Seismic Wave Propagation in Stratified Media Seismic Wave Propagation in Stratified Media BRIAN KENNETT Published by ANU E Press The Australian National University Canberra ACT 0200, Australia Email: [email protected] This title is also available online at: http://epress.anu.edu.au/seismic_citation.html National Library of Australia Cataloguing-in-Publication entry National Library of Australia Cataloguing-in-Publication entry Author: Kennett, B. L. N. (Brian Leslie Norman), 1948- Title: Seismic wave propagation in stratified media / Brian Kennett. Edition: New ed. ISBN: 9781921536724 (pbk.) 9781921536731 (pdf.) Subjects: Seismology. Seismic waves. Seismic tomography. Dewey Number: 551.220287 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior permission of the publisher. Cover design by ANU E Press. Printed by University Printing Services, ANU First edition © 1983 Cambridge University Press This edition © 2009 ANU E Press Table of Contents Preface page ix 1 Introduction 1 1.1 Seismic signals 1 1.2 Seismogram analysis 5 1.3 Seismic waves 6 1.3.1 The effect of prestress 6 1.3.2 Material anisotropy 7 1.3.3 Attenuation 9 1.4 Heterogeneity 14 1.5 Stratified models 16 1.6 Preview 17 2 Coupled Equations for Seismic Waves 20 2.1 Depth dependent properties 21 2.1.1 Coupled second order equations and propagation invariants 25 2.1.2 Recovery of spatial displacement from b vector 27 2.2 Fundamental and propagator matrices 30 2.2.1 The propagator matrix 32 2.2.2 Propagators and sources 35 3 Stress-Displacement Fields 37 3.1 A uniform medium 37 3.2 A smoothly varying medium 43 3.3 Uniform approximations for a smoothly varying medium 46 3.3.1 An asymptotic expansion 51 3.3.2 Interaction series 52 3.3.3 Relation to eigenvector decomposition 54 3.A Appendix: Transverse isotropy 57 4 Seismic Sources 59 4.1 Equivalent forces and moment tensor densities 59 4.2 The representation theorem 62 4.2.1 Source representation 64 4.2.2 Relaxation sources 67 v Table of Contents 4.3 The moment tensor and source radiation 68 4.3.1 A small fault 70 4.3.2 Radiation into an unbounded medium 72 4.3.3 Influences on radiation patterns 76 4.4 The source as a stress-displacement vector discontinuity 77 4.5 Wavevector representation of a source 79 5 Reflection and Transmission I 81 5.1 Reflection and transmission at an interface 81 5.1.1 SH waves 82 5.1.2 Coupled P and SV waves 83 5.1.3 The variation of reflection coefficients with slowness 87 5.2 A stratified region 90 5.2.1 The wave-propagator 90 5.2.2 Displacement matrix representations 93 5.2.3 Generalisation of reflection matrices 96 5.2.4 Reflection matrices for spherical stratification 99 5.A Appendix: Unitary relations for reflection and transmission 100 6 Reflection and Transmission II 103 6.1 Reflection and transmission matrices for composite regions 103 6.1.1 Superimposed stratification 103 6.1.2 Generalisation of addition rules 104 6.1.3 Interpretation of addition rules 105 6.2 Reflection from a stack of uniform layers 108 6.2.1 Recursive construction scheme 108 6.2.2 Comparison between recursive and propagator methods 110 6.3 Reflection matrices for piecewise smooth models. 111 6.3.1 Reflection from a gradient zone 112 6.3.2 Recursive construction scheme 119 6.3.3 Gradient zone calculations 120 6.4 The time dependence of reflections from a stratified region 123 6.A Appendix: Mixed solid and fluid stratification 127 7 The Response of a Stratified Half Space 129 7.1 The equivalent surface source representation 129 7.2 A source at depth 133 7.2.1 Treatment via free-surface reflection matrices 133 7.2.2 Explicit representation of free-surface reflections 139 7.3 Recovery of the response in space and time 140 7.3.1 The spectral method 142 7.3.2 The slowness method 148 7.3.3 Examples of complete theoretical seismograms 154 8 The Seismic Wavefield 158 8.1 Controlled source seismology 158 vi Table of Contents 8.1.1 Reflection studies 159 8.1.2 Refraction studies 161 8.2 Ranges less than 1500 km 163 8.2.1 Strong ground-motion 164 8.2.2 Local events 164 8.2.3 Regional events 167 8.3 Body waves and surface waves 168 8.3.1 Body waves 169 8.3.2 Surface waves 172 8.4 Long range propagation 174 9 Approximations to the Response of the Stratification 177 9.1 Surface reflections 177 9.1.1 Surface multiples 181 9.1.2 Alternative approach for surface reflections 185 9.2 Split stratification 186 9.2.1 Propagation in the upper zone 188 9.2.2 Deeper propagation 190 9.2.3 Sources at depth 193 