modeling after coordinate transformation

θ = [ a1 b˜1 a3 b˜3 ]T

(10.52)

The SSH measurement noise comes from a range of error sources of different spatial scales, including imperfect corrections of tropospheric and ionospheric effect,

10.3 Tsunami Reconstruction Table 10.8 The y-axis intercepts b˜1 and b˜3 of the two linear equations for line 1 and line 3 when the origin of the coordinate system is moved to the connection point between the two modeled half waves

351

Station # or satellite

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b˜3 (cm)

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127.5

−36.0

21,413

158.6

−31.3

Envisat

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−74.4

Jason-1

178.6

−103.3

lunar and solar tides, ocean dynamic heights, and geoid errors. To simplify the theoretical analysis, the measurement noise is approximately modelled by a Gaussian random variable. It is useful in the future to use a better noise model to perform the analysis. Note that when using the measurements as shown in Figs. 10.6, 10.7 or 10.21 by satellite altimeters for simulation studies, the addition of a Gaussian random variable would be sufficient, since most of the error components are already included in the measurements. Also, the measurement errors along the track would be correlated to some degree. For simplicity in analysis, the measurement errors are assumed uncorrelated and the case of correlated measurement errors requires further investigation. Theoretical Derivation Given the parameter vector defined in (10.52) and assuming that the measurement error samples are uncorrelated Gaussian random variables with a zero mean, we have the joint conditional probability density function (i.e. the likelihood function) of the measurement vector:  N  N   (u i − μ˜ i )2 1 exp − (10.53) p(u|θ) = √ 2σ 2 2π σ i=1 where σ is the STD of the measurement noise and μ˜ i is defined as ⎧ ˜ ⎪ ⎪ a1 iδ + b1 , ⎨ −a1 iδ, μ˜ i = ⎪ a iδ + b˜3 , 3 ⎪ ⎩ 0,

k0 ≤ i < k1 k1 ≤ i < k2 k2 ≤ i ≤ k3 elsewher e

(10.54)

where δ is the sampling distance interval and 

   b˜1 b˜1 k0 = r − , k1 = r − a1 δ 2a1 δ     b˜3 b˜3 k2 = r − , k3 = r − (a1 + a3 )δ a3 δ

(10.55)

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10 Tsunami Detection and Parameter Estimation

Here r (x) is the operation of rounding x to the nearest integer. k0 and k1 are negative integers, while k2 and k3 are positive integers. The log likelihood function is then given by N  √ (u i − μ˜ i )2 ln p(u|θ) = −N ln( 2π σ ) − 2σ 2 i=1

(10.56)

Since the noise is assumed as a Gaussian random variable and the covariance matrix is constant, the Fisher information matrix (FIM) can be determined by Fm,n =

∂μT −1 ∂μ  , m, n = 1, 2, 3, 4 ∂θm ∂θn

(10.57)

where  = σ 2 I ∈ R N ×N is the covariance matrix with I the identity matrix and μ = [ μ˜ 1 μ˜ 2 ... μ˜ N ]T

(10.58)

Note that if the noise STD is not a constant but geolocation-dependent, then the STD variation should be taken into account. Equation (10.57) can be further written as Fm,n =

1 ∂μT ∂μ , m, n = 1, 2, 3, 4 σ 2 ∂θm ∂θn

(10.59)

It can be readily shown that ⎧  ⎨ iδ, k0 ≤ i < k1 ∂ μ˜ ∂ μ˜ i 1, k0 ≤ i < k1 i = −iδ, k1 ≤ i < k2 , = ˜ ⎩ 0, elsewher e ∂a1 0, elsewher e ∂ b1   ∂ μ˜ i iδ, k2 ≤ i ≤ k3 ∂ μ˜ i 1, k2 ≤ i ≤ k3 , = = 0, elsewher e ∂ b˜3 0, elsewher e ∂a3 With some mathematical manipulations, we can obtain

(10.60)

