Geophysical Interpretation using Integral Equations

Geophysical Interpretation using Integral Equations

Geophysical Interpretation using Integral Equations L. ESKOLA Head of the Geophysics Department, Geological Survey of Finland 1~lll SPRINGER-SCIENCE+BUSINESS MEDIA, B.v.

First edition 1992 1992. L. Eskola Typeset in 10/12 pt Times by Pure Tech Corporation, India ISBN 978-94-010-5045-6 ISBN 978-94-011-2370-9 (ebook) DOI 10.1007/978-94-011-2370-9 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored, or transmitted, in any form or by any means, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing" Agency in the UK, or in accordance with the terms of licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries conceming reproduction outside the terms stated here should be sent to the publishers at the London address printed on this page. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. A catalogue record for thls b k is available rrom the British Library Library of Congress Cataloging-in-Publication data available

Contents Preface Introduction 1 General matters concerning integral equations 1.1 Demonstration of an integral equation solution 1.2 Classification of integral equations 1.3 Numerical solution 1.3.1 The method of moments 1.3.2 The Galerkin method 1.3.3 The point-matching method 2 Elements of electrostatics and potential theory 2.1 Differential representation of electrical potential 2.2 Integral representation of electrical potential 2.3 Primary current electrode 2.4 Volume charge distribution 2.5 Surface charge distribution 2.6 Electrical double layer ix xi 4 6 6 7 7 9 9 12 15 16 17 18 3 Electrical methods 21 3.1 Introduction 21 3.2 Resistivity of rocks 22 3.3 Resistivity method 23 3.3.1 Introduction 23 3.3.2 Integral equations for charge density 24 3.3.3 Integral equations for potential 30 3.3.4 Integral equation for perfect conductor model 33 3.3.5 Integral equations for two-dimensional earth models 36 3.3.6 Integrodifferential equations for thin conductor models 42

Vi Contents 3.4 Magnetometric resistivity 48 3.5 Mise-a-la-masse method 52 3.5.1 Integral equations for charge density 52 3.5.2 Integral equation for potential 55 3.5.3 Integral equation for perfect conductor model 57 3.6 Surface polarization 59 3.6.1 Introduction 59 3.6.2 Solution for perfect conductor models with surface polarization 60 3.6.3 Scaling of surface polarization models 64 3.7 Induced polarization 66 3.7.1 Introduction 66 3.7.2 Information-theoretical relations concerning resistivity dispersion 66 3.7.3 Modelling of IP anomalies 69 3.7.4 Applications 70 3.8 Self-potential 71 3.8.1 Origin of potentials 71 3.8.2 Integral equations for mineralization potential 73 3.8.3 Integral equations for electrofiltration potential 78 3.9 Electrical anisotropy 82 3.9.1 Definition 82 3.9.2 Physical significance 83 3.9.3 Integral equation for perfect conductor in an anisotropic environment 87 3.9.4 Integral equation for an anisotropic body in an anisotropic environment 91 4 Elements of magnetostatics 94 4.1 Introduction 94 4.2 Integral representation of magnetic potential 94 4.3 Volume distribution of poles 97 4.4 Surface distribution of poles 99 4.5 Volume distribution of dipoles 100 5 Magnetic methods 102 5.1 Magnetic properties of rocks 102 5.2 High-susceptibility models 103 5.2.1 Three-dimensional models 103 5.2.2 2~-dimensional models 106 5.2.3 Thin sheet model 107 5.3 Demagnetization and low-susceptibility models 110 5.3.1 Introduction 110 5.3.2 The demagnetization factor 111

Contents VB 5.3.3 Low-susceptibility models 113 5.4 Numerical applications 116 5.5 Effect of remanence 120 6 Electromagnetic methods 6.1 Introduction 6.2 Boundary-value problems for electromagnetic fields 6.2.1 Electric field 6.2.2 Magnetic field 6.3 Green's dyadics for electromagnetic boundary-value problems 6.4 Volume integral equations for three-dimensional electromagnetic fields 6.4.1 Electric field 6.4.2 Magnetic field 6.5 Volume integral equations for two-dimensional electromagnetic fields 6.5.1 Introduction 6.5.2 The E1- field 6.5.3 The Ell field 123 123 124 124 126 127 131 131 136 139 139 139 142 6.6 Surface integral equations for electromagnetic fields 144 6.6.1 Three-dimensional model 145 6.6.2 2~-dimensional model 148 6.7 Integral equation solution for electromagnetic fields in a thin conductor model 152 7 Seismic methods 158 7.1 Introduction 158 7.2 Integral formulas for elastic wave fields in an anisotropic medium 159 7.3 Integral formulas for elastic wave fields in an isotropic medium 163 7.4 Separation of elastic wave fields into a compressional and a rotational mode 168 7.5 Integral formulas for acoustic wave fields in the frequency ~m~n 1~ 7.6 Integral formulas for acoustic wave fields in the time domain 172 7.7 Applications 174 7.7.1 Scattering 174 7.7.2 Migration 175 Appendix A 177 Green's function for scalar potential in a two-layer half-space

viii Contents Appendix B Green's function for scalar potential in a half-space with a vertical contact Appendix C Green's function for scalar potential in an anisotropic half-space Appendix D Electric Green's dyadic for a half-space below the ground surface References Index 180 182 184 186 189

