Geophysical Interpretation using Integral Equations

Transcription

1 Geophysical Interpretation using Integral Equations

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

3 First edition L. Eskola Typeset in 10/12 pt Times by Pure Tech Corporation, India ISBN ISBN (ebook) DOI / 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

4 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 The method of moments The Galerkin method 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 Electrical methods Introduction Resistivity of rocks Resistivity method Introduction Integral equations for charge density Integral equations for potential Integral equation for perfect conductor model Integral equations for two-dimensional earth models Integrodifferential equations for thin conductor models 42

5 Vi Contents 3.4 Magnetometric resistivity Mise-a-la-masse method Integral equations for charge density Integral equation for potential Integral equation for perfect conductor model Surface polarization Introduction Solution for perfect conductor models with surface polarization Scaling of surface polarization models Induced polarization Introduction Information-theoretical relations concerning resistivity dispersion Modelling of IP anomalies Applications Self-potential Origin of potentials Integral equations for mineralization potential Integral equations for electrofiltration potential Electrical anisotropy Definition Physical significance Integral equation for perfect conductor in an anisotropic environment Integral equation for an anisotropic body in an anisotropic environment 91 4 Elements of magnetostatics Introduction Integral representation of magnetic potential Volume distribution of poles Surface distribution of poles Volume distribution of dipoles Magnetic methods Magnetic properties of rocks High-susceptibility models Three-dimensional models ~-dimensional models Thin sheet model Demagnetization and low-susceptibility models Introduction The demagnetization factor 111

6 Contents VB Low-susceptibility models Numerical applications Effect of remanence Electromagnetic methods 6.1 Introduction 6.2 Boundary-value problems for electromagnetic fields Electric field Magnetic field 6.3 Green's dyadics for electromagnetic boundary-value problems 6.4 Volume integral equations for three-dimensional electromagnetic fields Electric field Magnetic field 6.5 Volume integral equations for two-dimensional electromagnetic fields Introduction The E1- field The Ell field Surface integral equations for electromagnetic fields Three-dimensional model ~-dimensional model Integral equation solution for electromagnetic fields in a thin conductor model Seismic methods Introduction Integral formulas for elastic wave fields in an anisotropic medium Integral formulas for elastic wave fields in an isotropic medium Separation of elastic wave fields into a compressional and a rotational mode Integral formulas for acoustic wave fields in the frequency ~m~n 1~ 7.6 Integral formulas for acoustic wave fields in the time domain Applications Scattering Migration 175 Appendix A 177 Green's function for scalar potential in a two-layer half-space

7 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

8 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

9 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 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

10 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

11 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 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 Green's dyadic for an elastic wave field was constructed by Morse and Feshbach in 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.