10.3 Tsunami Reconstruction

353

 δ2 F1,1 = k2 (k2 − 1)(2k2 − 1) + k 0 (k 0 + 1)(2k 0 + 1) 6σ 2 k0 − k1 1 F2,2 = , F4,4 = (k3 − k2 + 1) 2 σ2 σ δ2 F3,3 = (k3 (k3 + 1)(2k3 + 1) − k2 (k2 − 1)(2k2 − 1)) 2 6σ 1 δ F1,2 = F2,1 = (k1 + k0 − 1)(k1 − k0 ) 2 2 σ 1 δ F3,4 = F4,3 = (k3 + k2 )(k3 − k2 + 1) 2 2 σ F1,3 = F3,1 = F1,4 = F4,1 = F2,3 = F3,2 = 0 F2,4 = F4.2 = F3,4 = F4.3 = 0

(10.61)

where k i is equal to the absolute value of ki . Although the parameter vector θ completely defines the wave, it is useful to define the wave directly by use of heights (h 1 and h 2 ) and lengths (L 1 and L 2 ) of the two modeled half waves. That is, alternatively, the parameter vector can be defined as η = [ h 1 L 1 h 2 L 2 ]T

(10.62)

The relationship between the parameters in θ and those in η was already given by (4), but the height and length parameters are functions of the linear shape parameters. Let the linear shape parameters be functions and the height and length parameters be variables. Then, the formulas can be derived as 2h 1 , b1 = 2h 1 L1 2h 1 h 2 2h 1 h 2 L 2 a3 = , b3 = − 2h 1 L 2 − h 2 L 1 2h 1 L 2 − h 2 L 1 a1 =

(10.63)

Therefore, the FIMs of the two parameter vectors are related by Fη (η) = JT Fθ (θ)J

(10.64)

where the elements of Fθ (θ) are given in (10.61) and J is the Jacobian matrix which can be derived as: ⎤ ⎡ 2 1 − 2h 0 0 L1 L 21 ⎥ ⎢ ⎥ ⎢ 2 0 0 0 (10.65) J=⎢ ⎥ ⎣ −2h 22 L 1 κ 2h 1 h 22 κ 4h 21 L 2 κ −4h 21 h 2 κ ⎦ 2h 22 L 1 L 2 κ −2h 1 h 22 L 2 κ −4h 21 L 22 κ 2h 1 h 22 L 1 κ

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10 Tsunami Detection and Parameter Estimation

∂θi where [J]i, j is defined as ∂η and κ = (2h 1 L 2 − h 2 L 1 )−2 . j The CRLB for the kth parameter of vector θ and that of vector η are respectively given by

C R L B(θk ) = [Fθ−1 (θ)]k,k , C R L B(ηk ) = [Fη−1 (η)]k,k

(10.66)

Under the assumption that the height estimation errors of the two half waves are uncorrelated and so are the length estimation errors, the variances of the estimation errors of the wave height and wavelength satisfy σh2 = σh21 + σh22 , σ L2 = σ L21 + σ L22

(10.67)

Thus, the CRLBs for wave height and wavelength estimation are simply given by C R L B(h) = C R L B(h 1 ) + C R L B(h 2 ) C R L B(L) = C R L B(L 1 ) + C R L B(L 2 )

(10.68)

Note that the square root of the CRLB is usually used to benchmark the STD of parameter estimation error. Numerical Results The parameters of the modeled waveform from the Tsunami data observed by satellite Envisat (see Tables 10.5 and 10.8) are used to numerically calculate the CRLB. The noise STD is selected to be between 10 and 70 cm. Figure 10.36 shows the square root of CRLB to benchmark the estimation error STD of the linear equation coefficients {a1 , b1 , a3 , b3 }. As expected from (10.64), the square root of CRLB increases linearly with the STD. The square root of CRLB for slope a3 is more than twice that for slope a1 . This may be caused by the fact that the height of the second half wave is only about half of the height of the first half wave, making the slope a3 more difficult to estimate. The same reason could apply to the phenomenon that the square root of CRLB for intercept b3 is greater than that of intercept b1 . Figure 10.37 shows the square root of CRLB for the modeled wave heights and lengths {h 1 , L 1 , h 2 , L 2 , h 2 , L}. The square root of CRLB for height h 1 is slightly greater than that for height h 2 , although h 1 is more than two times h2 . The square root of CRLB for base length L 2 is about 3 times that for base length L 1 , whereas L 1 is about 1.4 times base length L 2 , which might also indicates the impact of the significant difference between the two heights.

