Natural Electromagnetic Fields in Pure and Applied Geophysics (Springer Geophysics) 3030380963, 9783030380960

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Springer Geophysics

Kalyan Kumar Roy

Natural Electromagnetic Fields in Pure and Applied Geophysics

Springer Geophysics

The Springer Geophysics series seeks to publish a broad portfolio of scientific books, aiming at researchers, students, and everyone interested in geophysics. The series includes peer-reviewed monographs, edited volumes, textbooks, and conference proceedings. It covers the entire research area including, but not limited to, applied geophysics, computational geophysics, electrical and electromagnetic geophysics, geodesy, geodynamics, geomagnetism, gravity, lithosphere research, paleomagnetism, planetology, tectonophysics, thermal geophysics, and seismology.

More information about this series at http://www.springer.com/series/10173

Kalyan Kumar Roy

Natural Electromagnetic Fields in Pure and Applied Geophysics

123

Kalyan Kumar Roy Department of Earth Sciences Indian Institute of Engineering Science and Technology (IIEST) Howrah, West Bengal, India

ISSN 2364-9119 ISSN 2364-9127 (electronic) Springer Geophysics ISBN 978-3-030-38096-0 ISBN 978-3-030-38097-7 (eBook) https://doi.org/10.1007/978-3-030-38097-7 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Sponsored by Earth System Sciences, Department of Science and Technology (DST), New Mehrauli Road, New Delhi—110016.

Dedicated to: (i) Prof. V. K. Gaur, Distinguished Professor, I.I.A.P., Bangalore (ii) Prof. S. K. Sen, Distinguished Professor, I.I.T. Kharagpur (iii) Late Prof. K. Naha, Distinguished Professor, I.I.T. Kharagpur (iv) Late Prof. A. K. Saha, Distinguished Professor, Presidency University, Kolkata

Preface

Four professors, Prof. V. K. Gaur, Prof. S. K. Sen, the late Prof. K. Naha, and the late Prof. A. K. Saha, requested the author to join them in their endeavour for crustal studies in and around the Singhbhum Archaean Craton in eastern India—using magnetotellurics as one of the geophysical tools. Accordingly, the Earth System Sciences, Department of Science and Technology (DST), New Delhi, sponsored research projects along with one set of magnetotelluric-field equipment in 1984. Quite a few MT traverses, along with deep DC resistivity dipole–dipole soundings, were taken in various parts of the Singhbhum Orissa Iron Ore Craton and Eastern Ghat Proterozoic Mobile Belt with German-manufactured MMS02E equipment. The other geophysical data available for consultation were the gravity data of the late Prof. R. K. Verma, retired Professor of Geophysics, Indian School of Mines, Dhanbad, and deep-seismic sounding data across Mahanadi graben from the late Dr. K. L. Kaila and his coworkers at the National Geophysical Research Institute Hyderabad India. After retirement from IIT in Kharagpur, India, the author thought of writing a monograph primarily meant for undergraduate and postgraduate and research students based on this work done under DST-sponsored projects on magnetotellurics. After more than a decade of classroom teaching a course on ‘Electrical Methods in Geophysics’ and research experience with related electrical methods, the author finally planned to write a book where all aspects of applications of earth’s natural electromagnetic field in solving geophysical problems would come under one umbrella, based on the work of many other scientists in the MT community. Accordingly, DST approved the book proposal. But because of author’s severe health problems, there was considerable delay in completion of the manuscript. Although this book is written for postgraduates and research students of Indian universities, senior and advanced readers and researchers may find at least some points of interest to them. Very significant developments in (i) theory, (ii) instrumentation, (iii) data processing software, and (iv) inversion software have increased the trustworthiness of these low-resolution passive electromagnetic tools. These tools have now become very important scientific instruments to study at least as far as the earth’s upper and lower mantle boundary. Geophysicists from academic and industrial circles keep magnetotellurics and geomagnetic depth sounding as the ix

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items for their programmes on crust–mantle studies and oil exploration. Higher frequency components of these natural signals have applications in groundwater, mineral, and geothermal exploration. The author is grateful to Dr. Nandini Nagarajan and Dr. Kumerendra Mallick, retired scientist of the National Geophysical Research Institute, Hyderabad, India, who kindly agreed to go through the entire manuscript and then suggested some improvements. Dr. Sudha Agrahari, Assistant Professor, Department of Geology and Geophysics, IIT, Kharagpur has helped the author by providing many useful references from the central library of IIT Kharagpur, reviewed Chap. 9 of the book, and suggested some minor changes. Her contribution is gratefully acknowledged. The author is grateful to his elder daughter Dr. Baishali Roy, Conoco Phillips Company, Houston, Texas, USA, who provided the opportunity for the author to use Rice University’s Fondren Library, Houston, Texas, for an extended period. She purchased many books for consultation. The author is extremely grateful to the authorities of the Fondren Library for their all help to an outsider, including the borrowing books and Ph.D. theses from other university libraries. The author is grateful to his younger daughter Dr. Debanjali Roy, who made many research papers available from the library of the University of Miami, Florida, USA. The author is grateful to his student Dr. Kajal Kumar Mukherjee, Senior Geophysicist, Geological Survey of India, Kolkata, who borrowed many books and journals from the library of the Geological Survey of India, Kolkata for the present work. The author is grateful to Dr. Ranjit Kumar Majumdar, retired Professor of Jadavpur University, Kolkata, who helped the author by issuing books from the Jadavpur University Library, Kolkata. The author is grateful to Dr. Arkoprovo Biswas, ex-Research Scholar of the Department of Geology and Geophysics, IIT, Kharagpur, and Assistant Professor, Geophysics, Banaras Hindu University, Banaras, India, who provided many books and journals from IIT, Kharagpur, Central Library. Some results of magnetotelluric survey done in the department were the combined efforts of my research students Dr. C. K. Rao; Dr. Ajay Kishore Singh, Senior Scientist, Indian Institute of Geomagnetism, Mumbai, India; Dr. S. Srivastava, AFPRO, GIT, Ranchi, Jharkhand, India; and Dr. Suman Dey, Genpact, Gurgaon, Haryana, India. The author gratefully acknowledges their contributions. Dr. C. K. Rao helped the author providing some useful literature on magnetotellurics. I want to specially thank all my Ph.D. students for their contributions in the department’s academic programmes. The author gratefully remembers the teaching of Late Prof. P. K. Bhattacharyya, who introduced the subjects of electrotellurics, magnetotellurics, and geomagnetic depth sounding to the author and his classmates as early as 1962–1963. The author benefitted from discussions of some points on electrotellurics and magnetotellurics with Dr. A. Adam of the Hungarian Academy of Sciences, Sopron, in the early 1970s and in 2003. The author had the opportunity to interact with some of the stalwarts of geophysics, e.g. Dr. T. Madden of MIT, USA; Dr. S. H. Ward, University of Utah, USA; Dr. J. R. Wait, University of Arizona, USA; and Dr. L. Vanyan, Moscow, Russia, on certain points related to the present topics.

Preface

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The Department of Science And Technology, New Delhi, funded the purchase of the magnetotelluric equipment from METRONIX, MMS02E in Braunschweig, Germany. Their engineers were very helpful to assure that the equipment was working properly in the field, visiting IIT Kharagpur, at least three times. That is gratefully acknowledged. The author is immensely grateful to the authorities of the Geological Survey of India, Kolkata for sanctioning the joint field-oriented projects for two consecutive years and permitting their officers Dr. L. K. Das, Dr. D. C. Naskar, Dr. K. K. Mukherjee, Mr. H. Das, and others to be participating officers in the major field programmes. The author is grateful to the Director General of Geological Survey of India, Kolkata, for permitting many GSI technical staffs to join hands with the field crew of IIT Kharagpur, to give logistic support in these efforts of intense and joint fieldwork. The author is grateful to the supporting staff of the Department of Geology and Geophysics, IIT Kharagpur, who helped us in many ways to conduct the magnetotelluric fieldwork in Singhbhum Archaean Craton and Proterozoic Eastern Ghat Mobile Belt (EGMB) in eastern India. I am thankful to my wife for her infinite patience while encouraging me to complete this work, and she was extra solicitous about my health. Prof. Kalyan Kumar Roy Retired Professor Department of Geology and Geophysics Indian Institute of Technology, West Bengal, India Present Address Visiting Professor Department of Earth Sciences Indian Institute of Engineering Science and Technology (IIEST) Shibpur, Howrah, West Bengal, India Contacts for Communication Prof. Kalyan Kumar Roy Tollygunge, Kolkata, India

Contents

1

General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Preliminaries on Electromagnetic Waves and Their Application in Geophysical Investigations . . . . . . . . . . 1.3 Geomagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Magnetic Field of Internal Origin . . . . . . . . 1.3.2 Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . 1.3.3 Nondipole Field of Internal Origin . . . . . . . . 1.3.4 Inclination and Declination of the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Nondipole Time Varying Magnetic Field of External Origin . . . . . . . . . . . . . . . . . . . . . . 1.4 Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Solar Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Sunspot Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Solar Quiet Day (Sq) Variations . . . . . . . . . . . . . . . . . 1.8 L Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Equatorial ElectroJet (EEJ) and Polar Electrojet (PEJ) . 1.10 D, Dst and DS Variations . . . . . . . . . . . . . . . . . . . . . 1.11 Solar Flare Effect (SFE) . . . . . . . . . . . . . . . . . . . . . . 1.12 Magnetic Storms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Bay Type Variations . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Magnetic Substorms . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Interaction Between the Sun and the Earth . . . . . . . . . 1.16 Magnetosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Van Allen Radiation Belt . . . . . . . . . . . . . . . . . . . . . 1.19 Ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.20 Ring Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.21 1.22 1.23

Magnetotail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomagnetic Field Variations . . . . . . . . . . . . . . . . . . . . . Classifications and Causes of the Various Pulsations and Micropulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.23.1 Classification by Jacobs and Sinno (1960) . . . . . 1.23.2 Classifications by Madam Troitskaya (1960) . . . 1.23.3 Classification by Benioff’s (1960) . . . . . . . . . . . 1.23.4 Classification by Tepley and Wentworth (1962) . 1.23.5 Classification by Vladimirov and Kleimenova (1962) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.24 High-Frequency Natural Electromagnetic Signals, Spherics 1.25 Dead Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.26 Complex Structure of Natural Source . . . . . . . . . . . . . . . . 1.27 Earth’s Natural Electromagnetic Field as a Subject . . . . . . 1.27.1 Electrotelluric Method (T) . . . . . . . . . . . . . . . . . 1.27.2 Magnetotelluric Method (MT) . . . . . . . . . . . . . . 1.27.3 Geomagnetic Depth Sounding (GDS) . . . . . . . . . 1.27.4 Magnetometer Array Studies (MA) . . . . . . . . . . 1.27.5 Magnetovariational Sounding (MVS) . . . . . . . . . 1.27.6 Audiofrequency Magnetotelluric Method (AMT) 1.27.7 Sea-Floor Magnetotelluric Method (SFMT) . . . . 1.27.8 Marine Magnetotellurics (MMT) . . . . . . . . . . . . 1.27.9 Audiofrequency Magnetic Method (AFMAG) . . . 1.28 Controlled Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28.1 Controlled-Source Audiofrequency Magnetotelluric Methods (CSAMT) . . . . . . . . . . 1.28.2 Controlled-Source Marine Electromagnetics (CSEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28.3 Long-Offset Electromagnetic Transients (LOTEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28.4 Radio Magnetotellurics (RMT) . . . . . . . . . . . . . 1.29 Coverage of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Electrical Conduction in Rocks . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Expression of Electrical Conductivity for an Homogenous and Isotropic Medium Due to a Point Source of Current . . . . . . . . . . . . . . 2.2.2 Specific Resistivity or Conductivity . . . . . . . . . 2.2.3 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . .

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Electrical Permittivity and Displacement Current . . . . . . . . 2.3.1 Dielectric Constant . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Electric Displacement w and the Displacement Vector D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Tensor Electrical Permittivity . . . . . . . . . . . . . . . Magnetic Induction and Magnetic Permeability . . . . . . . . . 2.4.1 Magnetic Induction . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Magnetic Permeability . . . . . . . . . . . . . . . . . . . . Principal Methods of Electrical Conduction . . . . . . . . . . . 2.5.1 Electronic Conduction (Conduction of Current Through Metals) . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Conduction of Current Through Semiconductors . 2.5.3 Conduction of Current Through Solid Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Conduction of Current Through Dielectric Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Electrolytic or Ionic Conduction . . . . . . . . . . . . Factors that Control the Electrical Conductivity of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Porosity of Rocks . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Conductivity of Pore Fluids . . . . . . . . . . . . . . . . 2.6.3 Size and Shape of Pore Spaces . . . . . . . . . . . . . 2.6.4 Conductivity of Mineral Inclusions . . . . . . . . . . 2.6.5 Size and Shape of Mineral Grains . . . . . . . . . . . 2.6.6 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.7 Frequency of Excitation Current . . . . . . . . . . . . 2.6.8 Ductility and Degree of Partial Melt in Rocks . . 2.6.9 Electrical Conductivity of Various Types of Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.10 Chemical Activity and Oxygen Fugacity . . . . . . 2.6.11 Dependence of Electrical Conductivity on Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.12 Dependence of Electrical Conductivity on Volatiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.13 Major Geological Zones of Weaknesses . . . . . . . Piezoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell’s Geoelectrical Conductivity Models . . . . . . . . . 2.9.1 Soft Rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Hard Rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3 Ellipsoidal Grains . . . . . . . . . . . . . . . . . . . . . . . 2.9.4 Alternating-Current Conduction . . . . . . . . . . . . .

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2.10 Resistivities of Metals and Metallic Minerals . . . . . . . . . 2.11 Semiconducting Minerals . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Electrical Conductivity of Some Common Metallic Ores . 2.13 Some Common Geological Good and Bad Conductors . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Signal 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22

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Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Block Size . . . . . . . . . . . . . . . . . . . . . . . . . Editing of Time Series . . . . . . . . . . . . . . . . . . . . . . . . . Moving Average Algorithm . . . . . . . . . . . . . . . . . . . . . . Trend Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Series for Discrete Time-Period Signals . . . . . . . Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sinc Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Dimensional Fourier Transform . . . . . . . . . . . . . . . Aperiodic Function and Fourier Integral . . . . . . . . . . . . . Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . Shannon’s Sampling Theorem . . . . . . . . . . . . . . . . . . . . Linear Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filters and Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-Correlation and Autocorrelation . . . . . . . . . . . . . . 3.22.1 Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . 3.22.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . 3.22.3 Properties of the Autocorrelation and Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . Autopower and Cross-Power Spectra . . . . . . . . . . . . . . . 3.23.1 Energy-Density Spectrum of Aperiodic Signals 3.23.2 Power-Density Spectrum of Periodic Signals . . 3.23.3 Autopower Spectra . . . . . . . . . . . . . . . . . . . . . 3.23.4 Cross-Power Spectra . . . . . . . . . . . . . . . . . . . . Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.24.1 General Definition . . . . . . . . . . . . . . . . . . . . . 3.24.2 Geophysical Noise . . . . . . . . . . . . . . . . . . . . . 3.24.3 Geological Noise . . . . . . . . . . . . . . . . . . . . . . 3.24.4 Coherent Noise . . . . . . . . . . . . . . . . . . . . . . . .

Contents

3.24.5 Incoherent Noise . . . . . . . . . . . . . . . . . . . 3.24.6 Correlated Noise and Uncorrelated Noise . 3.24.7 White Noise . . . . . . . . . . . . . . . . . . . . . . 3.24.8 Man-Made Noise . . . . . . . . . . . . . . . . . . 3.24.9 Natural Noise . . . . . . . . . . . . . . . . . . . . . 3.24.10 Sensor Noise . . . . . . . . . . . . . . . . . . . . . 3.24.11 An Example of Noise Power . . . . . . . . . . 3.25 Robust Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.25.2 Median . . . . . . . . . . . . . . . . . . . . . . . . . 3.25.3 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.25.4 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . 3.25.5 Point Defects . . . . . . . . . . . . . . . . . . . . . 3.25.6 Nonstationarity . . . . . . . . . . . . . . . . . . . . 3.25.7 Quantile . . . . . . . . . . . . . . . . . . . . . . . . . 3.25.8 Breakdown Point . . . . . . . . . . . . . . . . . . 3.25.9 Non-Gaussian Distribution . . . . . . . . . . . 3.26 Robust Processing . . . . . . . . . . . . . . . . . . . . . . . . . 3.26.1 Downweighting of Outliers . . . . . . . . . . . 3.26.2 M-Estimator . . . . . . . . . . . . . . . . . . . . . . 3.26.3 Variable Weight M-Estimator . . . . . . . . . 3.26.4 Siegel’s Repeated Median Estimator (Siegel 1982; Smirnov 2003) . . . . . . . . . . 3.26.5 Field Example . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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128 128 129 129 129 130 130 131 131 133 133 134 135 135 135 136 138 138 138 139 140

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Electrotellurics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Basics of Electrotellurics . . . . . . . . . . . . . . . . . . . . . 4.3 Comparison of Electrotelluric and Magnetotelluric Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Nature of the Telluric Field . . . . . . . . . . . . . . . . . . . 4.5 Electrotelluric Method . . . . . . . . . . . . . . . . . . . . . . . 4.6 Potential Measuring Probes . . . . . . . . . . . . . . . . . . . 4.6.1 Electrode Potential . . . . . . . . . . . . . . . . . . 4.6.2 Non-polarisable Electrodes . . . . . . . . . . . . 4.7 Field Recording . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Electrotelluric Data Analysis . . . . . . . . . . . . . . . . . . 4.8.1 Interconnections Between Base and Mobile Station Telluric Field Vectors . . . . . . . . . . 4.8.2 Relative Ellipse Method . . . . . . . . . . . . . . 4.8.3 Triangle Method . . . . . . . . . . . . . . . . . . . . 4.8.4 Polygon Method . . . . . . . . . . . . . . . . . . . .

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4.8.5 Amplitude Ratio Method (Linear Polarisation) 4.8.6 Absolute Ellipse Method . . . . . . . . . . . . . . . . 4.9 Electrotelluric Boundary Value Problem . . . . . . . . . . . . 4.9.1 Electrotelluric Field over an Asymmetric Anticline . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Electrotelluric Profile Across a Step Fault . . . 4.10 Downward Continuation of Telluric Field Data . . . . . . . 4.11 Field Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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179 186 188 191 195 196

Magnetotellurics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Plane-Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Advancing Electromagnetic Wave . . . . . . . . . . . . 5.2.2 Plane-Wave Incidence on the Surface of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Skin Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Magnetotellurics for 1D Layered Earth: A Few Aspects of the Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Magnetotelluric Four-Layered Apparent Resistivity and Phase Curves . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Magnetotellurics Is a Low-Resolution Tool . . . . . 5.4.3 For a Certain Class of 1D Models, MT Fails to Resolve the Significant Subsurface Resistivity Contrasts Even Approximately When the Resistivity Contrast Is More Than Ten . . . . . . . . . 5.4.4 Magnetotelluric Signals Can See a Target that Is at a Depth Beyond Its Skin Depth . . . . . . . . . . 5.4.5 Granite Window Is a Must for Deep Magnetotelluric Surveys Because Two-km Thick Conducting Sediments on Top Can Reduce the Sensitivity of the Magnetotelluric Signals up to 300 km from the Surface and Deep Inside the Upper Mantle . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 Magnetotellurics Is a Suitable Geophysical Tool for Detecting Sediments Sandwiched Between Flood Basalt and Crystalline Basement . . . . . . . . . . . . . 5.5 Magnetotelluric Field Work and Field Data . . . . . . . . . . . . 5.5.1 Field Data Acquisition . . . . . . . . . . . . . . . . . . . . 5.5.2 Signal Strength Versus Frequency or Period . . . . . 5.5.3 Number of Degrees of Freedom Versus Frequency or Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.5.4 5.5.5

5.6 5.7 5.8

5.9 5.10

5.11 5.12

5.13 5.14

5.15 5.16 5.17 5.18

Coherencies . . . . . . . . . . . . . . . . . . . . . . . . . . . Different Components of the Impedance Tensors Versus Period . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 Processed Fourier Spectra . . . . . . . . . . . . . . . . . 5.5.7 Processed Field Data . . . . . . . . . . . . . . . . . . . . . Concept of Optimum Mathematical Rotation in Magnetotellurics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept of E and H Polarisation (TE and TM Mode) . . . . Estimation of MT Tensor Components . . . . . . . . . . . . . . . 5.8.1 Estimation of MT Tensors Using Coherencies . . 5.8.2 Estimation of MT Impedance Tensors Using Single-Station MT Data . . . . . . . . . . . . . . . . . . 5.8.3 Remote Reference Magnetotellurics . . . . . . . . . . 5.8.4 Robust Estimator . . . . . . . . . . . . . . . . . . . . . . . TE and TM Mode MT . . . . . . . . . . . . . . . . . . . . . . . . . . MT Tensor Decomposition . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Eggar’s Eigenstate Decomposition . . . . . . . . . . . 5.10.2 Bahr’s Tensor Decomposition . . . . . . . . . . . . . . 5.10.3 Groom Bailey Decomposition . . . . . . . . . . . . . . 5.10.4 Groom and Bailey’s Twist and Shear . . . . . . . . . 5.10.5 Jone’s Decomposition . . . . . . . . . . . . . . . . . . . . Tipper Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotation Invariant Parameters in Magnetotellurics . . . . . . . 5.12.1 Field Apparent Resistivity Phase Curves Using Rotation Invariant Parameters . . . . . . . . . . . . . . Magnetotelluric Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13.1 Magnetotelluric Phase Tensor . . . . . . . . . . . . . . Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14.1 Anisotropy in Magnetotelluric Domain . . . . . . . . 5.14.2 Phase Splitting in Magnetotellurics . . . . . . . . . . 5.14.3 Magnetotelluric Phase Above 90° . . . . . . . . . . . Galvanic and Inductive Distortion . . . . . . . . . . . . . . . . . . Magnetotelluric Current Channelling . . . . . . . . . . . . . . . . Magnetotelluric Strikes . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionality Indicator . . . . . . . . . . . . . . . . . . . . . . . . . 5.18.1 One-Dimensional Structure . . . . . . . . . . . . . . . . 5.18.2 Two-Dimensional Structure . . . . . . . . . . . . . . . . 5.18.3 Three-Dimensional Structure . . . . . . . . . . . . . . . 5.18.4 Dimensionality Indicator from Phase . . . . . . . . . 5.18.5 Dimensionality Indicator from Eigenstate Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.18.6 Swift Skew as Dimensionality Indicator . . . . . . .

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5.18.7

5.19 5.20

5.21 5.22

5.23

Complex Domain Plot of the Impedance Tensor as a Dimensionality Indicator . . . . . . . . . . . . . . . 5.18.8 Impedance Ellipse as a Dimensionality Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Domain Plot of the Impedance Tensors and Rotational Invariance Tensors . . . . . . . . . . . . . . . . . . . Static Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20.1 Curve Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20.2 Statistical Averaging . . . . . . . . . . . . . . . . . . . . . . 5.20.3 EMAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20.4 Use of Auxiliary Tools . . . . . . . . . . . . . . . . . . . . 5.20.5 Use of Constraining Parameters . . . . . . . . . . . . . . 5.20.6 Use of Well Log Data . . . . . . . . . . . . . . . . . . . . . 5.20.7 Higher Current Dipole Length . . . . . . . . . . . . . . . 5.20.8 Static Shift-Free Magnetotelluric Parameters . . . . . Magnetotelluric Designs . . . . . . . . . . . . . . . . . . . . . . . . . . Location of the MT Study Area in Eastern Part of the Indian Subcontinents Where a Few Magnetotelluric Observations Were Taken for Qualitative and Semi-Quantitative to Quantitative Interpretations . . . . . . . . . . . . . . . . . . . . . . . . Qualitative Signatures: An Important Sector of Magnetotelluric Data Interpretation . . . . . . . . . . . . . . . . . . 5.23.1 Qualitative Signature of a Rift Valley or Major Continental Fracture . . . . . . . . . . . . . . . . . . . . . . 5.23.2 UD (Phi Determinant) Pseudosection Can Depict the Subsurface with Greater Clarity (Ranganayaki 1984) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23.3 Qualitative Magnetotelluric Signatures of Faults . . 5.23.4 Qualitative Magnetotelluric Signatures of Sukinda Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23.5 Pseudo 3D Pseudosections of Rotation-Invariant Phases Across the Sukinda Thrust . . . . . . . . . . . . 5.23.6 Some of the Rotation-Invariant Parameters Are Heavy-Weight Parameters . . . . . . . . . . . . . . . . . . 5.23.7 Various MT-Parameter Pseudosections from Field Data Across the Sukinda Thrust . . . . . . . . . . . . . 5.23.8 Qualitative Signatures in Bhar’s Telluric Vectors Across Sukinda Thrust . . . . . . . . . . . . . . . . . . . . 5.23.9 Induction Arrows Show the Major Fractured Zones in Archean–Proterozoic Collision Zone . . . 5.23.10 Rotation-Invariant Parameters Less Affected by Static Shift . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23.11 Profiles and Pseudosections from Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

Semi-Quantitative to Quantitative Signatures from MT Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.24.1 One Dimensional Inversion of Magnetotelluric Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.24.2 Two-Dimensional Inversion and 2D Model . . . 5.24.3 2D and Pseudo-3D Models of the Mahanadi Graben . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.25 Field Example of Processed Magnetotelluric Data . . . . . . 5.25.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.25.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.25.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.25.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.26 Application of MT in Earth Sciences . . . . . . . . . . . . . . . 5.26.1 Major Breaks in the Crust and Upper Mantle . . 5.26.2 Detectability of Moho . . . . . . . . . . . . . . . . . . . 5.26.3 MT for Estimating Asthenosphere Temperature, as Well as for Mapping High Heat Flow Areas 5.26.4 MT for Oil Exploration . . . . . . . . . . . . . . . . . . 5.26.5 MT for Mapping Convergent and Divergent Plate Margins . . . . . . . . . . . . . . . . . . . . . . . . . 5.26.6 MT for Earthquake Prediction . . . . . . . . . . . . . 5.26.7 MT Can Estimate Permafrost Thickness . . . . . . 5.26.8 MT for Groundwater Exploration . . . . . . . . . . . 5.26.9 Marine MT . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6

Auxiliary Tools for Magnetotellurics . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Audiofrequency Magnetotellurics (AMT) . . . . . . . . . . . . 6.2.1 Source Characteristics . . . . . . . . . . . . . . . . . . . 6.2.2 Nature of AMT Signals . . . . . . . . . . . . . . . . . . 6.2.3 Field Procedure . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Qualitative Interpretation . . . . . . . . . . . . . . . . . 6.2.5 Quantitative Interpretation . . . . . . . . . . . . . . . . 6.2.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Controlled-Source Audiofrequency Magnetotellurics (CSAMT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Skin Depth and Effective Penetration Depth . . . 6.3.3 CSAMT Pseudosections for Theoretical Model and Field Data . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 CSAMT Sources . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Field Survey . . . . . . . . . . . . . . . . . . . . . . . . . .

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333 333 334 334 337 338 339 344 344

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Contents

6.3.6 Interpretation . . . . . . . . . . . . . . . . . . . . Long-Offset Electromagnetic Transients (LOTEM) 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.4.2 LOTEM Data Acquisition . . . . . . . . . . . 6.4.3 LOTEM Theory (Strack 1984) . . . . . . . 6.4.4 Data Processing . . . . . . . . . . . . . . . . . . 6.4.5 Nature of Forward and Inverse Problem Responses of LOTEM Data . . . . . . . . . . 6.4.6 Applications . . . . . . . . . . . . . . . . . . . . . 6.5 Radiomagnetotellurics (RMT) . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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362 362 365 367

Geomagnetic Depth Sounding (GDS) . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Separation of External and Internal Field . . . . . . . . . . . . . 7.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Separation of Normal and Anomalous Fields . . . . . . . . . . 7.5 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Solution of Laplace Equation in Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 When Potential Is a Function of All Three Coordinates, i.e. Radial Distance, Polar Angle and Azimuthal Angle, i.e. / = f(r, h, w) . . . . . . 7.5.3 Associated Legendre Polynomial . . . . . . . . . . . . 7.5.4 Geomagnetic Potential . . . . . . . . . . . . . . . . . . . 7.5.5 Geomagnetic Field . . . . . . . . . . . . . . . . . . . . . . 7.6 Magnetometer Array Studies . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Recording of Geomagnetic Data . . . . . . . . . . . . 7.6.2 Examples of Magnetometer Arrays: Example from South Africa . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Example from India . . . . . . . . . . . . . . . . . . . . . 7.6.4 Magnetogram . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.5 Processing Geomagnetic Data . . . . . . . . . . . . . . 7.6.6 Single-Site Transfer Function . . . . . . . . . . . . . . 7.6.7 Hypothetical Event Analysis . . . . . . . . . . . . . . . 7.7 Induction Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Parkinson’s Arrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Wiese Arrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Schmucker’s Concept of Transfer Function and Induction Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Z/H Pseudosection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 Difference Induction Arrows . . . . . . . . . . . . . . . . . . . . . . 7.13 Complex Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.13.1

Definition and Significance of Complex Demodulation . . . . . . . . . . . . . . . . . . . . 7.13.2 Relationship to Power Spectra . . . . . . . . 7.13.3 Computational Procedure . . . . . . . . . . . 7.14 Geomagnetic Depth Sounding . . . . . . . . . . . . . . . 7.14.1 Approach A . . . . . . . . . . . . . . . . . . . . . 7.14.2 Approach B . . . . . . . . . . . . . . . . . . . . . 7.14.3 Approach C . . . . . . . . . . . . . . . . . . . . . 7.15 Audiofrequency Magnetic Method (AFMAG) . . . . 7.16 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9

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Marine Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Marine Magnetotellurics . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Marine Magnetotellurics for Solid-Earth Geophysics . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Marine Magnetotellurics (MMT) for Oil Exploration . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Marine Controlled-Source Electromagnetics (CSEM) . . 8.4 Field Examples of Variation of Electrical Conductivity with Depth in Marine Environments . . . . . . . . . . . . . . 8.5 Magnetometric Resistivity Method (MMR) . . . . . . . . . 8.6 MOSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Self Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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405 405 406 406 407 410 411 413 416 416

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Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Two- and Three-Dimensional Problems . . . . . . . . . . . . 9.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Concept of Virtual Work and the Energy Method in the Magnetotelluric Domain (Coggon’s Model) . . . . . . . . . . . . . . . . . . . . . 9.3.2 Formulation Steps . . . . . . . . . . . . . . . . . . . . . 9.3.3 Minimisation of the Integrals . . . . . . . . . . . . . 9.3.4 Galerkin’s Method in a Finite Element Magnetotelluric Domain . . . . . . . . . . . . . . . . 9.3.5 Finite Element Formulation for the Helmholtz Wave Equation . . . . . . . . . . . . . . . . . . . . . . . 9.3.6 Element Equations . . . . . . . . . . . . . . . . . . . . 9.3.7 TM-Mode Magnetotellurics . . . . . . . . . . . . . . 9.3.8 TE-Mode Magnetotellurics . . . . . . . . . . . . . .

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9.3.9 Global Matrix Formulation . . . . . . . . . . . . . . 9.3.10 Isoparametric Elements in Finite Elements . . . 9.4 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Some Mathematical Physics Preliminaries Related to Finite Difference Modeling in Magnetotellurics . . . . . . . . . . . . . . . . . . . . 9.4.2 Finite Difference Modeling . . . . . . . . . . . . . . 9.4.3 Fomenko and Mogi’s Finite Difference Model (2002) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Calculation of the H Field . . . . . . . . . . . . . . . 9.4.5 Boundary Conditions (Mackie et al. 1993) . . . 9.4.6 Boundary Conditions (Fomenko and Mogi’s Model 2002) . . . . . . . . . . . . . . . . . . . . . . . . 9.4.7 Preconditioning of Matrix . . . . . . . . . . . . . . . 9.4.8 Divergence H Correction . . . . . . . . . . . . . . . . 9.4.9 Divergence J Correction . . . . . . . . . . . . . . . . 9.4.10 Advantages in This Approach (Fomenko and Mogi 2002) . . . . . . . . . . . . . . . . . . . . . . 9.5 Integral Equation Method . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Integral Equation as a Mathematical Tool . . . 9.5.2 Formulation of an Electromagnetic Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Green’s Function in the Vector Potential Field Solution of Helmholtz Electromagnetic Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 Three-Dimensional Boundary Value Problem . 9.6 Thin-Sheet Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Ranganayaki and Madden’s Model (1980) . . . 9.6.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Hybrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Different Combinations . . . . . . . . . . . . . . . . . 9.7.3 Hybrid Formulation (Lee, Pridmore and Morrison) . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Inversion of Geophysical Data . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Basic Framework of Geophysical Data Inversion . . . . . . 10.2.1 Construction of an Inverse Problem Algorithm and Convergence . . . . . . . . . . . . . . . . . . . . . . 10.3 Tikhnov’s Regularisation Philosophy . . . . . . . . . . . . . . .

