Seismic Migration: Imaging of Acoustic Energy by Wave Field Extrapolation, A. Theoretical Aspects PDF
Preview Seismic Migration: Imaging of Acoustic Energy by Wave Field Extrapolation, A. Theoretical Aspects
FURTHER TITLES IN THIS SERIES 1 F.A. VENING MEINESZ THE EARTH'S CRUST AND MANTLE 2 T. RIKITAKE ELECTROMAGNETISM AND THE EARTH'S INTERIOR 3 D. W. COLLINSON, K.M. CREER and S.K. R UNCORN METHODS IN PALAEOMAGNETISM 4 M. BATH MATHEMATICAL ASPECTS OF SEISMOLOGY 5 F.D. STACEYand S.K. BANERJEE THE PHYSICAL PRINCIPLES OF ROCK MAGNETISM 6 L. CIVETTA, P. GASPARINI, G. LUONGO and A. RAPOLLA PHYSICAL VOLCANOLOGY 7 M. BATH SPECTRAL ANALYSIS IN GEOPHYSICS 8 O. KULHANEK INTRODUCTION TO DIGITAL FILTERING IN GEOPHYSICS 9 T. RIKITAKE EARTHQUAKE PREDICTION 10 N.H. RICKER TRANSIENT WAVES IN VISCO-ELASTIC MEDIA 11 W.L. PILANT ELASTIC WAVES IN THE EARTH 12 A.J. BERKHOUT SEISMIC MIGRATION Imaging of acoustic energy by wave field extrapolation 13 V.C. DRAGOMIR, D.N. GHITÄU, M.S. MIHÀILESCU andM.G. ROTARU THEORY OF THE EARTH'S SHAPE Developments in Solid Earth Geophysics 14A S E I S M IC M I G R A T I ON IMAGING OF ACOUSTIC ENERGY BY WAVE FIELD EXTRAPOLATION A. THEORETICAL ASPECTS A.J. BERKHOUT Department of Applied Physics Delft University of Technology Delft, The Netherlands Second revised and enlarged edition ELSEVIER SCIENTIFIC PUBLISHING COMPANY Amsterdam — Oxford — New York 1982 ELSEVIER SCIENCE PUBLISHERS B.V. 1, Molen werf P.O. Box 211,1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, N.Y. 10017 First edition, 1980 Second revised and enlarged edition, 1982 Second impression, 1984 Library of Congress Cataloging in Publication Data Berkhout, A. J., 19^0- Seismic migration. (Developments in solid earth geophysics ; ikA- ) Contents: A. Theoretical aspects — Includes bibliographical references and index. 1. Seismic waves. I. Title. II. Series: Developments lkA in solid earth geophysics ; ! y etc. QE538.5.BVT 1982 551.2 2 82-18376 ISBN 0-MM*2130-0 (v. IUA) ISBN 0-444-42130-0 (Vol. 14A) ISBN 0-444-41799-0 (Series) © Elsevier Science Publishers B.V., 1984 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, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./Science & Technology Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands. Special regulations for readers in the USA — This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be ob- tained from the CCC about conditions under which photocopies of parts of this publica- tion may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. Printed in The Netherlands To Mabel, Carina and Martin VII PREFACE PREFACE TO THE FIRST EDITION The objective of any acoustical echo technique is the collection of information on the internal structure of a medium, avoiding destructive penetration. Examples of media which are being investigated intensively with acoustic waves are: the earths subsurface, transition zones of waterbottoms, human organs and tissues, construction materials etc. One of the major problems with echo-acoustical systems is the improvement of spatial resolu- tion. Particularly, in many different applications much effort is being spent to increase the lateral resolution of the method. Ultimate lateral resolution is determined by wave length. Due to the relatively low propagation velocities of acoustic waves: air : ~ 340 m/s water, human tissue : ~1 500 m/s sedimentary rock : ~3 000 m/s metamorphic rock, steel : ~6 000 m/s, small wave lengths are obtained at relatively low frequencies and, therefore, in principle the ultimate lateral resolution of acoustic echo systems is high. In many practical realizations the actual lateral resolution is far below the ultimate limit. This may be due to unfavourable acquisition conditions or limited data processing facilities. Tech- niques to improve lateral resolution are applied in different fields under different names. For example: 1. Focussing (ultra-sonic imaging) 2. Image reconstruction (tomographic imaging) 3. Aperture synthesis (radio astronomy) 4. Migration (seismic imaging). Seismic migration is never applied as a real-time technique but is carried out digitally in the processing centre. The first digital application was reported by D. W. Rockwell in 1965. In the early seventies an important extension to the migration theory was proposed by J. F. Claer- bout by introducing the wave equation for compressional waves into the theory. Claerbout