9.3 Approximate integration techniques 195 9.3.1 Reflectivity methods 196 9.3.2 Approximations for specific seismic phases 201 9.3.3 Teleseismic P and S phases 205 10 Generalized ray theory 209 10.1 Generation of generalized ray expansions 209 10.1.1 Uniform layer models 210 10.1.2 Piecewise smooth media 212 10.2 Ray Selection and generation 214 10.3 Slowness results for generalized rays 217 10.4 The Cagniard method 218 10.4.1 The Cagniard path 221 10.4.2 First-motion approximations 225 10.5 The Chapman method 228 10.6 Attenuation and generalized rays 233 11 Modal Summation 234 11.1 The location of the poles 235 11.2 SH wave dispersion: Love waves 236 11.2.1 A layer over a half space 236 11.2.2 Love waves in a stratified medium 237 11.3 P-SV wave dispersion: Rayleigh waves 240 11.3.1 P and S waves evanescent 240 11.3.2 S propagating, P evanescent 241 11.3.3 Propagating P and S waves 243 vii Table of Contents 11.4 Dispersion curves 245 11.4.1 Dispersion behaviour of an upper mantle model 245 11.4.2 Surface waves and channel waves 247 11.4.3 Computation of surface wave dispersion 251 11.4.4 Variational results 252 11.5 Theoretical seismograms by modal summation 255 11.5.1 Mode branch contributions 258 11.5.2 Examples of modal synthesis 261 11.6 Separation of body wave phases 266 11.A Appendix: Modal residue contributions 268 Appendix: Table of Notation 271 References 275 Index 285 viii Preface The primary sources of data in seismology are seismic records at the Earth’s surface of natural or man-made events. The object of modern seismic analysis is to extract as much information as possible from these surface records about the nature of the seismic parameter distribution with the Earth and the source which generated the waves. One of the most important techniques which has been developed in recent years is the construction of theoretical seismograms, as an aid to structural and source studies. In order to model such seismograms we must take into account the generation of seismic waves by the source, the passage of these waves through the Earth, and their subsequent detection and recording at the receiver. In this book I have endeavoured to present a unified account of seismic waves in stratified media. The emphasis is on the propagation of seismic waves in realistic earth models, and the way in which this can be understood in terms of the reflection and transmission properties of portions of the stratification. With this approach I have tried to show the interrelation between the major methods used for the calculation of theoretical seismograms, and to indicate the circumstances in which they are most useful. The theoretical techniques developed in this book are applicable to a wide range of problems with distance scales which vary from a few kilometres in geophysical prospecting, to many thousands of kilometres for seismic phases returned from the Earth’s core. These applications are illustrated by using examples taken from reflection and refraction seismic work, as well as earthquake studies. I have assumed an acquaintance with the basics of elastodynamics see, e.g., Hudson (1980), and the elements of geometrical ray theory. I have not repeated material which is available in many other sources. Thus, there is no discussion of the classical Lamb’s problem for a uniform half space, but I present a physically based description of the more general problem of the excitation of seismic waves in a stratified half space. Very low frequency seismic wave problems involve the properties of the whole Earth; these are best studied in terms of the free oscillations of the Earth and are not ix Preface treated here. A description of this approach may be found in the book by Lapwood & Usami (1981). This book is a revised version of an essay which shared the Adams’ Prize, in the University of Cambridge, for 1979–1980. The material has grown out of lectures for graduate students given at the University of California, San Diego, and for the Mathematical Tripos Part III at the University of Cambridge. The manuscript was prepared whilst I was a Visiting Fellow at the Cooperative Institute for Research in the Environmental Sciences, University of Colorado, Boulder, and I am very grateful for the generous provision of facilities during my stay. I would like to thank my research students N.J. Kerry, T.J. Clarke and M.R. Illingworth, who have done much to shape my ideas on seismic wave propagation, for their help and criticism.