10.4 Tsunami Parameter Estimation This section studies the estimation of Tsunami propagation direction and speed as well as Tsunami wavelength with GNSS-R based SSH measurements.

10.4 Tsunami Parameter Estimation

355

SRCRLB (cm/km)

0.4

a1 a3

0.2

0 10

20

30

40

50

60

70

50

60

70

Noise STD (cm)

SRCRLB (cm)

60

b1 40

b3

20 0 10

20

30

40

Noise STD (cm)

Fig. 10.36 Square root of CRLB (denoted by SRCRLB) for the slopes (a1 and a3 ) and y-axis intercepts (b1 and b3 ) of the linearize wave shape

SRCRLB (cm)

30

h1 20

h2 h

10 0

10

20

30

40

50

60

70

50

60

70

Noise STD (cm)

SRCRLB (km)

150

L1 100

L2 L

50 0

10

20

30

40

Noise STD (cm)

Fig. 10.37 Square root of CRLB (denoted by SRCRLB) for the heights (h 1 , h 2 and h) and lengths (L 1 ,L 2 and L) of the modeled two half waves

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10 Tsunami Detection and Parameter Estimation

10.4.1 Tsunami Propagation Direction and Speed Estimation The basic idea is to make use of multiple ocean-surface specular reflection tracks of GNSS signals, which cross the Tsunami wave and are adequately close to each other, to estimate Tsunami wave propagation direction and speed [9]. In particular, a specific point on the wave measured along the multiple reflection tracks is selected and detected for direction and speed estimation. Both the up zero-crossing point of the leading edge and the wave crest point can be considered as the specific point. The position of the crest point can be readily estimated by finding the maximum magnitude of the wave based on the SSH estimates. Typically, the crest point would not change dramatically in the presence of SSH estimation error. Although the zerocrossing point may change significantly when the SSH measurements are noisy, a technique can be used to reduce the error. In addition to smoothing, leading edge curve fitting can be applied to determine the zero-crossing point position. Specifically, a linear fitting can be performed on the middle segment of the leading edge, such as between 10 and 90% of the SSH at the crest. In the case where there are only a few points on the leading edge, interpolation can be used. The extension of the resulting line segment will cross the horizontal axis and the crossing point is treated as the desired specific point. Since the zero-crossing point determination will incur more computation time, the crest point is preferable and will be later employed in the simulation. Nevertheless, it would be useful to evaluate the performance difference between the two different options for the specific point selection and detection.

10.4.1.1

Theoretical Formulas

Consider the case where four reflection tracks cross the Tsunami wave. Each track crosses a specific point of the wave at different surface position and time instant. Suppose that the first track crosses the specific wave point at surface point p1 at time instant t1 ; the second one at point p2 at time instant t2 ; the third one at point p3 at time instant t3 ; and the fourth one at p4 at time instant t4 , as illustrated in Fig. 10.38. The coordinate system is on the East-North 2D plane with the horizontal axis on the East direction and the vertical axis on the North direction. The wave propagation direction is represented by the azimuth angle going clockwise from North to the wave propagation direction and denoted as θ . For clarity, the GNSS-R reflection tracks are not shown and they are not required to have a specific included angle with the wave propagation direction. The four dashed lines perpendicular to the propagation direction are not the reflection tracks, but each of them is used to indicate the positions of the same specific wave points along the wave cross section at a specific time instant. It is assumed that the wave travels at a constant speed over the duration from t1 to t4 and the time instants satisfy t1 < t2 < t3 < t4 which can be readily arranged based on measurements. The distance between each pair of the four points can be more than a few hundred kilometres, so that the Earth surface curvature should be taken into account when