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10.4

10.5 10.6 10.7 10.8 10.9 10.10

10.11

10.12

10.13 10.14

10.15 10.16

10.17 10.18

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Fréchet Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Parker’s Definition . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Zdhanov’s Definition . . . . . . . . . . . . . . . . . . . . . Major Methodologies for Inversion . . . . . . . . . . . . . . . . . . Basis Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krylov Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . Least Squares and Weighted Least Squares Estimator . . . . . 10.10.1 Least Squares and Ridge Regression Estimator . . . 10.10.2 Weighted Least Squares and Weighted Ridge Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Backus–Gilbert Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.2 Backus-Gilbert Formulation . . . . . . . . . . . . . . . . . 10.11.3 Backus–Gilbert Fréchet Kernel (Oldenburg 1979) . . . . . . . . . . . . . . . . . . . . . . . . 10.11.4 Field Example . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Source Code for Inversion: One Field Example of Application Multiple Source Codes for Inversion of Same Data Set: Application of Marginal Probability Density Function for Model Parameter Estimation . . . . . . . . Two-Dimensional Inversion . . . . . . . . . . . . . . . . . . . . . . . . Occam Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.14.1 2D Occam Inversion Formulation (deGroot–Hedlin and Constable 1990) . . . . . . . . . . . . . . . . . . . . . . 10.14.2 REBOCC Inversion . . . . . . . . . . . . . . . . . . . . . . 10.14.3 REBOCC Formulation (Siripunvaraporn and Egbert 2000) . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of Steepest Descent . . . . . . . . . . . . . . . . . . . . . . . . Conjugate Gradient Method . . . . . . . . . . . . . . . . . . . . . . . . 10.16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.16.2 Important Steps in Conjugate Gradient Method . . 10.16.3 Conjugate Gradient Method as a Direct Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.16.4 The Conjugate Gradient Method as an Iterative Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.16.5 Computation of Alpha and Beta . . . . . . . . . . . . . Rapid Relaxation Inversion (RRI) (Smith and Booker 1991) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10.19 Joint Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 10.19.2 Joint Inversion of Magnetotellurics and DC Dipole-Dipole Resistivity Data (Sasaki 1989) . 10.19.3 Joint Inversion of Resistivity and Induced Polarisation Sounding Data (Roy 2014) . . . . . 10.19.4 Joint 2D Resistivity and Seismic Inversion . . . 10.19.5 Joint Inversion of Seismic Refraction and Magnetotelluric Sounding Data (Synthetic Model) . . . . . . . . . . . . . . . . . . . . . 10.20 Appraisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

Chapter 1

General Introduction

Abstract In this chapter, very brief scenarios of the complex interaction of solar radiations with the Earth’s magnetosphere are presented and the intricate origin of the (i) sferics, (ii) pulsations and micropulsations, (iii) variations of the extraterrestrial magnetic fields, (iv) Van Allen Radiation belts, (v) ionosphere, (vi) ring current, (vii) magnetopause, (viii) magnetic sheath, (ix) magnetotail and (x) magnetic storms are discussed. The various branches of geophysics, evolved because of this time varying natural source electromagnetics, are also discussed in reasonable detail. They are (i) telluric current methods (T), (ii) magnetotellurics (MT), (iii) audiofrequency magnetotellurics (AMT), (iv) geomagnetic depth sounding (GDS), (v) magnetometer array studies (MA), (vi) magneto-variational sounding (MVS), marine MT (MMT) and (vii) the audio-frequency magnetic method (AFMAG). A few controlled-source auxiliary tools that are used along with MT and AMT are: (i) controlled-source audio-frequency magnetotellurics (CSAMT), (ii) long/offset transient electromagnetics (LOTEM), (iii) marine controlled electromagnetic (CSEM) and radiomagnetotellurics (RMT). Brief discussions on mathematical modelling and inversion are included to touch upon the interpretational aspects of geophysical data. Keywords Magnetosphere · Geomagnetic field · Geophysical tools

1.1 Introduction In this monograph, the author mostly restricts his discussion to the application of Earth’s natural electromagnetic field in geophysics. A very limited number of topics on controlled sources are covered here. The nondipole time-varying geomagnetic field of extraterrestrial origin is the basis of this subject. The subject on ‘Electromagnetic Induction inside the Earth’ has developed at a very faster pace. Some of these geophysical tools are based on the measurement of both electrical and magnetic fields. Some are based on the measurement of only magnetic or electric fields. Therefore, an attempt has been made to bring the different aspects of this subject under one umbrella, keeping in mind the needs of postgraduate and research students. The topics covered are (i) the nature of the nondipole time-varying geomagnetic field of extraterrestrial origin, (ii) electrical conduction through rocks, (iii) electromagnetic signal © Springer Nature Switzerland AG 2020 K. K. Roy, Natural Electromagnetic Fields in Pure and Applied Geophysics, Springer Geophysics, https://doi.org/10.1007/978-3-030-38097-7_1

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1 General Introduction

processing, (iv) electrotellurics (T), (v) magnetotellurics (MT), (vi) Audiofrequency magnetotellurics (AMT), (vii) controlled-source audiofrequency magnetotellurics (CSAMT), (viii) long offset electromagnetic transients (LOTEM), (ix) radio magnetotellurics (RMT), (x) geomagnetic depth sounding (GDS), (xi) magnetometer array studies (MA), (xii) magneto variational sounding (MVS), (xiii) the audiofrequency magnetic method (AFMAG), (xiv) marine magnetotellurics (MMT), (xv) controlledsource marine electromagnetics (CSEM), (xvi) marine direct-current electrical methods (marine DC), (xiv) mathematical modelling in electromagnetics and (xv) Inversion of electromagnetic data. Depths of investigations of these tools range from a very shallow level (a few meters) to deep inside the upper and lower mantle using signals from the sunspot cycle of eleven-year periodicity. These tools are used for (i) groundwater, (ii) petroleum, (iii) mineral, (iv) geothermal exploration on one hand and for studying crust-mantle heterogeneities on the other. The major energy supplier for all these activities is the Sun from the outside and the considerable amount of energy coming from the partially molten mantle through the mantle convection which keeps the geodynamic process alive. Fortunately for geophysicists, electrical conductivity is the most sensitive geophysical parameter, and its extreme values range from 108 to 10−5 m. Sharp changes in electrical resistivity or conductivity make the detection of subsurface complexity possible to a certain extent. The next greatest natural advantage in favour of geophysicists is the existence of the very wide frequency spectrum of the Earth’s natural electromagnetic field in nature. That makes deeper investigation inside the Earth possible through magnetotelluric and geomagnetic depth sounding. The third greatest natural advantage in favour of geophysicists is the longer period signals, which are meant for deeper investigation, have more energy and power. There is one natural disadvantage also. The temperature of the Earth increases with depth: The higher the temperature, the higher will be the electrical conductivity and the attenuation of an electromagnetic signal as it propagates downward through the Earth’s formation. As a result, higher frequency signals with better resolving power die down quickly. Only relatively longer period signals with poorer resolving power penetrate deep inside the Earth. In spite of this limitation, MT and GDS have become the most used and not so expensive geophysical tools for deeper probing inside the earth.

1.2 Preliminaries on Electromagnetic Waves and Their Application in Geophysical Investigations Magnetic fields originate when an electrical charge moves. Electromagnetic fields originate when the current due to the flow of charge is time varying. Electromagnetic waves start propagating in all directions with the speed of light. In the words of Feynman et al. (1964): “A caterpillar turns into a butterfly”. When an alternating current flows through any good electrical conductor, e.g. a wire loop, a linear conductor or through any other medium like the ionosphere in the upper atmosphere,

1.2 Preliminaries on Electromagnetic Waves and Their Application …

3

electromagnetic waves originate. These conductors are transmitters, i.e. the sources of electromagnetic waves. Electromagnetic waves are transverse waves, i.e. the direction of vibration of particles in a medium, through which the wave travels, is at right angles to the direction of propagation of the wave. Electric and magnetic field vectors of the EM waves are also mutually perpendicular (Fig. 1.1). Electromagnetic (EM) waves can travel through any medium like air, gas and a vacuum, but not through a perfect conductor. Electromagnetic waves attenuate rapidly in a good conductor because the eddy currents are stronger and the opposing EMF is stronger and attenuates the signal. In a medium of finite conductivity, say, for example, Earth’s sediment covers, crust and upper mantle, the velocity of EM waves is significantly retarded by several orders of magnitude.

Fig. 1.1 Transverse nature of electromagnetic wavelets orthogonal to the direction of propagation; mutually orthogonal electric and magnetic field components

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1 General Introduction

The broad spectrum of electromagnetic waves ranges from 1023 Hz to cycles/per 11 years (sunspot cycle). 16 kHz to sunspot cycle is the range for geophysicists. All the following applications work within this frequency range: audiofrequency magnetotellurics (AMT), magnetotellurics (MT), geomagnetic depth sounding (GDS), magnetometer array studies (MA), (magneto-variational sounding (MVS), (marine magnetotellurics) (MMT), electrotemagnetotellurics (CSAMT) and controlled-source electromagnetics (CSEM). The only high-frequency applications working beyond this range are (i) ground-penetrating radar (GPR), (ii) electromagnetic propagation tool (EPT) in borehole geophysics and (iii) gamma rays both in surface and borehole geophysics. The depth of penetration of electromagnetic waves is controlled by skin depth (to be defined later): the lower the frequency of EM waves, the greater is the depth of penetration and the lower is the resolving power. Similarly, the lower the electrical conductivity of earth or any other conductive media, then the greater will be the depth of investigation or depth of penetration. For oil, mineral and groundwater exploration, one needs a penetration depth of only a few meters to a few kilometers. In solid-earth geophysics, on the other hand, one needs penetration depth of EM waves as great as the radius of the Earth. Electromagnetic signals of longer and longer period, which are usually called variations, enter deep inside the Earth with poorer and poorer resolving power. Fortunately, the energy level of the Earth’s natural electromagnetic field is very high (on the order of 1023 ergs). The higher the period of the signal then the higher will be the energy level of the signal although it does not strictly follow this linear trend because of presence of more than one dead band in the frequency spectrum. Further discussion on this point is found in Chap. 5. In summary, this makes the magnetotelluric and magneto-variational sounding possible to study the Earth’s crust and upper mantle. As mentioned, most of the geophysical EM jobs are done in the audiofrequency range and subaudiofrequency range with very low frequency variations. Hence, for most geophysical problems, conduction currents dominate displacement current. It is negligible in this frequency range. The EM wave originates from Earth’s natural electromagnetic field, propagating through the Earth in a diffusive manner and not as a wave. Hence, conduction current and diffusion of EM waves through the Earth are important to geophysicists. Highly conducting zones offer greater opposition to the propagation of electromagnetic waves. As a result, high-frequency signals with greater resolving power die down quickly. In ocean-floor geophysics most of the high-frequency signals are attenuated by a three to four-km thick saline-water column. Therefore to study the oceanic crust and mantle, the starting frequency is of the order of cycles per 12 h and may reach as long as cycles per eleven years sunspot cycles. A series of researcher such as Filloux (1973, 1974, 1977), Constable (1990), Constable et al. (1998), Constable and Cox (1996) and Utada (2015) among others have developed the subject of marine EM. They developed the tools to measure the electrical conductivity of the oceanic crust and upper mantle. The frequency or period range for this sea floor magnetotellurics varies from cycles per 12 h to cycles per eleven years. Very recently, Constable (1990), Constable and Cox (1996) and

1.2 Preliminaries on Electromagnetic Waves and Their Application …

5

Constable et al. (1998) developed a new technology for studying the possibility of detection of hydrocarbons on shallow-to-deep ocean floors using magnetotellurics. Technology has been developed to amplify very feeble signals of a few or a fraction of a microvolt in a noise-free marine environment and to use them for mapping the seabeds suitable for accumulation of oil.

1.3 Geomagnetic Fields Detailed research work on the history of development of geomagnetism and palaeomagnetism were published by Marrill and McElhinny (1983). In this brief discussion, the excerpts to be highlighted from their work are: (i) People knew about geomagnetism as early as the 6th century B.C.; (ii) the earliest magnetic compass appeared in China as early as the 1st century A.D.; (iii) the first observations on magnetic declination were made in China in 720 A.D.; (iv) magnetic inclination was discovered by George Hartmann in 1544; (v) Henry Gellibrand first discovered the variation of declination of the Earth’s magnetic field; (vi) in 1546 Gerhard Mercator first realized that Earth’s magnetic pole lies on the surface of the earth, and he could fix these poles; (vii) Alexander von Humboltd made the first global magnetic survey and established that the intensity of the magnetic field varies with latitude; the field is strongest at the pole and weakest at the Equator; (viii) In 1600 Willium Gilbert first proposed that Earth as a whole acts like a big magnet; (ix) in 1838 Gauss first proposed the mathematical formula of the Earth’s magnetic field. He could pinpoint the position of the geomagnetic poles. These positions are the best fitting dipoles cutting the surface of the Earth. William Gilbert (1540–1603) in his treatise ‘De magnet’ first mentioned the existence of the magnetic field of Earth and that its origin lies in the interior of the Earth. The stable geomagnetic field of global presence originates due to the dynamo current in the Earth’s core composed most probably of iron, nickel and some lighter elements. Both iron and nickel are good conductors. The observation regarding the rotation of the magnetic needle (load stone) along a definite direction led to the discovery of the Earth’s magnetic field. Magnetite ores could be mined even in the 16th and 17th century to record deflections via magnetite needles. Now it is understood that the geomagnetic field originates due to the dynamo current in the core of the Earth, and it can be approximated as a magnetic dipole. It is a reasonably steady field with very slow variations. Earth has magnetic north and south poles, as well as a magnetic equator.

1.3.1 Magnetic Field of Internal Origin The most important part of the magnetic field on the surface is the Earth’s dipole field (Figs. 1.2 and 1.3). The magnetic poles of the Earth are defined as the locations

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1 General Introduction

Fig. 1.2 Electric and magnetic dipoles (Lowrie 1997)

Fig. 1.3 Global contour of isoclinal lines of dipole origin (Lowrie 1997)

where the inclination of the magnetic field is ±90°, i.e. where the direction of the magnetic field is either vertically upward or downward. An isoclinals map of the world for the year 1980 shows that the exact location of the North Pole is 77.3° North and 258° East. But the location of the South Pole is 65.6° South and 139.4° East, i.e. the North and South Poles are not exactly antipodal or on the opposite side

1.3 Geomagnetic Fields

7

of the globe. Therefore the geomagnetic field is more complicated than a perfect dipole field (Lawrie 1997).

1.3.2 Magnetic Dipole Pertaining to the Earth’s magnetic field of dipole origin (see Fig. 1.2), the important parameters are twofold. The first is the composition of the core is likely to be made up of iron, nickel and some lighter elements. Geochemical study of the meteors and meteorites, seismic velocities at the inner and outer cores and other physical properties of the earth, such as density. support this view point. Secondly, the core with its viscosity, electrical conductivity and temperature, nearly 3000 °C at the liquid outer core supported the origin of the dipole field. At high temperature and pressure, core electrical conductivity is around 3–5 × 105 mho/m which is more or less of the same order of a good electrical conductor. The dipole field of the earth resembles that of an uniformly magnetised sphere.

1.3.3 Nondipole Field of Internal Origin The part of the field of internal origin, about 5% of the total field, obtained by subtracting the field due to inclined geocentric dipole from the total field is collectively known as the nondipole field (Fig. 1.4a, b). The non-dipole field is believed to originate because of interaction of the irregular core fluid flow due to rough surface of the core mantle boundary and the turbulence in the flow created due to that (Lowrie 1997).

1.3.4 Inclination and Declination of the Magnetic Field Figure 1.5 shows that the directions of the geographical north and the magnetic north are different. The angle between them is the declination D. The angle made by the total field with the horizontal plane is the inclination of the field I. Therefore, we can write T2 = H2 + Z2

(1.1)

where T is the total field, H and Z are respectively the horizontal and vertical component of the magnetic fields. I is the inclination angle made by the total field with the horizontal component. Here in Fig. 1.5

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1 General Introduction

Fig. 1.4 a, b Nature of global nondipole field of internal origin (Lowrie 1997)

H = T cos I

(1.2)

z = T sin I

(1.3)

tan I = Z/H

(1.4)

HX = H cos D

(1.5)

HY = H sin D

(1.6)

1.3 Geomagnetic Fields

9

Fig. 1.5 Vectors showing the inclination and declination of the total magnetic field

Isogonic, isoclinic and isodynamic maps are respectively the contour maps of equal declination, equal inclination and equal values of H or Z. Inclination of the Earth’s magnetic field is downward throughout the entire Northern Hemisphere and its inclination is upward throughout the Southern Hemisphere. The geomagnetic field in Fig. 1.5 is showing the magnetic lines of forces. It shows the geographic north, magnetic north inclination and declination of the total magnetic field. Magnetic poles are those where the magnetic field is vertical. Geomagnetic poles are the extension of the magnetic dipole axis on the surface of the earth.

1.3.5 Nondipole Time Varying Magnetic Field of External Origin The nondipole field of external origin due to interaction of the solar flares with the magnetosphere and the Ionosphere is a time and space variant geomagnetic vector field. On each point on the earth’s surface, as any other vector quantity, three components are measured to completely specify the geomagnetic field. One can record three components continuously as is done in any magnetic observatory or at discrete time instant. The former constitutes the time function and the discrete values form a time series. The geomagnetic field consists of the main field and the time varying oscillations. The period of the earth’s magnetic field of dipole origin is so large that, for all practical purposes, it can be considered as stable and uniform. Therefore, the time varying components are mostly of extraterrestrial origin.

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1.4 Solar Radiation The continuously emitted solar radiations in the form of electromagnetic waves and particles are the chief sources of energy entering into our atmosphere. The amount of energy received at the top of the atmosphere is affected by four factors: (i) solar output, (ii) the Sun–Earth distance, (iii) the altitude of the sun, and (iv) length of day. A part of the energy comes in the form of geothermal energy, responsible for mantle convection and lithospheric plate movements and a part comes from radioactive disintegration. In general equatorial and low latitude areas receive more solar heat than the polar regions. Heat received by the surface of the earth has strong latitude and altitude dependence.

1.5 Solar Energy Solar energy originates from nuclear reactions within the sun’s hot core (16 × 106 K), and is transmitted to the sun’s surface by radiation and hydrogen convection. Visible solar radiation (light) comes from a ‘cool’ (~6000 K) outer surface layer called the photosphere. Temperatures rise again in the outer chromospheres (10,000 K) and corona (106 K), which is continually expanding in space. The outflowing hot gases (plasma) from the sun, referred to as the solar wind (with a speed of 1.5 × 106 km/h), interact with the earth’s magnetic field and upper atmosphere. The energy emitted is proportional to the fourth power of the absolute temperature of the body (T). The total solar output to space, assuming a temperature of 5760 K for the sun, is 3.84 × 1026 W, but only a tiny fraction of this is intercepted by the earth, because the energy received is inversely proportional to the square of the solar distance (150 million km). Satellite measurements since 1980 indicate a value of about 1366 W/m2 . The wavelength range of solar (short-wave) radiation and the infra-red (longwave) radiations are emitted by the earth and atmosphere. For solar radiation, about 7% is ultraviolet (0.2–0.4 μm), 41% visible light (0.4–0.7 μm) and 52% near infra-red (>0.7 μm); (1 μm = 1 micrometre = 10−6 m) (Barry and Chorley 2003).

1.6 Sunspot Cycle Sunspots are dark (i.e. cooler) areas visible on the sun’s surface. Although sunspots are cool, bright areas of activity known as faculae, that have higher temperatures, surround them. The net effect is for solar output to vary in parallel with the number of sunspots. Magnetic storm activity invariably increases when the sun spots are numerous because solar flare activity is closely associated with the occurrence of sunspots. Hence the bursts of solar ion clouds sent earthwards can be expected to be more frequent in periods when sunspots are more numerous. Thus the eleven year

1.6 Sunspot Cycle

11

periodicity of the Earth’s current variations is explained. Sunspot number and position change in a regular manner, known as sunspot cycles. Satellite measurements during the latest cycle show a small decrease in solar output as sunspot number approached its minimum, and a subsequent recovery. Sunspot cycles have periods averaging 11 years (Fig. 1.6).

Fig. 1.6 Sunspot cycles (Barry and Chorley 2003)

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1.7 Solar Quiet Day (Sq ) Variations Solar quiet day variations are those where solar emissions, which are primarily responsible for variations of the magnetic field, are minimum. Geomagnetic field remains more or less stable for a few days at a stretch. These days are known as solar quiet days and these systematic variations are known as Sq variations. The variations are periodic over solar quiet days. Their magnitudes are dependent upon the season of the year and latitude and are also stronger in the summer than in the winter. These fields are found to be stronger in higher latitude than in the equatorial zone. Earth’s main field and the conducting ionosphere constitute the ionosphere dynamo and create current in the E-layer. The major part of the Sq variations comes from the ionospheric currents and their variations. The other 10% of the Sq variations come from the compression of the magnetosphere. This happens due to the interaction of the solar flares with the magnetosphere and solar winds cutting across the ionosphere. If the Sun remains quiet (weaker-level solar emissions), the geomagnetic field also remains quiet. These are periodic over the solar days, and they are dependent on latitude and the seasons of the year. Night-time variations are negligible. The variations are stronger at higher latitudes. The strength of these fields is about 20–100 nT. All permanent geomagnetic observatories globally are recording these variations. Sq has a strong correlation with sunspot cycle and therefore has a variation with a periodicity of eleven years or more. Alldredge (1977a, b), Bloxham and Gubbins (1985), Bloxham et al. (1989), Slaucitajis and Winch (1965), Campbell (1989) and Bannister and Gough (1978) have provided detailed descriptions of these secular variations of the Earth’s geomagnetic field. These research articles also treated the other variations of the geomagnetic field. The magnetic potentials of this field is obtained as a series of spherical harmonics Therefore, one method of study of the geomagnetic field is spherical harmonic analysis in which the potential of a particular magnetic field can be expressed by a series of spherical harmonic terms. The diurnal (S1 ), the semidiurnal (S2 ), eight-hour (S3 ) and six-hour (S4 ) components correspond respectively to Pm n where n = 2, 3, 4 and 5 and n − m = 1. Here P is Legender’s polynomial discussed in Chap. 7.

1.8 L Variations S. Chapman (1919) did the pioneering work on solar and lunar variations. His is still considered important contributions to this topic. Malin (1969, 1970) also studied this topic in some detail. Geomagnetic variations that associated with lunar times (one lunar day = 24.84 solar hours) are called L variations. These are much weaker variations, and the strength of this field is in the range of 1–10 nT. It is interesting to note that daytime L variations are stronger than night-time variations. At the magnetic observatory at Hyderabad, the general Sq values on an average are found to be of

1.8 L Variations

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the order of 50 nT when L variations have their magnitude 2 nT. Each phase has four major harmonics L1 , L2 , L3 and L4. Their origin, the same as that of Sq , has been attributed 90% to the ionospheric dynamo and 10% to the compression of the magnetosphere.

1.9 Equatorial ElectroJet (EEJ) and Polar Electrojet (PEJ) Chapman and Rajarao (1965), Sethia et al. (1980), Kane (1973), Iyer et al. (1976), Duncan (1960), Rastogi and Rajaram (1971), Rastogi et al. (1973) and many others have studied equatorial electrojets in great detail. Magnetic observatories near the magnetic equator record the daytime enhancement of Sq and L by a factor of two or more. The peak values are reached at noontime. The magnitude of the associated magnetic field are of the order of 100 nT for Sq and 10 nT for L. The enhancement of the current is along a narrow strip at an altitude of about 100 km asm. These are known as equatorial electrojets, and such an enhanced flow of current in the polar region is known as the polar electrojets (PEJ).

1.10 D, Dst and DS Variations Magnetic storms that are due the solar-particle radiation and are characterized by a decrease in the horizontal component, especially in low and mid-latitudes, are D-variations (introduced by S. Chapman). A typical magnetogram during the storm may show a few or all of the events. On the basis of their time dependence, geomagnetic storms can be divided into two categories: Dst (storm time component) and DS (local time component). Dst is often called DR since it is believed to be caused by ring currents. The entire storm is composed of several individual events designated polar substorms in auroral zones. These are nothing but the bay disturbances associated with the equatorial electrojet.

1.11 Solar Flare Effect (SFE) It has been established through rocket measurements that solar X-rays. Ionize the ionosphere, making it more conducting, thereby more absorbent of radio waves. These solar eruptions, known as solar flares, causes radio fade out. The magnetic effect of solar flares, called SFE (solar flare effect) is also referred to as geomagnetic crochet. It can last for two minutes to two hours. The magnetic field is enhanced by 5–30 nT. But in equatorial observatories, the SFE can increase to 100 nT. The SFE owes its origin to ionospheric dynamo theory, and the current system is being enhanced in both in D and E layers of the ionosphere.

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1.12 Magnetic Storms Rothwell and Mellwain (1960), Peddington (1960), Tsurutani et al. (1997) and Desseler et al. (1960) have addressed this topic in some detail, and some of the salient points are touched upon here. The variations in the magnetic field are observed due to enhancement in solar flares. Both rays and particles are emitted at an enhanced rate that causes an increase in the strength of the magnetic field and causes magnetic storms. The strength of the magnetic field rises to a certain peak and then decreases gradually. Magnetic storms have certain periodicity. It is possible to predict the onset of magnetic storms. Disturbance day variations of both horizontal and vertical components of the magnetic field cause auroras at higher latitudes. Enhancement of the magnetic fields creates night-time glow on the sky known as aurora borealis and aurora australis respectively in the northern and southern polar regions. Magnetic storms that are due to the solar radiations and which are characterised by a decrease in the horizontal components at low and midlatitudes are D variations. Magnetic storms form a separate class of disturbed geomagnetic fields. When any of the elements of the magnetic fields depart significantly from their quiet day levels, the amplitude of these departures can range from a few gamma to thousands of gamma. Magnetic storms are divided into those with an abrupt onset or sudden commencement (SSC) or those with gradual onset. In the first case, against a background of quiet behaviour, all elements describing the magnetic field depart simultaneously within one or two minutes at all the stations over the entire Earth. The beginning of the storm is particularly well expressed by the horizontal components of the magnetic field which decrease by tens of gamma. In the second case, with the gradual onset of the storm, it may be difficult to note the exact time of the storm. The initial time of commencement noted at various magnetic observatories may differ by an hour or two. Magnetic storms were first noticed in the 19th century as the large-scale depression of the horizontal components of the Earth’s magnetic field globally and simultaneously in many geomagnetic observatories in the world. It was later realised that this is due to the interaction of the disturbance day (Dst phase) solar wind with the magnetosphere of the Earth. Magnetic storms were first observed in the equatorial and low-latitude areas as a depression, especially in the horizontal components of the magnetic field. Magnetic storms generally has three phases: (i) Initial Phase (ii) Main Phase and (iii) Recovery Phase. In the initial phase, the magnetic field suddenly rises within a timespan of a few minutes. This is referred to as a Storm Sudden Commencement (SSC) and is caused by the rapidly increased solar-wind pressure in the magnetosphere during disturbance day variations (Desseler et al. 1960; Nagata 1952; Oguti 1956). The initial phase of the storm lasts a few hours and is the period during which the Dst solar flare remains at a high level. SSC starts after that. The second phase is the main phase in which the strength of the solar wind decreases by a few hundred nanotesla and reaches its minimum value and the ring current increases. As a result the horizontal components of the geomagnetic field decreases. The main phase of the storm lasts about 5–10 h. During this period, the energy from the solar wind is stored in the magnetosphere. The final phase is the

1.12 Magnetic Storms

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Fig. 1.7 Nature of variation of the magnetic field during a magnetic storm (Gubbins and Bervera 2007)

recovery phase, and during this period the Dst returns back to its pre-storm value. This phase takes 10–15 h. These three phases continues 24–30 h. In the recovery phase, on the other hand, the magnetic field gradually increases and the ring current decays.

1.13 Bay Type Variations The rapid decrease of the horizontal component of the magnetic field immediately after the sudden enhancement phase and the reduction of the same field more gradually during the recovery phase, giving rise to the signature of a coastal area surrounding a bay in shoreline areas. That is why they are termed “magnetic bays” or bay-type variations (Fig. 1.7).

1.14 Magnetic Substorms Akasofu (1964), Kamide (1998) and Lyon (1996) discussed this topic in considerable detail. A few points are mentioned here. A magnetospheric substorm again has three phases, i.e. (i) growth (ii) expansion and (iii) recovery phases. During the growth phase, the energy from the solar wind is transferred to the magnetotail. During the expansion phase, there will be an explosion of stored energy in the magnetotail, some energy will go to ring current and a part will be released in the ionosphere. In the recovery phase, the magnetic field relaxes back again to its original position. The total duration of the magnetospheric substorm is about 2–4 h. A solar eruption causes solar flares, a solar wave phenomenon that propagates with the speed of light and reaches the Earth in eight minutes. The corpuscular flux composed of solar particles propagates at a velocity between 1,000 and 3,000 km/s and takes 20–40 h to reach the Earth’s surface. Therefore, SFE can be regarded as the forerunner of a magnetic storm.

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Fig. 1.8 The nature of the presumed geomagnetic field showing the magnetic lines of force and the magnetosphere

1.15 Interaction Between the Sun and the Earth The nearly stationary magnetic field of Earth, which originate from the dynamo current in the core, generates a magnetic field in large adjacent regions of space, i.e. several thousand kilometers above the surface of the Earth and in the upper atmosphere. This region, known as the magnetosphere, exists on both the sides of the axis of the magnetic dipole (Fig. 1.8). The region distance above the surface of the Earth in which the magnetic field exists is the magnetosphere. In the absence of any kind of disturbing field, the shape of the magnetosphere would have been as shown in Fig. 1.8. The Sun continuously emits gamma rays, hard and soft X rays, near- and far ultraviolet rays, near and far infrared rays, protons and electrons, neutrons and alpha particles in the form of a solar plasma of variable intensity. The solar wind interacts with the magnetic field of the Earth to compress the magnetic field above the Earth’s surface to a certain extent on the dayside. Due to complex interactions of the solar flares of variable intensity and cosmic rays from interplanetary space with the magnetosphere, quite a few important phenomena occur in the upper atmosphere. Details are beyond the scope of this book. Very brief descriptions follow are given about; (i) the magnetosphere, (ii) radiation belts, (iii) the ionosphere (iv) ring current and (v) the magnetotail.

1.16 Magnetosphere Cowley et al. (2003) and Ohtani et al. (2000) have discussed this topic in considerable detail. The Earth’s magnetosphere is the region of space above the Earth’s surface in which the magnetic field exists. It is the space within which the solar wind (rays and particles) interacts with the magnetic field that originated from the dynamo

1.16 Magnetosphere

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current in the earth’s core. It extends from 100 km above the Earth’s surface to about ten earth radii (nearly 63,800 km) in the sunward side (day side) and to several hundred earth radii on the dark side (night side) of the Earth. The zone where the magnetic field ceases to exist is known as the magnetopause. So the magnetopause separates interplanetary space and the magnetosphere. Due to the compression of the magnetosphere by the pressure and interaction with the solar wind, the extent of the magnetosphere is only 10 Earth radii. That boundary is known as the bow shock (Fig. 1.9). On the dark side the earth, the magnetosphere is composed of magnetosheath and magnetotail as shown in Fig. 1.9. It contains high-energy solar plasma. The solar flare is supersonic. At magnetopause level, a collisionless bow shock is formed in the solar wind. At the bow shock, the solar wind–magnetosphere interaction starts, and the solar wind becomes thermalized. At this level, the supersonic solar flare becomes subsonic and continues to flow around the magnetosphere to form a magnetosheath plasma. A part of it forms a part of the magnetotail, and the remaining part joins the undisturbed solar plasma. The particles and rays on the sun side generate several current systems and radiation belts. A very important and prominent component of those are ring currents, the Van Allen radiation belt and the ionosphere. Due to the interaction of the solar flares of variable intensity and the magnetosphere, time-varying

Fig. 1.9 Interaction of solar flares with the magnetosphere; creation of the compressed magnetosphere, bow shock, magnetopause, magneto sheet and magnetotail and the journey of the solar wind towards outer space on the night side; entry of cosmic rays from outer space on the day side

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electromagnetic fields of very wide bands of frequencies are generated. The complex interaction of the variable solar flares with the magnetic lines of forces in the magnetosphere, interactions of electrons and protons in the form of a solar plasma with the magnetic lines of forces in the magnetosphere and the origin of pulsations and micropulsations are discussed briefly in this chapter. From the observed time lag between the occurrence of a solar flare and the beginning of the enhanced magnetic field, the velocities of the streams are found to vary over a wide range. Depending on their velocity and their associated magnetic fields, these particles are trapped at different heights in the magnetosphere. After being trapped, they are guided earthward along the lines of forces in spiral paths. On their way, they, along with their movements in the ring, bounce within the northern and southern auroral zones until their energies die down and they lose their capability to be reflected back from the auroral zone. At this stage, the electrons and protons start leaking from the ring towards the Earth along the magnetic lines of forces (Fig. 1.9). The arrival of electrons and protons in the ionosphere from the magnetosphere temporarily intensifies the lines of forces and increases the intensity of the dynamo currents in the ionosphere. As a result, magnetic fluctuations, which we call a magnetic storm (the Sd phase) are observed. And many phases of the micropulsations are explained by this theory. In 1961, U.S. Explorer XII added some sensational information to our existing knowledge. The data show that the region trapping energetic particles is more or less continuous over a vast zone, beginning about 600 miles above the earth and extending outward a 30,000–40,000 miles in the plane of geomagnetic equator. This entire region is known as the magnetosphere (Fig. 1.9).

1.17 Cosmic Rays Lockwood and Shea (1961), Dorman (2006) and Alfven and Falthammar (1963) discussed cosmic rays in some details. Primary cosmic rays are positively charged particles. The energy range of these particles is on the order of 1018 eV (an electron volt is a unit of energy that is equivalent to the energy acquired by an electron that is accelerated through a potential difference of one volt. One electron volt is equivalent to an energy of 1.6 × 10−9 J). These cosmic rays are interplanetary particles and are known to originate from the interstellar atoms that have been accelerated to high energy levels in the outer heliosphere. These are the principal sources of high-energy heavy ions trapped in the radiation belt. Anomalous cosmic rays in outer space are the principal sources of high-energy heavy ions (of the order of 0–50 MeV/nuc, trapped in the radiation belts). These high-energy ions are H, He, C, N, Ne and Ar.

1.18 Van Allen Radiation Belt

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1.18 Van Allen Radiation Belt Van Allen (1959), Van Allen and Frank (1959) and Van Allen and Lin (1960) discovered these two doughnut-shaped radiation belts in the upper atmosphere. The Van Allen radiation belts are conducting zones and are ionised by the highly penetrating gamma rays and x rays. There are inner (r = 1.5 to 2.2a) and outer (r = 3 to 6a, where a is the radius of the earth) radiation belts as shown in the Fig. 1.10. Van Allen radiation belts are zones of high energy ions and electrons and remain trapped by the magnetic lines of forces. In the absence of any perturbation, the particles and rays remain permanently trapped In these two inner and outer Van Allen radiation belts. Solar flare particles and solar wind plasma were recognised immediately as potential sources for ions and electrons in the radiation belt. To populate these belts, the particles migrate across the magnetic shell and reach the innermost regions of the radiation belts. The primary radiation-belt particles are electrons, protons and alpha particles. Nuclear interaction between inner zones and atoms in the upper atmosphere produce energetic H and He nuclei. Sources of radiation-belt particles include the decay of cosmic neutrons, the solar wind, the ionosphere and anomalous cosmic rays. A remarkable experimental verification of this theoretical concept was done in 1958 by the U.S. satellites Explorer I and Explorer III. A Geiger–Muller Counter carried by these satellites found that a large region around the Earth contains a

Fig. 1.10 The geomagnetic field created by the inner and outer Van Allen radiation belt, ring currents and the ionosphere

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very high intensity of protons and electrons trapped in the geomagnetic field. The maximum radiation intensity is found in two belts.

1.19 Ionosphere Mitra (1946, 1948), Kelley (1989), Ramanathan (1960), Young et al. (1982), Strahler (1965), Schunk and Nagy (2000), Rastogi et al. (1973), Rastogi and Rajaram (1971) and Appleton (1946) reported more detailed studies of the ionosphere. The ionosphere is an extensive zone in the upper atmosphere where there is a higher concentration of ionised gases in the form of electrons and protons that come from the solar flares in the form of a solar plasma. These regions of higher ion concentration totally surround the Earth, and they form the interface between the atmosphere and outer space. The discovery of the ionosphere came from the reflection of radio waves transmitted from the ground. Only a presence of electrons and protons layers can explain the observed reflectivity of the radio waves. Later these observations and inferences were verified through space-research programs using rockets and satellites. There are three layers in the ionosphere, namely, the D, E and F layers. The altitude range of the F layer is 150–500 kms. The F region is sometimes divided into the F1 and F2 layers during daytime. The altitude of the E region varies from 90 to 150 km, while the altitude of the D region varies from 60 to 90 km. During night-time, the D, E and F regions coalesce to form only E region as shown in Fig. 1.11b. Figure 1.11a shows the various layers of the upper atmosphere and the region above the ozonosphere from where the ionosphere starts. Figure 1.12 show the variation of electron density with height in the ionosphere. The peak plasma density as high as 106 /cm3 occur near noontime. The factor that limits the plasma-density values is the recombination rate, i.e. the rate at which ions and electrons recombine to form neutral molecules or atoms. The ionosphere is a strong conductor so, when it oscillates in the Earth’s magnetic field due to Earth tides, it generates electromagnetic waves that propagate towards the surface of the Earth. Moreover, the Earth’s surface and the ionosphere create an Earth–ionosphere wave guide that is responsible for the propagation of the sferics originating due to thunderstorm activity. Ions in the Ionosphere are known to be accelerated to energies of the order of 1–10 keV by electric fields parallel to the magnetic lines of forces and emanating from the auroral zones. The sun continuously emits visible light, far and near ultraviolet rays, infrared rays, soft and hard X-rays, gamma rays, cosmic rays and solar plasmas (electrons and protons). These radiations, especially gamma rays, X-rays and ultraviolet rays, are absorbed by the molecules and atoms of oxygen and nitrogen in the upper atmosphere. And during this process, each molecule or atom loses an electron and become a positively charged molecule or atom known as an ion. Due to this process of ionisation, a region, from about 50 to 250 miles above the Earth, where the concentration of positive ions and negative electrons is the most dense, is known as the ionosphere.