Preface Along with the general development of numerical methods in pure and applied sciences, the ability to apply integral equations to geophysical modelling has improved considerably within the last thirty years or so. This is due to the successful derivation of integral equations that are applicable to the modelling of complex structures, and efficient numerical algorithms for their solution. A significant stimulus for this development has been the advent of fast digital computers. The purpose of this book is to give an idea of the principles by which boundary-value problems describing geophysical models can be converted into integral equations. The end results are the integral formulas and integral equations that form the theoretical framework for practical applications. The details of mathematical analysis have been kept to a minimum. Numerical algorithms are discussed only in connection with some illustrative examples involving well-documented numerical modelling results. The reader is assumed to have a background in the fundamental field theories that form the basis for various geophysical methods, such as potential theory, electromagnetic theory, and elastic strain theory. A fairly extensive knowledge of mathematics, especially in vector and tensor calculus, is also assumed. In Chapter 1 the concept of the integral equation is introduced using a simple direct current boundary-value problem. Definitions for various classes of integral equation, and numerical methods used in solving integral equations are briefly discussed. Chapter 2 gives a summary of the elements of potential theory as applied to electrostatic problems. Chapter 3 is concerned with electrical methods in applied geophysics, involving the resistivity and magnetometric resistivity methods as well as the mise-a-la-masse, induced polarization, and self-potential methods. Chapter 4 is concerned with the elements of potential theory applied to magneto static problems, the treatment being completely analogous to that presented for electric potential problems in the preceding two sections, while Chapter 5 considers the magnetic methods of applied

x Preface geophysics. Chapter 6 deals with integral equations for electromagnetic induction problems. Finally, Chapter 7 gives integral formulas for elastic and acoustic wave fields to serve as a mathematical framework for forward and inverse seismic modelling. The idea of writing this book was originally suggested to me by Professor D. S. Parasnis during the EAEG Meeting in Berlin in 1989. I also wish to express my gratitude to him for the scientific editing of the manuscript. I am grateful to my employer, the Geological Survey of Finland, for allowing me to write this book as part of my normal work. I also thank my colleagues and friends who helped in many ways. Of these, Dr Heikki Soininen, Mr Matti Oksama, Mr Pekka Heikkinen and Mr Hannu Hongisto were kind enough to make critical comments on the manuscript, Mrs Sisko Sulkanen and Mr Markku Eskola drew the illustrations, and Dr Peter Ward nursed my English. Lauri Eskola Espoo, Finland

Introduction Geophysical model calculations are important in evaluating field techniques and interpreting geophysical anomalies. Integral equations, which form the subject of this book, are well suited to the modelling of anomalies obtained by electric, magnetic, electromagnetic and seismic methods. The spatial behaviour of static electric and magnetic fields is described by Poisson's equation or by a particular form of it known as Laplace's equation. Solutions to particular problems are specified by boundary conditions which in tum depend upon the physical and geometrical structure of a given model. In addition to the static fields which they describe exactly, Poisson's and Laplace's equations also describe, with a high degree of accuracy, certain types of time-varying field. An example of this is the induced polarization method, which utilizes such low frequencies of source current that the effect of electromagnetic induction is negligible. The behaviour of the dynamic electromagnetic and elastic fields in space and time is described by electromagnetic and elastic wave equations and relevant boundary conditions. All solutions of the boundary-value problems described above fall into one of two classes, analytic and numerical. An analytic solution is obtained in the form of an algebraic equation in which values of the parameters defining the field can be substituted. Analytic methods have the advantage that a general solution can be obtained, from which it is in principle possible to gain an overall picture of the effect of various parameters. A numerical solution takes the form of a set of numerical values of the function, and the effect of different parameters must be calculated separately for each set of values. Hence, an overall picture can often be achieved, but at the expense of a great amount of computation. Analytical solutions to boundary-value problems are restricted to certain simple geometries of earth models in which the discontinuity surfaces of the medium are coincident with constant coordinate surfaces. For modelling more complex earth structures, effective numerical methods are available. There are

xu Introduction two basic approaches to numerical modelling, differential equation methods and integral equation methods. A third approach, the hybrid method, is a combination of the differential and integral equation methods. In the differential equation approach, the entire earth is modelled on a grid, a feature which makes differential equation methods preferable for modelling complex geological structures. Integral equation solutions involve more complex mathematics, but their advantage is that unknown source functions are required only for the anomalous regions. Thus, the integral equation approach is preferable in modelling one or a few small anomalous bodies. A decisive step towards the practical integral equation modelling of geophysical anomalies was taken in transforming the boundary-value problems defined by the partial differential equations and boundary conditions into integral formulas. A basic integral formula for the electromagnetic field was developed by Stratton and Chu in 1939, and Green's whole-space dyadic was given by Levine and Schwinger in 1950. Integral formulae for acoustic waves were derived for the frequency domain by Helmholtz as early as 1860, and for the time domain by Kirchhoff in 1883. Green's dyadic for an elastic wave field was constructed by Morse and Feshbach in 1953. The very intensive development of integral equations and their numerical solutions into effective tools for geophysical interpretation has taken place within the last thirty years in close connection with the development of fast digital computers. A major objective of present-day research is the further development of the flexibility and efficiency of the modelling algorithms for three- and two-dimensional structures excited by three-dimensional sources.