10.4 Tsunami Parameter Estimation

357

North

p4

θ 3,4 θ 2,3

θ

Wave propagation direction

θ p3

p2

θ1,2 θ θ

p1

East

Fig. 10.38 Geometry between Tsunami wave propagation direction and the first crossing points of reflection tracks on the Tsunami wave in the North-East 2D plane

calculating the distance. Without loss of generality, denote the 2D position coordinates of a pair of surface points as (αi , βi ) and (α j , β j ) where {αk } are latitudes and {βk } are longitudes. Then, according to the Vincenty formula [6], the shortest surface distance is calculated by  L i, j = R × arctan

(cos α j sin β)2 + (cos αi sin α j − sin αi cos α j cos β)2 sin αi sin α j + cos αi cos α j cos β (10.69)

where R is radius of the Earth, which can be approximated to be 6,378,137 m, and β = β j − βi

(10.70)

The corresponding azimuth angles (θi, j ) are calculated by  θi, j = arctan

cos α j sin β cos αi sin α j − sin αi cos α j cos β

 (10.71)

where if θi, j is negative, then θi, j + 2π is the azimuth angle. Using the geometry, the distance equation satisfies vw t j,i = L i, j | cos(θ − θi, j )|

(10.72)

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10 Tsunami Detection and Parameter Estimation

where vw is the wave propagation speed, t j,i = t j − ti where t j > ti , and | · | is the operation of taking the absolute value. Consider points k and  which either are completely different from points i and j or share one point with points i and j. Then, a similar distance equation can be obtained as vw t,k = L k, | cos(θ − θk, )|

(10.73)

where t > tk . There are four different cases when removing the operation of | · | to solve (10.72) and (10.73). When cos(θ − θi, j ) and cos(θ − θk, ) are both positive or negative, dividing (10.73) by (10.72) and re-arranging the equation produce tan θ =

L k, t j,i cos θk, − L i, j t,k cos θi, j L i, j t,k sin θi, j − L k, t j,i sin θk,

(10.74)

That is, the estimate of the wave propagation direction is produced as  θˆ = arctan

Lˆ k, tˆj,i cos θˆk, − Lˆ i, j tˆ,k cos θˆi, j Lˆ i, j tˆ,k sin θˆi, j − Lˆ k, tˆj,i sin θˆk,

 (10.75)

where “ˆ” is used to denote the estimate of the relevant parameter. In the case where cos(θ − θi, j ) is positive but cos(θ − θk, ) is negative, or vice versa, (10.75) becomes 

Lˆ k, tˆj,i cos θˆk, + Lˆ ab tˆ,k cos θˆi, j θˆ = arctan − Lˆ i, j tˆ,k sin θˆi, j + Lˆ k, tˆj,i sin θˆk,

 (10.76)

The undesirable angle estimate can be readily rejected by substituting the angle estimate into (10.72) and (10.73) and calculating the residual: ˆ 2 r esidual = (|tˆj,i | − |L i, j cos(θˆ − θˆi, j )|)2 + (|tˆ,k | − |L k, cos(θˆ,k − θ)|) (10.77) The angle estimate which produces the smaller residual is accepted as the estimate of the direction angle. There are six different pairs among the four points, producing six distance equations. That is, fifteen combinations of two distance equations can be formed to produce fifteen estimates of the wave propagation direction. The average of these estimates may be taken as the final estimate of the direction. However, the median of these estimates can be a better estimate, as evidenced by simulation results using real Tsunami wave data. In the case where the four specific points cover a large region such as over the wave front of several hundred kilometres and they are near the surface area beneath which the Tsunami is originated, the estimated direction can be treated as the mean direction or the direction in the middle between points p1 and p4 .

10.4 Tsunami Parameter Estimation

359

Substituting the direction estimate into the six distance equations and considering parameter estimation error yield six wave propagation speed estimates: vˆw =

Lˆ p,q | cos(θˆ − θˆp,q )| tˆq, p

(10.78)

Similarly, the median of the six estimates can be treated as the final estimate of the propagation speed. Clearly, the direction and speed estimation errors come from three error sources, time interval estimation, distance estimation and azimuth angle estimation. However, all the three error sources are related to the errors in the estimation of the four specific points ( p1 , p2 , p3 , and p4 ). That is, these point position errors are the dominant error in determining the direction and speed. Thus, the selection of such a specific point on the wave is important. Typically, the propagation direction and speed of the wave along a reflection track would be very similar to those along another reflection track provided that the two tracks are close to each other. On the other hand, the wave parameters in one region might be quite different from those in another far away region. Therefore, it is best to use the measurements related to close reflection tracks so as to generate more accurate wave parameter estimation for the surface region of interest. However, in order to increase the probability of detection of a Tsunami wave, measurements associated with reflection track(s) with larger distances to other tracks should be used to cover a larger surface area.