1.19 Ionosphere

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Fig. 1.11 a The layering of the upper atmosphere up to the lower part of the ionosphere; (i) the troposphere, (ii) tropopause, (iii) stratosphere, (iv) stratospause, (v) mesosphere, (vi) mesopause, (vii) thermosphere (Barry and Chorley 2003). b Altitudes of the D, E and F layers of the ionosphere and the charge densities of the various layers; various layers coalesce to one layer at night; charge densities at different heights in outer space (Mitra 1946)

Electrons thus ejected are free to travel in the form of an electric current. Thus we can think of the ionosphere and the Earth as electrical conductors with an insulating atmosphere of 60 km thickness in between. The Sun and the Moon by their gravitational attraction produce tidal motion in the ionosphere. This tidal motion gives rise to electric currents that, in turn, affect changes in the Earth’s magnetism at ground level. These currents are known as dynamo currents because they are generated in the same way as the electric currents generated by a dynamo. Some, of course, believe that these current systems arise from fluctuating ionospheric winds that blow the ionised air across the lines of forces of the geomagnetic field. The electric field is thereby generated to drive electric currents in eight current whorls in the ionosphere. These current whorls are the telluric current whorls discussed in Chap. 4. However, the tidal current hypothesis appears to be more appropriate to explain the solar (Sq ) and lunar (L) quiet day variations.

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Fig. 1.12 Shows variation of electron density with height in the upper atmosphere (Mitra 1946)

1.20 Ring Current The following researchers, as well as many others, have contributed significantly to upper-atmospheric physics: Akasofu and Chapman (1961), Desseler et al. (1961), Liemohn et al. (1999), Liemohn et al. (2001), Jordonova et al. (1996, 2003), Christofilos (1959), Takahashi et al. (1990), Fok et al. (2001), Frank (1970), Hamilton et al. (1988), Kistler et al. (1989), Kozyra et al. (1987), Moore et al. (2001) and Cladis and Francis (1985). Radiation belt particles represent significant energy storage in the magnetosphere (2.1015 to 2.1018 J). This establishes a current encircling the earth in a toroidal form (Fig. 1.10) and which is responsible for global depression of the magnetic field on the Earth’s surface. These are ring currents. The bulk energy densities of the ring-current particles is contained within an energy range around 85 keV. Protons, helium and oxygen together form the ring currents encircling the Earth. The intensity of the ring current varies with the variation of the intensity of the solar flares. But it never decays to vanishingly low levels, and it is always there with the Earth. The altitude of the ring current may extend outward from the region a little above the ionosphere to an altitude of 5 Re, i.e. five times the radius of the earth and beyond. It is linked to the ionosphere and inner and outer Van Allen radiation belt. Due to trapping of the charged particles from the solar flares by the magnetosphere, the strength of the geomagnetic field and ring current vary. Singer (1957) was the first to suggest that the main phase-ring current was due to the motion of the particles trapped in the Earth’s magnetic field. He suggested that the decay of ring current is due to the scattering of particles, either by field changes

1.20 Ring Current

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or atomic collisions. Dessler et al. (1960) and Wentworth et al. (1959) were the first to note the importance of charge exchange in removing the ring current. Charge exchange has been accepted as the standard procedure for the decay of ring current. These charges are mostly H+, He+ and O+. To test the trapping of electrons in the magnetic field surrounding the earth, the U.S. Atomic Energy Commission (Christofilos 1959) exploded an atom bomb far above the earth’s atmosphere to release a greater quantity of electrons into the Earth’s magnetic field. Reports from instruments carried by U.S. satellite Explorer IV showed that the electrons quickly spread completely around the Earth into a shell-like form. The U.S. space program, therefore, gave a better footing for the ringcurrent hypothesis. Ring currents are observed both in the inner and outer Van Allen radiation belts. Explorer IV found evidence of the ring current beyond five times the Earth’s radius.

1.21 Magnetotail On the dark side, the earth’s magnetic field is stretched out in an elongated geomagnetic tail that extends several hundred Earth radii. The geomagnetic tail’s field lines originate in high geomagnetic latitudes near the auroral zones at the geomagnetic poles. A geomagnetic tail consists of roughly oppositely directed field lines separated by a neutral sheet with a nearly zero magnetic field. Surrounding the neutral sheet, a plasma of hot particles exists. The second large energy-storage region in the magnetosphere is the extended geomagnetic tail. The plasma-sheet particles and the extended geomagnetic field lines constitutes the magnetotail.

1.22 Geomagnetic Field Variations The geomagnetic field of the Earth is composed of two parts. The first part, which is the main field, originates from the dynamo current in the outer core and is a stable magnetic field varying very slowly. This field constitutes about 94% of the geomagnetic field. The other part is a smaller field of purely extraterrestrial origin and constitutes only about 6% of the Earth’s field. It is time varying at a relatively faster rate and superimposed on the stable magnetic field of the earth. This time-varying part, which is considered as noise in magnetic prospecting, is the signal of interest in natural-source electromagnetic methods. This rapidly varying magnetic field induces eddy current inside the Earth and generates the natural electromagnetic field of the earth with a very wide spectrum. The periods range from a few milliseconds to several days/months/years. Variable solar flares with rays of various intensities and particles generate timevarying magnetic fields. These magnetic fields are superimposed on the permanent

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and stable geomagnetic field of the Earth as mentioned. Details of the major longterm variations of the Earth’s magnetic field are the central point of discussion in Chap. 7.

1.23 Classifications and Causes of the Various Pulsations and Micropulsations It is believed that the geomagnetic pulsations observed on the surface of the Earth are mainly due to the ionosphere’s current system and the variable trapping of charged particles along the magnetic lines of force in the magnetosphere. The trapped charged particles spiral around the magnetic lines of forces because the Lorentz force acts at right angles to the direction of propagation of the charged particles (Fig. 1.13). The variations in the intensity of the micropulsations are due to the variations in the number of particles coming from the Sun and participating in the ring current’s circulation (D phase). Such micropulsations are the superposition of the effect of the charged particles moving in these periodic orbits on the terrestrial magnetic fields. Geomagnetic micropulsations are temporal variations in the frequency range of a fraction of a mill hertz to several hertz. Systematic study of micropulsations on a global scale was initiated during the International Geophysical Year (1957−1958). They are classified as Pc (continuous pulsations), Pi (irregular pulsations), Pp (pearltype pulsations), Pg (giant pulsations) etc. Jacobs and his co-workers (1960a, b, 1963, 1965), Troitskaya (1960), Yungul (1962), Benioff (1960), Tepley and Wentworth (1962), Vladimirov and Kleimenova (1962) and others have studied the behaviour of these micropulsations and tried to classify them on the basis of certain criteria, the classification differing from author to author. Criteria considered by the various investigators are physical associations, genesis, periodic characteristics, time and probability of occurrences.

Fig. 1.13 a Trapping of electrons and protons in the form of plasma and coming from the solar wind by the geomagnetic lines of forces; b spiral movement of the electrons and protons

1.23 Classifications and Causes of the Various Pulsations …

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1.23.1 Classification by Jacobs and Sinno (1960) Geomagnetic micropulsations are divided into two classes: continuous pulsations Pc and irregular pulsations Pi . (a) ‘Pc ’: These continuous pulsations are series of pulses lasting for many hours with periods ranging from 10 to 60 s and with amplitude of the order of 0.1 gamma. The maximum occurrence is during morning hours. Micropulsations Pc type have a predominantly quasi periodic form. Pcs are divided into six ranges of rapid variations with periods from 0.2 to 1,000 s. The period ranges of these continuous pulsations are: (i) PC1 → 0.2–5.0 s; (ii) PC2 → 5.0–10.0 s; (iii) PC3 → 10.0–45.0 s; (iv) PC4 → 45.0–150.0 s; (v) PC5 → 150–600.0 s; and (vi) PC6 → 600.0 s. (b) ‘Pi ’: Pulsations Pi are irregular in character. They are damped trains of pulsations and are observed to last for nearly an hour. The period is 40 s to a few minutes, and the amplitude is nearly 0.5 gamma. The maximum occurrence is before midnight. There are three period ranges for Pi as: (i) Pi1 → 1.0–40.0 s; (ii) Pi2 → 40.0–150.0 s; and (iii) Pi3 → 150 s. Jacobs and Sinno (1960) have identified two more trains of micropulsations, PI and Pg . PI are damped trains of pulsations appearing as several series together, and each series lasting for about 10–20 min and the whole phenomena lasting for about an hour. Pg are the giant pulsations of amplitudes up to tens of gamma appearing near the auroral zones. The period is longer than that of ‘PC4 ’ (200–300 s), and the duration is on the order of an hour.

1.23.2 Classifications by Madam Troitskaya (1960) Troitskaya identified four classification of the micropulsations. They are SIP, PP, PC and IPDP. This classification is based on the association of pulsations with physical phenomena. (a) SIP: Short irregular pulsations composing microstructures of other macroscopic disturbances. The period ranges from six to ten secs, and they show extremely high correlation with auroras at high latitudes (Lat. −65° to 70°). (b) PP: Pulsations of ‘pearl’ type have periods of one–four seconds. They are distinguished by their regular form and are found to be correlated with cosmic ray intensity. The time series looks like a pearl necklace. (c) PC: Continuous pulsations occurring during daytime only, with period less than 15 s. (d) IPDP: Intervals of pulsations diminishing in period (period one–ten secs). They usually occur ‘during the main phase of a magnetic storm and coincide with a number of other upper atmospheric disturbances, like propagation of red aurora

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in low latitude, bursts of X-rays in the stratosphere and sharp increases in the cosmic noise absorption.

1.23.3 Classification by Benioff’s (1960) Benioff gave Type-A, Type-B and Type-C micropulsations. Most of these pulsations form part of the substorm, which can occur two to three times a day. This is similar to Troitskaya’s in many aspects. The subgroups are: (a) Type A: Oscillations are nearly sinusoidal in shape (period 0.3–2.5 s). Such wave trains usually exhibit pearl-type structures. (b) Type B: Oscillations have nearly sinusoidal forms with period two–eight secs. (c) Type C: This type is similar to Mme. Troitskaya’s SIP type.

1.23.4 Classification by Tepley and Wentworth (1962) Tepley (1961) defined the term hydromagnetic emissions or ‘hm’ as signals in the frequency range of 0.5–3.0 c/s and oscillating at a single frequency, which may however vary slowly with time. Thus the pearls or Type-A oscillations including a number of frequencies may represent a number of distinct hm emissions. This classification is done into two groups: (a) ‘hm’ emissions: They have well-defined frequency bands and often exhibit a regular structure, with sinusoidal waveform. (b) Noise bursts: These signals are characterised by broad spectral-energy distribution, with no observable distinct frequencies, but with much superposition.

1.23.5 Classification by Vladimirov and Kleimenova (1962) They classified the micropulsations in the range of 0.5–100.0 c/s according to probability of occurrences. (a) Largest number of micropulsations is found in eight-ten c/s range. They may be observed at all hours of a day. b) Oscillations in the interval of 0.5–1.5 c/s are observed infrequently, usually before sunrise. c) Oscillations in 3.5 c/s range registered least frequently.

1.23 Classifications and Causes of the Various Pulsations …

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Moreover, numerous authors have tried to classify the micropulsations obtained from magnetic observatories situated all over the world and tried to establish a suitable explanation for their origin. Only a small fraction of this entire problem is treated in this chapter.

1.24 High-Frequency Natural Electromagnetic Signals, Spherics Ward (1959) and Strangway and Vozoff (1969) studied these high-frequency electromagnetic signals in detail. Within the frequency range of interest in audiofrequency magnetotellurics (AMT) and audiofrequency magnetic method (AFMAG), only the EM signals of atmospheric origin are important. The AMT signals range from 1.0 Hz to 16 kHz and are created by global thunderstorm activity in the lower atmosphere. These signals from the thunderstorm and lightning activity are concentrated in the tropics and are referred to as ‘spherics’ or ‘atmospherics’. Cloud-to-ground thunder is the principal component. It is surrounded by several cloud to cloud thunders. The EM wave propagates around the globe along the earth−ionosphere waveguide of thickness 50–60 kms. There are three storm centers located in Brazil, Central Africa and Malaysia, which have on an average of 100 storm days per year. The geographical distribution of these storm centers is such that there is a storm in progress in one of the centers during every hour of the day. Therefore, detectable high-frequency EM signals are always available somewhere around the globe. These signals propagate around the world, trapped in the waveguide formed between the ionosphere and the Earth’s surface. The energy is reflected back and forth between the lowermost layer of the ionosphere and the ground surface with the spherics attaining their peaks in the early afternoon, local time. Some of the high-frequency manmade signals from broadcasting stations or defense installations or powerlines get mixed with the thunderstorm signals.

1.25 Dead Band There exists a minimum frequency around 1 Hz, below and above which the amplitude of the signal rises with frequency. The energy of a signal increases with longer periods or decreases in frequency. This frequency range of 1–10 Hz, where the energy level is minimum, is called the dead band in magnetotellurics.

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1.26 Complex Structure of Natural Source Long-period variations, short-period variations, pulsations, micropulsations and spherics frame the complete structure of the Earth’s natural electromagnetic field of extraterrestrial origin. Detailed spherical harmonic analysis shows that 94% of the Earth’s magnetic field is of internal origin. Eighty percent of that is of deeper Earth’s core dipole origin; the remaining 20% is a mixture of shallower non dipole origin. The time-varying electromagnetic field of extraterrestrial origin is superimposed on the Earth’s stable magnetic field. Detailed analysis of the Earth’s natural electromagnetic field has revealed that the electromagnetic induction on the surface of the Earth is an extremely complex phenomenon. No single source can generate these electromagnetic fields over such a wide spectrum. Today, we can now summarise the origin of the Earth’s natural electromagnetic field, assembling bringing all the factors into one. But, even then, a couple of points might have escaped the notice of the author. These factors are: (i)

(ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv)

the interaction of solar flares with the magnetosphere; creation of compressed magnetosphere, Van Allen radiation belts, ionosphere, ozonosphere and magnetopause on the dayside and magnetotail on the night side Variable emissions of solar plasma in the form of electrons and protons and their subsequent entrapment by the magnetic lines of force on the Earth ionisation of various layers of ionosphere and Van Allen radiation belt by highly penetrating X-rays and gamma rays tidal oscillation of the ionosphere in the earth’s magnetic field ring currents at greater height at magnetopause level and their variations solar and lunar variations Earth’s rotation around the Sun Earth’s revolution on its own axis sunspot activities thunderstorm activities and propagation of high-frequency electromagnetic wave through the Earth–ionosphere waveguide variations of solar emissions during day and night-time variation of solar emission during summer and winter variation during solar quiet and disturbed days onset and recession of magnetic storms formation and movements of equatorial and polar electrojets, etc.

This is the complex scenario of the Earth’s natural electromagnetic field.

1.27 Earth’s Natural Electromagnetic Field as a Subject

29

Fig. 1.14 Structure of the subjects concerning Earth’s natural electromagnetic field

1.27 Earth’s Natural Electromagnetic Field as a Subject Subjects based on Earth’s natural electromagnetic field have several scientific branches: (i) electrotellurics (T), (ii) magnetotellurics (MT), (iii) audiofrequency magnetotellurics (AMT), (iv) magneto variational sounding (MVS), (v) geomagnetic depth sounding (GDS), (vi) magnetometer array (MA) studies, (vii) SeaFloor magnetotellurics (SFMT) and marine magnetotellurics (MMT) and (viii) the audiofrequency magnetic method (AFMAG) (Fig. 1.14). In GDS, MVS, MA and AFMAG, only the magnetic components of time-varying electromagnetic field are measured. In magnetotellurics, audiofrequency magnetotellurics and sea-floor magnetotellurics, both the electric and magnetic field components are measured. In electrotellurics, only the electric component of the time-varying electromagnetic field is measured. In this monograph, we shall primarily discuss certain aspects of these topics. A few controlled-source electromagnetics closely associated with the subjects based on natural sources are briefly included. They are controlled-source audiofrequency magnetotellurics (CSAMT), long offset electromagnetic transients (LOTEM), radio magnetotellurics (RMT) and controlled-source marine electromagnetics (CSEM).

1.27.1 Electrotelluric Method (T) Boissonas and Leonardon (1948), Porstendorfer (1954, 1959, 1961a, 1961b), Berdichevskiy (1960), Antal 1961, Vero (1959, 1960), Yungul (1962, 1968, Gugunava and Chernyavsky 1964), Tuman (1951), Combs and Wilt (1976), and Kunetz

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(1957) developed this particular branch for the application of the Earth’s natural electromagnetic field as a subject. In the electrotelluric method, only the electrical components of the Earth’s natural EM fields are recorded continuously in a multichannel recorder for a period of six to eight hours at the base and mobile stations. The recorded time series are then analysed in the time domain to obtain the electrical nature of the Earth’s sedimentary cover above the upper crust. This can be used as a reconnaissance tool for oil and mineral exploration. It can also be used for shallow crustal studies. It was one of the very early geoelectrical tool for exploration in which natural electromagnetic field was used.

1.27.2 Magnetotelluric Method (MT) Tikhonov (1950), Cagniard (1953), Vozoff (1972), Berdichevskiy (1960), Keller and Frischknecht (1966), Zhdanov (1994), and many others have treated this subject. In the magnetotelluric method, both the electrical (EX (t), EY (t)) and magnetic (HX (t), HY (t) and HZ (t)) components of the geoelectric and geomagnetic micropulsations/pulsations/short-period variations are measured continuously in the form of time series for a period depending upon the nature of the problem. It may be for ten to 14 h to as long as 30–40 days. These time series in digital form undergo rigorous signal processing, and all five components are brought from the time domain to the frequency domain, i.e. EX (ω), EY (ω), HX (ω), HY (ω) and HZ (ω). Transfer functions are computed and the impedances are estimated from EX (ω)/HY (ω) and EY (ω)/ HX (ω). The Hz (ω) component is required to have a measure of lateral heterogeneities. It is a very powerful tool for studying sedimentary covers for oil exploration, especially in flood basalt-covered areas, as well as for studying the crust mantle silicates. MT, along with seismic methods, have become the most powerful tools for studying the crust and upper mantle.

1.27.3 Geomagnetic Depth Sounding (GDS) Schmucker (1964, 1970, 1971, 1974), Roberts (1983, 1984, 1986a, b), Alldredge (1977a, b), McDonald (1957) and many others have studied this aspect of the geomagnetic field for the nature of the variations of electrical conductivity deep inside the Earth. In GDS, long-period variations of the geomagnetic field are collected mostly from geomagnetic observatories. The range of the periodic variations in the geomagnetic field can be: (i) cycles/minute, (ii) cycles/ten minutes, (iii) cycles/hour, (iv) cycle/twelve hours, (v) cycles/day, (vi) cycles/month, (vii) cycles/year or (viii) cycles/eleven years. From the extensive records of the geomagnetic fields obtained by the geomagnetic observatories or field stations, some parts are selected for processing and analysis. In GDS, only the magnetic components are measured. The external source field induces the internal field. Therefore, the first stage of processing is to

1.27 Earth’s Natural Electromagnetic Field as a Subject

31

separate the external and internal fields. In the next stage, the normal and anomalous parts of the internal field are separated. Long-period variations collected from permanent geomagnetic observatories are used for geomagnetic depth sounding (GDS) (Schmucker 1974) to find out the electrical conductivity of upper and lower mantle.

1.27.4 Magnetometer Array Studies (MA) Ingham (1983), Gough (1981, 1989), Frazer (1974), Gough and Ingham (1983), Egbert and Booker (1993), Alabi (1983), DeBeer et al. (1982), and Bannister and Gough (1978) studied this subject in considerable detail. In magnetometer array studies, only the long-period magnetic components within the range of ten minutes to 24 h are measured using Gough and Reitzel variometers. The magnetic signals are recorded continuously for a period of one month in about 50 to 100 or even more stations scattered over a large area separated by about 10–100 km depending upon the nature of the problem. Deep-seated lateral inhomogeneities and current channeling are detected by this tool. Large-scale conductors at upper-mantle depth can be delineated. It is a good tool for studying tectonic set up.

1.27.5 Magnetovariational Sounding (MVS) Pringle et al. (2000), Ingham (1983, 1985) Arora and Reddy (1991), Dmitriev (1977) studied this subject in some detail. Magneto-variational sounding is a subset of GDS. The main difference between GDS and MVS lies in the frequency or period range of the geomagnetic signal. GDS periods are much longer than MVS periods. As a result, a spherical Earth model is chosen for GDS and a plane Earth model for MVS. The impedances are obtained from the ratio of the two magnetic fields. It can be used for studying the electrical conductivity of the upper mantle and has become a substitute for magnetotelluric sounding where it is difficult to measure the electrical component of the magnetotelluric field because of high cultural noise.

1.27.6 Audiofrequency Magnetotelluric Method (AMT) Strangway and Vozoff (1969), Strangway et al. (1973), Strangway and Koziar (1979) and Goldstein and Strangway (1975) developed the Audiofrequency Magnetotelluric (AMT) method, which is the high frequency part of the magnetotellurics. The frequency range of operation is 1−16 kHz. The signals originate from thunderstorm activities. Both electric and magnetic components of the electromagnetic field are measured on the surface, the same way it is done in magnetotellurics. Both

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scalar and tensor magnetotelluric measurements are possible. It is a tool suitable for studying the first ten km of the Earth’s crust. It is an effective tool for mineral and geothermal exploration, as well as shear zone/contact zone/collision zone studies.

1.27.7 Sea-Floor Magnetotelluric Method (SFMT) Filloux (1973, 1974, 1977), Chave and Cox (1982), Chave (1983, 1984, 1985), Chave and Filloux (1984), Chave et al. (1989), Chave et al. (1996), Chave and Luther (1990) and Utada (2015) have developed this subject. Here electric and magnetic fields are measured at the bottom of the sea. Only EM signals that can survive after penetrating five km of seawater can be used for sea-floor magnetotellurics. Therefore the starting periods for SFMT is about cycles/twelve hours. One records noise-free long-period signals, which can see the electrical conductivity structure of the Earth down to 1,000 km so it is a very powerful tool for studying the convergent and divergent plate margins.

1.27.8 Marine Magnetotellurics (MMT) Steve Constable (1990), Constable and Cox (1996), Constable and Weiss (2006), Constable and Srnka Leonard (2007), Cox et al. (1971), Cox (1980, 1981) and Cox et al. (1986) were the chief architects of this subject. It is a fairly recent geophysical tool designed for ocean-bed oil exploration. MMT for oil exploration is an immensely important subject in the 21st century. Vast oceanic regions have a huge untapped reserves of oil and gas, including gas hydrates. There have been some success using MMT and CSEM (controlled-source electromagnetics) in ocean-bed oil exploration.

1.27.9 Audiofrequency Magnetic Method (AFMAG) Ward (1959, 1960, 1966) developed this particular tool for mineral exploration. Audiofrequency components of the magnetic field that originate in thunderstorms are normally measured. The magnetic vector of the electromagnetic field is considered horizontal in the homogeneous ground; it is however inclined when the ground is inhomogeneous. Thus the angle of inclination of the vector is a measure of inhomogeneity.

1.28 Controlled Sources

33

1.28 Controlled Sources 1.28.1 Controlled-Source Audiofrequency Magnetotelluric Methods (CSAMT) Zonge and Hughes (1991) are Zonge et al. (1986) are the pioneer workers in this field. In controlled-source audiofrequency magnetotelluric methods (CSAMT), a generator is used to create artificial manmade em signals. Electric and transverse magnetic fields are measured at a distance from the source where plane wave approximation is roughly valid. Its range of applicability is less than that of audiofrequency magnetotellurics. AMT can see down to ten kms from the surface, while as CSAMT can see two–thee kms.

1.28.2 Controlled-Source Marine Electromagnetics (CSEM) Constable (1990) and Constable and Cox (1996) developed the technology for marine CSEM once they had discovered the totally noise-free and low-impedance environment at the ocean bottom. The field survey can be done with completely different field logistics and at a much faster rate. Currents as high as 2,000 amp can be fed through the low-impedance ocean bottom, and the entire field set up can be dragged by the ship along the ocean bottom. Since the gas- and oil-bearing zones are much more resistive, the oil bearing horizons can be detected.

1.28.3 Long-Offset Electromagnetic Transients (LOTEM) Strack (1994) was the chief architect behind development of this subject. LOTEM is one of the methods of electromagnetic transient sounding to measure the response of the Earth to the energisation of the ground caused by sending current pulses. The Absence of this primary excitation is the first step towards the origin of “Time Domain Electromagnetics or Electromagnetic Transients”. LOTEM is “Long-Offset Transient Electromagnetics”. Its primary aim is to develop a suitable transient electromagnetic tool to see deep inside the Earth by: (i) increasing the transmitter–receiver distance within the range of 5–100 km; (ii) increasing the range of pulsating direct current through the transmitter from 5 to 100 amp and beyond and using a 10–100 Kwt truck-mounted generator; (iii) increasing capability of measuring signals down to microvolt range; and (iv) using theories developed by Kaufmann and Keller (1983), Wait (1951), and Strack (1994).

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1.28.4 Radio Magnetotellurics (RMT) Radiomagnetotellurics is a recently invented subject developed by a Swedish group of scientists (Bastani and Pederson 2001; Linde and Pederson 2004) in which VLF transmitters are used as sources instead of natural sources. Measuring probes, e.g. nonpolarisable electrodes for electric field and induction coils for the magnetic fields, are kept far enough away such that plane-wave assumptions are approximately valid. Separation between the electrodes are kept to five m for measuring Ex and Ey. Induction coils are 0.4 m in diameter for measuring Hx, Hy and Hz. Cagniard’s theory of magnetotellurics is applicable here. The frequency range of operation is 10–250 kHz.

1.29 Coverage of This Book The main aim of this book is to bring all aspects of the subject under one umbrella so various aspects of pure and applied geophysics occur within this brief discussion. Those who were the pioneer workers in developing these subjects come to the forefront. It is written for students and researchers who already have some preliminary familiarity with the subject.

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Kunetz, G. 1957. Anwendung Statisticher Eigenschaften der Erdstrome in der Practischen, Geophysik, Freiberger Forschungshefte C-32: 5–19. Liemohn, M.W., J.U. Kozyra, V.K. Jaranova, G.V., Khazanov, M.F., Thomson and T.E. Cayton. 1999. Analysis of early phase ring current recovery mechanism during geomagnetic storm. Geophysical Research Letter 25: 2845. Liemohn, M.W., J.U. Kogyra, M.F. Thomson, J.F. Roeder, G. Lu, J.E. Borovsky, and T.E. Cayton. 2001. Dominant role of the asymmetric ring current in producing the storm time Dst . Journal of Geophysical Research 106: 10883. Linde, N., and L.B. Pederson. 2004. Characterisation of fractured granite using radiomagnetotelluric (RMT) data. Geophysics 69: 1155–1165. Lockwood, J.A., and M.A. Shea. 1961. Variations of the cosmic radiations in November 1960. Journal of Geophysical Research 66 (10): 3083–3093. Lowrie, W. 1997. Fundametals of geophysics. Cambridge: Cambridge University Press. Lyons, L.R. 1996. Substorms fundamental observational features, distinction from other disturbances, and external triggering. Journal of Geophysical Research 101: 13011. Malin, S.R.C. 1969. The effect of sea on the lunar variation of the vertical component of the geomagnetic field. Planetary and Space Science 17: 487–490. Malin, S.R.C. 1970. Separation of lunar geomagnetic variations in parts of ionospheric and oceanic origin. Geophysical Journal of the Royal Astronomical Society 21: 447–455. Marrill, R.I., and M.W. McElhinny. 1983. The earth’s magnetic field, its history. London: Origin and Planetary Perception Academic Press. McDonald, K.L. 1957. Penetration of the geomagnetic secular field through a mantle with variable conductivity. Journal of Geophysical Research 62: 117–141. Michael, C., and Kelley. 2009. The earth’s ionosphere, plasma physics and electrodynamics, 2nd ed. Amsterdam: Elsevier. Mitra, S. K. 1948. The upper atmosphere. The Royal Asiatic Society of Bengal. Mitra, S.K. 1946. Geomagnetic control of region F2 of the ionosphere. Nature 158: 668–669. Moore, T.E., M.O. Chandler, M.C. Fok, B.L. Giles, D.C. Delcourt, J.L. Horwitz, and C.J. Pollock. 2001. Ring current and internal plasma sources. Space Science Reviews 95 (1/2): 555–568. Nagata, T. 1952. Distribution of SC* . Reports on Ionospheric Research Japan 6: 13–30. Oguti, T. 1956. Notes on the morphology of SC. Reports on Ionospheric Research 10: 81–90. Ohtani, S.I., R. Fujii, M. Hesse, and R.L. Lysak. 2000. Magnetospheric current systems. Geophysical monograph, vol. 118, Washington D.C., American Geophysical Union. Peddington, J.H.. 1960. Geomagnetic storms theory. Journal of Geophysical Research, 6593–6106. Porstendorfer, G. 1954. Tellurik, Grundlagen and Anwendungen, Freiber, Forsch, C−16. Berlin: Akademie-Verlag. Porstendorfer, G. 1959. Direkte Autzeichnungen telluricher vector diagram und ihre Anwendungen in Bergbangebieten. Gerland. Beitr. ZurGeophysik 68: 295. Porstendorfer, G. 1961. Grundlagen, Metechmik und neue Einsatzmoglich keiten, Frieb. Forsch, C−107, Akademie-Verlag, Berlin. Porstendorfer, G. 1961. Tellurik Grundlagen und Anwendungen, Freiberger Forschung Shefte, c-16, Akademie-Verlag Berlin. Pringle, D., M. Ingham, D. McKnight, and F. Chamalaun. 2000. Magnetovariational sounding across the southern Island of New Zealand; difference induction arrows and the southern Alps Conductor. Physics of the Earth and Planetary Interiors 119: 285–298. Ramanathan, K.R., R.V. Bonsle, and S.S. Degaonkar. 1960. Effect of electron–Ion collisions in the F region of the ionosphere on the absorption of cosmic radio noise at 25 Mc/s at Ahmedabad: changes in absorption associated with magnetic storms. Journal of Geophysical Research 66 (9): 2763–2771. Rastogi, R.G., and G. Rajaram. 1971. Electrojet effect on the equatorial F-region during magnetically quiet and disturbance days. Indian Journal of Pure and Applied Physics 9: 531–536. Rastogi, R.G., R.P. Sharma, and V. Shodhan. 1973. Total electron content of the equatorial ionosphere. Planetary and Space Science 21: 713–720.

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Roberts, R.G. 1983. Electromagnetic Evidence for lateral Inhomogeneities within the Earth’s Upper Mantle. Physics of the Earth and Planetary Interiors 33: 198–212. Roberts, R.G. 1986a. The deep electrical structure of the Earth. The Geophysical Journal of the Royal Astronomical Society 85: 583–600. Roberts, R.G. 1984. The long period electromagnetic response of the Earth. Geophysical Journal of the Royal Astronomical Society 78: 547–572. Roberts, R.G. 1986b. Global electromagnetic induction. Surveys In Geophysics 8: 339–374. Rothwell, P., and C.E. Mellwain. 1960. Magnetic storms and the Van Allen radiation belts: observations from explorer IV. Journal of Geophysical Research 65: 799–806. Schmucker, U. 1964. Anomalies of geomagnetic variations in the south western United States. Journal of Geomagnetism and Geoelectricity 15 (4): 193–221. Schmucker, U. 1970. Anomalies of geomagnetic variations in the South Western United States, Bull Scripps Institute of Oceanography, University of California, LaJolla, 165 pages. Schmucker, U. 1971. Interpretation of Induction anomalies above nonuniform surface layers. Geophysics 36 (1): 156–165. Schmucker, U. 1974. ‘Erdmagnetische Tiefensondierung miet langperiodischen Variationen’ in Protocall uber das colloquium ‘Erdmagnetische Tiefensondierung’ in Grafrath, Bayern, 113–342. Schunk, R.W., and A.F. Nagy. 2000. Ionosphere: physics plasma physics and chemistry. Cambridge, U.K: Cambridge University Press. Sethia, G., R.G. Rastogi, M.R. Deshpande, and H. Chandra. 1980. Equatorial electrojet control of the low latitude ionosphere. Journal of Geomagnetism and Geoelectricity 32: 207–216. Singer, S.F. 1957. A new model of magnetic storms and aurorae. Transactions of the American Geophysical Union 38: 175–190. Slaucitajis, L., and D.E. Winch. 1965. Some morphological aspects of geomagnetic secular variation. Planetary and Space Science 57: 151–155. Strack, K.M. 1994. Exploration with deep transient electromagnetics. Amsterdam: Elsevier Scienfic Publising Company. Strahler, A.N. 1965. The earth sciences. John Weatherhill, Inc. Strangway, D.W., J.D. Redman, and D. Maclin. 1989. Shallow electrical sounding in the Precambrian crust of Canada and United States in Magnetotelluric Methods, ed. K. Vozoff. SEG reprint series, 720–748. Reprinted from “The Continental Crust and its mineral deposits, 273–301. Strangway, D.W., and A. Koziar. 1979. Audiofrequency magnetotelluric sounding—A case history at the Cavendish Geophysical test range. Geophysics 44 (8): 1429–1446. Strangway, D.W., and K. Vozoff. 1969. Mining exploration with natural electromagnetic field mining and groundwater geophysics, Geological Survey of Canada. Economic Geology Report 26: 109– 122. Strangway, D.W., C.W. Swift Jr., and R.C. Holmer. 1973. An application of audiofrequency Magnetotellurics (AMT) to mineral exploration. Geophysics 38: 1159–1175. Takahashi, S., T. Iyemori, and M. Takada. 1990. A simulation of the storm time ring current. Planet and Space Science 38: 1133. Tepley, L.R. 1961. Observation of hydromagnetic emissions. Journal of Geophysical Research 66: 1651. Tepley, L.R., and R.C. Wentworth. 1962. Hydromagnetic Emissions, X-ray Bursts and Electron Bunches, I. Experimental Results II. Theoretical Interpretation. Journal of Geophysical Research 67: 3317–3335. Tikhonov, A. V. 1950. Determination of the electrical characteristics of the deep strata of the earth’s crust. Dokl. Akad. Nauk, 73: 275–297 (Eng. trans. in Magnetotelluric methods, SEG, Geophysics. Reprint Sr. 5 (ed. Vozoff, K.), 1986). Troitskaya, V.I. 1960. Pulsations of the earth’s electromagnetic field with periods of 1 to 15 seconds and their connection with phenomena in the high atmosphere. Journal of Geophysical Research 66 (1): 5–18. Tsurutani, B.T., W.D. Gonzalez, W. Kamide, and J.K. Arballo. 1997. Magnetic storms. Washington: D.C., American Geophysical Union.