and his group at the Stanford University have given an invaluable contribution to the modern way of migration. In addition, the excellent work of Stolt, Gazdag, Schneider, Larner and Hatton, Rocca and many others has also contributed to our extensive knowledge on migration today. VIII In this book the migration theory is derived from first principles. In this respect it serves the purpose of a textbook. However, an appreciable amount of material is new and has not yet been published. Thus, the second half of the book in particular has some of the characteristics of a research monograph. By introducing the propagation matrices and the scattering matrix, an elegant formulation of the forward modeling and migration theory has been obtained, par- ticularly for pre-stack data. The elements of the propagation matrices follow from the wave equation under consideration. The migration theory presented in this book may thus be ex- tended to any type of wave defined by a scalar wave equation. Under certain conditions matrix multiplications can be replaced by convolutions. As most geophysicists are very familiar with the theory of convolution and deconvolution, ample use is made of this property. I am indebted to many friends and colleagues for their assistance. Particularly, the criticism of Prof, van Wulfften Palthe was of great help to me. His support was also essential in getting many of the mathematical formulations in good order. The inspiring discussions with Elio Poggiagliolmi on the seismic aspects of acoustic wave theory and on the limits of resolution were very helpful; I am grateful for many of his sugges- tions. I have had many fruitful conversations with Don Rockwell and Don Larson. Particularly, the discussions on the implementation aspects were most important to me. The exchange of ideas with the staff of the ultra-sonic imaging laboratory at the Thorax Cen- trum of the Erasmus University in Rotterdam taught me how to consider migration from the point of view of focussing. Thanks are also due to Jan Ridder, Diemer de Vries and Len van de Wal who were of great help to get many of the 'bugs' out of the text. I also appreciate the help of Bart de Jong who generated the graphs in the finite-difference chapter. Finally, I would like to thank Mrs. Wilma van Dam for her ever continuing enthousiasm to type and retype the manuscripts over and over again. I also appreciate the help of Mr. A. S. G. de Knegt of the draughting department, Mr. A. R. Suiters of the photographic department and the professional support of Mrs. Angel- ina de Wit of Qualitype Services in preparing the final version of the manuscript. Dr. A. J. Berkhout Delft, February 1980 IX PREFACE TO VOLUME Ά' OF THE SECOND EDITION Encouraged by many reactions from the seismic industry, I have spent a considerable amount of effort to revise and update the original edition. The first four chapters obtained only minor changes, but chapters 5-11 were thoroughly rewritten. The notation has been improved and a number of new aspects were included, a.o. Chapter 5: Kirchhoff integral for inhomogeneous fluids. Chapter 6: Discussion on the discretization of seismic models; inclusion of multiples in the forward problem. Chapter 7: Inverse scattering problem in terms of pre-stack migration and multiple elimina- tion; migration of individual records. Chapter 8: Recursive wave-number mapping techniques. Chapter 9: Three-dimensional migration by two-dimensional summation operators. Chapter 10: Spatial velocity derivatives in finite-difference schemes; stabilization of explicit schemes; combination of wave-number methods and finite-difference techniques. The original edition contained an appreciab e amount of unpublished material and, therefore, it contained the characteristics of a research monograph. However, most of the material in the second edition has already been published and some of it may be considered well established. Considering the large amount of material, it was decided to produce the second edition in two volumes. This volume (A) contains, like the original edition, the theoretical aspects of migration. Practical aspects and examples will be treated in volume B. I hope that this new edition may serve asa helpful textbook toall of them whotruly wish toobtain a thorough understanding on migration techniques, currently being used in the seismic industry. Many thanks are due to Messrs De Graaff, Ridder, Van Riel and Wapenaar, who patiently carried out the proof-reading and who generated many of the new figures. Finally, I would liketothank Mrs. HannekeMulderforhersecretarialsupport. Ialsoappreciatethe help of Mr. A. S. G. de Knegt of the draughting department, Mr. A. R. Suiters of the photographic department and the professional support of Mrs. Gerda Boone of 'Gebotekst' in preparing the final version of the manuscript. Dr. A. J. Berkhout, Delft, May 1982. 