10.4.1.2

Kalman Filter Based Smoothing

It is expected that the surface height measurements using the GNSS-R technique would be noisy especially when using the C/A code. In order to estimate the Tsunami wave height, propagation direction and speed, and wavelength accurately, the raw surface height measurements need to be smoothed. For simplicity, the noisy GNSS-R based SSH measurements are modelled as the sum of the station buoy sensor based relative sea level measurements and a zero-mean Gaussian random variable with two different standard deviations of 10 cm and 20 cm, respectively. Such an accuracy may be achieved using high-directivity antenna and advanced signal processing technique. For convenience, the sampling period of the SSH estimation is assumed to be 2 s. Note that with a longer integration time, the wave height estimation will have a systematic decrease, making the measured tsunami wave profile look smoother. As a consequence, the estimation of Tsunami parameters will be affected, although the impact would not be as serious as that on the SSH estimation. It would be useful to do further investigation on this issue. Since GNSS-R based (especially with C/A code) SSH measurements would be noisy, filtering is highly desirable to reduce the noise and measurement error. Here the linear Kalman filter (KF) is employed to smooth the modelled GNSS-R based SSH measurements which are corrupted by significant Gaussian noise. An alternative

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10 Tsunami Detection and Parameter Estimation

0.8

0.8

0.6

0.6

Relative Sea Level (m)

Wave Amplitude (m)

approach is to use the wavelet based noise mitigation to reconstruct the waveform as studied in the preceding section. It is expected that the wavelet denoising would outperform KF, so the KF based results may be treated as the upper bound of the method using wavelet denoising. Figure 10.39 through Fig. 10.41 show the original, modelled GNSS-R based, and KF smoothed SSH measurements when the standard deviation of the Gaussian noise is 20 cm. The dashed line represents the modelled GNSS-R based SSH measurement by adding the station measurements with a zeromean Gaussian random variable of standard deviation 20 cm. Tables 10.9 and 10.10 show the estimated wave heights and wavelengths (or wave duration of a single cycle) when the SSH estimation error standard deviation is 10 cm and 20 cm, respectively. A total of 2000 error samples associated with a specific error STD are generated for each SSH estimation. The maximum of the SSH estimates

0.4 0.2 0 -0.2 -0.4 -0.6

0

200

400

600

800

1000

GNSS-R based smoothed

0.4 0.2 0 -0.2 -0.4 -0.6

0

200

400

600

800

1000

Distance (km)

Distance (km)

Fig. 10.39 (left) Original SSH measurements obtained at station #21401; (right) simulated SSH measurements (dashed) using real data and smoothed measurements (black solid) using a linear KF

Table 10.9 Error statistics (cm) of wave height estimation SSH Error STD = 10 cm Mean

SSH Error STD = 20 cm STD RMSE mean

STD RMSE

10.2

18.8

#21,401

7.7

12.8

14.2

23.6

#21,413

10.9

9.0

14.1

17.8

18.9

26.0

#21,418 −37.5

8.5

38.5

−33.9

22.6

40.7

Table 10.10 Error statistics (km) of wavelength estimation SSH Error STD = 10 cm Mean