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Chapter 2

Electrical Conduction in Rocks

Abstract In this chapter, we discuss the various ways that electrical currents are conducted through rocks, namely: (i) electronic conduction or conduction through metals; (ii) conduction through semiconductors; (iii) conduction through solid electrolytes; (iv) Ionic conduction or conduction through liquid electrolyte; and (v) conduction through dielectrics due to displacement current in mega- and gigahertz ranges. Also highlighted are dependences of electrical conductivity on: (i) porosity and permeability of rocks; (ii) conductivity of the pore fluid; (iii) size and shape of the mineral grains in the rocks; (iv) conductivity of the mineral grains; (v) temperature; (vi) pressure; (vii) frequency of the exciting current; (viii) ductility and degree of partial melt; (ix) oxygen fugacity; (x) volatiles; (xi) the Hall effect (xii); and the piezoelectric effect. Some experimental results are included, and Maxwell’s theory of electrical conduction is covered briefly. Keywords Electrical conduction · Rocks · Theory · Experiments

2.1 Introduction Electrical conduction through rocks subsumes several phases: (i) electronic conduction; (ii) electrolytic or ionic conduction; (iii) semiconductor-type conduction; (iv) solid electrolyte-type conduction; and (v) conduction through displacement currents due to time-varying alternating currents. Electrical conductivity is defined as the ease with which an electric current (direct or alternating current) flows through a medium. Resistance and resistivity are respectively the reciprocal of conductance and conductivity. The unit of conductance and conductivity are respectively “mho” and “mho/meter”. The Siemen is also used as a unit of electrical conductivity. The units of resistance and resistivity are respectively “ohm” and ohm-meter. The resistance offered by the ground is always a complex quantity in the presence of reactive component. Since geophysicists deal with the conduction of current through the Earth, reactive parts will exist. Here pure resistance is a real component. Both capacitance and inductance are reactances that generate imaginary components. Imaginary components are as real as real components. But the reactive components, leading or

© Springer Nature Switzerland AG 2020 K. K. Roy, Natural Electromagnetic Fields in Pure and Applied Geophysics, Springer Geophysics, https://doi.org/10.1007/978-3-030-38097-7_2

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2 Electrical Conduction in Rocks

lagging by 90° from the real components in the phase domain, are imaginary components. The complex resistance offered by the ground is called the impedance of the circuit. The expression for the impedance is Z=

1 R+iωL

1 + iωC

(2.1)

where R is the resistance, L the Inductance, C the capacitance, and ω the angular √ frequency (= 2πf), where f is the frequency in cycles/second or Hertz and i = − 1. The electrical equivalent of a geological body is simply a resistance, capacitance and inductance network where capacitance and resistance are in parallel path with a circuit. I is the current flowing through the circuit. Here V = IZ = I(X + iY)

(2.2)

The potential difference is a complex quantity. It has amplitude and phase in the form of  A = X2 + Y2 φ = tan−1

Y X

(2.3)

where A and φ are respectively the amplitude and phase of the complex impedance Aeiφ . Figure 2.1 is a simple L, C, R circuit model of the Earth. Direct current cannot flow through a capacitor but can flow through an inductor. At high frequency, current can easily pass through the capacitor in the form of displacement current but finds it difficult to pass through the inductance. This is because the value of Lω becomes very high as ω → ∞. Therefore, for flow of alternating current, displacement current (discussed in Sect. 2.4 dominates at very high frequency, i.e. in the mega- and gigahertz ranges. Conduction current dominates at DC/audio/subaudio AC frequencies. Conduction current is controlled by electrical conductivity (discussed in Sect. 2.2). It is guided by the first basic electromagnetic Fig. 2.1 Model of the L, C, R circuit of earth

2.1 Introduction

43

equation J = σE. Displacement current is controlled by the electrical permittivity or dielectric constant and high-frequency alternating current (discussed in Sect. 2.3). For very highly resistive rocks, direct currents find it difficult to pass through but electromagnetic waves can easy pass without much attenuation. Therefore, the electromagnetic method is a better geophysical tool for studying hard rock areas. In this section, the second basic equation of electromagnetics, i.e. D = mE becomes useful. The third basic equation of electromagnetics is B = μH. Here the electromagnetic propagation constant in Helmholtz’s equation contains the parameter μ which is associated both with electrical conductivity σ and electrical permittivity ∈ in low and high frequencies, respectively. Free-space magnetic permeability js μ0 = 4π × 10−7 H/meter. Further details are given in Sect. 2.4. Of all the physical parameters used by the geophysicists, e.g. density, magnetic permeability, moduli of elasticity and density bound seismic velocities, electrical conductivity is the most sensitive parameter. The ratio of the extreme values of electrical conductivities is on the order of 1015 –1020 . Resistivities of dry laboratory samples of granites or granodiorites is on the order of 106 –108 ohm-m. Resistivity of an insulator like porcelain, at room temperature, is of the order of 1012 –1014 ohm-m. Metallic copper resistivity is of the order of 1.6 × 10−8 ohm-m (Keller and Frischknecht 1966). There are far too many factors that control the electrical conductivity of a formation with too many variables in geology of the different areas. The factors which control the electrical conductivity of rocks: (i) rock types; (ii) conductivity of mineral inclusions; (iii) conductivity of pore fluids; (iv) porosity of rocks; (v) size and shape of the pore spaces; (vi) size and shape of mineral inclusions; (vii) interconnectivity of mineral grains and pore spaces; (viii) temperature; (ix) frequencies of excitation current; (x) ductility, i.e. the degree of partial melt in rocks; (xi) pressure; (xii) oxygen fugacity; (xiii) hydrogen-ion concentration; and (xiv) geological zones of weaknesses, such as shear zones, fractures, fissures, lineaments, major contacts between the rocks of different geological ages, continent–continent borders and ocean– continent-plate boundaries and (xv) contact mineralization. Since the geophysical tools based on Earth’s natural electromagnetic field are needed for shallow to very deep (sediments, crust, mantle and core) exploration, this has been the subject of study of a broad group of researchers, including: Shankland (1975, 1981), Shankland and Duba (1990), Shankland and Waff (1974, 1977), Shankland and Ander (1983), Shankland et al. (1981, 1993), Duba et al. (1974, 1990), Constable and Duba (1990), Constable et al. (1992), Haak (1980), Haak and Hutton (1986), Kariya and Shankland (1983), Olhoeft (1977), Lastoviskova (1981, 1983, 1987), Keller and Frischknecht (1966), Dvorak (1973), Duba and Constable (1993), Duba et al. (1974), Olhoeft et al. (1974), Constable and Duba (1990), Hirsch et al. (1993), Hirsch and Shankland (1993). Silicates (felsics and others) have been studied on the high-temperature–pressure electrical conductivity of crust mantle rocks and minerals. Electrical conductivity has a very strong dependence on temperature. An insulator at room temperature can become a good conductor at 1200–1300 °C. Therefore, high-pressure/temperature laboratory studies on electrical conductivity and viscoelastic properties go together with high- pressure and temperature petrological laboratory studies on crust mantle silicates. Very many geological problems

44

2 Electrical Conduction in Rocks

can be handled by electrical methods. Earth’s natural electromagnetic field can be exploited to address some of these problems. In this book, we shall restrict our discussion mostly to these topics. A few controlled-source electromagnetic problems relevant to these topics are also addressed. Phase transition of silicates (olivine-spinel transition shows sharp changes in electrical conductivity. More seismic discontinuities in the upper mantle at 420- and 520-km depths are reported in IUGG 2007 in Perugia, Italy. Petrological changes of a rock with temperature and the corresponding changes in electrical conductivity with degree of partial melt has been established by Shankland and Waff (1977). It is an established fact now that higher electrical conductivity and higher ductility with rise in temperature go together.

2.2 Electrical Conductivity In this section, we shall define the electrical conductivity, the basic parameter for any kind of geoelectrical studies and one of the three basic equations underpinning →

→

electromagnetic theory, i.e. J = σ E. Here J is the current density amp/meter2 , E is the electric field in volt/meter and σ is the electrical conductivity in mho/meters or Siemens.

2.2.1 Expression of Electrical Conductivity for an Homogenous and Isotropic Medium Due to a Point Source of Current Potential is a function of radial distance r, i.e. φ = f(r). For a point source of direct current in an homogenous and full isotropic space, potential will be a function of r, i.e., φ = f(r) only and is independent of θ and ψ, the polar and azimuthal angles respectively in a spherically polar coordinate. Here φ is the scalar potential, and r is the radial distance from the point source. The Laplace equation reduces to the form (Roy 2007)   ∂ 2 ∂φ r =0 ∂r ∂r ⇒ r2

∂φ = C1 ∂r

(2.4) (2.5)

where C1 is a constant. From Eq. (2.4), we can get (Fig. 2.2) φ = C2 −

C1 r

(2.6)

2.2 Electrical Conductivity

45

Fig. 2.2 Spherical polar coordinates

where C2 is another constant. This is the potential at a point at a distance r from the source due to a point source of current. Since the potential will be zero at r = ∞, C2 = 0, and the potential reduces to φ = −

C1 r

(2.7)

Since  → C1 → E = −grad φ = − 2 and I = J · n · ds r  →  C1 C1 → = σ E · n · ds = −σ ds = −σ 2 4πr2 r2 r I = −4πσC1 . Hence C1 = − 4πσ

→

and φ=

Iρ 1 I 1 , = = , 4πσ r 4π r

(2.8)

where ρ is the resistivity of the medium and ρ = σ1 . This is the expression for potential at a point at a distance ‘r’ from a point source in an homogeneous and isotropic full space. The solid angle subtended at the source point is 4π. For an atmosphere–Earth boundary (Fig. 2.3), when the point source is on the surface of the Earth, the solid angle subtended will be 2π, and the potential at a point will be φ=

Iρ 1 · . 2π r

(2.9)

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2 Electrical Conduction in Rocks

Fig. 2.3 Direct current generated bipole field on the surface of the earth

2.2.2 Specific Resistivity or Conductivity In direct current or very low-frequency alternating current, electrical conduction is dependent upon σ, the electrical conductivity of the body. The reciprocal of electrical conductivity is electrical resistivity. Electrical resistivity of a medium is defined as the resistance offered by the two opposite faces of a unit cube. Here R = ρ L/A

(2.10)

where R is the resistance in ohms, and L and A are respectively the length and cross sectional area of the medium in meters and meters2 respectively for the flow of current. The unit of ρ and the specific resistivity of a medium is in ohm-meters. Similarly C=σ

A L

(2.11)

where C is the electrical conductance of a medium in mho. σ is the electrical conductivity, and its unit is mho/meter or Siemen. Current I can be taken as the rate of flow of charge, i.e. I = dq . At any point in a medium, I cannot be defined, but J can dt be defined. A small area S is chosen normal to the flow of current. The amount of current that flows through the face in a time t is given by (Fig. 2.4) I = qv

S · l t

(2.12)

where l is the distance travelled by the charges, and qv is the volume density of charge. Hence

2.2 Electrical Conductivity

47

Fig. 2.4 Flow of direct current through two opposite faces of a rectangular parallelepiped solid

I l = qv S t →

(2.13)

→

⇒ J = qv v →

(2.14)

→

where J is the current density and v is the velocity. The expression for the current is given by  I=

→

→

J · n · ds

(2.15)

→

where n is the normal vector.

2.2.3 Ohm’s Law Ohm’s law is defined, temperature remaining constant, as the potential generated between the two points of a conductor that has direct proportionality with the current flowing through the ground. So I = (φ1 − φ2 )

1 R

(2.16)

where I is the current flowing through this medium, φ1 and φ2 are the potentials at two points in the medium and R, the constant of proportionately, is the resistance offered by the ground. We can now write S A =σ . L l

(2.17)

S · (−φ) · σ I

(2.18)

C =σ Here I =

48

2 Electrical Conduction in Rocks

⇒

φ I =− ·σ S l →

→

⇒ J =σ E.

(2.19)

− → − → This is the differential form of Ohm’s law. Both J and E are vectors, and σ is a scalar in a homogeneous and isotropic medium. It is a tensor in an inhomogeneous and anisotropic medium. The tensor σ in a Cartesian coordinate is given by ⎤ σxx σxy σxz σ = ⎣ σyx σyy σyz ⎦ σzx σzy σzz ⎡

(2.20)

The conductivity tensor has a simple form if two of the orthogonal coordinate directions are selected to lie in the direction of maximum conductivity and minimum conductivity (the principal direction of the conductivity tensor). Here ⎤ σxx 0 0 σ = ⎣ 0 σxy 0 ⎦ 0 0 σzz ⎡

(2.21)

changes to a diagonal matrix where the off diagonal terms are zero. In isotropic minerals or rocks, the three principal values of conductivity remain the same. Isotropic minerals and rocks are very much in the minority. Minerals in which a property such as conductivity depends on direction are termed anisotropic. If the directional dependence remains in the atomic or molecular level, then the anisotropy is termed as intrinsic anisotropy. Aggregates of otherwise anisotropic minerals may show directional dependence and appear to be anisotropic in bulk. Such behaviours are commonly known as ‘structural’ anisotropy. In some rocks and minerals that exhibit uniformity of structure in a plane, two of the three principal values of the conductivity are equal. Such materials are termed gyrotropic. The electrical conductivity in some cases show linear property, but the break-down voltage in air and discharge of thunderstorms are non-linear. Electrical properties of geological bodies are dependent on σ, μ and ∈. In the low-frequency range, as well as in the case of DC, it is dependent on σ. In the highfrequency region, it is both dependent on σ and ∈; in the gigahertz range, ∈ dominates σ. Magnetic permeability μ is generally taken as free-space magnetic permeability in most of the geological terrain unless the area is an iron-ore zone. In magnetite or titanomagnetite areas, free space μ is not equal to μ0 .

2.3 Electrical Permittivity and Displacement Current

49

2.3 Electrical Permittivity and Displacement Current 2.3.1 Dielectric Constant If two opposite faces of an insulator (Fig. 2.5) are charged with potential difference φ applying an external electric field, the charges on the two opposite faces will be given by φC = q

(2.22)

where C is the capacitance of a dielectric, and φ is the voltage across the two faces. Capacitance between the two plates can be defined as the charges needed for a unit-potential difference between the two opposite faces of a capacitor. The unit of capacitance is farad. This name was chosen to honour Michael Faraday. The unit farad = 1 C/volt. More practical units are microfarad (1μf = 10−6 f) and picofarad (1pf = 10−12 f). Equation (2.22) can be written as, l · C · A

  φ q = l A

(2.23)

Here l is the distance between the two opposite faces of a capacitor where ∈ = C A . The capacitance of a body per unit length and unit cross section is termed the electrical permittivity or electrical capacitivity of a medium. Potential per unit width of the →

capacitor is the field E.  →

So E = φl and D = qA is the charge per unit area. It is termed the dielectric flux density. It is also termed the displacement vector. Its unit is coulomb/meter2 . In a vacuum, ∈ = ∈0 = 8.854 × 10−12 farad/meter. When an electric field is applied between two opposite faces of a dielectric material, the potential generated between the two opposite faces of the dielectric depends upon capacitance of the dielectric. Dielectric constant is given by ∈k = ∈0 ∈r where ∈0 = 36π 1x 109 and is the free space electrical permittivity of the medium. ∈r is the relative electrical permittivity of the body with respect to the free-space value. Fig. 2.5 Charging of a cube-shaped dielectric (capacitance) across the two opposite faces

50

2 Electrical Conduction in Rocks

2.3.2 Electric Displacement ψ and the Displacement Vector D Faraday’s experiment on movement of electrostatic charges in different spherical shells has shown that a sphere with charge q, when placed within another spherical shell without touching it and when the outer sphere is momentarily grounded and the inner sphere is removed, the charge on the outer shell is found to be exactly the same as that in the inner sphere but of opposite sign. This is true for all sizes of the sphere and for all dielectric constants of the media. There is a displacement of charges from the inner sphere to those in the outer sphere. The amount of displacement depends only upon the magnitude of the charge q. Thus, the displacement is in coulombs, i.e. ψ = q. The electric displacement per unit area at any point on a spherical surface →

of radius ‘r’ is the electric displacement density D. It is a vector because there is a definite direction for this displacement. So, →

D=

q ψ = 2 4πr 4πr 2

(2.24)

The unit of D is in coulombs/meter2 . The displacement per unit area at any point depends upon the direction of the area, and it is normal to the surface elements. This displacement is along the direction of the field in a homogeneous and isotropic dielectrics. Therefore, we can again write, from Eq. (2.24), →

D= →

q → · r 4πr2 →

D =∈ E .

(2.25) (2.26)

The vector D is also called displacement vector. This is the second basic equation in electromagnetic theory related to electrical conductivity. Maxwell recognised that there must be a constitutive Eq. (2.26) that relates the electric field intensity to the displacement and that defines another property of the medium ∈ known as the dielectric permittivity. Like conductivity, dielectric permittivity is also a tensor since most rocks and minerals are anisotropic. In contrast to electrical conductivity, the dielectric permittivity has a well-defined value even in the absence of matters, and this value is ∈o = 8.854 × 10−12 F/meter in free space. At very high frequencies, σ changes to (σ + iω ) and contributes to electrical conduction.

2.3.3 Tensor Electrical Permittivity In an anisotropic dielectric, the electrical permittivity becomes a tensor and the connecting relation between D and E can be expressed as

2.3 Electrical Permittivity and Displacement Current

51

Dx = ∈11 E x + ∈12 E y + ∈13 E z D y = ∈21 E x + ∈22 E y + ∈23 E z Dz = ∈31 E x + ∈32 E y + ∈33 E z

(2.27)

and in the matrix form ⎤ ⎡ ⎤⎡ ⎤ ∈11 ∈12 ∈13 Ex Dx ⎣ D y ⎦ = ⎣ ∈21 ∈22 ∈23 ⎦⎣ E y ⎦ Dz ∈31 ∈32 ∈33 Ez ⎡

(2.28)

2.4 Magnetic Induction and Magnetic Permeability 2.4.1 Magnetic Induction Magnetic flux B is the number of magnetic lines of forces created by a magnetic field source. Its unit is the Weber. Magnetic-flux density or magnetic induction is the number of lines of forces cut per unit area. Its unit is weber/meter2 or tesla. The direction of B is at right angles to the plane of the loop when the flux cut is maximum. When the loop is in the same plane and is oriented along the direction of the flux, the flux cut will be minimum. The higher the magnetic permeability or magnetic susceptibility, the higher will be the concentration of magnetic-flux lines. Materials that becomes magnetized in the presence of a magnetic field are called magnetic substances. Magnetic-flux density and magnetic induction are vectors. The magnetic flux can be written as  → → (2.29) φ = B · n · ds s

where B is the magnetic-induction vector, n is the direction of the normal to the plane of a coil (see Fig. 2.6) or the magnetic substance; ds is a small element of the area that was induced.

2.4.2 Magnetic Permeability Magnetic induction is connected to the magnetic-field strength by the relation B = μH where μ is the magnetic permeability. A magnetic body, when placed in an external magnetic field H, becomes magnetized and generates a field H of its own. Magnetic Induction is defined as the total

52

2 Electrical Conduction in Rocks

Fig. 2.6 Magnetic flux in a solenoid

field within this body and is given by B = H + H = H + 4πKH = (1 + 4πK) H = μH.

(2.30)

Here H is the magnetic field. Its unit is amperes/meter or ampere turns/meter or teslas. The more practical unit is the nanoteslas or gammas. K is the magnetic susceptibility, and it is expressed in CGS units. Faraday’s law of electromagnetic induction states that the induced emf is proportional to the rate of change of magnetic flux (see Fig. 2.7), i.e. φ=

dI ∂N =L ∂t dt

(2.31)

where N is the number of lines of forces, and L is the constant of proportionality. So the induced emf is proportional to the rate of change of current and also the flux. The constant of proportionality is the self-inductance L. We can write

(a)

(b)

(c)

n

Fig. 2.7 The generation of electric current in a coil when a bar magnet (a) is approaching towards (b) or receding from (c) a single-term coil or a solenoid

2.4 Magnetic Induction and Magnetic Permeability



T

53



φ dt =

LdI = L

dI = LI

(2.32)

0

and magnetic flux T φdt = LI.

ψ=

(2.33)

0

From Eq. (2.33), we can write I ψ =L A A where A is the area of the coil and magnetic induction. We can write from Eq. (2.34)

ψ A

(2.34)

= B, and it is the magnetic flux density or

  l I ψ =L A A l

(2.35)

⇒ B = μ H.

(2.36)

This is the third basic equation of electromagnetics. Here, μ = L Al , i.e. the inductance per unit length and unit cross section is the magnetic permeability (in henrys/meter). The value of the free-space magnetic permeability is μ0 = 4π × 10−7 henry/meter. The unit of inductance is a henry. An inductance on a coil generates extra impedance in the circuit. Magnetic permeability is also a tensor in an inhomogeneous and anisotropic medium, and it is a scalar in a homogenous and isotropic medium. In low-frequency electromagnetics, the square of the propagation constant U2 = iωμσ, and in the very high-frequency range U2 = −ω2 μ and in the several kilohertz to several megahertz range U2 = iωμ (σ + Iω ). That’s how μ contributes in a complex electrical conduction domain in time-varying electromagnetic field.

2.5 Principal Methods of Electrical Conduction 2.5.1 Electronic Conduction (Conduction of Current Through Metals) Many of the metals are the most conducting substances through which electrical current can pass with minimum electric pressure applied to the opposite faces of

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2 Electrical Conduction in Rocks

Fig. 2.8 Electrical conduction model for metallic conductors

a conductor. Copper, gold, iron, lead etc. are very good conductors. Some of the metallic minerals are also very good conductors. Electronic conduction takes places through metals and metallic minerals where very small energy gaps exist between the valence band and the conduction band (see Fig. 2.8). Very weak electric pressure can tear off the electrons from the outermost orbit, and the current starts flowing through the conduction band. These conduction band electrons travel without much resistance in a perfect conductor. However imperfections and defects in the crystals generate resistance in the conduction bands. Most of the metallic minerals and metallic ores are included in this group. Conduction through metals and metallic minerals are always electronic. Tables of highly conducting metals, metallic minerals and sulphides are available in Keller and Frischknecht (1966). For a metal, a rise in temperature increases the vibration of the atoms, and as a result it increases the resistance offered to electrons travelling from the valence band to the conduction band. Therefore, for perfect metals, the higher the temperature, the higher will be the resistivity or lower will be the conductivity. Figure 2.8 is a sketch of the energy gap, expressed in electron volts, needed for transfer of electrons from the valence to the conduction band.

2.5.2 Conduction of Current Through Semiconductors The next mode of conduction of current is flow through semiconductors. Semiconductors are solids through which the mode of conduction of current is through electrons and holes (electron vacancies). Although the mode of conduction is electronic, the number of electrons available in the conduction band for current flow are much less, and the energy required to raise the electrons from the valence band to the conduction band will be much higher. Figure 2.9 shows the conceptual model of the conduction of current through semiconductors. It shows that, in comparison to the level of energy needed for metallic conduction, the energy required will be much higher. In semiconductors, the number of electrons available for current conduction is given by

2.5 Principal Methods of Electrical Conduction

55

Fig. 2.9 Electrical conduction model for semiconductors

ne ∝ e−E/KT

(2.37)

where ne is the number of electrons available in the conduction band, and E, in electron volts, is the energy required to raise the electrons from the valence band to the conduction band where they are free to move through the lattice. K is the Boltzman constant (8.6165 × 10−5 electron volts per kelvin), and T is the absolute temperature in degree Kelvin. The expression for electrical conductivity at a certain temperature is given by σ = σ0 exp(−(E + PV))/kT

(2.38)

Here P is the pressure in kilobars, and V is the activation volume. This relationship suggests that electrical conductivities of semiconductors are very much dependent upon temperature. It is a thermally activated process. At high temperatures, the current + conduction takes place due to jumps of electrons and holes. The higher the temperature, the higher will be the electrical conductivity. In other words, the temperature coefficient will change sign from negative to positive in semiconductors. Amongst the geological materials, metallic minerals, sulphides and oxides are semiconductors. The number of electrons in the conduction band available at a particular temperature varies over a wide range. Therefore electrical conductivity also varies over a wide range, i.e. from nearly a metallic mineral to nearly an insulator. Figure 2.9 shows a sketch of the nature of valence–conduction model for semiconductors showing the increase in the energy gap. For tables of geological semiconductors, see Keller and Frischknecht (1966).

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2 Electrical Conduction in Rocks

2.5.3 Conduction of Current Through Solid Electrolytes The next form of conduction of current through solids is through solid ions and vacancies, which are the charge carriers. Impurities and defects in the crystals enhance the electrical conductivity, reducing the gap of the energy barrier. Figure 2.10 is a sketch of the mode of conduction current in solid electrolytes where the energy barrier is much higher. Impurities and defects in the crystals reduce the energy gap marginally. Most of the rock forming minerals, felsic rocks and other silicates conduct electricity by ionic conduction and face tougher barriers in the sense that much energy will be required to bring the valence ions into a position where they can jump to the conduction band to initiate flow of current. Generally the crust’s mantle silicates have some of the important rock forming minerals through which current flow is mostly ionic and through vacancies, and it flows through the defects in the crystals. It is also a thermally activated process. Ionic movement is temperature dependent. The number of ions jump per unit time increases with increasing temperature and is given by the relationship nji α e−Ui/KT

(2.39)

where nji is the number of ion/vacancies jump per unit time for the ith species of the mineral constituent. U1 is the height of the potential energy barrier for the ith species across which the jumps takes place, K is the Boltzman constant and T is the absolute temperature. An insulator at room temperature can become a good conductor at 1,200–1,400 °C. Direct current find it difficult to flow. Difficult to flow through hard and compact granitic upper crustal rocks. But timevarying electromagnetic waves, which can travel through a vacuum, can diffuse through the highly resistive upper crustal rocks without much attenuation and can reach the deeper rocks inside. That is the starting point of the application of the Earth’s natural electromagnetic fields for geophysical exploration. Figures 2.16, 2.17 and 2.18 show the nature of the variation of the electrical conductivities of the crust Fig. 2.10 Electrical conduction model for solid electrolytes

2.5 Principal Methods of Electrical Conduction

57

Fig. 2.11 Energy gap between conduction and valence bands in dense minerals as a function of composition and crystal structure (Nitsan and Shankland 1976)

mantle silicates based on the published laboratory-measured high-temperature and high-pressure electrical conductivity data (references cited above). Figure 2.11 shows the nature of variations of the energy barriers of a few rock forming minerals. Duba and Shankland (1990) proposed that, in the presence of several mineral constituents in a rock, the geometric mean of the electrical conductivities of the mineral constituents should be taken as 1/3

= (σ1a σ1b σ1c )1/3 e−A1/kT . σ1a e−A1/kT · σ1b e−A1/kT · σ1c e−A1/kT

(2.40)

In general, mafic rocks are more conducting than felsic rocks. Fayalites are more conducting than forsterites. Amongst the mafic rocks, fine-grained mafic rocks are more conducting than coarse-grained mafic rocks (Kariya and Shankland 1983). Electrical conductivity increases with an increase iron, alkali and Titanium oxide (Tio2 ) in the oxides and decreases with the increase in the content of silica (SiO2 ) and magnesium oxides (MgO) (Lastovickova 1981). See the tables of rock forming minerals and their order of electrical conductivities from Keller and Frischknecht (1966).

2.5.4 Conduction of Current Through Dielectric Displacement Conduction of displacement current through dielectrics occurs only in the case of the propagation of high-frequency alternating current. In Maxwell’s second electromagnetic Eq. (2.41), the ∂D/∂t is the displacement current component part.

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2 Electrical Conduction in Rocks

Fig. 2.12 Total loss tangent versus frequency at several temperature. Black dots represent the data, and the solid lines represent the best theoretical fits (Olhoeft et al. 1974)

→

∂B , (i) curl E = − ∂t →

(2.41)

→

∂D , (ii) curl H = J + ∂t →

→

(2.42)

From Maxwell’s equations, the Helmholtz electromagnetic wave equations can be derived. If the electromagnetic field is a harmonically varying field, i.e. E = Eeiωt and H = Heiωt , the Helmholtz wave equations →

∂ 2E ∂E +μ∈ 2 ⇒ ∇ E = μσ ∂t ∂t 2

→

(2.43)

change to the form (Fig. 2.12) →

→

→

(i) ∇ 2 E = iωμ σ E +i2 ω2 μ ∈ E →

→

= iωμ(σ + iω ∈) E = γ2 E

(2.44)

For √ the magnetic  field in Eq. (2.42), E will be replaced by H. Here γ = iωμ(σ + iω ∈) is termed the propagation constant. For the audio or √subaudio frequency range, the displacement current is negligible. √ Therefore, γ = iωμσ. In the megahertz and several hundred kilohertz range, γ = iωμ(σ + iω ∈ ), and both conduction and displacement currents have contributions in the intermediate range where neither conduction nor displacement currents can be neglected. The ratio of the real and imaginary parts ωσ∈ ≈ 1. This is obtained from the complex propagation constant and is expressed as the loss factor or loss tangent. In the gigahertz range, √ γ = iω μ ∈, i.e. displacement currents dominate conduction currents. Conduction currents may be negligibly small. The frequency dispersion of the dielectric constant is very high. Only in radio magnetotellurics (RMT) and the highest frequency zone

2.5 Principal Methods of Electrical Conduction

59

of audiomagnetotellurics (AMT), electrical conductivity has very little contributions from displacement current.

2.5.5 Electrolytic or Ionic Conduction Electrolytic conduction takes place in a liquid due to movement of cations and anions in the opposite direction. In a bath of electrolyte, the flow of current is controlled by the equation: I = AF(C1 v1 + C2 v2 + C3 v3 + · · · Cn vn )

(2.45)

where I is the current flowing between the anode and cathode in an electrolytic bath, and Cn and vn represent the concentration and mobility of each species of ions. Mobility of an ion (anion or cation) is the velocity under unit potential gradient between the positive and negative electrodes in the bath. A is the cross sectional area through which the current flows. F is the Faraday number, i.e. 96,500 coulomb. Electrolytic current conduction through geological medium takes place mostly through pores or void spaces filled with electrolytes. Porosity of a rock is defined as Porosity φ =

Pore volume × 100. Total volume

It is a dimensionless fraction expressed as a percentage. It is a primary porosity or intergranular porosity generally found in argillaceous sedimentary rocks. Secondary porosity comes from the solution cavities, vugs in carbonate rocks like limestone, dolomite etc. Fractures, fissures generate secondary porosities in all kinds of rocks, e.g. sedimentary, metamorphic and igneous rocks. Sedimentary rocks can be highly porous. Porosity can be as high as 45% in shale or clay even with minimum permeability, i.e. the ease with which a fluid can flow through a porous medium. Permeability also has some minor contribution to the electrical conductivity of rocks. Sandstones and porous limestones and dolomites can have porosities of 35% at the most. For clean sandstones, the rock matrix are silica grains which are insulators. Therefore, current is mostly conducted through the pore fluids. With the simplifying assumption that the grains are spherical in shape and of same size, the porosity will be constant; it will be 47.6% for square packing, and this is the maximum porosity possible in pure sandstone. Porosity will be 26% for hexagonal packing provided each grain touches each other. In sedimentary rocks, the porosity will be of the order of 20%, in general. When the silica grains are of different sizes, the porosity will be less than 20%. For hard and compact sediments, the porosity may vary from 10 to 20%. For unconsolidated sediments, the porosity can be as high as 40%. Porosity is a variable parameter. Differences in grain size, angularities, packing and degrees of cementation control the porosity of a formation.

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2 Electrical Conduction in Rocks

Interconnected porosity which contributes to fluid permeability is termed effective porosity, which is a fraction of the total porosity. Secondary porosity in the form of solution cavities are predominant in carbonate rocks. Fractures and fissures are prevalent in hard rocks, be they igneous, metamorphic or sedimentary. Igneous and metamorphic rocks with very low porosity (1% or less) are designated as hard rocks. Current conduction is related to porosity, and it follows Archie’s law; the relationships are: F=

a φm

(2.46)

where F is the formation factor and φ is the porosity of a formation. Here the formation factor F can be written as F=

Ro Rw

(2.47)

Here, Ro is the resistivity of a formation where pore space is occupied by water of resistivity Rw where Ro =

a Rw . φm

(2.48)

and m is the cementation factor, whose value varies from 1.8 to 2.2 and a is a proportionality constant whose value varies from 0.62 to 1.0. When a part of the pore space is occupied by water and a part is occupied by hydrocarbons or any other fluids, then the resistivity of a medium changes from Ro to Rt . Rt is the true resistivity of a formation when a part of the pore space is occupied by water and a part is occupied by oil and gas. Water saturation is Sw . Both oil and gas are more resistive than the saline water. Water is generally saline at depths where we search for oil. Water saturation of a formation is given by Sw = (Pore Space Occupied by Water/Total Pore Space) × 100 in %. Here we assume three models of a rectangular parallelepiped (Fig. 2.13a, b, c) with the same cross-sectional area A1 . The first one is filled with water. Current passes through the cross section A1 and length L1 . In the second model, the rectangular parallelopiped is filled with sandstone of porosity φ, and the effective cross-sectional area available for the flow of current is A2 which is filled with water. Here A2 /A1 = φ. L2 is the effective length the current has actually travelled to move through the tortuous path. In the third model, a part of the pore space is occupied by water, and the remaining part is occupied by oil or gas. So the effective cross sectional area available for the flow of current is A3 . Here A3 /A2 = Sw and A3 /A1 = Sw φ. Now the resistance offered by the two opposite faces in the first model is

2.5 Principal Methods of Electrical Conduction

61

Fig. 2.13 a, b, c. a A rectangular parallelepiped block model filled totally with water. b A rectangular parallelepiped block model of sandstone filled partly with sand grains and partly with water. c A rectangular parallelepiped block model of a sandstone filled partly with sand grains, partly with water and partly with oil

r1 =

L1 · RW A1

(2.49)

The resistance offered by the two opposite faces in the second model is r2 =

L2 · RW A2

(2.50)

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2 Electrical Conduction in Rocks

Here r2 L2 A1 = · r1 L1 A2

(2.51)

where L2 /L1 is the tortuosity of the path. Rw is the resistivity of the formation water. We can write r2 =

L1 · Ro A1

(2.52)

We assumed that the current passes through the water path only. Ro = resisitivity of the formation filled with water of resistivity RW . Now The formation factor is F=

Ro r2 L2 1 = = · RW r1 L1 φ

(2.53)

Since tortuosity also varies inversely with φ, the formation factor F α φ12 . This is Archie’s law. Archie’s formula is given by F = a/φm

(2.54)

where ‘m’ is the cementation factor and ‘a’ is the constant of proportionality. Humble’s formula is given by F=

0.62 φ2.15

(2.55)

where a = 0.62 and m = 2.15. These two equations are the relationships connecting the formation factor and porosity. Now, since A3 = SW φ A1

(2.56)

φ r3 L3 · RW L3 1 . = = · r2 SW · φ L2 · RW L2 SW

(2.57)

Rt r3 L3 1 = = · . Ro r2 L2 SW

(2.58)

∴

A2 = φ, A1

A3 = SW , A2

Now

Since the ratio

L3 α 1 L2 SW

2.5 Principal Methods of Electrical Conduction

63

Rt 1 = −n ⇒ SW = (Ro /Rt )1/n Ro SW

(2.59)

The value of n (saturation exponent) = 2 ∴ SW =

Ro = Rt

FRW Rt

(2.60)

Rt is the true resistivity of a formation. For groundwater exploration, SW = 100%.

2.6 Factors that Control the Electrical Conductivity of the Earth A brief discussion on the various factors that control the electrical conductivity of rocks is given in this section.

2.6.1 Porosity of Rocks Since the silica grains are infinitely resistive, most of the currents flow through the pore fluid. Therefore, electrical conductivity has a direct relationship with grains the porosity of a formation which is again dependent on the nature of packing of the silica grains in the rock Dakhnov’s formula for resistivity of a porous rock is (Fig. 2.14) √ 1 + 0.25 3 1 − α  ρ1 ρ= 1 − 3 (1 − α)2

(2.61)

where ρ1 and α are respectively the matrix grain resistivity and porosity. Sedimentary rocks are mostly porous and the electrical conduction takes place mostly through the pore spaces that are filled with formation water. For hard rocks the percentage of pore space is very low, and therefore the electrical conductivity is also very low unless one encounters the metallic minerals. The conductivity is very low for metallic inclusions when the matrix is non-conducting (σ = 10−5 –10−10 mho/m), and the metallic inclusions are not connected. When the formation liquid is highly saline, the conductivity of the formation may come up to 10−4 –10−2 . In the oil bearing formation, the conductivity may be of the order of 10−1 –10−2 mho/m. For gas, the conductivity is nearly zero and resistivity is very high.

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2 Electrical Conduction in Rocks

Fig. 2.14 a, b, c, d, e, f, g, h, i. a A rectangular model of sand stone with uniform grain size and square packing; pore space is occupied by water. b A rectangular model of shale sandstone where the pore space is occupied by both water and clay. c A rectangular model of assorted sand grains of different sizes; part of the pore space is occupied by both sand and water. d A rectangular model of carbonate rocks, e.g. limestone, dolomite, or anhydrite with solution cavities. e A rectangular model of clay or shale with very fine-grained sand and the major part is occupied by clay minerals (alumina) and water. f A rectangular model of hard rock with cracks filled with water. g A rectangular model of fractured hard rock with interconnected secondary pore spaces occupied by water. h A rectangular model of hard rock with non-interconnected pore spaces occupied by water and surrounded by insulating hard rock matrix. i A rectangular model of a hard rock with interconnected conducting mineral

2.6.2 Conductivity of Pore Fluids The higher the electrical conductivity of the pore fluid, the higher will be the electrical conductivity of the rocks provided the pore spaces are interconnected and the current can flow bypassing an insulating to almost insulating rock matrix (Fig. 2.15).

