Seismic True-Amplitude Imaging Jörg Schleicher Martin Tygel Peter Hubral SEG Geophysical Developments Series No. 12 Robert H. Stolt, volume editor Stephen J. Hill, series editor SOCIETY OF EXPLORATION GEOPHYSICISTS The international society of applied geophysics Tulsa, Oklahoma, U.S.A. Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/4158418/9781560801672.fm.pdf by guest on 24 September 2021 “1127fm” — 2007/6/26 — page i — 16:59 — Stage II — #1 ISBN 0-931830-41-9 (Series) ISBN 1-56080-143-3 (Volume) Copyright © 2007 Society of Exploration Geophysicists P.O. Box 702740 Tulsa, OK U.S.A. 74170-2740 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transcribed in any form or by any means, electronic or mechanical, including photocopying and recording, without prior written permission of the publisher. Published 2007 Printed in the U.S.A. Library of Congress Cataloging-in-Publication Data Schleicher, Jörg. Seismic true-amplitude imaging / Jörg Schleicher, Martin Tygel, Peter Hubral. p. cm. -- (SEG geophysical developments series ; no. 12) Includes bibliographical references and index. ISBN 1-56080-143-3 (volume) -- ISBN 0-931830-41-9 (series) 1. Seismic reflection method. 2. Earth--Internal structure. I. Tygel, M. II. Hubral, Peter. III. Title. IV. Series. QE538.5.S34 2007 551.110284--dc22 2007025618 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/4158418/9781560801672.fm.pdf by guest on 24 September 2021 “1127fm” — 2007/6/26 — page ii — 16:59 — Stage II — #2 To Sergey Goldin, whose pioneering ideas have been a great source of inspiration to us And to all who contribute to the advancement of geophysics without receiving the deserved recognition Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/4158418/9781560801672.fm.pdf by guest on 24 September 2021 “1127fm” — 2007/6/26 — page iii — 16:59 — Stage II — #3 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/4158418/9781560801672.fm.pdf by guest on 24 September 2021 “1127fm” — 2007/6/26 — page iv — 16:59 — Stage II — #4 Contents About the Authors ........................................... xii Foreword ................................................. xv Volume Editor’s Preface ....................................... xvii Authors’ Preface ............................................ xix Acknowledgments ........................................... xxv List of Symbols and Abbreviations ............................... xxvii Variables and symbols ...................................... xxvii Latin lowercase letters .................................. xxvii Latincapitalletters..................................... xxxi Calligraphic capital letters ................................xxxvii Greek lowercase letters .................................. xxxix Greek capital letters .................................... xli Other symbols ........................................ xliv Indices and accents ........................................ xliv Subscripts ........................................... xliv Superscripts ......................................... xliv Mathematical accents ................................... xlv Operational symbols ....................................... xlvi Chapter 1: Introduction ....................................... 1 True-amplitude Kirchhoff migration ............................. 1 True-amplitude Kirchhoff demigration ........................... 9 True-amplitude Kirchhoff imaging .............................. 12 Additional remarks on true amplitude ............................ 16 Overview ............................................... 17 Chapter 2: Description of the Problem ............................ 21 Earth model ............................................. 21 Macrovelocity model ................................... 22 Wavemode selection .................................... 22 Coordinate system ..................................... 22 Measurement configurations .................................. 23 Measurement surface ................................... 23 Measurement configuration ............................... 24 Data-space description .................................. 29 Hagedoorn’s imaging surfaces ................................. 30 The diffraction-traveltime, or Huygens surface .................. 31 The isochronous surface ................................. 31 Hagedoorn’s imaging conditions ............................ 32 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/4158418/9781560801672.fm.pdf by guest on 24 September 2021 “1127fm” — 2007/6/26 — page v — 16:59 — Stage II — #5 vi Contents Mapping versus imaging .................................... 33 Migration and demigration: Mapping ........................ 33 Generalized Hagedoorn’s imaging surfaces .................... 34 Unified approach: Mapping ............................... 36 Seismic-reflection imaging ............................... 46 Summary ............................................... 53 Chapter 3: Zero-Order Ray Theory .............................. 55 Wave equations ........................................... 55 Ray ansatz .............................................. 57 Homogeneous medium .................................. 57 Inhomogeneous medium ................................. 58 Time-harmonic approximation ............................. 59 Time-domain expressions ................................ 60 Validity conditions ..................................... 61 Eikonal and transport equations ................................ 63 Acoustic case ......................................... 63 Elastodynamic case .................................... 65 Rays as characteristics of the eikonal equation ...................... 69 Slowness vector ....................................... 69 Characteristic equations ................................. 71 Rayfields .............................................. 72 Ray coordinates ....................................... 72 Transformation from ray coordinates to global Cartesian coordinates . 72 Ray Jacobian ......................................... 73 Solution of the transport equation .............................. 73 Solution in terms of the ray Jacobian ......................... 73 Point-source solutions ................................... 75 Caustics ................................................ 78 Computation of the point-source solution ......................... 79 Homogeneous medium .................................. 79 Inhomogeneous medium ................................. 79 Ray-centered coordinates .................................... 80 Transformation from ray-centered to global Cartesian coordinates ..... 81 Transformation from ray to ray-centered coordinates .............. 82 Ray Jacobian in ray-centered coordinates ...................... 83 Ray-tracing system in ray-centered coordinates .................. 83 Paraxial and dynamic ray tracing ............................... 85 Paraxial ray tracing ..................................... 85 Dynamic ray tracing .................................... 86 Paraxial approximation .................................. 88 Initial conditions for dynamic ray tracing ...................... 89 Ray-centered propagator matrix Πˆ .......................... 90 Rays at a surface ..........................................˜ 91 Vector representations ................................... 91 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/4158418/9781560801672.fm.pdf by guest on 24 September 2021 “1127fm” — 2007/6/26 — page vi — 16:59 — Stage II — #6 Contents vii Surface representation ................................... 93 Transformation from local Cartesian coordinates to ray-centered coordinates ............................ 93 Transformation from local to global Cartesian coordinates .......... 95 Relationship between the slowness-vector representations .......... 96 Surface-to-surface propagator matrix Tˆ ....................... 99 Rays across an interface .....................................˜ 101 Boundary conditions .................................... 101 Dynamic-ray-tracing matrices ............................. 102 Ray Jacobian across an interface ............................ 103 Primary reflected wave at the geophone .......................... 104 Ray amplitude at the geophone ............................. 104 Complete transient solution ............................... 108 Summary ............................................... 109 Chapter 4: Surface-to-Surface Paraxial Ray Theory ................... 111 Paraxial rays ............................................. 111 Traveltime of a paraxial ray .................................. 113 Infinitesimal traveltime differences .......................... 113 Surface-to-surface propagator matrix ......................... 116 Paraxial traveltime ..................................... 120 Ray-segment decomposition .................................. 124 Chain rule ........................................... 124 Ray-segment traveltimes ................................. 128 Meaning of the propagator submatrices .......................... 130 Propagation from point source to wavefront .................... 130 Propagation from wavefront to wavefront ...................... 131 Fresnel zone ............................................. 133 Definition ........................................... 134 Time-domain Fresnel zone ................................ 136 Projected Fresnel zone .................................