SSH Error STD = 20 cm STD RMSE mean

STD RMSE

#21,401 −12.4

38.8

40.7

−25.6

64.7

69.6

#21,413 −11.7

46.1

47.6

−30.4

80.7

86.3

#21,418 −6.1

77.9

78.1

−43.4

97.8

107.0

10.4 Tsunami Parameter Estimation

361

is first located and then the single cycle of the wave is determined by searching for the zero up-crossing on both sides of the position of the estimated maximum SSH. To handle the situation where there is a long tail with a magnitude close to zero but up zero-crossing does not occur on either the wave front or the rear, the wave starting point is assumed if the SSH estimate decreases to a small positive number such as 6 cm or the wave ending point is assumed if the SSH estimate increases to a small negative number such as−6 cm. The wavelength is simply estimated as the distance between the starting and ending points, while the wave height is estimated as the difference between the maximum SSH estimate and the minimum SSH estimate between the two points. One observation is that the estimation error associated with station #21,418 is significantly higher than those related to the other two stations. This may be due to the specific shape of the wave, the long vibrating tail on the trailing edge and the fact that the narrow negative spike is significantly reduced by smoothing. On average, the wave height and wavelength estimation error is still significant partially because the long trailing edge with small negative magnitudes can be easily distorted by SSH measurement errors. The error can be reduced by manually checking each individual waveform after a sequence of SSH estimates with significant magnitudes are observed.

10.4.1.3

Results for Propagation Direction and Speed Estimation

Assume that there are four surface reflection tracks which are parallel to each other for simulation simplicity. The distance between each pair of neighbouring tracks is randomly selected to be between 50 and 200 km. The wave propagates at a speed of 0.2 km/s towards the Northeast with an azimuth angle of 45 deg, while the specular scattering point travels at a speed of 7.4 km/s towards the West with an azimuth angle of 270 deg. The reflection tracks cross the specific point (crest) on the wave randomly selected on the tracks of over 500 km long. The selection of the parameter values is arbitrary within the range of possible realistic values. Based on the geometry of the reflection tracks and the wave propagation direction, the time interval between each pair of the four points can be calculated. The estimation errors of the crest positions in the simulation for wave height and wavelength estimation discussed in the preceding subsection are used to simulate the position errors of the four specific points. Three different waveforms and two different STDs of the SSH estimation error are all considered in the evaluation of the method. Figure 10.42 shows the wave propagation direction estimation results using the median based method, while Fig. 10.43 shows the results when normal average based method is used. The original Tsunami waveform shown in the left of Fig. 10.41 is used and the error STD of the SSH estimation is set to be 20 cm. It can be seen that the median based method performs much better than the normal averaging based method. The performance gain of the median based method may come from the fact that the large estimation errors are mostly discarded by the median operation.

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10 Tsunami Detection and Parameter Estimation

Table 10.11 Error statistics (deg) of wave propagation direction estimation SSH Error STD = 10 cm Mean

SSH Error STD = 20 cm STD RMSE Mean

STD RMSE

#21,401 −0.34

4.84

4.85

−0.77

7.35

7.39

#21,413 −0.26

4.18

4.19

−0.65

6.62

6.66

#21,418 −0.32

4.21

4.22

−0.28

3.61

3.62

Table 10.12 Error statistics (m/s) of wave propagation speed estimation SSH Error STD = 10 cm Mean

SSH Error STD = 20 cm STD RMSE Mean

STD RMSE

#21,401 −0.43

13.1

13.1

1.2

21.6

21.7

#21,413 −0.13

12.9

12.9

0.4

19.7

19.7

#21,418 0.12

12.0

12.0

0.3

11.6

11.6

Figure 10.44 shows the corresponding results of the propagation speed estimation using the median based method. Except for a few relatively large errors, the speed estimates are basically around the ground truth (0.2 km/s) with an error within ± 0.05 km/s. Table 10.11 shows the error statistics of the direction estimation in terms of mean, STD and RMS of error, while Table 10.12 shows the corresponding wave propagation speed estimation results. The average of the RMSE of direction estimation is 4.4 deg and 5.9 deg when the SSH error STD is 10 cm and 20 cm respectively. Meanwhile, the average of the RMSE of speed estimation is 12.7 m/s and 17.7 m/s respectively.