2.6.3 Size and Shape of Pore Spaces If the volume percentage of pore space is higher, the probability of their interconnectivity increases. If the pore spaces are elongated in shape, there is every likelihood

2.6 Factors that Control the Electrical Conductivity of the Earth

65

Fig. 2.15 Variation of formation factor with porosity of marine sands (taken from Jackson et al. 1978)

that the pore spaces will be interconnected, and the permeability of the pore fluid will increase. Fluid permeability through rocks also has contributes to enhancing electrical conductivity. If the porosity is low in percentage and the pore spaces are not interconnected and they are surrounded by the insulating matrix, direct current electrical conductivity may not increase with the increase in electrical conductivity of the pore fluid. For shale or clay, both the grain size and the pore size will be small, and the fluid permeability is zero. Even then, the electrical conductivity will be high because a major portion of the rock matrix is conducting. Along with silica (SiO2 ), alumina (Al2 O3 ) are also present which will provide the conducting path for the flow of current. In general, the shale or clay are highly porous, and porosity may be higher than that of sandstone. On top of that, it has a conducting matrix. Therefore, impermeable wet shale or clay are highly conducting.

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2 Electrical Conduction in Rocks

2.6.4 Conductivity of Mineral Inclusions For rocks that are practically nonporous, i.e., porosity cannot play any role, electrical conductivity of the mineral grains will dictate the overall conductivity of the rocks. Whether the minerals are metallic or rock-forming silicates or semiconductor type sulphides and oxides, without interconnectivity of the conducting path, the overall electrical conductivity of the rock will not increase.

2.6.5 Size and Shape of Mineral Grains If the mineral grains are highly conducting and they are interconnected having a needle-like shape, electric current can completely overlook the 99.9% insulating matrix and pass through the 0.1% conducting path so the rock may be branded as a conducting rock. Therefore, the interconnectivity of mineral grains is very important. Otherwise, the same rock will be an insulating rock. Graphite is a common accessory minerals found in metasediments primarily obtained as a regional metamorphism of carbonaceous black shale. Graphite may also remain attached as grain boundary graphites in a lower crustal amphibolites facies or granulite facies metamorphic rocks. Graphite is a good conductor and the presence of interconnected graphites are found in the field where the lower crust is exposed. It can significantly increase the electrical conductivity, although the volume percentage of graphite is negligibly small. For example, if the mineral grains are plate like as in the case of muscovites or biotites, the conducting path for the flow of current will be lost.

2.6.6 Temperature Temperature has a profound effect on electrical conductivity of the rock. It has already been mentioned that rock forming sulphides, oxides and silicates (felsics and mafics) have temperature dependent electrical conductivity. For example, an insulator at room temperature can become a good conductor at 1,300–1,400 °C. Figures 2.16, 2.17 and 2.18 show the nature of variations of electrical conductivity of the crust mantle rocks and minerals (silicates) with temperature. Based on this property, only magnetotellurics has become one of the important geophysical tools for mapping magma chambers: (i) in high heat-flow areas, (ii) in volcanic cones, (iii) in mid-oceanic ridges and (iv) in subduction zones. Even hot springs are good targets for electrical conductivity mapping by MT. Figure 2.18 shows the nature of variations of the electrical conductivity of the crust-mantle silicates based on the high-pressure temperature laboratory-measured data on electrical conductivity collected by the authors cited in Sect. 2.1. Silicates (felsics and mafics) have temperature-dependent electrical conductivity. If ρ(θ) is the resistivity of a rock at temperature θ, then we

2.6 Factors that Control the Electrical Conductivity of the Earth

67

Fig. 2.16 Variation of electrical conductivity of olivine with temperature (taken from Constable et al. 1992)

Fig. 2.17 Variation of electrical conductivity of granite, basalt and gabbro within the temperature range of 500–1,000 °C (taken from Kariya and Shankland 1983)

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2 Electrical Conduction in Rocks

Fig. 2.18 Compilation of high-temperature laboratory-measured electrical-conductivity data for crust mantle silicates based on published results

2.6 Factors that Control the Electrical Conductivity of the Earth

69

Fig. 2.19 Variation of dielectric constant of a material with frequency at different temperatures (taken from Olhoeft et al. 1974)

have ρθ =

ρ18◦ 1 + α(θ − 18◦ )

(2.62)

where α = 0.025.

2.6.7 Frequency of Excitation Current It has already been discussed that displacement current through dielectrics becomes dominant at very high frequency (in the megahertz and Gigahertz range) alternating currents. In geophysics, ground-Penetrating Radar (GPR), electromagnetic propagation tool (EPT) in well logging and radiomagnetotellurics (RMT) are the tools where displacement current part is important. This is so because the dielectric constant has very strong frequency dispersion (Fig. 2.19).

2.6.8 Ductility and Degree of Partial Melt in Rocks Ductility is a property that measures to what degree a hard solid gradually loses its hardness with a gradual increase in temperature. In the first phase, a solid becomes

70

2 Electrical Conduction in Rocks

Fig. 2.20 Variation of electrical conductivity of a rock samples with varying degrees of melt fraction (taken from Shankland and Waff 1977)

a semisolid. With the gradual increase in temperature, it then becomes a highly viscous liquid. Viscosity gradually goes down with a gradual increase in temperature. Electrical conductivity has strong dependence on ductility. It is again controlled by the degree of partial melt of a crust-mantle rocks (Shankland and Waff 1977). Detection of the lithosphere–asthenosphere boundary is one of the important targets in magnetotellurics (Fig. 2.20).

2.6.9 Electrical Conductivity of Various Types of Rocks Felsic green schist facies granites and granodiorites with less than 1% porosity is the most resistive of rocks. Laboratory samples of dry granites can have a resistivity on the order of 106 –108 ohm-m. Laboratory samples of mafic–to-ultramafic lower crust to upper mantle rocks with higher concentration of ferromagnesian minerals can have resistivities on the order of 104 –106 ohm-m. The resistivity of felsic or mafic metamorphic rocks will be less than their igneous counterparts. Volcanic rocks like basalts and andesites are less resistive than igneous rocks. High-grade granulite facies rocks are relatively less resistive than low-grade green schist facies rocks. Sedimentary rocks are the least resistive. Hard and compact carbonate rocks like

2.6 Factors that Control the Electrical Conductivity of the Earth

71

limestone and dolomite are more resistive amongst the sedimentary rocks. Shale or clay is the least resistive rock. Sandstone and porous limestone are a little more resistive than shale.

2.6.10 Chemical Activity and Oxygen Fugacity Electrical conductivity depends upon the chemical activity of several ionic species and tend to increase with increasing iron content. The changeable valence state from Fe++ to Fe+++ can change the crystal electronic structure and hence the electrical conductivity. Electrical conductivity has some dependence on oxygen fugacity or partial pressure of oxygen. For hurzburgite and olivine, changes in the partial pressure of oxygen (O2 ) increased the electrical conductivity. Figure 2.21 shows the nature of dependence of electrical conductivity on oxygen fugacity.

Fig. 2.21 Dependence of electrical conductivity on oxygen fugacity (taken from Duba and Constable 1993)

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2 Electrical Conduction in Rocks

Fig. 2.22 Variation of electrical conductivity with temperature and pressure (taken from Shankland et al. 1993)

2.6.11 Dependence of Electrical Conductivity on Pressure Electrical conductivity has some dependence on pressure. Figure 2.22 shows the nature of dependence of electrical conductivity on pressure. In general, the pressure effect is much less than the temperature effect.

2.6.12 Dependence of Electrical Conductivity on Volatiles Amongst the volatiles in crust mantle rocks, e.g. hydrogen (H2 ), water (H2 O), carbon dioxide (CO2 ), carbon monoxide (CO), nitrogen (N2 ) and argon (Ar), the contribution of water is maximum towards enhancing the electrical conductivity of the crustmantle rocks (Haak and Hutton 1986). Hydrogen ions can also increase the electrical conductivity. Karato (1990) hypothesised that solubility and diffusivity of hydrogen in upper-mantle olivine can significantly enhance the electrical conductivity of the crust-mantle silicates. Figure 2.23 graphs the variation of electrical conductivity with hydrogen-ion concentration. Chemically bound volatiles like H2 O, CO2 and SO2 increase conductivity under high pressure and temperature.

2.6.13 Major Geological Zones of Weaknesses Fractures, fissures, suture zones, lineaments, shear zones, contacts of rocks of different geological ages, faults planes, thrusts and overthrusts are generally the weak zones in a geological formation and having higher electrical conductivities relative to the major geological formations of the adjacent areas or host rocks.

2.7 Piezoelectric Effect

73

Fig. 2.23 Variation of electrical conductivity of dry olivine with temperature at different degree of presence of hydrogen ion concentrations; i.e. the role of hydrogen in the electrical conductivity of the upper mantle (taken from Karato 1990)

2.7 Piezoelectric Effect Piezoelectric phenomena is observed within a single crystal or in aggregate of crystals (rocks) with an appropriate internal structure. The fundamental features of minerals and rocks that exhibit piezoelectricity is due to the absence of a centre of symmetry. When the piezoelectric effect is present, there is a linear relationship between the components of the polarisation intensity vector and the mechanical stress vector. Because the piezoelectric effect occurs only in anisotropic media, the modulus of piezoelectricity d, which is a physical property characterising a rock or mineral, relates the polarisation vector P to the mechanical-strain tensor S and is written as P = dS

(2.63)

The polarisation is a vector with three components, and the strain is a tensor with nine components. So the modulus of piezoelectricity have 27 components. Because of the symmetry of some of the components, the modulus of piezoelectricity can be written in terms of 18 components as

74

2 Electrical Conduction in Rocks S11

S22

S33

S23

S31

S12

P1

d111

d122

d133

d123

d131

d112

P2

d211

d222

d233

d223

d231

d212

P3

d311

d322

d333

d323

d331

d312

2.8 Hall Effect Edwin Hall in 1879 observed that, when an electric current passes through a conductor placed in a magnetic field, a potential proportional to the current and to the magnetic field is developed across the conductor in a direction both perpendicular to the current and to the magnetic field. This effect is known as Hall Effect. It is given by EH = R B × J

(2.64)

where EH is the additional electric field generated by the Hall effect, and R is the Hall coefficient—a property of the material. B and J are respectively mutually orthogonal magnetic and electric fields. Hall coefficients are largest in metals and small in other types of conductors. With the measurement he made, Hall was able to determine for the first time the sign of the charge carriers in a metallic conductor and semiconductor.

2.9 Maxwell’s Geoelectrical Conductivity Models Maxwell (1892) tried to find a theoretical model of electrical conductivity of rocks based on certain simplifying assumptions. It gives a rough qualitative to semiquantitative idea about the nature of variations of electrical conductivity of rocks with variations of porosity, conductivity of the pore fluid and conductivity of the matrix. In the case of a soft sedimentary rocks, the variations of porosity conducting fluid in the pore space are important. Matrices are generally of very high resistivity. In the case of a hard rock, the variation in conductivity in the presence of conducting mineral grains with their relative abundance are presented. The simplifying assumptions are that the grains are spherical in shape in one case and ellipsoidal in the other. Electrical conductivity in presence of alternating current is also presented based on this model. From electromagnetic theory (Stratton 1941), we know that for a sphere of conductivity σ2 , if placed in a medium of conductivity σ1 , then the current density inside the sphere is (Figs. 2.24 and 2.25) σ1 − σ2 − Jo → − → Jo → J2= σ2 + 2σ1

(2.65)

2.9 Maxwell’s Geoelectrical Conductivity Models

75

Fig. 2.24 Diagram of variations of current density in a spherical conductor in a uniform field

Fig. 2.25 Diagram of the variation of current density within a spherical model of composite mixture of a rock with different percentages of conducting mineral grains

where J0 is the current density of the host rock. When a part of this body is filled with a material of conductivity σ2 , and the rest part is filled with materials of conductivity σ1 inside the spherical body. The current density will be σ1 − σ − → − → Jo J2= σ + 2σ1

(2.66)

This is the main assumption of Maxwell and the current density inside the body is − → − → J =βJ2

(2.67)

where J = J2 for β = 1. Here β is a fraction and is given by β=

Volume occupied by the grain Total volume of the rock

Here, α + β = 1, where α and β are respectively the porosity and the partial volume occupied by the grains. Now if we have several grains having a great enough distance between them such that the initial assumption of current density J0 is valid. Therefore, σ1 − σ2 σ1 − σ =β σ + 2σ1 σ2 + 2σ1

(2.68)

Here σ is the electrical conductivity of the composite mixture of materials present in the rocks. The value of σ can be determined by:

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2 Electrical Conduction in Rocks

σ = σ1

σ2 (1 + 2β) + 2σ1 (1 − β) σ2 (1 − β) + σ1 (2 + β)

(2.69)

This expression can be written in terms of porosity α by putting β = 1 − α, so σ = σ1

(3 − 2α)σ2 + 2ασ1 α σ2 + (3 − α)σ1

(2.70)

This is the expression for average conductivity of a rock that contains spherical grains. Now we are interested to see how electrical conductivity is controlled by porosity in the case of a soft rock and the presence of volume percentage of conducting mineral grains β in the case of a hard rocks.

2.9.1 Soft Rock Here the assumptions are σ1 is the electrical conductivity of the pore fluid, and σ2 is the electrical conductivity of the matrix. When σ2 σ1 , the expression for electrical conductivity (Eq. (2.70)) reduces to σ = σ1

2α . 3−α

(2.71)

Here electrical conductivity is independent of the conductivity of the grains, and it is totally controlled by the conductivity of the pore fluid and porosity. Graphical Plot α

0.01

0.05

0.1

0.25

0.40

0.60

0.80

0.90

σ σ1

0.007

0.03

0.07

0.13

0.36

0.50

0.71

0.90

Figure 2.26 shows the nature of variation in electrical conductivity with an increase in porosity. That is, in a sedimentary rock, where the pore space is filled up with a Fig. 2.26 Graph of the dependence of electrical conductivity of the soft rock on pore fluid and porosity

2.9 Maxwell’s Geoelectrical Conductivity Models

77

Fig. 2.27 Graph of the variation of electrical conductivity of a hard rock on variation of electrical conductivity of the spherical grains and their concentration

liquid, the electrical conductivity is completely controlled by the first few percentages of porosity. Only a small change in porosity can cause significant change in the electrical conductivity.

2.9.2 Hard Rock In the case of a hard rock, igneous or metamorphic rock, we assume that σ2 σ1 . In this case, σ1 stands for the conductivity of the matrix, and σ2 stands for the conductivity of the spherical grain conducting minerals. Equation 2.70 reduces to σ = σ1

1 + 2β 3 − 2α ≈ σ1 . 1−β α

(2.72)

When conducting minerals are not interconnected, the conductivity of the rock will be controlled by the conductivity of the matrix. In the second case, when the grains are highly conductive, as in the case of a sulphide ore body, we have the second formula. Figure 2.27 shows the nature of variation of electrical conductivity with a gradual increase in β. Graphical Plot β

0.01

0.05

0.1

0.4

0.6

0.8

0.9

σ σ1

1

1.2

1.3

3

5.6

13

25

The effect is negligible unless the concentration of the grain is very high, of the order of 80–90%.

2.9.3 Ellipsoidal Grains Next we consider the case of an ellipsoidal grain. The electrical conductivity will − → depend upon the direction of the current density J . Let us take the case of ellipsoidal grains. Figure 2.28 show the three axes a, b, c in a ellipsoid. We assume that the two

78

2 Electrical Conduction in Rocks

Fig. 2.28 A model of an ellipsoidal grain

horizontal axes are equal. The third vertical axis is either too big or too small in comparison to other two horizontal axes. (i) When a = b c → one gets needle-like grains as in tourmaline or graphite. (ii) When a = b c → one gets plate like grains as in mica.  

a2 The eccentricity e e = 1 − c2 exists either in the ac or bc plane and e = 0 in the ab plane. Now, in the case of spheroidal grains, the conductivity will be different when the current moves along the c direction. The general expression for the electrical conductivity is given by σ(K) = σ1

α(nk − 1)σ1 + [nK − (nK − 1)α]σ2 (nK − α)σ1 + ασ2

(2.73)

Here, k = 1, 2, 3. The values of n1 , n2 and n3 are given by n1 = n2 =

2e3 e − (1 − e2 ) tan h−1 e

(2.74)

and n3 =

2e3 (1 − e2 )(tan h−1 e − e)

From here, we can deduce the special cases.

(2.75)

2.9 Maxwell’s Geoelectrical Conductivity Models

79

Case I. When e = 0, then n1 = n2 = n3 = 3. σ(1) = σ(2) = σ(3) = σ1

2ασ1 + (3 − 2α)σ2 . (3 − α)σ1 + ασ2

(2.76)

Then we can define the average conductivity as σ=

σ(1) + σ(2) + σ(3) 3

(2.77)

Case II. When c a, i.e. when e → 1, then n1 = n2 = 2 and n3 = ∞. Hence σ(1) = σ(2) = σ1

ασ1 + (2 − α)σ2 (2 − α)σ1 + ασ2

σ(3) = α σ1 + (1 − α)σ2

(2.78) (2.79)

Now we shall take these two cases, i.e. (i) when the matrix is conducting and (ii) when grains are conducting. Here we have (a) when σ1 σ2

σ(1) = σ(2) = σ1

α 2−α

(2.80)

and σ(3) = α σ1 . σ=

4−α σ(1) + σ(2) + σ(3) = ασ1 3 3(2 − α)

(2.81)

(b) σ2 σ1 , σ(1) = σ(2) =

2−α .σ1 α

σ(3) = (1 − α)σ2 . So the average conductivity is given by σ=

1 2 2−α · σ1 + (1 − α)σ2 3 α 3

(2.82)

80

2 Electrical Conduction in Rocks

≈

1 (1 − α)σ2 3

(2.83)

So in the case of needle-like bodies, the next most important thing is when the matrix is conducting σ = f(σ1 ) and when the grain are conducting σ = f(σ2 ). Therefore, in a rock if we have a low conducting matrix, the presence of a small percentage of conducting elongated grains can significantly increase the conductivity of rock due to the short-circuiting effect. Even 2–3% of inclusion of needle-like conducting grains are sufficient for a significant increase in conductivity if they are interconnected. Case III. e → i α for plate-like bodies, we get the values of n1 , n2 and n3 as follows

n1 = n2 = ∞, n3 = 1 σ(1) = σ(2) = α σ1 + (1 − α)σ2 σ1 σ2 (1 − α)σ1 + ασ2

σ(3) =

(2.84)

(a) When σ1 σ2 σ(1) = σ(2) = ασ1 σ2 = 0. σ(3) = 1−α Hence the average conductivity is σ=

2ασ1 3

(2.85)

(b) Now, if σ2 σ1 σ(1) = (1 − α)σ2 σ(3) = 0 Hence, the average conductivity σ=

2(1 − α) σ2 3

(2.86)

Here, also in a plate-like bodies, the conductivities are controlled by σ2 . The following figures show the nature of the variation of overall conductivity with the

2.9 Maxwell’s Geoelectrical Conductivity Models

81

variation of the conductivities of the two components we are dealing with in this section. For α = 0.25 (i.e. for 25% porosity), [A] e → 0 σ1 σ2 [B] e → 1 σ1 σ2 [C] e → iα σ1 σ2

σ2

σ1

σ2

σ1

σ2

σ1

σ = 0.18 σ1 σ = 0.11 σ1 σ = 0.18 σ1 σ = 0.25 σ2 σ = 0.17 σ1 σ = 0.25 σ2

(2.87)

When the matrix is conducting, the conductivity of a rock does not depend upon the shape of the grain, and in every case it remains the same. When the grains are conducting, the conductivity of a rock with spherical grains will be smaller than the rocks with needle-like or plate-like bodies. Therefore, for rocks it is very essential to ascertain whether or not the rocks contain elongated grains.

2.9.4 Alternating-Current Conduction Displacement current can pass through the dielectrics when the field is time varying. Therefore, J = σ E changes to J = σ E + ∂∂tD , and for a harmonic field it is J = (σ + iω)E. This is the modified Ohm’s law when the displacement current is not negligible and the conductivity is a complex quantity. Here we can assume that σ2 changes to σ2 and σ1 changes to σ1 . Substituting these values in Eq. (2.71), we get σ  = σ1

(3 − 2α)σ2 + 2ασ1 ασ2 + (3 − α)σ1

(2.88)

We can write Eq. (2.88) as (σ + iω) = (σ1 + ω1 )

(3 − 2α)(σ2 + iω2 ) + 2α(σ1 + iω1 ) α(σ2 + iω2 ) + (3 − α)(σ1 + iω1 )

(2.89)

Let (i) (σ + iω) = X + iY = Zeiφ

(2.90)

(ii) (σ1 + iω1 ) = A + iB = C eiφ1

(2.91)

(iii) (3 − 2α)(σ2 + iω2 ) = M + iN = P eiφ2

(2.92)

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2 Electrical Conduction in Rocks

(iv) 2α(σ1 + iω1 ) = R + iS = T eiφ3

(2.93)

(v) (σ2 + iω2 )α = U + iV = Weiφ4

(2.94)

(vi) (3 − α)(σ1 + iω1 ) = E + iF = Geiφ5

(2.95)

Then we can write Z eiφ = Ceiφ1

Peiφ2 + Teiφ3 Weiφ4 + Geiφ5

(2.96)

Equation (2.90) can further be simplified to Z eiφ = Z k eiφk

(2.97)

Using the standard technique of separation of real and imaginary parts of complex variables, one can obtain expressions for σ and . Case A: We are interested in some special cases, i.e. when the matrix grains are of very low conductivity. Pore fluids are of very high electrical conductivity, i.e. (a) When σ1 σ2 , we get σ1 = σ1 and σ2 = iω2 , so the conductivity of the medium is σ + iω =

(3 − 2α)iω2 + 2ασ1 iω2 + (3 − α)σ1

(2.98)

This is much more simplified version of Eq. (2.89), and the real and imaginary parts can be written as σ = σ1

2ασ12 (3 − α) + ω2 22 α(3 − 2α) σ12 (3 − 2α)2 + ω2  2 α 2

(2.99)

9(1 − α)σ1 2 σ12 (3 − α)2 + ω2  2 α 2

(2.100)

and  = σ1

These two are the expressions for the electrical conductivity and electrical permittivity when the pore fluid is highly conducting. Now Case I When the frequency of the signal is zero, i.e. for the static DC case (ω → 0) σ (0) =

2α σ1 3−α

(2.101)

2.9 Maxwell’s Geoelectrical Conductivity Models

83

and (0) =

9(1 − α) (3 − α)2

(2.102)

Case II When the frequency is infinitely high, i.e. for ω → ∞ σ (∞) =

3 − 2α σ1 α

(2.103)

and (∞) = 0

(2.104)

As ω increases, the dielectric constant or electrical permittivity decreases. For α = 0.25, i.e., for 25 % porosity, σ (0) = 0.18σ1

(2.105)

σ (∞) = 10 σ1

(2.106)

So there will be a significant change in electrical conductivity with frequency. In case of electrical permittivity, (0) = 0.08

(2.107)

(∞) = 0

(2.108)

Case: B When the highly conducting mineral are present in the rocks in the form of mineral inclusion or in the form of insulating matrix rock, as in the case of granitic rock, we can consider the following cases, i.e. when σ2 σ1 , we get σ1 = iω1 σ2 = σ2

(2.109)

Equation (2.89) changes to the form σ + iω = iω1

(3 − 2α)σ2 + 2αiω1 ασ2 + (3 − α)iω1

(2.110)

The real and imaginary part are respectively given by σ =

9(1 − α)ω2  2 σ22 + (3 − α)2 ω2 12

α 2 σ22

(2.111)

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2 Electrical Conduction in Rocks

and =

1 σ22 α(3 − 2α) + ω2 12 2α(3 − α) α 2 σ22 + 3 − α)2 ω2 12

(2.112)

Case-I: When ω → 0 σ (0) = 0

(2.113)

3 − 2α α

(2.114)

σ (∞) = σ2

9(1 − α) (3 − α)2

(2.115)

(∞) = 1

2α (3 − α)

(2.216)

(0) = 1 Case-II: When ω → ∞

For regular and spherical grains and for 25% pore spaces, i.e. for α = 0.25 σ (0) = 0

(2.117)

(0) = 10 1

(2.118)

σ (∞) = 0.9

(2.119)

(∞) = 0.181

(2.120)

Therefore, in the case of a solid hard rock, the increase in electrical conductivity is relatively less in comparison to the rate of decrease in dielectric constant as ω increases. It can be shown that 2α 3 − 2α > α 3 − 2α For α = 0.25, i.e. for 25% porosity of a rock, σ (0) = 0.18σ1

(2.121)

σ (∞) = 10σ1

(2.122)

∞(0) = 0.08

(2.123)

2.9 Maxwell’s Geoelectrical Conductivity Models

85

(∞) = 0

(2.124)

Therefore, electrical conductivity increases with frequency but dielectric constant decreases with frequency.

2.10 Resistivities of Metals and Metallic Minerals (1) (2) (3) (4)

Sodium → 4.3 × 10−8 ohm-m Iron → 9.0 × 10−8 ohm-m Copper → 1.6 × 10−8 ohm-m Gold → 2.0 × 10−8 ohm-m

2.11 Semiconducting Minerals (1) Chalcopyrite → Fe2 S3 . Cu2 S → 150–9000 × 10−6 (2) Chalcocite → Cu2 S → 80–100 × 10−6 (3) Galena → PbS → 6.8 × 10−6 to 9.0 × 10−2 at normal temperature and pressure.

2.12 Electrical Conductivity of Some Common Metallic Ores σ Chalcopyrite

10–103

Pyrite

103 –104

Pyrrhotite

104 –105

Anthracite

104 –102

Coal

10−2 –10−6

Magnetite

102 –104

Galena

103 –105

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2 Electrical Conduction in Rocks

2.13 Some Common Geological Good and Bad Conductors Good conductors

Bad conductors

Graphite

Hematite

Pyrrhotite

Zinc Blende

Pyrite

Braunite

Chalcopyrite

Sphalerite

Galena

Chromite

Magnetite

References Anderssen, R.S., J.F. Devane, Gustfason, and D.E. Wincj. 1979. The qualitative character of the global electrical conductivity of the earth. Physics of the Earth and Planetary Interiors 20: 15–21. Brace, W.F., and A.S. Orange. 1968. Further studies of the effects of pressure on electrical resistivity of the rocks. Journal of Geophysical Research 73 (16): 5407–5420. Constable, S., and A.G. Duba. 1990. Electrical conductivity of olivine: a dunite and the mantle. Journal of Geophysical Research 95 (B5): 6967–6978. Constable, S., T.J. Shankland, and A.G. Duba. 1992. The electrical conductivity of an isotropic olivine mantle. Journal of Geophysical Research 97 (B3): 3397–3404. Duba, A.G., and C. Constable. 1993. The electrical conductivity of Lherzolite. Journal of Geophysical Research 98: 11885–11899. Duba, A.G., Schock, E.L. Arnold, and T.J. Shankland. 1990. An apparatus for measurement of electrical conductivity to 1500 degrees at known oxygen fugacity. Geophysical Monograph Series, American Geophysical Union 56: 207–209. Duba, A.I., H.C. Heard, and R.N. Schock. 1974. Electrical conductivity of olivine at high pressure and under controlled oxygen fugacity. Journal of Geophysical Research 79 (11): 1667–1673. Dvorak, Z. 1973. Electrical conductivity of several samples of olivinites, peridotites and dunites as a function of pressure and temperature. Geophysics 38 (1): 14–24. Haak, V. 1980. Relation between electrical conductivity and petrological parameters of the crust and upper mantle. Surveys In Geophysics 4: 57–69. Haak, V., and R. Hutton. 1986. Electrical resistivity in continental lower crust. In The nature of the lower continental crust, No. 24, ed. J.B. Dawson, D.A. Carswell, J. Hall, and K.H. Wedepohl, 35–49. Geological Society Special Publication. Hermance, J.F. 1979. Electrical conductivity of materials containing partial melts. Geophysical Research Letters 6 (7): 613–616. Hirsch, L.M., and T.J. Shankland. 1993. Quantitative olivine defect model: Insights on electrical conduction, diffusion, and the role of Fe content. Geophysical Journal International 114: 21–35. Hirsch, L.M., T.J. Shankland, and A.G. Duba. 1993. Electrical conduction and polaron mobility in Fe-bearing olivine. Geophysical Journal International 114: 36–44. Hyndman, R.D. 1988. Dipping seismic reflectors, electrically conductive zone and trapped water in the crust over a subducting plates. Journal of Geophysical Research 93 (B11): 13391–13404. Jackson, P.D., D. Taylor Smith, and P.N. Stanford. 1978. Resistivity-porosity-particle shape relationships for marine sands. Geophysics 43 (6): 1250–1268. Jones, A.G. 1982. Electrical conductivity of the continental lower crust, Chapter 3, geological survey of Canada contribution no. 17492.

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Karato, S. 1990. The role of hydrogen in the electrical conductivity of the upper mantle. Nature 347: 272–273. Kariya, K.A., and T.J. Shankland. 1983. Electrical conductivity of dry lower crustal rocks. Geophysics 48 (1): 52–61. Keller, G.V., and F.C. Frischknecht. 1966. Electrical methods of geophysical prospecting. New York: Pergamon Press. Lastovickova, M. 1981. Electrical conductivity of garnets and garnet bearing rocks. Gerlands Beitrage zur Geophysik 90 (6): 529–536. Lastovickova, M. 1983. Laboratory measurements of electrical conductivity of rocks and minerals. Geophysical Surveys 6: 201–213. Lastovickova, M. 1987a. Electrical conductivity of some rocks from the Indian subcontinent, Studio. Geoph et Geodetika 31: 60–72. Lastovickova, M. 1987b. Electrical conductivity of some minerals at high temperature and for an extended period. Physics of the Earth and Planetary Interiors 45: 204–208. Madden, T.R., G.A. Latorraca, and S.K. Park. 1993. Electrical conductivity variations around the Palmdale section of the San Andreas fault zone. Journal of Geophysical Research 98 (B1): 795–808. Maxwell, J.C. 1892. A treatise on electricity and magnetism, 3rd ed., vol. 1, Chapter 9. Oxford, UK: Clarendon. Mitchell, A.J., and Landisman. 1971. Electrical and seismic properties of the Earth crust in the south western great plains of the USA. Geophysics 36 (2): 363–381. Nitsan, U., and T.J. Shankland. 1976. Optical properties and electronic structure of mantle silicates. Geophysical Journal of the Royal Astronomical Society 45: 59–87. Olhoeft, G.R. 1977. Electrical conductivity of the water saturated basalt, preliminary results, USGS Technical Report, Denver, Colorado 80225. Olhoeft, G.R., A.L. Frissilo, and D.W. Strangway. 1974. Electrical properties of lunar soil sample 15301, 38. Journal of Geophysical Research 79 (11): 1599–1604. Rai, C.S., and M.H. Manghnani. 1978. Electrical conductivity of ultramafic rocks to 1820 Kelvin. Physics of the Earth and Planetary Interiors 17: 6–13. Roy, K.K. 2007. Potential theory in applied geophysics. Germany: Springer. Schock, R.N., A.G. Duba, and T.J. Shankland. 1989. Electrical conduction in olivine. Journal of Geophysical Research 94 (B5): 5829–5839. Schwarz, G. 1990. Electrical conductivity of the Earth’s crust and upper mantle. Surveys In Geophysics 11: 133–161. Shankland, T.J. 1975. Electrical conduction in rocks and minerals: parameters for interpretation. Physics of the Earth and Planetary Interiors 10: 209–219. Shankland, T.J. 1981. Electrical conduction in mantle materials. Evolution of the Earth, Geodynamics Series 5: 256–263. Shankland, T.J., and M.E. Ander. 1983. Electrical conductivity, temperature and fluids in the lower crust. Journal of Geophysical Research 88 (B11): 9475–9484. Shankland, T.J., and A.G. Duba. 1990. Standard electrical conductivity of isotropic and homogenous olivine in the temperature range of 1200 degree to 1500 degree centigrade. Geophysical Journal International 103: 25–31. Shankland, T.J., and H.S. Waff. 1974. Conductivity in fluid bearing rocks. Journal of Geophysical Research 79 (32): 4863–4868. Shankland, T.J., and H.S. Waff. 1977. Partial melting and electrical conductivity anomalies in the upper mantle. Journal of Geophysical Research 83 (33): 5409–5417. Shankland, T.J., O’Connell, and H.S. Waff. 1981. Geophysical constraints on partial melt in the Upper mantle. Reviews of Geophysics and Space Physics 19 (3): 394–406. Shankland, T.J., J. Peyronneau, and J.P. Poirier. 1993. Electrical conductivity of Earth’s lower mantle. Nature 366: 453–455. Shur, M. 2004. Physics of semiconductor devices. New Delhi: Prentice Hall of India. Stratton, J.A. 1941. Electromagnetic field theory. New York: McGraw Hill.

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Tozer, D.C. 1979. The interpretation of upper mantle electrical conductivities. Tectonophysics 56: 147–163. Xu, Yousheng, Brent T. Poe, T.J. Shankland, and D.C. Rubie. 1998a. Electrical conductivity of olivine, Wadsleyite and Rinwoodite under upper mantle conditions. Science 280: 1415–1418. Xu, Yousheng, C. Mc Cammon, B.T. Poe. 1998b. The Effect of alumina on the electrical conductivity of silicate perovskite. Science 282: 922–924 (Reprint Series of the American Association for the Advancement of Science). Zhdanov, M.S., and G.V. Keller. 1994. The geoelectrical methods in geophysical exploration. Amsterdam: Elsevier.

Chapter 3

Signal Processing

Abstract In this chapter, we briefly introduce a few topics of signal processing that are quite used by the geophysicists, e.g. (i) Fourier series, (ii) the Fourier transform, (iii) the discrete, Fourier transform, (iv) the fast Fourier transform, (v) Shannon’s sampling theorem, (vi) linear filter, (vii) convolution, (viii) Z transform, (ix) autocorrelation, (x) cross correlation, (xi) autopower spectra, (xii) cross power spectra, (xiv) noise, (xv) robust statistics and robust processing. Keywords Signal processing · Robust statistics · Geoelectricity

3.1 Introduction Any scientific data collected on the surface of the Earth, from the marine environment (off-shore and deep marine), upper atmosphere (low-flight imagery-airborne geophysics, high-flight imagery-satellite borne remote-sensing data), borehole-(well logging) are geophysical signals. These scientific signals provide some information about the Earth. They are called gross earth functionals. They may be time varying or time invariant. A signal is defined as any physical quantity that varies with time or space or any other independent variable that contains scientific information having units and dimensions that not just pure numbers. Mathematically, a signal is a function of more than one independent variable. Scientifically it is a response of a medium due to a particular type of excitation from a source of a variable nature. These responses are the gross earth functionals or scientific data we are talking about. These data have a direct relationship with the physical fields, e.g. gravity, magnetic, electrical, electromagnetic, seismic, thermal, radioactive fields etc. They vary either with time or space or any physical property of the earth or the origin of the signals. These signals are contaminated by noise to a greater or lesser extent when they are collected from the geological field or from the surface of the Earth. Because the environment is polluted by noise of many types, they are discussed in detail in Sect. 3.24. That is precisely the reason why signal processing is so important in geophysics. Moreover, signal processing as such is a vast subject. Scientists of various disciplines, such as physics, electrical communication, electrical power, geophysics, astrophysics, aerospace etc., require these methods to process data. Many textbooks © Springer Nature Switzerland AG 2020 K. K. Roy, Natural Electromagnetic Fields in Pure and Applied Geophysics, Springer Geophysics, https://doi.org/10.1007/978-3-030-38097-7_3

89

90

3 Signal Processing

are available for postgraduate and research students, e.g. Childers and Durling (1980), Claerbout (1976), Kesavamurthy and Narayana Iyer (2009), Proakis and Malonakis (1996), Smith (2010), Manolakis and Ingle (2012), Ingle and Proakis (2000), Huber (1981), Draper and Smith (1968), Hawkins (1980) and Lathi (2007), Jenkins and Watts (1969), Jain (1995), Mitra (2001), Patra and Mallick (1980). In this chapter, a few points from these references and needed for this subject are touched upon very briefly. In geophysics, there are many fields that are time varying, especially, e.g. earthquake seismology, seismic methods for geophysical exploration and electromagnetic fields in geophysics. Therefore, time series analysis both in the time and frequency domains is an integral part of geophysics. Take, for example: (i) Ex (t), the x-component of the electric field, is a function of time and (ii) Ex (ω); the x-component of the electric field, is in the frequency domain (a function of angular frequency (ω); and (iii) HZ (x, y), is the vertical component of the earth’s magnetic field at different points on the surface of the Earth. Here, x and y are space variables. The data are collected from naturally existing electromagnetic fields. These naturally existing fields of global presence are the subject matter of this discussion. Therefore, the presence of a field, natural or artificial, is required for the generation of signals that are associated with one or more than one physical property of Earth, irrespective of whether they are periodic or aperiodic. Earth’s natural electromagnetic-field-dependent geophysical tools are based on time series analysis. The important steps needed in geophysical signal processing are: (i) editing of the time series and removal of the noisy sectors; (ii) moving average algorithm; (iii) detrending algorithm; (iv) autocorrelation; (v) cross correlation; (vi) the Fourier transform; (vii) the discrete Fourier Transform and fast Fourier transform; (viii) integral transform; (ix) auto and cross power spectra; (x) windows; (xi) filters; (xii) Shannon’s sampling theorem; (xii) aliasing; (xiii) convolution; (xvii) noise removal; and (xviii) robust processing. These points are touched upon very briefly.

3.2 Selection of Block Size The time series signals can be divided into various sectors. The processing is done by dividing the time series into blocks. Superposition is done in the frequency domain. For magnetotellurics (MT) and geomagnetic depth sounding (GDS), analyses are done in the frequency domain. For electrotellurics, most of the data analyses are done in the time domain. Here, different sections of the time series or different parts of the frequency spectrum can be taken at a time for further analysis. Stacking of the spectra from different blocks is the common practice before further analysis.

3.3 Editing of Time Series

91

3.3 Editing of Time Series Different parts of the entire time series collected from the field must be scrutinized thoroughly, and the bad sectors then removed, as far as practicable. We analyse the time series in which the data quality is good. Manual editing is a must if the data quantity is not enormously large. Machine learning and editing is done these days to remove bad data as far as practicable. The remaining portions of noises are removed at the data processing stage by robust processing. There should be a correlation between the corresponding signals.

3.4 Moving Average Algorithm The moving average filter operates by averaging a number of points from the input signal to produce each point in the output signal. In equation form, this is M −1 1  x(i + j) M j=0

(3.1)

x(1) + x(2) + x(3) + x(4) + x(5) 5

(3.2)

y(i) = where y(3) =

As an alternative, the group of points from the input signal can be chosen symmetrically around the output point. For instance, in a ten-point moving average filter, j can range from 0 to eleven (one-side averaging) or −5 to +5 (symmetrical averaging). Symmetrical averaging demands that M be an odd number (Fig. 3.1).

3.5 Trend Elimination Here any systematic deviation of the signals from the x-axis has been removed. The mean value (bias) is set to zero and a straight line in the data (trend), which differs from the x-axis, is removed. Here A is the time series (raw data with trend), and B is the time series without a trend. The effect of trend elimination is shown in Fig. 3.2.

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Fig. 3.1 Graph showing how the moving average uncovers the signal from the scattered data

Fig. 3.2 Graph showing the trend elimination from a time series

3.6 Fourier Series

93

3.6 Fourier Series The mathematical representation of any periodic function in physics or geophysics is practically the origin of the Fourier series and the Fourier integral (Kreyszig 1993; Pipes 1958). Any periodic function that satisfies Dirichlet’s conditions, i.e. (i) the signal x(t) has a finite number of discontinuities in any period, (ii) the signal x(t) contains a finite number of maxima and minima, (iii) the signal x(t) is absolutely integrable in any period, i.e. the signal, which appears in a geophysical field in the form of a time series, can be represented by a Fourier series. Later on, it was realized that even in an aperiodic function that can be expressed in terms of a Fourier integral only the limit of integration changes from some finite value to infinity. Let f(x) is a periodic function of period 2π , then it can be represented as 1 an = π 1 bn = π

+π f (x) cos nxdx

(3.3)

−π

+π f (x) sin nxdx

(3.4)

−π

These an and bn are known as Fourier coefficients and Eqs. (3.3) and (3.4) are the Fourier series.

3.7 Complex Fourier Series Using well known trigonometric relationships, einx = cos nx + i sin nx

(3.5)

e−inx = cos nx − i sin nx,

(3.6)

we get, after a few steps of algebraic simplifications, f(x) =

∞ 

cn einx

(3.7)

f (x)e−inx

(3.8)

−∞

and 1 cn = 2π

+π −π

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3 Signal Processing

Equations (3.7) and (3.8) are respectively the complex Fourier series and complex Fourier coefficient.

3.8 Fourier Series for Discrete Time-Period Signals For a periodic sequence x(n) with period N, i.e. x(n) = x(n + N) for all n, the Fourier series representation for x(n) consists of N harmonically related exponential functions ei2πkn/N for k = 0, 1, 2, … N – 1 and can be expressed as x(n) =

N −1 

ck ei2πkn/N

(3.9)

k=0

where {ck } are the coefficients of the series representation. It can be shown, within a few steps of algebraic simplification, that ck =

N −1 1  x(n)e−i2πkn/N . N

(3.10)

k=0

Equation (3.9) is the discrete time series, and ck (Eq. (3.10)) are the Fourier coefficients.

3.9 Integral Transforms Integral transform is a very powerful and important topic in mathematical physics and geophysics, e.g. Zhdanov (1988), Kreyszig (1993), Pipes (1958), Ghosh (1970). There are many forms of integral transforms, i.e. the (i) Fourier transform, (ii) Laplace transform(iii), Hilbert transform, (iv) Walsh transform, (v) Henkel transform, (vi) Z-transform, (vii) Hurtley transform, (viii) Abel transform, (ix) Melin transform and many others. With these mathematical tools, one can take the problem from one variable to the other where the problem will be easier to solve or it becomes scientifically more meaningful or acceptable. The problem is solved in that domain, and through an inverse transform one can come back to the domain of origin. This process has some similarity with the conformal transformation in complex variables where one can go from one plane to another where the problem will become easily solvable. One can then return to the original plane to present the results. An integral transform simply means unique mathematical operations through which a real or a complex-valued function f is transformed into another function F as a set of data that can be observed or measured experimentally. In electrical and electromagnetic methods, the original function f(t) may represent a signal that

3.9 Integral Transforms

95

is a function of time. The Fourier transform F(ω) of f(t) represents the frequency spectrum of the signal f(t), and it is physically useful as the time representation of the signal itself. It is often more important for the geophysicists to work with F rather than f. Conversely, given the frequency spectrum F(ω), the original signal can be reconstructed. The integral transform has wide application in (i) electrical, electromagnetic and seismic and other methods in geophysics. In this chapter, we shall briefly discuss certain aspects of integral transforms, e.g. (i) the discrete Fourier Transform and (iii) the fast Fourier transform. In magnetotellurics, the electric fields Ex(t), Ey(t) and magnetic fields Hx(t), Hy(t), Hz(t) measured in the time domain are transformed to frequency-domain response Ex(ω), Ey(ω), Hx(ω), Hy(ω), Hz(ω) using the fast Fourier transform. Most of the data analyses and inversions are done in the frequency domain, although the raw field data are collected in the time domain in the form of a time series. In geomagnetic depth sounding (GDS), most of the qualitative diagnostics are done when the data are in time domain, e.g. identification of magnetic storms, solar quiet days, spherics or atmospherics due to thunder storm activities, pulsations, micropulsations, other short- or long-period variations etc. In electrotellurics, most of the data analyses are in the time domain only. Very recently some frequency-domain data analysis has started in electrotellurics. For depth sounding, i.e. to find out the depth of the lower mantle and upper mantle boundaries, e.g. the 670-km discontinuity, frequency domain (GDS) data are used.

3.10 Fourier Transform The Fourier transform is one several integral transforms used in geophysics. Using this transform, we transfer the geophysical time-domain data in the form of time series to frequency-domain data at discrete frequencies for further analysis. Many linear boundary value or initial value problems in geophysics or mathematical physical problems can be solved using the Fourier transform, Fourier cosine transform or Fourier sine transform. The expression for Fourier transform is +∞ g(ω)eiωt d ω F(t) =

(3.11)

−∞

Here t is in the time domain, and ω is in the frequency domain. This is the integral by which we can change time-domain data to frequency-domain data which we do regularly in data analysis. The equation is 1 g(ω) = 2π

+∞ F(t)e−iωt dt −∞

(3.12)

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This integral is known as the inverse Fourier transform. It changes the signal from the frequency domain to the time domain while keeping the information content unaltered. It is a symmetrical problem.√Many authors try to enhance the symmetry of the problem by writing the factor 1/ 2π rather than having no factor before the integral (Pipes 1958). Let g(x) be an aperiodic function of space variable ‘x’; then the transformation is given by +∞ g(x)e−2πif (x) dx G(f) =

(3.13)

−∞

In the Fourier transform, the variables from one domain to the other are reciprocals of each other. If the initial data are in the time domain, then the Fourier transform will be in the frequency domain. If the initial data are in the space domain, then the Fourier transform will be in the reciprocal of space domain. G(f) is in general a complex quantity and can be expressed as G(f) = A(f ) + iB(f ), where A(f) and B(f) are respectively the real and imaginary components. The condition for the transform are (a) g(x) has finite number of discontinuities (b) g(x) is integrable (Figs. 3.3 and 3.4). The inverse Fourier transform converts the function back to its own domain from the frequency representation and is stated as Fig. 3.3 A sinusoidal signal in time domain

Fig. 3.4 A spike in the frequency domain spectrum at one particular frequency after Fourier transform operation

3.10 Fourier Transform

97

∞ g(ω) =

G(f )e

2πif (x)

∞ =

df

−∞

  G(f )df e2πif (x)

(3.14)

−∞

The equation points to the fact that an aperiodic function can be synthesised by an infinite aggregate of sinusoids of all possible frequencies. The span of the frequency-domain response is generally termed as the spectrum. The time-domain to frequency-domain transfer is termed spectral analysis. Signals of all possible frequencies present in the time-domain data are reflected in the total spectrum. g(x) and G(f) are the different modes of representation of the same quantity; they are known as the Fourier transform pair and denoted as g(x) ↔ G(f ). If F is the Fourier transform operator, then F−1 is the inverse Fourier transform operator. Both are linear integral operators depending on the nature of the problem, for example, in electromagnetics ‘x’ and f are respectively replaced by ‘t’ and ‘ω’ and the Fourier transform pair is defined as ∞ F{f (t)} = F(ω) =

f (t)e−iωt dt

(3.15)

−∞

and −1

F {F(ω)} = f(t) =

1 2π

∞ F(ω)eiωt dω

(3.16)

−∞

Thus a time-varying periodic signal in the form of a time series in time domain changes to a spectrum within a wide frequency range because infinitely many sinusoids come together in a time series. In the frequency domain, one gets a naturally gifted spectrum of a wide frequency range from the Earth’s natural electromagnetic field. In short, the Fourier transform maps a function (or signal) of time t to a function of frequency ω. There are two major advantages of transforming the time-domain field data to the frequency domain. Firstly, their representation in the frequency domain is often more easily interpretable. Secondly, the mathematical relationships that describe the filtering process are simplified, making the effect of the filter much easier to understand. The amplitude and phase spectra of the normal complex frequency-domain representation is given by X(f) = U(f) + iV (f )

(3.17)

In terms of amplitude X(f) and phase ∅(f ), X(f) is called the amplitude spectrum (Eq. 3.19) and ∅(f ) (Eq. 3.20) is the phase spectrum.   X(f) = X (f ) ei∅(f )

(3.18)

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where X(f) =



(U 2 + V2 )

(3.19)

V (f ) U (f )

(3.20)

and ∅(f ) = arctan

An example of such a Fourier transform pair is the sinc function rectangular box car function discussed in the next section.

sin x x

and the

3.11 Sinc Function A box car function in the time domain generates a sinc function in the frequency domain. Let   1 for |t| ≤ T /2 (3.21) w(t) = 0 for |t| ≥ T /2 The Fourier transform of this rectangular function is ∞ W(f) =

T

w(t)e−i2πft dt =

−∞

2

e−i2πft dt

(3.22)

− T2

T

2 =2

cos(2π ft)dt 0

sin 2π ft T /2 sin(π fT ) = | =T πf o π fT This

sin x x

(3.23)

is the sinc function (Fig. 3.5a, b).

3.12 Two-Dimensional Fourier Transform The Fourier transform can be extended to functions with more than one independent variable. Let g(x, y) be an aperiodic function of real variables x and y.

3.12 Two-Dimensional Fourier Transform

99

Fig. 3.5 a Boxcar function in the time domain; b sinc function in the frequency domain (Lathi 2007)

∞ ∞ g(x, y)dxdy ≤ const < ∞

If

(3.24)

−∞ −∞

Then the two-dimensional Fourier transform is ∞ ∞ G(kx · ky ) =

g(x, y)e−i2π (kxx +kyy ) dxdy

(3.25)

−∞ −∞

The two-dimensional inverse Fourier transform for the arbitrary space or time variables x and y is given by ∞ ∞ g(x, y) =

G(kx · ky )ei2π (kxx +kyy ) dkx dky

(3.26)

−∞ −∞

If x and y represent two spatial variables, then G(kx · ky ) describes g(x, y) in a two-dimensional wave-number domain. The wave numbers are number of cycles

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per unit distance. For the three-dimensional problem, the wave number vector k has the components kx , ky , kz . The absolute value of k is the reciprocal of the wave length λ.

3.13 Aperiodic Function and Fourier Integral In a spectral representation of an aperiodic function 1 Xn = T

T

x(t)e−i2πnft dt for n = 0, ±1, ±2, ±3,

(3.27)

0

if a function x(t) satisfies the Dirichlet’s condition within an arbitrary interval and if +∞ x(t)dt converges, then x(t) can be expressed as the Fourier integral the integral −∞

+∞ x(t) = X (f )ei2πft df

(3.28)

−∞

where +∞ X(f) = x(t)e−i2πft dt

(3.29)

−∞

Here f represents frequency in cycles per second or Hertz. Since ω = 2π f , we can write Eqs. (3.28) and (3.29) as 1 x(t) = 2π

+∞ X (ω)eiωt dω

(3.30)

−∞

where +∞ X(ω) = x(t)e−iωt dt.

(3.31)

−∞

If x(t) is absolutely integrable, then the Fourier integral Eq. (3.28) converges for all the real values of f. X(f) is then a representation of x(t) in the frequency domain. X(f) is a complex-valued function and is a complex spectrum of x(t). Equation (3.27) represents the aperiodic function x(t) in the form of a spectrum with a wide frequency

3.13 Aperiodic Function and Fourier Integral

101

band. X(f) is the spectral density of x(t). We can write X(f) = F(x(t) and x(t) = F−1 (X(f)). Therefore the Fourier transform is a powerful tool to solve problems in the diversified field by transforming a function from its function domain to the frequency domain. Let g(x) be an aperiodic function of the space variable x. Then the transform is given by ∞ G(f) =

g(x)e−i2πfx dx

(3.32)

−∞

Here G(f) is a complex quantity and can be split into G(f) = A(f) + iB(f) where A(f) and B(f) are respectively the real and imaginary components. Here, in the case of space variables, also the conditions for the transform are (i) g(x) has finite number of discontinuities (ii) g(x) is integrable. The inverse Fourier transform converts the function back to its own domain from the frequency representation as follows ∞ g(x) =

∞ G(f )e

−∞

i2πfx

df =

G(f )df ei2πf (x) dx

(3.33)

−∞

An aperiodic function can be represented by an infinite aggregate of sinusoids of all possible frequencies. Also, in the space domain, g(x) and G(f) are different modes of representation of the same quantity, and they are termed the Fourier transform pair and denoted by g(x) ↔ G(f)

3.14 Discrete Fourier Transform In the discrete Fourier transform (DFT), a continuous time-series signal is discretised at regular intervals, say T as shown in Fig. 3.6. Now these discretized digitised data retrieve the entire information from the continuous time series collected from the field. We denote a continuous data record x(t) by x(nT) in the discretised domain where T represents the sampling interval. The data are periodic and complex having real and imaginary parts, and it repeats itself after travelling a complete circle in the complex domain, as, for example, if we take the total number of data N = 8. Then the positions of x and W (defined in this section) will be x(0), x(1), x(2), x(3), …

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Fig. 3.6 Discretisation of a continuous time series in time domain; discretisation interval

x(7) and W(0), W(1), W(2), … W(7). In other words, it will go from N = 0 to N − 1. After that, the process will be repeated as shown in the Fig. 3.7. Consider the following representation N −1 1  X(mf) = A(mf ) = x(nT )e−i2πmfnT N n=0

=

Fig. 3.7 Circular shifting approaches; sequence x(n) using circular mapping

N −1 1  x(nT )e−i2πmn/N N n=0

(3.34)

(3.35)

3.14 Discrete Fourier Transform

103

=

N −1 1  x(nT )W −mn N n=0

(3.36)

where the definitions of the above terms are known. W is termed the twiddle factor. Here, f = 1/(nT ) is the discrete form. The coefficients X(mf) = A(mf ) are the discrete Fourier coefficients, and their plots are known as Fourier spectrum X(nT). The coefficients are in general complex and periodic. The real and imaginary coefficients can be combined as the sum of the squares and plotted as the magnitude squared which is frequently called the power spectrum. Now let us call the term of the equation N −1 1  x(nT ) N n=0

(3.37)

N −1 1  x(nT )W −n N n=0

(3.38)

 1 x(0) + x(T )W −1 + x(2T )W −2 + · · · N

(3.39)

X(0) = X(f) = X(f) =

X((N − 1)f) = X((N − 1)f) =

N −1 1  x(nT )W −(N −1)n N n=0

 1 x(0) + x(T )W −(N −1) + x(2T )W −2(N −1) + · · · N

(3.40) (3.41)

The following properties are of interest. Note that W −K = [W −(N −K) ] *

(3.42)

where * denotes the complex conjugate [W −(N −k) ] * = [e−i2π(N −k)/N ] *

(3.43)

= ei(2π/N )(N −k) = ei2π · e−i2πk/N = W −k

(3.44)

*  W −1 = W −(N −1)

(3.45)

*  W −2 = W −(N −2)

(3.46)

Thus

which in turn implies that

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X(f) = X ∗ ((N − 1)f)

(3.47)

X(2f) = X ∗ ((N − 2)f)

(3.48)

X(N/2 − 1)f ) = X ∗ ((N/2 − 1)f)

(3.49)

provided N is even. Now X(nT) =

N −1 

X (mf )W mn

(3.50)

 N −1 1  −mn x(nT )W W +lm N n=0

(3.51)

m=0

which can be verified as follows X(IT) =

N −1 



m=0 N −1 

 W

m(l−n)

=

m=0

N if I = n(mod(N)) 0 if I = n(mod(N))

(3.52)

which is an orthogonality condition. We, therefore, obtain X(Nf) = =

N −1 1  x(nT )W −nN N n=0

(3.53)

1 [x(0) + x(T )W −N + x(2T )W −2N · · · ] N

(3.54)

1 [x(0) + x(T ) + x(2T ) · · · ] N

(3.55)

=

= x(0) since W −N = W 0

(3.56)

X(Nf) = x(0)

(3.57)

Thus

Similarly X((N + 1)f ) = X (f ) X((N + 2)f ) = X (2f ) and so on. Thus the discrete spectrum is repeated or extended due to sampling. That is the magnitude of the DFT. This is called the periodic extension of the spectrum.

3.14 Discrete Fourier Transform

105

For N = 8, X(2f) =

1 [1 + W −2 + W −4 + W −6 ± · · · ] 8

=0 X(3f) =

(3.58) 1 [1 + W −3 + W −6 + W −9 ± · · · ] 8 √ 1 = [1 + i(1 − 2)] 8

(3.59) (3.60)

3.15 Fast Fourier Transform The fast Fourier Transform is based on an algorithm meant to drastically reduce the computation time for the discrete Fourier transform. Cooley and Tukey (1965) first presented this idea. This algorithm utilises the symmetry properties of the trigonometric functions to achieve a considerable saving of computer time. This is done by breaking a large matrix into a large number of small matrices while keeping the result same. To calculate the discrete Fourier transform of the time series Xj , j = 0, 1, 2, … N − 1, it is necessary to break the complex domain data into smaller and smaller parts until one needs to compute a two-by-two matrix. This reduces the computation time very significantly. Therefore, the number of data points should be of an even number, and it has to be a power of two. Therefore, the number of data points should be made equal to 2, 4, 8, 16, 32, 64, 128 … 4096 … To prepare the discrete digital data set compatible for using FFT, zero padding may be necessary on both sides of the data set so that total number of data is a power of two. The FFT operates by decomposing an N point time-domain signal to two-by-two time-domain signals so that computations are done at the two-by-two stage. An eight-point signal is decomposed through three different stages. The first stage breaks the eight-point signals into two signals each consisting of four points. The second stage decomposes the data into four signals of four points. This process continues until there are n signals each of one point. There are log2 N stages required in this decomposition, as, for example, a 16-point signal needs four stages, a 512-point signal needs seven stages and 4,096 points needs twelve stages. Binary coding is done for each signal (Fig. 3.8). Using the symmetry and periodicity property of the time-domain sinusoidal and cosinusoidal signals, we can write the following equations (Manolakis and Ingle 2012; Childers and Durling 1980) WNkn = WNk(n+N ) = WN(k+N )n (periodicity in k and n)

(3.61)

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Fig. 3.8 Different stages of breaking the matrix to reduce the computation time; eight-point decimation in the FFT algorithm (Proakis and Malonakis 1989)

WNk(N −n) = WN−kn = (WNkn ) * . (complex conjugate symmetry)

(3.62)

The real and imaginary parts of W give real and imaginary parts of seven additional terms of W−n for N = 2k values around the unit circle with no complex multiplication. For example, consider the case for N = 32 where W−1 = e−i2π/N = 1 −

i2π = α + iβ N

(3.63)

where α = cos 2π/N and β = − sin 2π/N . The Cooley and Tukey (1965) algorithm is based on the division (or decimation) of time series into smaller and smaller sequences until the DFT is performed on N records of one point each. The decimation in time can be accomplished by first dividing the records into two time series each of N/2 samples. Here N must be a highly composite number capable of being written as the product of many smaller numbers. Here, in their algorithm N = 2k , although it can be generalized to any other base. The decimation in time is accomplished by taking alternate values from the new sequence of the form

3.15 Fast Fourier Transform

107

Y(nT) = x(2nT) (even index samples)

(3.64)

Z(nT) = x(2n + 1)T (odd index samples)

(3.65)

and

For n = 0, 1, 2, … (N/2 − 1). Alternatively the data may be divided into first half and second half of the data record. DFT X(mf) =

N −1 1  x(nT )W −mn N n=0

(3.66)

involves N2 complex multiplication and approximately N2 additions which is readily seen by expanding the summation and observing that there are N complex multiplications and N − 1 additions for each ‘m’. But, since there are N such terms, approximately N2 computations are required. Thus if we apply this equation to the two records of length N/2, we need 2(N/2)2 multiplications. Decimating again, we obtain four records which require a total of 4(N/4)2 multiplications. Continuing to divide the records k times (N = 2k ), we get N records of one point each requiring a total of 2k (N/2k )2 = N multiplication and no addition. For the FFT, the butterfly type of computation is suggested. For N = 8, Eq. 3.62 changes to the matrix form of Eq. (3.67) (Manolakis and Ingle 2012). ⎤ ⎡ 1 X [0] ⎢ X [1] ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎢ X [2] ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎢ X [3] ⎥ ⎢ 1 ⎥=⎢ ⎢ ⎢ X [4] ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎢ X [5] ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎣ X [6] ⎦ ⎣ 1 X [7] 1 ⎡

1 W8 W82 W83 W84 W85 W86 W87

1 W82 W84 W86 1 W82 W84 W86

1 W83 W86 W8 W84 W87 W82 W85

1 W84 1 W84 1 W84 1 W84

1 W85 W82 W87 W84 W8 W86 W83

1 W86 W84 W82 1 W86 W84 W82

⎤ ⎤⎡ 1 x[0] ⎥ ⎢ W87 ⎥ ⎥⎢ x[1] ⎥ 6 ⎥⎢ W8 ⎥⎢ x[2] ⎥ ⎥ ⎥ ⎥⎢ W85 ⎥⎢ x[3] ⎥ ⎥ ⎢ ⎥ W84 ⎥⎢ x[4] ⎥ ⎥ ⎢ ⎥ ⎢ x[5] ⎥ W83 ⎥ ⎥ ⎢ ⎥ W82 ⎦⎣ x[6] ⎦ x[7] W8

(3.67)

Changing the order of the matrix columns and the elements of the right-hand side vector in the same way does not alter the final result. Thus we can put the columns for the samples x[0], x[2], x[4], X[6] (even) first and then the columns for x[1], x[3], x[5], x[7] (odd). If we use the identity W84 = −1 and rearrange the matrix Eq. (3.67) by grouping even and odd terms, we obtain

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⎤ ⎡ 1 X [0] ⎢ X [1] ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎢ X [2] ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎢ X [3] ⎥ ⎢ 1 ⎥=⎢ ⎢ ⎢ X [4] ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎢ X [5] ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎣ X [6] ⎦ ⎣ 1 X [7] 1 ⎡

1 W82 W84 W86 1 W82 W84 W86

1 W84 1 W84 1 W84 1 W84

1 W86 W84 W82 1 W86 W84 W82

1 W8 W82 W83 −1 −W8 −W82 −W83

1 W83 W86 W8 −1 −W83 −W86 −W8

1 W85 W82 W87 −1 −W85 −W82 −W87

⎤ ⎤⎡ 1 X [0] ⎥ ⎢ W87 ⎥ ⎥⎢ X [1] ⎥ ⎢ ⎥ W86 ⎥⎢ X [2] ⎥ ⎥ ⎥ ⎥⎢ W85 ⎥⎢ X [3] ⎥ ⎥ ⎥⎢ −1 ⎥⎢ X [4] ⎥ ⎥ ⎢ ⎥ ⎢ X [5] ⎥ −W87 ⎥ ⎥ ⎢ ⎥ −W86 ⎦⎣ X [6] ⎦ X [7] −W85

(3.68)

Now it is possible to partition the N = 8 DFT matrix into 2 × 2 matrix with 4 × 4 blocks. If we use the identity W82 = W4 , we can write the matrix Eq. (3.67) as 

XT XB



 =

W4 D8 W4 W4 −D8 W4



xE xo

 (3.69)

where W is a 4 × 4 DFT matrix and the diagonal matrix D8 are given by ⎡

1 ⎢1 W4 = ⎢ ⎣1 1

1 W4 W42 W43

1 W42 1 W42

⎤ ⎡ 1 1 0 0 ⎢ 0 W8 0 W43 ⎥ ⎥ and D8 = ⎢ ⎣0 0 W2 W42 ⎦ 8 W4 0 0 0

⎤ 0 0 ⎥ ⎥ 0 ⎦ W83

(3.70)

Here the subscripts T, B, E and O are respectively top, bottom, even and odd entries into the vectors respectively. If we define the N/2 point DFTs for N = 8, then we get XE = W N2 xE

(3.71)

XO = W N2 x0

(3.72)

The N-point DFT as in Eq. (3.69) can be expressed as XT = XE + DN XO

(3.73)

XB = XE − DN XO

(3.74)

That provides all the ingredients for the computation of DFTs as shown in Fig. 3.9.

3.16 Dirac Delta Function

109

Fig. 3.9 Diagram showing the format for computation of DFT coefficients in an FFT algorithm (Manolakis and Ingle 2012)

3.16 Dirac Delta Function A function x(t) that has a certain value at t = t0 and is zero everywhere, i.e. at t = t0 and with unit amplitude is the Dirac delta function or the unit-impulse response, i.e. when δ(t) = 0 for t = t0 ,

(3.75)

mathematically we can write the Dirac delta function as +∞ δ(t − to)x(t)dt = x(to)

(3.76)

−∞

and +∞ δ(t)dt = 1

(3.77)

−∞

when t = t0 . The delta function is a pulse whose width approaches zero as its amplitude approaches infinity while the area under the curve of the function describing the function remains constant. The Fourier transform of the delta function can be obtained by substituting e−i2πft for x(t):

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3 Signal Processing

. F(δ(t)) = F(δ(t − to)) .. to = 0 +∞ . δ(t − to)e−i2πft dt .. to = 0 = −∞

. = e−i2πft .. to = 0 = 1

(3.78)

The spectrum of the delta function is constant for all the frequencies. The spectrum of the function +∞ δ(t − to)e−i2πft dt = e−i2πft o = 1 F(δ(t − to)) =

(3.79)

−∞

and the phase spectrum ∅(f ) = −2π fto

(3.80)

3.17 Shannon’s Sampling Theorem A continuous periodic analog signal is digitised at some discrete and equal time interval t in the process of analog-to-digital conversion, and, if it is possible to reconstruct the analog signal from the limited number of digitized data, then the sampling of data is done correctly. For all the information present in the analog signal can be retrieved from the limited and finite number of digitised data, then the sampling must be done properly. In that case, only one analog signal can be produced from the digitised data. With the increase in frequency of the sinusoidal or cosinusoidal signal, it becomes increasingly more difficult to digitise the data properly as mentioned. The digitised data can generate a sinusoidal signal of altogether different frequency. This phenomenon of a sinusoid changing frequency is called aliasing. In that case, the digital data is no longer uniquely related to a particular analog signal. Shannon’s sampling theorem starts from here. Shannon’s fundamental sampling theorem says that, if a signal g(t) has a frequency spectrum ranging from zero to some maximum frequency fmax = ω, then the signal g(t) can be completely determined from the values of g(t) sampled at a series of instants separated by t where (Ghosh 1970) t =

1 2fmax

(3.81)

3.17 Shannon’s Sampling Theorem

111

Consider a signal g(t) having a spectrum G(f), then ∞ g(t) =

G(f )e2πift df

(3.82)

g(t)e−2πft dt

(3.83)

−∞

where ∞ G(f) = −∞

For a finite frequency band 0 to ω, one gets ω g(t) =

G(f )e2πift df

(3.84)

−ω

If the sampling rate is 2ω, then at the sampling instant T=

n = nt 2ω

(3.85)

therefore, ω g(t) = g(nt) =

G(f )e−i2πfnt df

(3.86)

−ω

We defined G(f) in the region −ω ≤ f ≤ ω, and we can expand G(f) in the Fourier series form in the interval G(f) =

∞ 

Cn e−2πifnt dt

(3.87)

G(f )e−i2πfnt dt

(3.88)

−∞

where the Fourier coefficients are 1 Cn = 2ω

∞ −∞

Comparing this equation with the previous one Cn =

1 · g(nt), 2ω

(3.89)

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it indicates that, once we have the values of the function at the sample point, we can find the coefficients Cn in the signal spectrum, and we can, therefore, reconstruct the signal. Substituting G(f) =

∞ 1  g(nt)e−i2πfnt, 2ω −∞

(3.90)

we get 1 g(t) = 2ω

∞  ∞

 g(nt)e

i2πf (t−nt)

df

(3.91)

−∞

−∞

Interchanging the summation and integration, we get  ∞ 1  g(t) = g(nt) ei2πf (t−nt) df 2ω −∞

(3.92)

 i2πω(t−nt)  ∞ e − e−i2πω(t−nt) 1  g(nt) = 2ω −∞ i2π (t − nt)

(3.93)

∞ 1  sin 2π ω(t − nt) = g(nt) 2ω −∞ π (t − nt)

(3.94)

ω

−ω

Therefore, g(t) =

∞ 

g(nt)

−∞

sin 2π ω(t − nt) 2π ω(t − nt)

(3.95)

Thus it is seen that the value of g(t) can be obtained from knowledge of the function at the sample point in the form of a sinc function. In short, the following steps are to be remembered: (a) Fixation of the sampling Interval is

∇x =

1 1 or fc = 2fc 2x

(3.96)

where fc is the Nyquist frequency. In order to avoid distortion of the spectrum, t must be chosen so that the largest frequency in x(t), i.e. fmax is smaller than the 1 or t < 2f 1 Nyquist frequency of the spectrum, i.e. fmax < fny = 2t max

3.17 Shannon’s Sampling Theorem

113

(b) Reconstruction: A band-limited function sampled according to the condition stated is completely recovered through the sample values. The process of reconstruction of the function at the intermediate points is effected by replacing the sample values by the sinc function. The sinc function truly refines the original function. This is gr (x) =

∞  −∞

g(m · x)

sinπ (x − mx)/x π (x − m · x)/x

(3.97)

where gr (x) is the reconstructed function. g(m · x) are sample values of the original function at the sample points x, 2x, 3x mx. (c) h(x) = Pulse response of the function (d) Operations in the frequency domain: (i) B(f) = Spectrum of sinc function × H (f ) (ii) H(f) = Frequency characteristics of the linear filter response (iii) G(f) = Frequency representation of the sinc response. The sinc response can be recovered from B(f) by applying the inverse Fourier transform.

3.18 Linear Filter A linear filter is defined by its response to an input of a Dirac delta-unit impulse function, δ(x). This function is the limiting form of an even rectangular function where the height or the amplitude tends to become infinite at the base. Equations 3.77 and 3.79 signifies that the total area under the curve is one. Graphically, it is represented as a spike of unit height. A physical system is said to be linear when an excitation g(x) applied to it and the response function given by it can be expressed by a linear equation, i.e. the graph between the two quantities should be a straight line. The important properties of the linear system are (Ghosh 1970): If f1 (x), f2 (x) and f3 (x) are respectively the response to the excitation g1 (x), g2 (x) and g3 (x), then f1 (x) + f2 (x) + f3 (x) should be the combined response to the combined excitation of (g1 (x) + g2 (x) + g3 (x)). This is known as the principle of superposition, and it is inherent in any linear system. (ii) If the excitation g(x) is multiplied by a real constant “C”, then the response will be multiplied by the same constant for all the values of “C” and g(x) (iii) A linear system is said to be x-invariant if the excitation, shifted by x0 , is a real constant, and the response will also be shifted by the same constant. That is, f(x − x0 ) will be the response of the excitation g(x − x0 ). (i)

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3 Signal Processing

(iv) A linear system is generally termed a filter when the excitation and the response to the system represents the input and the output. A linear filter is defined by its response to an input of a Dirac delta-unit impulse function δ(x). This function is the limiting form of an even rectangular function where the height of the amplitude tends to become infinite ∞ as the base δ(x) approaches zero. It is an integrable function such that −∞ δ(x)dx = 1. This equation signifies that the total area of the function under the curve is one as mentioned earlier. (v) The response of this filter to an input of this spike function is known as the unit-impulse function of a filter h(x). Its job is to weigh an input with its pulse response to produce an output. The output is thus expressed in terms of the applied input and the characterising function show the Dirac delta function.