10.4.2 Tsunami Wavelength Estimation 10.4.2.1

Theoretical Formulas

Tsunami wavelength can be retrieved using the estimates of the wave propagation direction and speed as well as the specular reflection track associated with a specific GNSS satellite and the LEO satellite of interest [9]. As shown in Fig. 10.45, the wave propagation direction is Northwest with an azimuth angle equal to θ , while the reflection track (going from A to D) has an azimuth angle equal to θ R calculated by (10.71). Assume that at time instant t A a complete cycle of the main wave is located between line 1 and line 3, and the specular scattering point crosses the main wave rear at point A. After a time interval of t D A , the specular scattering point crosses the main wave front at point D (wave starting point) and then moves out of the main wave.

10.4 Tsunami Parameter Estimation

363

That is, at time t A + t D A the main wave is located between line 2 and line 4. Since the included angle between the wave propagation direction and the reflection track direction is acute, the wave shape is always expanded over the time axis and the wavelength or wave width is larger than the actual one as illustrated in Fig. 10.46. The solid waveform is part of the original Tsunami already shown in Fig. 10.40. Using the geometry produces L AD =

L w + vw t D A cos θinc

(10.79)

1

1

0.8

0.8

Relative Sea Level (m)

Wave Amplitude (m)

where L AD is the distance between the two surface points A and D calculated by (10.69), L w is the wavelength of the main wave, and θinc = θ R −θ is the included angle between the wave propagation direction and the reflection track moving direction. In this case the included angle is assumed to be acute. From (10.79), the wavelength of the wave when traveling over the region enclosing points A, B, C and D is calculated

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

GNSS-R based smoothed

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

0

200

400

600

800

-0.8

1000

0

200

400

600

800

1000

Distance (km)

Distance (km)

2

2

1.5

1.5

Relative Sea Level (m)

Wave Amplitude (m)

Fig. 10.40 (left) Original SSH measurements obtained at station #21413; (right) simulated SSH measurements (dashed) using real data and smoothed measurements (black solid) using a linear KF

1 0.5 0 -0.5 -1 -1.5

0

200

400

600

Distance (km)

800

1000

GNSS-R based smoothed

1 0.5 0 -0.5 -1 -1.5

0

200

400

600

800

1000

Distance (km)

Fig. 10.41 Left Original SSH measurements obtained at station #21,418; (right) simulated SSH measurements (dashed) using real data and smoothed measurements (black solid) using a linear KF

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10 Tsunami Detection and Parameter Estimation 30

Direction Estimation Error (deg)

20 10 0 -10 -20 -30 -40 -50 -60 -70

0

500

1000

1500

2000

Number of Sample Fig. 10.42 Median based wave propagation direction estimation error sequence 30 20

Direction Estimation Error (deg)

10 0 -10 -20 -30 -40 -50 -60 -70

0

500

1000

1500

2000

Number of Sample

Fig. 10.43 Averaging based wave propagation direction estimation error sequence

10.4 Tsunami Parameter Estimation

365

by L w = L AD cos θinc − vw t D A

(10.80)

As mentioned earlier, the Tsunami may just have one complete cycle in some cases, whereas several cycles may occur in other cases. In the presence of multiple cycles, it would be useful to determine the whole duration or width of the wave. The width can be determined in a similar way, but at time t A the whole wave is located between line 1 and line 3 and at time t A + t D A it is located between line 2 and line 4. Now, consider the case where the reflection track direction is reversed. That is, the specular scattering point first crosses the main wave front at point B at time instant t B . Then, after a time interval of tC B , the scattering point crosses the main wave rear at point C (wave ending point). In this case, the included angle between the wave propagation direction and the reflection track direction is obtuse. Thus, compared with the true wavelength, the measured wavelength in the absence of SSH estimation error can be either longer, shorter or exactly the same, depending on the included angle, the wave propagation speed and specular scattering point (or reflection track) moving speed. Figure 10.47 shows the case where the measured wavelength is shorter and thus the wave is compressed. Similarly, the Tsunami wavelength can be determined by L w = −L BC cos θinc + vw tC B

(10.81)

Speed Estimation Error (km/s)

0.1

0.05

0

-0.05

-0.1

-0.15 0

500

1000

Number of Sample

Fig. 10.44 Wave propagation speed estimation error sequence

1500

2000

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10 Tsunami Detection and Parameter Estimation