3.19 Convolution Convolution is an operation that produces the output of a linear time-invariant impulse response for a discrete-time system. It is given by a summation operation, and for a continuous system it will be an integral known as a convolution integral. A signal can be decomposed into a group of components called impulses. An impulse is a signal composed of all zeros, except a single nonzero point. In effect, impulse decomposition provides a way to analyse signals one sample at a time. When impulse decomposition is used, the operation is termed as a convolution. The problem in the analysis of any linear system, in the form of a continuous-time series, is determining the response (or output) of the system to a specified input or excitation. One such elementary signal is the impulse. The convolution method amounts to breaking up the signal, whatever be the level of complicacy, into a series of impulses very close to each other. If we know the response to an impulse, then the response to the whole series of impulses will be the summation of the constituent responses due to the individual impulses, taking into account their delay in occur. Thus the complete response to an arbitrary input is known if we know a priori the response of an impulse input. The summation that is carried out in this way results in the response to an arbitrary input. This process is called the convolution integral. Impulse response is present in a system when a delta function is the input. The impulse response to a unit impulse is given the symbol h(n). Mathematically the action of a linear filter can be represented by a convolution integral. The symbol ∗ is used to signify the convolution. Let us assume that the input g(x) is being convolved in the filter with the pulse response of the filter h(x). This can be stated as f(x) = g(x) ∗ h(x) ∞ g(τ )h(x − τ )d τ = −∞

(3.98)

3.19 Convolution

115

Convolution is a formal mathematical operation like addition, subtraction or multiplication. Convolution takes two signals and produces the third signal. In a linear system, convolution is used to describe the relationship between the three signals in use, i.e. input signal, the impulse response and the output signal. Suppose that a signal (b0 , b1 ) of length two is the input to the digital filter with an impulse response function (a0 , a1 ) also of length two. The output, which is equal to the convolution of the input with the impulse response function, is a wavelet of length three and is given by (c0 , c1 , c2 ) = (a0 b0 , a0 b1 + a1 b0 , a1 b1 )

(3.99)

Thus the convolution of two wavelets a = (a0 , a1 , a2 , . . .)

(3.100)

b = (b0 , b1 , b2 , . . .)

(3.101)

c = a ∗ b(c0 , c1 , c2 , . . .)

(3.102)

is defined to be

where the coefficient of c is given by the formula ct =

t 

as bt−s

(3.103)

s=0

The asterisk ∗ is used in this way, i.e. as a binary operation between the two discrete time functions, and it denotes convolution. To illustrate the use of this formula, suppose a = (a0 , a1 ) = (2, 1)

(3.104)

(b0 , b1 ) = (3, 4)

(3.105)

and

As mentioned already, each of these wavelets has two coefficients or has a length of two. Therefore c = a ∗ b = (c0 , c1 , c2 ) = (6, 11.4)

(3.106)

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3 Signal Processing

where c0 = a0 b0 = 2 × 3 = 6 c1 = a0 b1 + a1 b0 = 2 × 4 + 1 × 3 = 11 c2 = a1 b1 = 1 × 4 = 4

(3.107)

Convolution is a folding operation. To see how the convolution may be performed by folding, one needs to construct a table whose entries are the products of the wavelets ‘a’ and ‘b’ which are in the margin of the square and which are going to fold successively on the dotted lines and get the convolved wavelet. Convolution may also be regarded as an operation on series written on movable strips, and finally we slide this strips on one or more notches.

An input signal x(n) enters into a linear system h(n) resulting in an output signal, i.e. x(n) → Linear system (h(n)) → y(n)

(3.108)

x(n) ∗ h(n) = y(n)

(3.109)

The physical significance of the concept of convolution is a folding operation. The impulse function is displaced and then folded back and operated with the input to yield the output. The value of the integral remains the same whichever way the two functions being convolved is displaced and folded. Convolution is commutative. The next important property is that convolution in the time domain changes to multiplication in the frequency domain, i.e. X(f) = H(f) × Y (f ) or simply X(f) = H(f)Y(f)

(3.110)

3.19 Convolution

117

where X(f), Y(f) and H(f) are respectively the Fourier transforms of x(n), y(n) and h(n). Data analysis becomes advantageous because it gives a simple relation between input, output and impulse response. And, the filter characteristics are given by H(f) = X(f)/Y(f). We can characterise the input–output relationship of a linear system by the convolution operation involving the input and the impulse-response function of a system. A general signal x(t) can be resolved into a series of impulses. The graphical representation of the convolution integral involves four steps, i.e. (i) folding, (ii) displacement or dragging, (iii) multiplication and (iv) addition. Important properties of convolution are: (i) Convolution of two sequences in the time domain is equivalent to their products in the frequency domain; (ii) convolving any signal with a delta function results in exactly the same function, i.e. mathematically, it is x(n) ∗ δ(n) = x(n); (iii) A system that amplifies or attenuates has a scaled delta function, i.e. x(n) ∗ k δ(n) = kx(n); and (iv) A relative shift between the input and the output signals corresponds to an impulse response that is a shifted delta function, i.e. x(n) ∗ δ(n + s) = x(n + s), where s is the amount of shift.

3.20 Z Transform The Z transform of a finite wavelet is the polynomial of Z whose coefficients are the coefficients of the wavelet. In other words, Z transform of the wavelet B = (b0 , b1 , b2 , b3 , b4 , b6 , . . . bn)

(3.111)

B(z) = b0 + b1 z + b2 z 2 + · · · + bn z n ).

(3.112)

is the polynomial

The Z transform of an infinite wavelet is the power series in z whose coefficients are the coefficients of the wavelet. In other words, the z transform of the wavelet Eq. (3.111) is the polynomial Eq. (3.112). Convolution may be performed by multiplication of polynomials or power series. We write the Z transform as A(z) = a0 + a1 z or A(z) = 2 + z

(3.113)

B(z) = b0+ b1 z or B(z) = 3 + 4z

(3.114)

Multiplying the polynomial, we get a0 b0 + (a0 b1 + a1 b0 )z + a1 b1 z 2 .

(3.115)

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The resulting polynomial has coefficients equal to that obtained by convolution. Thus multiplication of polynomials corresponds to their coefficients. Convolution of two wavelets corresponds to the multiplication of their Z transforms.

3.21 Filters and Windows The main purpose of filters in signal processing is to eliminate or pick up certain frequency bands of the spectrum to examine their behaviours in greater details or to avoid certain frequency bands. It is done using windows. Windows are used to separate one band of frequencies from another. These windows are the: (i) Hanning window, (ii) Hamming window, (iii) Parzan window, (iv) Cosine window etc. These windows allow only a certain frequency band of the spectrum that is able to pass through, and the rest of the spectrum is obstructed by attenuating those frequency bands to negligible values. All frequencies below the cut-off frequency fc are passed with unity amplitude, whereas all the higher frequencies are blocked. The pass band is absolutely flat, the attenuation in the stop band is infinite and the transition between the two is infinitely small. The windows generally used in signal processing of geophysical data are: (i)

Rectangular window  w(n) =

1, 0 ≤ n ≤ M 0 otherwise

 (3.116)

(ii) Bartlett window (triangular) ⎫ ⎧ 2π M ⎨ M 0 ≤ n ≤ 2 M even ⎬ M w(n) = 0 is called the regularisation parameter (Zhdanov 2002) under the constraint m α → m t as α → 0 where m α = A−1 α d...

(10.17)

is the solution of the inverse problem, and m t is the true solution of Eq. (10.15). We assume that the solution of these inverse problems m α will asymptotically approach m t as α → 0. The operator R(d, α) is called the regularising operator in the vicinity of the element dt = Amt . Definition of the Regularising Operator: The operator R(d, α) dependent upon a scalar parameter α is called the regularising parameter in some neighbourhood of the element dt = Am t and, if there is a function α(δ) such that for any  > 0 a positive number, δ() can be found such that (i) μ D (dδ , dt ) < d() (ii) μm (m α , m t ) <  and (iii) where

m α = R(dδ , α(δ)).

(10.18)

10.3 Tikhnov’s Regularisation Philosophy

521

Here m α is a continuous function, and m α = R(dδ , α(δ)) → m t when α → 0 where μ D (dδ , dt ) and μm (m α , m t ) are respectively the distances between the two points mentioned in the data, and model abstract hyperspaces δ is the noise level in the data space,  is the small discrepancy between the true and regularised model parameters. In other words, with noisy data set uδ , the approximate model set is Zα = R(uδ , α), obtained with the aid of the regularising operator R(u, α) where α = α(δ, uδ ). Here, δ is the noise level, and α is the regularising parameter. Every regularising parameter defines a stable method of approximate construction of a solution of Eq. (10.15). The choice of the regularising parameter(s) should be consistent with the accuracy of assessment of δ the noise level in the data (α = α(δ)). This regularising parameter α is chosen in such a way that, when δ → 0 and uδ → u, then Z tends towards ZT or Ztrue . Thus the problem of finding an approximate solution of Eq. (10.15) is centered around getting a stable solution under minor perturbation in the data space and determining the regularisation parameter α from additional information related to the problem. This method of constructing the approximate solution is called the regularisation method. This is the basic philosophy of Tikhnov’s regularisation (Tikhnov and Arsenin 1977). For further detail, please consult Zhdanov (2002), who has shown, with great detail, the final working formula of the regularised inverse problem as (Figs. 10.4 and 10.5). −1  T 2   A Wd d+ ∝ Wm2 m apr m ∝ = A T Wd2 A+ ∝ Wm2

(10.19)

This is the regularised solution of the generalised weighted least squares problem. Here α is the regularisation parameter to be determined. Wd and Wm are weights in the respective data and model domain in the weighted least squares approach.

10.4 Fréchet Derivative 10.4.1 Parker’s Definition It is a derivative of a functional on a Hilbert Space. Let m be in a Hilbert space, and let F : x ⊆ H → R be a functional on some domain x in H, then F is said to be the Fréchet differentiable at element ‘m’ when we can find ‘D’ such that, for all  with  < ε F(m + ) = F(m) + (D, ) + R() and R()/ → 0 as  → 0

(10.20)

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10 Inversion of Geophysical Data

Fig. 10.4 Various initial choices give different answers, but they lie in the compactum if the data are adequate and accurate

In this definition, , D ∈ H and R is another functional on H. The element ‘D’ is called the Fréchet derivative of F at m. It plays a role of a gradient in ordinary multivariate calculus. We can write Eq. (10.20) as F(m + ) = F(m) + (D, ) + 0

(10.21)

To find ‘D’, we must perform a perturbation analysis of the forward problem, discovering how small variations about a fixed model changes the value of a particular measurement. Equation (10.21) changes to the form F(m + ) = F(m) +

N  i=1

i

∂F +R ∂m i

(10.22)

10.4 Fréchet Derivative

523

Fig. 10.5 Regularisation operator and its function in data and model space (Zhdanov 2002)

which follow from the Taylor’s series expansion.

10.4.2 Zdhanov’s Definition The operator A is called differentiable at some point x ∈ X . If there exists a linear bounded operator Fx, acting from X to Y such that A(x + δx) − A(x) = Fx (δx) + 0δx where

0δx → 0 when δx → 0 δx

(10.23) (10.24)

The operator Fx is called the Fréchet derivative of A at x and is written as F = A (x)

(10.25)

The expression Fx (δx) is called the Fréchet differential of A(x) at x and is written as Fx (δx) = δ A(x, δx), f

if there exists such a linear function Fx .

(10.26)

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10 Inversion of Geophysical Data

10.5 Major Methodologies for Inversion There are four major branches of inverse problems: (i) the linear problem; (ii) linearisable problem; (iii) weakly nonlinear problems and (iv) strongly nonlinear problems. In deterministic as well as in the stochastic domain, the procedural details of the inversion methodology are different. They include : (i) singular value decomposition (SVD); (ii) least squares (LS); (iii) weighted least squares (WLS); (iv) ridge regression (RR); (v) weighted ridge regression (WRR); (vi) underdetermined problem; (vii) Backus–Gilbert inversion (BG); (vii) Occam inversion; (viii) stochastic inversion; (ix) method of steepest decent; ix) conjugate gradient minimisation; (xii) Monte Carlo inversion; (xiii) genetic algorithm; (xiv) simulated annealing; (xv) artificial neural networks; (xvii) rapid relaxation inversion; (xviii) half-space inversion and (xix) joint inversion. Both creeping, creeping like ants, and jumping, random jumping like monkeys do exist in abstract space simultaneously in inversion. Members of the global optimisation methods, i.e. Monte Carlo inversion, genetic algorithm and simulated annealing are in the random jumping group. All members of the least squares are in the creeping group. Artificial neural network have both qualities.

10.6 Basis Function In mathematics, a basis function is an element of a particular basis for a function space. Every continuous function in the function space can be represented as a linear combination of a basis function, just as every vector in the vector space can be represented as a linear combination of a basis vector. Polynomial basis: The collection of quadratic polynomial with real coefficients has (1, r, r2 ) as a basis. Every quadratic polynomial can be written as a + bt + ct2 . A basis B of a vector space V over a field F is a linearly independent substitute of V. There are many different ways a mathematical signal can be represented. One of the ways is the linear combinations of the elementary time functions. These elementary time functions are basis functions. The functions ψ(t) are known as the basis functions of the signal expansions V(t) =



an ψn (t)

(10.27)

n

In fact, there are infinite number of basis functions in applied mathematics and are used in various branches of geophysics.

10.7 Subspace

525

10.7 Subspace A subspace is a vector space that is contained in another vector space so every subspace is a vector space but it is defined with respect to some other vector space of larger dimension. Suppose that W and V are two vector spaces of identical properties of vector additions and multiplication, then W is a subset of V (i.e. W ⊆ V).

10.8 Krylov Subspace In linear algebra, the Krylov subspace generated by the n × n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b, under the first ‘r’ powers of A starting from A0 , that is

 Kr (A, b) = span b, Ab, A2 b, A3 b, . . . Ar−1 b

(10.28)

This concept is named after Russian applied mathematician Alexei Krylov, who published this work in 1931. The basis for Krylov subspace is derived from the Caley–Hamilton’s theorem which implies that the inverse of a matrix can be derived in terms of the linear combination of its powers. Let A ∈ R n×n and v ∈ Rn . . . and v = 0 Then the sequence of the form v, Av, A2 v, … A5 v is called the Krylov sequence. The matrix of the form   Kk v, Av, A2 v, A3 v . . . , Ak−1 v is called the Kth Krylov matrix associated with A and v. The corresponding subspace  K k (A, v) = span(v, Av, A2 v, A3 v . . . Ak−1 v . . .

(10.29)

It is called the Kth Krylov subspace or the Krylov space associated A and v. Krylov subspace methods have become a very popular tool because of its inherent simplicity. It is very useful for large sparse matrices. Krylov subspace can be used both for direct and iterative solutions. Iterative methods take the upper hand over direct methods when the matrix is very large, as in the case of 3D FDM and FEM problems. It can be used to handle both solid and sparse matrices. As a result, it has entered into many branches of science and engineering including the method of conjugate gradient minimisation.

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10 Inversion of Geophysical Data

10.9 Singular Value Decomposition An n × m matrix can be decomposed into an n-th degree polynomial. The n-th degree polynomial λn + Cn−1 λn−1 + · · · + C1 λ = 0

(10.30)

have n roots. These roots are λ1 , λ2 , λ3 … λn which are the eigenvalues of the system matrix. They satisfy the equations Aui = λi ui

(10.31)

where ui is called the eigenvector. The set of eigenvalues form a diagonal matrix λ = diag (λ1 , λ2 , λ3 , … λn ). Every eigenvalue has an eigenvector. The eigenvector matrix is an orthogonal matrix, i.e. UUT = I and UT U = I, where I is the identity matrix. ‘n’ equations can be combined into a matrix equation AU = UA

(10.32)

where U is an orthogonal matrix. Therefore A = UλUT

(10.33)

This is singular value decomposition. Singular values are eigenvalues. A matrix can be decomposed in term of its eigenvalues and eigenvectors. For a rectangular matrix, A will be equal to A = UλVT

(10.34)

where U is an eigenvector matrix in the n-space, and V is the eigenvector matrix in the m space in an n × m system. Once an n × m matrix is decomposed in terms of its eigenvalues and eigenvectors, we can write the generalised inverse of A as A−g = νλ−1 uT .

(10.35)

P = A−g G

(10.36)

Hence

where A−g is the generalised inverse. Therefore P = vλ−1 uT G

(10.37)

10.9 Singular Value Decomposition

527

Equation (10.37) is the basic equation for inversion of geophysical data using singular value decomposition (Lanczos 1941) Equation (10.38) with all non-zero eigenvalues of the system matrix is given by uT G P = v λ−1 m×m m×m m×n n×1 m×1

(10.38)

If k < m, then eliminating zero eigenvalues, we get P =

uT G v χ −1 m×k k×k k×n n×1

(10.39)

Eliminating very small eigenvalues that cause instability, we get P =

uT G v χ −1 m×q q×q q×n n×1

(10.40)

where q (q < k < m) is the number of significant eigenvalues. Here k is the rank of the matrix.

10.10 Least Squares and Weighted Least Squares Estimator 10.10.1 Least Squares and Ridge Regression Estimator From Eq. (10.12), we get P = A−g G

(10.41)

where A−g is the generalised inverse of a rectangular matrix. It is observed that a generalised inverse matches with the least squares inverse and the least squares estimator is  −1 P = A−g G = P = AT A AG

(10.42)

Since the problem is an ill-posed problem because of noise-contaminated data collected from the field, the sensitivity matrix is generally an ill-conditioned matrix, i.e. some of the eigenvalues are zero or very small. Since the eigenvalue matrix is a diagonal matrix, the diagonal elements of the inverse of the eigenvalue matrix λ−1 . . . are (1/λ1 , 1/λ2 . . . 1/λn ) . . .. As a result, the inversion estimator becomes unstable. Marquardt (1963, 1970) proposed to add a coefficient K to the diagonal elements of the matrix (AT A)−1 such that the least squares estimator Eq. (10.42) changes to the

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10 Inversion of Geophysical Data

form  −1 P∗ = AT A + KI AT G

(10.43)

where K is the Marquardt’s coefficient, I is the identity matrix, P is the model modification vector, A is the sensitivity matrix and P* is the model modification vector with a reduced rate. Equation (10.43) is known as the ridge regression estimator or damped least squares estimator. It is called the damped least squares because the amplitude of the model modification goes down, i.e. P* < P. It has taken care of the presence of both zero and very small eigenvalues of the ill-conditioned sensitivity matrix. The ridge regression estimator is much more stable than the least squares estimator. It has both the qualities of the Newton–Rhapson method and gradient method. The Newton-Rhapson method converges very fast if the starting value is close to the actual answer. The system diverges when the initial guess is away from the real answer. In the gradient method, however, the convergence is possible even if the initial guess is considerably distant from the actual answer. But convergence is very slow near the actual answer. Ridge regression has the positive qualities of both these approaches, i.e. it converges very fast near the actual answer and it’s radius of convergence is reasonably high. This means that, even if the initial guess is poor, the system converges. It also means that, even if the initial guess is poor, i.e. the distance between the mPrior and mtrue is high, ridge regression can drag the model towards the actual answer. The larger the number of parameters, the lesser will be the radius of convergence. Data inadequacy and data inaccuracy have direct relationships with the radius of convergence. The choice of the value of K is dependent upon the interpreter; the starting value of K can be anything like 10.0, 1.00, 0.01 and 0.001, as suggested by Marquardt (1963). But, as the iterative solution converges, the value of K must be successively lowered until its value becomes negligible. Many interpreters used variance–covariance values instead of a pure number like Marquardt’s coefficient (Tarantola 1987; Menke 1984). K is also known as Marquardt and Levenburg coefficient. This stabilising parameter is also a part of the regularisation process. Other constraints, as discussed in Sect. 10.3, are also part of the regularisation process.

10.10.2 Weighted Least Squares and Weighted Ridge Regression In most the scientific work, we see that some of the experimental data in any experiment are less reliable than the others. This is quite common in geophysical field-data analysis. It means that the data variances are not all equal. In other words, the matrix Var (ε) (Variance (ε) is not in the form of Iσ2 where I is the identity matrix and σ2 is the variance (square of the standard deviation) in the data. But Var (ε) is diagonally dominated matrix with unequal diagonal elements. It happens in some problems that the off-diagonal elements of Var (ε) are not zero, i.e. the observations are correlated.

10.10 Least Squares and Weighted Least Squares Estimator

529

When either or both of these occur, the general least squares are not valid, and it is necessary to change the procedure for obtaining the estimator. Draper and Smith (1968) derived the expression for the weighted least squares and the estimator can be written as  P = (A T W −1 A)−1 AT W −1 G

(10.44)

and the weighted ridge-regression estimator can be written as  P = (A T W −1 A + K I )−1 AT W −1 G

(10.45)

Here the data are weighted by the inverse square root of the data variance–covariance matrix (Inman 1975). Zdhanov (2002) used both data-space and parameter-space weights and presented the parameter estimation formula using the regularisation parameter and dual minimisation as −1  T 2   A Wd d+ ∝ Wm2 m apr m ∝ = A T Wd2 A+ ∝ Wm2

(10.46)

where Wd and Wm are the data and parameter space diagonal weight matrices. Tarantola (1987) also used data and parameter weight matrices in the form of variance and covariance operators for dual minimisation. The covariance matrix in the weighted ridge-regression estimator is given by V = σ 2 (A T W −1 A)−1

(10.47)

GT W−1 G . n−m

(10.48)

where σ2 =

Here, σ2 is termed as the residual variance, and n and m are respectively the number of data points and number of parameters to be retrieved. The parameter standard error or the parameter uncertainty is defined as the square the diagonal √ root of√ elements of the parameter variance–covariance matrix. Thus, V11 , . . . V22 . . . etc. The parameter uncertainties are added to the retrieved model parameters, i.e. the estimated parameters will now be written as 

m1 ±



  V11 , . . . m 2 ± V22 , . . . etc.

The data variance–covariance matrix W is assumed to be a diagonal variance matrix with zero or negligible off-diagonal covariance part. Reciprocals of the standard deviations are data weights and given by

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W = (1/σ1 , 1/σ2 , 1/σ3 , 1/σ4 , . . . 1/σN ) where variance is square of standard deviation. Parameter variance–covariance matrix is also used for dual minimisation as will be discussed later.

10.11 Backus–Gilbert Inversion 10.11.1 Introduction The Backus–Gilbert’s approach of generalised linear inversion of geophysical data is a unique approach in many ways (Backus–Gilbert 1967, 1968, 1970). It does not enter into the Gauss–Newton approach for general linearised inversion like least squares and weighted least squares, nor does it come under the category of stochastic inversion, global optimisation or conjugate gradient minimisation. It can tell the finite resolving power of the gross earth data collected on the surface of the Earth in a more quantitative way. Secondly with this approach, one can find out one physical property of the Earth at a depth Z0 for any kind of geological terrain (igneous, metamorphic or sedimentary). The third important contribution is that it can give some idea about the depth of investigation. The most important Philosophy of the Backus– Gilbert approach should be such that maximum amount of information comes from a particular depth and very little information should come from above and below that depth. In other words, a kernel function should be constructed in such a way such that it should look approximately like the Dirac delta function. In reality, that does not happen. With a finite number of field data, “Gross earth functional” finite amount of resolution can be obtained. And that resolution gradually decreases with depth. Backus–Gilbert mathematically designed a kernel, termed the averaging kernel that has a certain width. Backus–Gilbert called it the spread function. They mathematically tried to minimise it, using a Lagrange multiplier. The auxiliary constraint needed and chosen for application of the Lagrange multiplier was the area of the averaging kernel A(Z 0 Z). Readers can go through the original papers of Backus–Gilbert cited above. The B-G formulation part is given briefly in Roy (2007), so it is not included here. For formulation and computation of B-G parameters, evaluation of Fréchet kernel is necessary. Oldenburg (1979) has computed the B–G Fréchet kernel both for DC resistivity and the magnetotelluric domain. Here, derivation of the Fréchet kernel for magnetotelluric domain is included.

10.11.2 Backus-Gilbert Formulation The important advantages of the B-G method are: (i) Layered-earth approximation is not required. The physical property say resistivity or density is a function

10.11 Backus–Gilbert Inversion

531

of depth z only at the point of measurement. Therefore, data collected over complicated Archean–Proterozoic terrain can also be inverted using the B–G approach; (ii) Depth of exploration of the DC resistivity, electromagnetic and magnetotelluric sounding can be estimated from the B–G spread function. B–G assumed that the earth functionals are linear, and they are Frechet differentiable. Therefore the linear earth functionals g1 , g2 , … gN (field data) can be connected to the earth model parameter through Fredhom’s integral of the first kind i.e. Z gi (m) =

Gi (z)m(z)dz

(10.49)

Z0

where Gi (z) is a known function of z because g is a known linear functional. So the inverse problem is what can we say about m(z) at z = z0 when all we know is a set of g1 , g2 , … gN collected on the surface of the Earth. We try to compute m Z0 which is the average value of m taken within a short interval z0 ±  z0 as practicable. These local averages are the model values at a particular depth z0 within the resolving length z 0 +  z 0 to z 0 −  z 0 . B–G defined the local average using the averaging Kernel in the form: z max

m z0 =

A(z0 , z)m(z)dz

(10.50)

z0

where averaging kernel is assumed to be unimodular, and it is (Fig. 10.6) z max

A(z0 , z)dz = 1

(10.51)

z

The averaging kernel ‘A’ is distributed in z, i.e. at depths where we shall try to compute m(z), we shall have to compute the averaging kernel to examine the quality of inversion based on the strength of the data. Data quality and adequacy improves the quality of A. Theoretically, the nature of A should be unimodular for practical problems when the data contain noise. It can show the multimodular nature, and the spread may increase very fast. Ideally, A (z0 , z) = δ (z – z0 ), where δ is the Dirac delta distribution. With only a finite number of field data available for computing m Z0 , we should not expect the local average to be so localised. We assume that the average m Z0 , whatever it turns out to be, depends only on g1 (m), g2 (m) … gN (m). Since m Z0 and g1 (m), g2 (m) … gN (m) have a linear relationship, we can write

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10 Inversion of Geophysical Data

Fig. 10.6 Backus-Gilbert (i) gross earth functionals (ii) spread of the averaging kernel (iii) depth of investigation

m Z0 =

N 

ai (z0 ) gi (m)

(10.52)

i=1

These coefficients depends on the depth z0 , and at each depth we have to find out these coefficients ai (z0 ) to compute m Z0 and the additional constraint is

10.11 Backus–Gilbert Inversion

533

A(z0 , z) =

N 

ai (z0 )Gi (z)

(10.53)

i=1

So we have to determine the ai (z0 ) at all depths so that it satisfies Eqs. (10.50– 10.52). Detailed derivation of B–G theory is partly available in Roy (2007).

10.11.3 Backus–Gilbert Fréchet Kernel (Oldenburg 1979) Oldenburg has derived the B–G Fréchet kernel, both for DC resistivity and for magnetotellurics. Here, only his derivation for MT is given. Here we assume a plane earth and adopt a Cartesian coordinate system with z direction pointing vertically downward. The conductivity σ is assumed to be isotropic and is a function of depth only, σ = σ (z). The general assumptions for MT: (i) the waves are plane waves and they are propagating vertically downward; (ii) displacement current is negligible; (iii) the magnetic permeability of the medium is equal to the free-space magnetic permeability, i.e. μ = μ0 . The electric field and magnetic field of the form E = E(0, Ey , 0) e−iωt and B = B(Bx , 0, 0) e−iωt . For linearly polarised plane waves, the electric and magnetic fields are respectively in the y and x direction. The angular frequency , where T is the time period of the signal. The Helmholtz electromagnetic ω = 2π T wave equation reduces to ∂2 Ey + iωμ0 σ (z)E y = 0 ∂z 2

(10.54)

where Ey = E y (z, ω) and Bx = Bx (z, ω). The source field can be considered to be a superposition of plane waves of various frequencies. Therefore, the timedomain recording of electric Ey (t) and magnetic Bx (t) field made on the surface of the Earth (z = 0) are processed to have the reliable estimates of Ey (ω) and BX (ω) for any frequency ω j . The ratio of these fields give the impedance of the medium Rj0 = R j (O j , ω j ) =

Bx (0, ω) j = 1... N E y (0, ω)

(10.55)

For any set of observed responses, it is required to construct a map of resistivity versus depth which will reproduce these responses to within a degree justified by their statistical error. To construct such a model, it must be known how the responses change when the conductivity structure is perturbed, i.e. the Fréchet kernel must be derived. This has already been done by Parker (1977a) who has shown that the MT response is Fréchet differentiable with respect to conductivity, even when that structure is discontinuous. Here the kernels were derived by Oldenburg (1979) using the standard perturbation technique. Since there is only x-component of the magnetic field and only y-component of the electric field, the subscripts are dropped and we

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10 Inversion of Geophysical Data

write B(z; ω) E(z; ω)

R(z; ω) = B(z; ω) = −

1 iω

∂ E(z; ω) ∂z

(10.56) (10.57)

Equation (10.54) changes to the form dR − iω R 2 − μ0 σ = 0 dz

(10.58)

This equation yields after integration x

δ R(0; ω) = ∫ −μ0 ( 0

E(z; ω) 2 ) δσ (z)dz E(0; ω)

(10.59)

Since electrical conductivity can change by several orders of magnitude, it is often convenient to choose m(z) = lnρ(z) = ln(

1 ) σ (z)

(10.60)

Therefore, the Eq. (10.59) can be written in the form x

δ R(0; ω) = ∫ −μ0 ( 0

E(z; ω) 2 ) δm(z)dz E(0; ω)

(10.61)

Since B and E are complex numbers, there ratio will also be a complex quantity that will have both amplitude and phase. So we can write R(0; ω) = R(0; ω)e−iφ(0) Now the final equations, after separating the real and imaginary parts, are ∂ 2 E(z; ω) + iωμσ (z)E(z; ω j ) = 0 ∂z 2

(10.62)

R j = R(0; ω) = R(0; ω)e−iφ(0)

(10.63)

 x  δ R 0; ω j = ∫ G j (z)δm(z)dz

(10.64)

0

This G j (z) is the Fréchet kernel in B–G inversion and is given by

10.11 Backus–Gilbert Inversion

535

    E 2 z; ω j G j (z) = μ0 σ (z)ω j R 0; ω j lm{   d E(0;ω j ) } E 0; ω j dz

(10.65)

x

δφ j (0) = ∫ H j (z)δm(z)dz

(10.66)

  E 2 z; ω j H j (z) = −μ0 σ (z)ω j Re{   d E(0;ω j ) } E 0; ω j dz

(10.67)

0

If we let ζ j (z) denote either the amplitude or phase kernel and T j the corresponding response, then z

δT j = ∫ ζ j (z)δm(z)dz j = 1, 2, . . . N

(10.68)

0

To carry out the model construction, we first choose a starting model m 0 (z) and compute the responses Tj . Let T be the difference T j0 − T j where T j0 is the observed datum. The N equations in (10.55) then can be solved to find a δm(z), and the new model m 1 (z) = m 0 (z) + δm(z) will have responses in closer agreement with the observations. This process can be continued until an acceptable model that produces the observations to within an accuracy determined by their statistical errors have been found. The rate of convergence will be monitored by the root mean square (rms) error ε  ε=

n 1  (δT j /T j0 )2 N j=1

(10.69)

The model construction process will be terminated when ε is below the prescribed limit. Structural appraisal is carried out by considering the averages of the model about any depth of interest. Backus–Gilbert (1970) showed that the only averages available to us take the form z

m(z 0 ) = ∫ m(z)A(z, z 0 )dz

(10.70)

0

where A(z, z0 ) =

N 

a j (z0 )G j (z)

(10.71)

j=1

As discussed earlier, A(z, z0 ) is the averaging kernel, and it is the window through which the earth model is seen.

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10 Inversion of Geophysical Data

Backus–Gilbert inversion is computer intensive and a relatively difficult inversion technique because each element in the system matrix is an integral to be evaluated numerically. Evaluation of the Frechet kernel also is a complex problem if the solution is not already available. But it has some intrinsic merit. No initial choice of the model parameter is necessary. Therefore it can be used for any kind of complicated geology. Any assumed physical property is a function of depth only.

10.11.4 Field Example One can see the strength of the data and the quality of inversion from the nature of the averaging kernel. One can directly see the depth of investigation of the data. Beyond a certain depth, the B–G spread function explodes and dictates the depth up to the data could see the subsurface with a certain amount of resolving power (Figs. 10.7, 10.8, 10.9, 10.10, 10.11, 10.12 and 10.13). Fig. 10.7 The magnetotelluric apparent resistivity data from the station Tangavilla near Joshipur over the Singhbhum granite batholith in eastern India

Fig. 10.8 The magnetotelluric phase data from the station Tangavilla near Joshipur over the Singhbhum granite batholith in eastern India

10.11 Backus–Gilbert Inversion

537

Fig. 10.9 Comparison of the B-G inversion result with that obtained by ridge regression and Schmucker ρ ∗ − g ∗ algorithm

Fig. 10.10 The nature of the averaging kernel showing the rapid increase of the B–G spread function with depth

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Fig. 10.11 Increase in the B-G spread function with depth and depth of investigation from field data

10.12 Multiple Source Code for Inversion: One Field Example of Application Multiple Source Codes for Inversion of Same Data Set: Application of Marginal Probability Density Function for Model Parameter Estimation Tarantola’s idea of estimating the model parameters from the marginal probability density function σ M (m) in the parameter space obtained from the conjunction of the state of information in a posteriori joint probability density function σ (d, m) is shown using the deep Schlumberger and azimuthal dipole DC sounding data collected over the Singhbhum Granite Batholith in eastern India by the author and his coworkers. To test the nature of response of σ M (m), a synthetic four-layer Schlumberger sounding data are taken for ρ1 = 5.0  m, ρ2 = 20  m, ρ3 = 100  m, ρ4 = 30  m, h1 = 5 m, h2 = 10 m and h3 = 40 m. Figure 10.14 shows the nature of responses for σ M (m) and the estimated model parameters. Next, we opted for studying the behaviors of σ M (m) for actual field data. We have chosen combined deep Schlumberger sounding (AB/2 = 1,000 m) and azimuthal dipole sounding (OO → the distance between the centres of current and potential dipole = 20 km) data. parameters (ρ1 ,

10.12 Multiple Source Code for Inversion: One Field Example …

539

Fig. 10.12 A posteriori marginal probability density functions for a four layer synthetic earth model and the estimated earth models from σ M (m)

ρ2 , ρ3 , ρ4 , h 1 , h 2 , h 3 ) for an assumed four-layer synthetic earth model and the inverted-layer parameters. One-dimensional inverted models. Figure 10.15 shows the combined Schlumberger and azimuthal dipole–dipole sounding data over the Singhbhum granite batholith, in eastern India. For Schlumberger sounding, half of the electrode separation (AB/2) continued up to 5 km and azimuthal dipole separation continued up to 20 km until the potential difference in potential dipole was readable and acceptable with a certain amount of confidence. Figure 10.16 shows that the a posteriori marginal probability density response patterns for the different layers, both for layer resistivities and thicknesses. σ M (m) may not give sharp peak for all the parameters. In this case, the thickness of the third layer (h3 ) is showing a flat top. Therefore, estimation of m from σ M (m) will be an ill- determined parameter. Figure 10.17 shows the 1D inverted models of the Khejurikota data by (i) stochastic inversion, (ii) genetic algorithm, (iii) simulated annealing and (iv) weighted least squares methods. It is always advisable to invert data with many source codes based on completely different statistics and mathematics. It is a field example of a 1D problem.