Fig. 10.45 Geometry of wave propagation direction and reflection track

Wave propagation direction

North Line 4 Line 3 Line 2

Reflection track D

Line 1

θR

C

B

A

East

θ

0.7

Original Measured

0.6

Wave Amplitude (m)

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 0

200

400

600

800

1000

1200

Distance (km)

Fig. 10.46 Illustration of the original Tsunami waveform and the expanded waveform observed by a satellite-borne sensor

where θinc = π +θ R −θ . It can be shown that whether the measured wave is expanded or compressed can be determined by L(vw , tC B , θinc ) < L w , L(vw , tC B , θinc ) = L w ,

wave shape is ex panded wave shape r emains the same

L(vw , tC B , θinc ) < L w ,

wave shape is compr essed

(10.82)

10.4 Tsunami Parameter Estimation

367

0.7

Original Measured

0.6

Wave Amplitude (m)

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 0

200

400

600

800

1000

Distance (km)

Fig. 10.47 Illustration of the original Tsunami waveform and the compressed waveform observed by a satellite-borne sensor

where L(vw , tC B , θinc ) =

vw tC B 1 + cos θinc

(10.83)

Equations (10.80) and (10.81) can be written in a unified form as L w = |L 1,2 cos θinc − vw tG F |

(10.84)

Considering parameter estimation errors, (10.84) can be written as Lˆ w = | Lˆ 1,2 cos θˆinc − vˆw tˆG F |

(10.85)

where tˆG F is the estimate of the time interval. In the case where multiple reflection tracks are available, multiple wavelength estimates can be produced in the same way when Tsunami wave travels at different regions. Simulation Results. In Sects. 10.3 and 10.4, wavelet denoisiing amd Kalman filtering were considered to smooth the SSH estimates so that wave height and wavelength estimation can be performed. However, it is only a special case where the waveform is simply corrupted by measurement noise, but neither expanded nor contracted. In practice, the wavelength measurement has to be corrected by use of the included angle between the wave propagation direction and the reflection track as well as the wave propagation

368

10 Tsunami Detection and Parameter Estimation 300

Wavelength Estimation Error (km)

200 100 0 -100 -200 -300 -400

0

500

1000

1500

2000

Number of Sample

Fig. 10.48 Wavelength estimation error sequence

Table 10.13 Error statistics (km) of wavelength estimation SSH Error STD = 10 cm Mean #21,401 −27.4

SSH Error STD = 20 cm STD RMSE Mean 44.6

52.3

−56.9

STD RMSE 70.5

90.6

#21,413 −28.9

52.7

60.1

−67.6

83.3

107.3

#21,418 −23.3

78.3

81.7

−75.5

88.6

116.4

speed and the specular scattering point moving speed. Wavelength estimation error comes from four error sources, original wavelength measurement error, wave direction estimation error, wave speed estimation error, and time interval estimation error. Figure 10.48 shows the estimation error when the SSH error STD equals 10 cm and the waveform observed at station #21,418 is used. Table 10.13 shows the wavelength estimation error statistics when using three waveforms and two SSH error STDs. Compared with the results shown in Table 10.10, it can be seen that the error is slightly increased since more error sources are involved.

10.5 Summary This chapter studied Tsunami detection and parameter estimation based on the simulation of sea surface height estimated by spaceborne GNSS-R. DDM is the main observation data of GNSS-R receiver carried by a LEO satellite such as TDS-1,

10.5 Summary

369

CYGNSS and Bufeng-1 satellites. It is useful to fully make use of the large amount of DDM data for Tsunami detection and parameter estimation. A number of researchers produced some promising preliminary results in the past few years [8]. Although investigation on GNSS-R based Tsunami detection started more than 10 years ago [6, 7], all the results using SSH and DDM for Tsunami detection up to now are produced by simulations. There are very few or no reports on the use of real GNSSR satellite data to detect real Tsunamis. Thus, further studies are vital to evaluate the feasibility of GNSS-R on Tsunami detection using real GNSS-R altimetric data. Much more efforts are required to make GNSS-R based Tsunami detection practical.

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