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10 Inversion of Geophysical Data

Fig. 10.13 Combined Schlumberger and azimuthal dipole deep resistivity sounding data around the village Khejurikota over the Singhbhum granite batholith for crustal studies and inverted models by stochastic inversion

Fig. 10.14 Behaviours of marginal probability density functions σ M (m) for seven parameters for a four layered-earth model from field data

10.12 Multiple Source Code for Inversion: One Field Example …

541

Fig. 10.15 Inversion of DC resistivity field data taken for crustal studies using (i) genetic algorithm, (ii) simulated annealing, (iii) stochastic inversion and (iv) weighted ridge regression Fig. 10.16 The method of steepest descent approaches the minimum in a zig-zag way in which the new search direction is orthogonal to the previous one

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10 Inversion of Geophysical Data

Fig. 10.17 In the method of steepest descent, the step size gradually decreases as the problem gradually approaches convergence

10.13 Two-Dimensional Inversion Smith and Booker (1991) developed the Rapid Relaxation Inversion (RRI) method applicable to both 2D and 3D data. Another magnetotelluric inversion scheme for 2D MT data based on the generalised RRI method was put forward by Siripunvaraporn and Egbert (2000) and presented a new and much more efficient variant on the Occam scheme. This scheme has been referred to as REBOCC, the reduced basis Occam’s inversion. Spichak and Popova investigated the possibility of solving the 3D inverse problem of magnetotellurics using the artificial neural network (ANN) approach. A new algorithm for computing regularised solutions for 2D magnetotelluric inverse problems using nonlinear conjugate gradients (NLCG) scheme was proposed by Rodi and Mackie (2001). De Groot–Hedlin and Constable (2004) developed a linearised algorithm to invert noisy 2D magnetotelluric data. A 2D earth model is parameterised by means of a grid of rectangular prisms each having uniform conductivity. The grid is terminated laterally by uniform layers and below by prisms elongated with depth. The grid is referred to as a regularisation mesh. The individual blocks are made smaller than the data-resolution length so that the location of the boundaries do not affect the final model. To perform the forward calculations required in the inversion scheme, deGroot–Hedlin and Constable (1990) used a finite element code described by Wannamaker et al. (1987). The program uses a rectangular array of nodes to perform the finite element calculations, which is called the finite element mesh. The finite element mesh must contain at least the regularisation mesh as a subset since there must be a node at every conductivity boundary. In normal practice, many nodes should be used across the conductivity boundary to ensure that the EM fields are computed correctly. However, the regularisation mesh contains a much larger number of conductivity regions than is normally used for 2D models based on the assumed geological structures. So, using several nodes for each conductivity element would be computationally expensive. The smooth inversion scheme will prevent large conductivity contrasts from appearing in the model, and several inversions have been conducted using no more nodes than defined by the regularisation mesh. However, it is usually desirable to insert more nodes within

10.13 Two-Dimensional Inversion

543

the regularisation mesh to maintain more accuracy in the forward modeling code, especially at the edge of the grid. To maintain the accuracy of the forward code, the spacing between the nodes should be approximately one-third of the skin depth. The EM field in a half space decreases exponentially with depth. So the usual practice in MT survey is to logarithmically space the frequencies at which the responses are computed. Various depth scaling for 1D smooth inversion were investigated by Smith and Booker (1988), and it is not surprising that a logarithmic depth scaling was found to fit the data most uniformly. Accordingly, the logarithmic depth scale is used for the node spacing and the block sizes. The ideal depth scale is structure-dependent and cannot be determined a priori for real data. Both the regularisation and finite-element meshes remain fixed between the iterations in inversion method. To determine the sizes of the resistivity blocks, an estimate of the resistivity is done by 1D inversion of the TE mode data.

10.14 Occam Inversion 10.14.1 2D Occam Inversion Formulation (deGroot–Hedlin and Constable 1990) To suppress model structures not required by the data, the model roughness must be minimised. For a 2D structure with the x-axis as the direction of strike direction, the model roughness may be given by  2 R1 = ∂ y m + ∂z m2

(10.72)

where m is the vector of model parameters, ∂ y is the roughness matrix in the y direction and for the horizontally adjacent blocks and ∂z is the roughening matrix along the vertically adjacent prisms. This is the expression for the first derivative of the roughness penalty. The penalty for the second derivative roughness is given by  2  2 R2 = ∂ y2 m  + ∂z2 m 

(10.73)

Since the model grid is terminated by uniform layer at the sides and uniform blocks below, first-derivative smoothing best matches the boundary conditions imposed by the forward code. Therefore, only the R1 roughness factor will be discussed. The vertical scale of the prisms is exponentially measured as a function of depth in order to coincide with the loss of resolving power with depth. This is equivalent to increasing the penalty for roughness as a function of depth. The horizontal block boundaries and the node spacings in the forward code extend to depth and are constrained by the requirement of having a fine mesh near the surface. If the data are represented by dj , j = 1, 2, … M, and one assumes that each of the data set has known variance σ j , it is the model’s ability to fit the data using the

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10 Inversion of Geophysical Data

two-norm measure χ 2 = Wd − WF(m)2

(10.74)

where F(m) are forward functional acting upon the discretised model m to produce a model response and W is the M × M diagonal weighting matrix, i.e. W = diag(1/σ1 , 1/σ2 . . . 1/σm )

(10.75)

To solve the minimisation problem, a Lagrange multiplier formulation is used and a stationary point is found for the unconstrained functional 2  U(m) = ∂ y m + ∂z m2 + μ−1 Wd − WF(m)2 − χ∗2

(10.76)

where μ−1 is the Lagrange multiplier. The functional U is minimised at points where the gradient with respect to the model is zero. Since the data functionals are nonlinear, the functional U is linearised and solved iteratively. For a starting model m1 , the first two terms of the Taylor’s series expansion give the following approximation [Fm 1 +  = F[m 1 ]] · · · + j1 

(10.77)

where j1 is the Jacobian matrix or M × N matrix of partial derivatives of F(m1 ) with respect to the model parameters. And  = m2 − m1

(10.78)

here  is a small perturbation about a starting model. If these equations are substituted back into Eq. (10.74), then the following expression is obtained: 2  2  U = ∂ y m + ∂z m2 + μ−1 Wd1 − Wj1 m1  − χ∗2

(10.79)

where d1 = d − F(m1 ) + j1 m1

(10.80)

Note that U is now linear up to m2 . Differentiating with respect to m2 to find the model that minimises U gives an iterative sequence for finding the model.     Mi+1 = μ ∂ yT ∂ y + ∂zT ∂z + (W ji )T (W ji )]−1 (W ji )T(Wdi)

(10.81)

A univariate search is conducted along μ at each iteration to find a model that minimises the misfit to the data until the desired tolerance is obtained where the model mismatch part is shown in Eq. (10.79). Here m0 is the starting model, m is the true model parameter, Cm is the model variance–covariance parameter and X2m is the model misfit function in the parameter domain. To solve this minimisation problem,

10.14 Occam Inversion

545

  −1 (d − F[m])T Cd−1 d − F[m] − X2∗ U(m, μ) = (m − m0 )T C−1 m (m − m0 ) + μ (10.82) we seek stationary points (with respect to both m and μ). Alternatively, we may consider the penalty functional W(m) such as

 −1 (d − F[m])T Cd−1 (d − F[m]) (10.83) W(m) = (m − m0 )T C−1 m (m − m0 ) + μ In Eq. (10.83), μ acts to “trade-off” between minimising the norm of data misfit and the norm of the model. When μ is large, the data misfit is de-emphasized, leading to a smoother model. In contrast, as μ tends to 0, the inverse problem becomes closer to the ill-conditioned least-square inversion problem, resulting in an erratic model (Parker 1980). Note that both U and W have the same minimum points with respect to variation of the model, i.e. δU = δm and δW = δm, where μ is fixed. Parker (1994) uses this to show that the stationary points of Eq. (10.84) can be found by minimising Eq. (10.82) for a series of μ values, and then choosing μ so that the misfit satisfies the constraint X2d = X2* . For linear F[m], this is straightforward, since in this case (for fixed μ) δU/δm = 0 is a linear system of equations which may be solved for m. Because F[m] is nonlinear, iterative solution methods are required. Rodi and Mackie (2001) provide a good review of several approaches, including a straightforward Gauss–Newton (GN) method. This approach is based on linearising F[m] with a Taylor series expansion,   F mk+1 = F[mk + δm] = F[mk ] + Jk (mk+1 − mk )

(10.84)

where k denotes the iteration number, and Jk = δF/δm is the N*M sensitivity matrix. Calculation of Jk , which describes the perturbations in the data due to changes in the model, is described in detail by Rodi and Mackie (2001), Mackie and Madden (1993a, b) and Rodi (1976). Substituting Eq. (10.84) in (10.83), we obtain W = (mk+1 − m0 )T C−1 m (mk+1 − m0 ) + μ−1 {(dk− Jk (mk+1 − m0 ))T Cd−1 (dk − Jk (mk+1 − m0 ))

(10.85)

The OCCAM approach, first proposed by Constable et al. (1987) (see also Parker 1994; deGroot–Hedlin and Constable 1990) is also based on linearising F[m] and then solving for the minimum points of Eq. (10.85). Differentiating Eq. (10.85) with respect to m and setting the result to zero leads to an iterative sequence of approximate solutions.   m −1 T −1 Jk Cd dk + m0 mk+1 (μ) = μC−1 m + k

(10.86)

The unique feature of the Occam approach is that the parameter μ is used in each iteration both as a step-length control and a smoothing parameter. That is, Eq. (10.85)

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is solved for a series of trial values of μ, and the misfit X2d ((mk+1 (μ)) for each μ is evaluated by solving the 2D forward problem. As for the linear problem, μ should be chosen so that the condition X2d = X2* is met. Usually, in the early iterations, the true misfit X2d is higher than the desired X2* for all possible μ. The Occam process thus chooses the model with the minimum misfit.

10.14.2 REBOCC Inversion In 1D problems, the number of data points is greater than the number of assumed model parameters or the number of layers, i.e. N  M. Sometimes to over parameterise the model, i.e. to make an overdetermined problem underdetermined, we make M > N . But for 2D/3D case, i.e. in the discretised domain, M is much larger than N(M  N). In REBOCC (Siripunvaraporn and Egbert 2000), the reduced basis Occams’s inversion is an efficient subspace algorithm for 2D/3D inversion of magnetotelluric data. Although Occam’s inversion gives a realistic model, the main disadvantage is its long computation time. One approach to reduce the computation time is to express the solution as a linear combination of the rows of the sensitivity matrix smoothed by the model covariance. In this way, the inverse problem shifts from the M-dimensional model space to the N-dimensional data space (M  N). This method is referred to as DASOCC, the data space Occam’s inversion (Siripunvaraporn and Egbert 2000). Since, generally, N  M, this technique eliminates much of the computation. In addition to this, another modification is done in the REBOCC scheme that makes it even faster than the DASOCC algorithm. Generally, MT data are smooth in period and also in space for closely spaced sites. Hence, the representers or the basis functions also vary slowly with period and site positions. Therefore, in the data-space approach, there is no need to use all of the sensitivities as basis functions. A subset is sufficient to get the model with all the details included in it. With this approximation, it is unnecessary to compute all the sensitivities, and the size of the system of equations gets reduced significantly. Thus REBOCC becomes much faster than Occam and is also faster than DASOCC. In addition to this, the memory requirement is also reduced so that large sets of data can be inverted with REBOCC. A large number of geophysicists, mathematicians and statisticians started working on the reduction of computation time for solution of an inverse problem. Computation of the forward problem and sensitivity matrix have a big role to play in this regard. Out of many efforts, only a few are mentioned in this section. Rapid relaxation inversion (Smith and Booker 1991), conjugate gradient method (Hestenes and Eduard 1952) and the method of steepest decent (Dieft and Zhou 1993). The concept of subspace inversion was introduced by Oldenburg et al. (1993) and Oldenburg and Ellis (1991) to reduce the computation time. REBOCC also is one of the very efficient members of this subspace inversion. Reduction of computation time in forward modelling and inversion was an important topic of research when the cost of computation was an important factor. A

10.14 Occam Inversion

547

remarkable breakthrough in this front for solving an inverse problem was achieved by Ghosh (1970) and Koefoed (1979), bringing filter theory to the 1D DC resistivity forward problem. Also, in 1D electromagnetic sounding, filter theory has applied. In the 2D/3D domains, attempts to reduce computation time in an inversion source code, including forward and inverse problems, so far are: (i) subspace Inversion; (ii) application of half Cholesky in the case of symmetric matrix in finite element problems; (iii) reducing the bandwidth of a sparse matrix; (iv) inversion approaches which don’t have any sensitivity matrices; (v) RRI as mentioned already and (vi) saving only the nonzero elements at their proper coordinates, keeping aside all zeros. That saves computer storage and computation time; (vii) in an inverse problem where each and every element is an integral in a sensitivity matrix, only two or three point Gauss quadratures are used for numerical integration to reduce the computation time; (viii) reduction of size of the matrix from model space to data space and then to subspace in an inverse problem. The data space has N data points and model space has M model parameters as discussed in the beginning of the chapter in the FEM/FDM/IEM domain. The central theme of Siripunvaraporn and Egbert (2000) is to reduce the M × M matrix in the model space to an N × N matrix in the data space and which is a considerably smaller matrix and reaches a Dasocc inversion stage. Now, using the concept of data subspace inversion (Oldenburg et al. 1993), an L × L sub space zone is selected with much smaller matrix (Siripunvaraporn and Egbert 2000) such that the concept of convergence X 2L ∼ = X ∗2 is valid. Then the reduced basis Occam subspace inversion will be a powerful inversion algorithm for the 2D/3D MT domain. This concept of data redundancy forms the basis for the Reduced Data Space Occam approach (REBOCC) and can significantly speed up the inversion process and decrease the memory requirements. A subset is sufficient to get the model with all the details in it. With this approximation, it is unnecessary to compute all the sensitivities and the size of the system of equations gets reduced significantly. Thus, REBOCC becomes much faster than Occam.

10.14.3 REBOCC Formulation (Siripunvaraporn and Egbert 2000) Rebocc inversion has three parts. The first part is in the model parameter domain the size of the matrix is (M × M) wher e M  N in 2D, 3D models. The second part is in the data space where the size of the matrix is (N × N ) where (N × N ) matri x  (M × M) matri x. The third part is the subspace inversion. Here the matrix (L× L) is the subspace matrix where L  N . The central message is that one does not need the entire information to construct the model. Even a small fraction of the information is good enough to achieve a reasonably good and acceptable model. That significantly reduces the computation time. In REBOCC inversion, the Earth is considered to be discretised into a series of M constant resistivity blocks where m = [m1 , m2 , …,

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mM ]T . Let there be N observed data d = [d1 , d2 ,…, dN ]T , with the uncertainties e = [e1 , e2 ,…, eN ]T . Then the fit of the theoretical model response F [m] to the observed data can be expressed as X2d = (d − F[m])T C−1 d (d − F[m])

(10.87)

where the superscript T represents the matrix transpose and Cd is the data covariance matrix, which in practice is a diagonal matrix (Tarantola1987; Roy 2007). To quantify the model structure, a model norm is considered in the general form: X2m = (m − m0 )T C−1 m (m − m0 )

(10.88)

where m0 is the prior model and Cm is the model covariance matrix, which characterises the expected magnitude and smoothness of resistivity variations relative to m0 . Now, the minimum structure inverse problem is to minimise X2m subject to X2d = X2* , where X2* is the desired level of misfit. To solve this minimisation problem, a Lagrange multiplier λ−1 is introduced resulting in an unconstrained functional U(m, λ) given by, −1 T −1 2 U(m, λ) = (m − m0 )T C−1 m (m − m0 ) + λ {(d − F[m]) Cd (d − F[m]) − X* }, (10.89)

for which stationary points are sought for with respect to both m and λ. Alternatively, the penalty functional Wλ (m) may be considered which is given by, −1 T −1 Wλ (m) = (m − m0 )T C−1 m (m − m0 ) + λ {(d − F[m]) Cd (d − F[m])}. (10.90)

In this equation, m = mtrue or mest . m0 is the initial choice of the model parameter. Cm−1 and Cd−1 are respectively the model variance–covariance and data variance– covariance matrices. d is the data in the data space, and F(m) is the connecting link between the data space and model space. It is the model response for the data and is a linear differential operator (Lanczos 1941) or the sensitivity matrix. λ−1 is the Lagrange multiplier. The Lagrange multiplier is a mathematical operator used to maximise or minimize a function. In Backus–Gilbert inversion (Backus–Gilbert 1968) or solving an underdetermined problem (Menke 1984), one has to use and determine the value of the Lagrange multiplier first and then proceed to the solution of the problem. But here in the entire Occam inversion group, λ acts to “trade off” between minimising the norm of data misfit and the norm of the model (Parker 1994). When λ is large, the data misfit is de-emphasized, leading to a smoother model. In contrast, when λ tends to zero, the inverse problem becomes closer to the ill-conditioned least-square inversion problem, resulting in an erratic model (Parker 1980). It is to be noted that both U and Wλ have the same stationary points with respect to variations of the model, i.e. ∂U/∂m = ∂Wλ /∂m, where λ is fixed. Parker (1994) used this to show that stationary points of Eq. (10.88) can be found by minimising

10.14 Occam Inversion

549

Eq. (10.90) for a series of λ values and then choosing λ so that the misfit satisfies the constraint X2d = X2* . If F[m] is linear, then this is straightforward. But, because F[m] is nonlinear for the MT inverse problem, iterative solution methods are required. Some of the approaches taken up by various workers are considered next, and then an outline of the REBOCC approach is given. Rodi and Mackie (2001) provide a good review of several approaches including the Gauss–Newton (GN) method, which is based on linearising F[m] with a Taylor series expansion,   F mk+1 = F[mk + δm] = F[mk ] + Jk (mk+1 − mk ),

(10.91)

where k denotes the iteration number, and Jk = (∂F/∂m)|mk is the N × M sensitivity matrix calculated at mk . Calculation of Jk , which describes the perturbations in the data due to changes in the model, is described in detail by Rodi and Mackie (2001), Mackie and Madden (1993a, b) and Rodi (1976). Substituting Eq. (10.91) in Eq. (10.90), we obtain  = (mk+1 − m0 )T C−1 W m (mk+1 − m0 )   T  −1 ˆ −1 ˆ dk − Jk (mk+1 − m0 ) × Cd dk − Jk (mk+1 − m0 ) , (10.92) +λ  is then quadratic in mk+1 , and thus where dˆ k = d − F[mk ] + Jk (mk − m0 ). This W can be minimised exactly for fixed λ. For numerical stability (Marquardt 1963), damping is generally required to control step size for each iteration. The system of equations to be solved for each iteration then becomes  −1  T −1  m Jk Cd (dk − F[mk ]) − λC−1 mk+1 −mk = λC−1 m +  k + εk I m (mk − m0 ) , (10.93) T −1 where the model space cross-product matrix  m k = Jk Cd Jk is an M × M positive semidefinite symmetric matrix, I is the identity matrix and εk is a damping parameter. It is to be noted that with the GN approach λ is fixed. Therefore, the algorithm will converge to a stationary point of Eq. (10.90), not Eq. (10.89.). To achieve the stationary point of Eq. (10.89) (with respect to both λ and m), the process would have to be repeated with different values of λ until the constraint X2d = X2* has been satisfied. REBOCC (Siripunvaraporn and Egbert 2000), the reduced basis Occam’s inversion, is a 2D inversion algorithm for magnetotelluric (MT) data, which is a more efficient variant on the Occam (deGroot-Hedlin and Constable 1990) scheme. The Occam approach is also based on linearising F[m] and then solving for the stationary points of Eq. (10.92). Differentiating Eq. (10.92) with respect to m and setting the result to zero leads to an iterative sequence of approximate solutions:

  m −1 T −1 ˆ Jk Cd dk + m0 . mk+1 (λ) = λC−1 m + k

(10.94)

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The unique feature of the Occam approach is that the parameter λ is used in each iteration, both as a step length control and a smoothing parameter. That is, Eq. (10.94) is solved for a series of trial values of λ, and the misfit X2d (mk+1 (λ)) for each λ is evaluated by solving the 2D forward problem. As for the linear problem, λ should be chosen so that the condition X2d = X2* is met. Usually, in the early iterations, the true misfit X2d is higher than the desired X2* for all possible values of λ. The Occam process thus chooses the model with the minimum misfit as the basis for the next iteration. The process is then repeated until the misfit reaches the desired level. Parker (1994) called this process of bringing the misfit down to the target level phase I. Once the misfit reaches the desired level, phase II begins by keeping the misfit at the desired level, but varying λ to search for the model with the smallest norm. Since the problem is nonlinear, the desired misfit may never be reached. However, in practice, improvement of the misfit from iteration to iteration can be expected, until a minimum is achieved. Both GN and Occam share similar computational steps. For each iteration, Jk must be calculated, and an M × M system of equations [Eq. (10.93) for GN and (10.94) for Occam], must be solved. These methods are thus very time consuming. Furthermore, these methods require much memory to store the sensitivity and cross-product matrices. These computational inefficiencies are the result of strong dependence on the model space dimension M. In the Data Space Occam Method (DASOCC), which is a variant on the Occam approach, the inverse problem is transformed from the model space into the data space, by expressing the solution as a linear combination of rows of the sensitivity matrix smoothed by the model covariance. This transformation reduces the size of the system of equations to be solved from M × M to N × N. Since the number of model parameters M is generally much larger than the number of data N, a significant decrease in both CPU time and memory can be achieved by this approach. Parker (1994) showed that the minimiser of Eq. (10.92) for iteration k can be expressed as a linear combination of rows of the smoothed sensitivity matrix Cm JkT , mk+1 − m0 = Cm JkT βk+1

(10.95)

  where βk+1 is an unknown expansion coefficient vector of the basis functions Cm JkT j ; j = 1, …, N, which are sometimes referred to as the “representers” of the linearised data functionals for iteration k (Parker 1994). Substituting Eq. (10.95) into (10.92), we obtain  T   −1 ˆ T n −1 n n ˆ  dk −  k βk+1 × Cd dk −  k βk+1 , W = βk+1  k βk+1 + λ (10.96) where  nk = Jk Cm JkT is the N × N data space cross-product matrix. Differentiating Eq. (10.96) with respect to β and rearranging, the unknown expansion coefficients can be obtained as −1  βk+1 = λCd +  nk dˆ k .

(10.97)

10.14 Occam Inversion

551

The inverse problem thus becomes a search for the N real expansion coefficients βk+1 , instead of the M-dimensional model, mk+1 . Exactly as for the standard Occam, we can solve for βk+1 , update the model and then check the misfit for various values of λ. We again choose λ to achieve the minimum misfit if this exceeds the desired level X2* (phase I) and use this model as the basis for the next iteration. Once the desired misfit is achieved, phase II begins to eliminate unnecessary features, while keeping the misfit at the desired level. The data-space formulation clearly shows that the solution is a linear combination of natural basis functions or representers. Each representer corresponds to a single data element (at a particular period, station, response and mode). Since the MT data are smooth and redundant, the representers vary slowly with period and site location for a given response and mode. These basis functions are thus highly redundant. Hence, an excellent approximation to the solution can be found in a subspace of much lower dimension (Parker 1994). This concept of data redundancy forms the basis for the reduced data space Occam approach (REBOCC) and can significantly speed up the inversion process and decrease the memory requirements. Before solving the inverse problem, a subset of L (out of N) data is selected for which representers will be calculated at each iteration. For an example, we could choose all data for every other period so that L = N/2. It is to be noted that L can typically be considerably smaller than this. For iteration k + 1, solutions of the form mk+1 = Cm GkT αk+1 + m0

(10.98)

are sought for where αk+1 is the L-dimensional unknown coefficient vector for the reduced basis, and Gk is the L × M subset sensitivity matrix. Now, to fit all of the data and to derive equations for αk+1 analogous to Eqs. (10.96) and (10.97), a linearised relationship between δm and dˆ k is required. In fact, this relationship does not strictly exist, unless all of the sensitivities are calculated. However, the data vary smoothly, and so a data value would be well approximated by interpolation of nearby data (e.g. adjacent frequencies from the same site). In the same way, the sensitivities vary smoothly with frequency and/or site location and can be interpolated from nearby sensitivities. We can thus express the approximation to the full sensitivity matrix Jk in terms of the subset sensitivity matrix Gk using an interpolation matrix B, of size N × L, i.e. Jk ≈ BGk .

(10.99)

Substituting Eqs. (10.98) and (10.99) into (10.92), we get  = αTk+1  lk αk+1 + λ−1 W



dˆ k − BΓ lk αk+1

T

  ˆ k − BΓ lk αk+1 , (10.100) d × C−1 d

where  lk = Gk Cm GkT is the L × L data subspace cross-product matrix.

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Following, C−1/2 B is decomposed into the N × N orthonormal matrix d   Q, whereT ¯ 0 , and R ¯ Q B = QR, where Q = Q Q = Q−1 and the N × L matrix R, i.e., C−1/2 d   ¯ ¯ and Q ¯ 0 have dimensions N × L and N × N − L respectively. = R|0 . Matrices Q ¯ is the square L × L upper triangular matrix, and 0 is the N − L × L zero Matrix R matrix, i.e. all elements of the matrix are zeros. Equation (10.100) then becomes T

 = αTk+1  lk αk+1 + λ−1 W

 T  − 1  l l −1 Cd 2 dˆ k − Q k αk+1 × Cd 2 dˆ k − Q k αk+1 (10.101)

where  −1 T ¯ αk+1 = R αk+1

(10.102a)

l ¯ T.  k = R lk R

(10.102b)

and

l

Inserting QQT = I in between  k and αk+1 on the right side of Eq. (10.101) and rearranging we get  T    l l l T −1 2 ¯ ¯  W = αk+1  k αk+1 + λ Xmin + dk −  k αk+1 dk −  k αk+1 ,

(10.103)

where ¯ T C−1/2 dˆ k , d¯ k = Q d and        ¯ −1/2 ˆ 2  −1/2 ˆ 2  ¯ T −1/2 ˆ 2 dk  = Cd dk  − Q Cd dk  X2min = Q 0 Cd

(10.104)

is the approximate minimum achievable total square misfit for the selected basis. If we use all representers (i.e. B = I), then X2min = 0. This corresponds to the fact that for a linear problem we can fit the data exactly if we use all representers. This will not be true for the nonlinear MT problem. Thus X2min only provides a very rough estimate of the magnitude of data misfit that might be achieved with the chosen reduced basis. In general, X2min is high in the early iterations and decreases to a constant in the later iterations.

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553

Differentiating Eq. (10.104) with respect to α¯ and setting the result to zero, the unknown expansion coefficients can be obtained in a form similar to Eq. (10.97),  l −1 d¯ k . αk+1 = λI +  k

(10.105)

Again, just as in the model and the data space methods, after solving Eq. (3.105), the model is updated using Eqs. (10.98) and (10.10.102a), and then the forward problem is solved to evaluate X2d . The procedure is repeated to find the appropriate λ. The outer loop of the iterative minimisation of Eq. (10.90) proceeds exactly as for Occam or for DASOCC. Forward modelling can be said to be the heart of inversion and thus must be reliable, fast and accurate. It is used in two parts of the inversion to compute the sensitivity matrix and to compute responses for calculating the misfit. A finite difference (FD) method as in Smith and Booker (1991) is applied in REBOCC for forward calculations.

10.15 Method of Steepest Descent The method of steepest descent is one of the tools for finding out the minimum of a function iteratively Dieft and Zhou (1993). Given a function f(x) ⊆ Rn that is differentiable at the starting point x0, it is the vector −∇ f (x0 ). To see this, let us consider the function ϕ(α) = f (x0 + αu)

(10.106)

where u is the unit vector, i.e. u = 1 Then by the chain rule of differentiation, we get ϕ (α) =

∂ f ∂ xn ∂f ∂f ∂ f ∂ x1 + ··· + = ∗ u1 + · · · + ∗ un ∂ x1 ∂α ∂ xn ∂α ∂ x1 ∂ xn

(10.107)

ϕ (α) = ∇ f (x0 )u = ∇f(x0 ) cos θ

(10.108)

and therefore

where θ is the angle between ∇ f (x0 ) and u. It follows that ϕ (α) is minimised when θ = π , which yields u=− and

∇ f (x0 ) ∇ f (x0 )

(10.109)

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10 Inversion of Geophysical Data

ϕ (0) = −∇f(x0 )

(10.110)

We can therefore reduce the problem of minimising a function of several variables to a single variable minimising the problem by finding the minimum of ϕ(α) for this choice of a t0 . That is we can find the value of α for α > 0 that minimises ϕ0 (α) = f (x0 − α f (x0 )

(10.111)

After finding the minimiser t0, we can set x1 = x0 −α0 ∇ f (x0 )

(10.112)

and this process will continue by searching for x1 in the direction of ∇ f (x1 ) to obtain x2 by minimising ϕ1 (α) = f (x1 − α∇ f (x1 )

(10.113)

and so on. This is the method of steepest descent. Given an initial guess x0 , the method computes a sequence of iterates xk where xk+1 = xk − αk ∇ f (xk )

(10.114)

for k = 1, 2, 3, … where tk > 0 minimises the function

ϕk (α) = f (xk − α∇ f (xk ))

(10.115)

Two important properties of the method of steepest descent are (i) Let f(x) = Ax − b within the space Rn and α ∗ > 0, be the minimiser of the function

ϕ0 (α) = f (x0 − α∇ f (x0 )) for α > 0

(10.116)

x1 = x0 − α ∗ ∇ f (x0 ), then f(x1 ) < f(x0 ),

(10.117)

Therefore, through the method of steepest descent, it is guaranteed that one can head towards the global minimum or the true answer at each iterative steps.

10.15 Method of Steepest Descent

555

(ii) Let f(x) be a function in space Rn that is continuously differentiable, and let XK+1 and XK are two consecutive iterates produced by the method of steepest descent for k > 0. Then the steepest descent directions are orthogonal. It means

∇ f (xk ) ∗ ∇ f (xk+1 ) = 0

(10.118)

It means the method of steepest descent searches independently the search directions for two consecutive iterates. In the method of steepest descent, we start at an arbitrary point x0 and slide down to the bottom of the paraboloid. We take a series of steps x1 , x2, … until we are satisfied that we are close enough to the solution Ax = b. When we take a step in which ϕ0 (α) decreases most quickly, it is the direction opposite to f’ (x1 ). Accordingly, this direction is −f (x1 ) = b − Ax1 . Our first step along the direction of steepest descent will fall somewhere on the line of steepest descent. A line search is a procedure that chooses α to minimise f along the line and we get x1 = x0 + αr0 . . . where r0 is the residual. From the basic calculus α minimises f (x) when the directional derivative T d T d d f(x 1 ) is equal to zero by the chain rule dα f(x1 ) = f (x1 ) dα X1 == f (x1 ) r0 . dα Setting this expression to we find that α should be chosen in such a way that r0 and f (x1 ) are orthogonal. To determine α, we note that f (x1 ) = −r1 . Using the orthogonal property of the residual vectors, we have r1T r0 = 0

(10.119)

(b − Ax1 )T r0 = 0

(10.120)

(b − A(x0 + αr0 ))T r0 = 0

(10.121)

(b − Ax0 )T r0 − α(Ar0 )T r0 = 0

(10.122)

(b − Ax0 )T r0 = α(Ar0 )T r0

(10.123)

r 0T r0 = αr0T (Ar0 )T

(10.124)

α=

r 0T r0 r0T (Ar0 )

Xi+1 = Xi + α i ri The program runs until it converges.

(10.125) (10.126)

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10.16 Conjugate Gradient Method 10.16.1 Introduction Conjugate gradient (CG) method is a mathematical tool from linear algebra typically meant for solving a large-size square, sparse or solid matrices (Hestenes and Eduards 1952) for an n × n matrix, A within the space R n×n . The conjugate gradient method is suitable when the matrix is very large, i.e. n ≥ 106 or more. It is meant for symmetric (A = AT ) and positive definite (xT Ax > 0) matrix. For the finite element method, one generally gets sparse, symmetric and positive definite matrix when the starting equations are either Helmholtz electromagnetic wave equation (∇ 2 H = γ 2 H) or Poisson’s equation (∇ 2 ϕ = − ρ ). For the finite difference method, when the differential equation changes to a difference equation, the matrix generated to estimate the potentials or fields at the nodal points may or may not be symmetric. For symmetric and positive definite matrices, Cholesky’s decomposition (CD) is often used. For n ≥ 106 or 104 , both CD and CG can be used with equal validity. But for n ≥ 6, CG is always prescribed. For solution of large matrix, CG can be used both for a direct or for an iterative approach. One good quality of CG is rapid convergence both for direct and iterative approaches. For CG, the solution gradually goes down along the maximum slope direction, but each of this grad vectors descent in two mutually perpendicular directions until it reaches the global minimum. CG can be used for global optimisation problems using iterative process. Therefore, CG can be used both for forward modeling and inversion, as well as both for direct and iterative approaches. For 2D and 3D inverse problems CG is generally used. In CG, one quadratic function f(x) =

1 T x Ax − xT b 2

(10.127)

where x e Rn . The implication behind this quadratic function is f (x) = Ax − b, and if it is zero then Ax = b and one reaches the global minimum. Since CG is used as a tool for inversion, the inverse problem does not get trapped into any local minimum. Inverse problem based on least squares may get trapped in a local minimum pocket if they are not properly regularised. The minimum point of this paraboloid (Fig. 10.18) generated from this quadratic function is the true solution Ax = b.

10.16.2 Important Steps in Conjugate Gradient Method (a) In an undulating surface of a paraboloid, we start from the top at x0 = 0 and come down along the direction of maximum slope, i.e. along the true dip direction or ∇ f (x) = Ax − b = 0 at the minimum point where the slope is zero.

10.16 Conjugate Gradient Method

557

Fig. 10.18 The paraboloid generated from the quadratic expression. The minimum point of the paraboloid is the solution of the problem

(b) Conjugate means orthogonal here. So conjugate gradient is an orthogonal gradient. Therefore, the search directions in different iterations or in the direct method are mutually orthogonal. Therefore the descending path is zig zag as shown in the figure. (c) Let the value of x at kth iteration is xk. The residual rk ) =b − Ax. When b − Ax = 0, one gets the solution for x. These residuals are also vectors and they are orthogonal vectors.

rk+1 · rk = 0

(10.128)

(d) The vector pk is the search direction. It is also a vector, and they are also orthogonal in different inner product space. Therefore

pk+1 · pk = 0

(10.129)

xk+1 = xk+1 + αk pk

(10.130)

(e) It generates

Here, αk is a scalar and it selects the step length in each iteration. (f) It converges at the most in n iterations. Often k can be less than n for very fast convergence.

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10.16.3 Conjugate Gradient Method as a Direct Approach Suppose we want to solve the following system of linear equations Ax = b

(10.131)

For the vector x where known, nn matrix is symmetric (i.e. AT = A) and positive definite (i.e. xT A x > 0 for all the nonzero vectors x in the space Rn ), and real and b is also a known vector. We want to find out the unique solution of the vector x. From matrix calculus and linear algebra, we know that two nonzero vectors u and v are conjugate with respect to A, when uT Av = 0

(10.132)

Since A is a square symmetric and positive definite matrix. The left hand-side defines an inner product as follows:   (u, v) = Au, v = uAT v = u, Av = uT Av

(10.133)

Two vectors are conjugate if and only if they are orthogonal with respect to the inner product. Being conjugate is a symmetric relationship. If u is a conjugate to v, then v is a conjugate to u. Suppose that P = { p1 , p2 , . . . pn }

(10.134)

is a n mutually conjugate vectors, and P forms the basis for Rn , and we can express the solution of x in Ax = b in the basis function form as x =

n 

αi pi

(10.135)

αi A pi

(10.136)

αi pkT Api

(10.137)

i=1

From Eq. (10.133), we can write Ax =

n  i=1

pkT



Ax =

n  i=1

pkT b

=

n  i=1

  αi pk pi A

(10.138)

10.16 Conjugate Gradient Method

559

pk b =

n  i=1

αk pk pk A

 pk b αk = pk pk A

(10.139)



(10.140)

Thus the equation Ax = b (Eq. 10.129) can be solved after getting n conjugate directions and the value of αk . Next we start with x0 = 0 and set p0 = r0 = b − Ax0 For k = 0, 1, 2, … (a) xk+1 = xk + αk pk

(10.141)

where αk is a scalar that dictates the step length in successive iterations for moving downhill and is equal to =

rkT rk pkT Apk

(10.142)

Equation (10.139) can be derived from Eqs. (10.141 and 10.142) (b) rk+1 = b − Axk+1 = rk − αk A pk

(10.143)

(c) pk+1 = rk+1 + βk pk

(10.144)

where βk =

T rk+1 rk+1

rkT rk

(10.145)

Equation (10.144) shows the connecting link between the search direction vector and the residual vector. Let rk be the residual after the kth iteration, i.e. rk = −b− Axk (Eq. (10.128)). Here, rk is the negative gradient of f(x) at x = xk , so the gradient descent method would be to move in the direction rk . Here we see that the directions of pk are conjugate to each other. It is also necessary that the search directions be built out of the present residue and all the previous search directions. Therefore, the conjugation constraint is an orthonormal type constraint. Hence the algorithm has some resemblance to Gram–Schimdt orgonalisation. Hence pk = r k −

 p T Ark k T p i Api i