1 INTRODUCTION The use of seismic methods has become indispensable in the search for oil and gas. This not only applies to the e x p l o r a t i on for new reservoirs, but also includes the e v a l u a t i on of discoveries and existing fields. Nowadays, appraisal and development drilling is fully guided by integrated know-how from seismic data and bore-hole information. SEISMIC OBJECTIVES The seismic reflection method is an acoustic imaging technique with the objective to collect information from the earth subsurface by measuring and analysing the response to seismic sources at the earth surface. In practical applications the desired information is primarily derived from three properties: 1. Arrival time The arrival time of a seismic event h^s been the only item of interest for many years. From arrival times the s t r u c t u r al property of the subsurface is derived. Moreover, information on the propagation velocity for the different geological layers can be obtained. 2. Ampii tude In the late sixties, the abnormally large amplitude of reflections from gas reservoirs in unconsolidated sands was recognized. This observation was the start of extensive research into the area of seismic amplitudes. Nowadays, seismic amplitudes play an essential role in the prediction of lithology and porefill from seismic data. 3. Character In most situations the character of a seismic event is determined by an interference pattern of several seismic reflections which cannot be separated, e.g. due to lack of bandwidth. Changes in the differential arrival times or amplitudes will change the shape of the event. The concept of character is mainly used by interpreters for qualitative correlation. Iterative modeling techniques aim at a quantitative analysis of interference patterns. 2 SEISMIC PROCESSING TECHNIQUES Since the early sixties, digital seismic p r o c e s s i ng has been playing an important role in exploration seismology. Particularly when it became clear that, apart from s t r u c t u r al details, 1 i t h ο 1 ο g y and ρ ο r e f i 1 1 information could be obtained from seismic data ('seismo- 1 stratigraphy ), much effort has been spent on the design of effective processing procedures. Nowadays, carefully processed seismic data is a valuable asset in the a ρ ρ r a i s al and d e v e l o p m e nt of oil and gas fields ('development seismology'). Seismic processing techniques can be subdivided into four categories: 1. Techniques to improve s i g n a l - t o - n o i se r a t io - averaging; - stacking of common-mid-point data; - long-period reverberation and multiple attenuation; - limitation of the temporal and/or spatial bandwidth. 2. Techniques to improve v e r t i c al r e s o l u t i on - least-squares inverse filtering (whitening); - least-squares prediction-error filtering (gapped decon); - wavelet deconvolution; - minimum entropy techniques. 3. Techniques to improve l a t e r al r e s o l u t i on - use of highly directional patterns (far-field imaging); - migration (near-field imaging). 4. Techniques to extract interpretational information from seismic data such as - reflection coefficients; - propagation velocities; - probability curves for lithology and porefill. Of all processing techniques in seismic exploration, m i g r a t i on has received by far the most attention within the last few years. Modern migration techniques give considerably better results in terms of l a t e r al r e s o l u t i on and t r ue a m p l i t u d e s. An important effect of 3 this new development is the revived interest of the seismologist in basic material such as the s c a l ar w a ve e q u a t i on and d i f- f r a c t i on t h e o r y. As a consequence, many principles that are common knowledge in acoustics and optics are now being evaluated by seismologists. Since the introduction of the f i n i te d i f f e r e n ce techniques by Claerbout (1970, 1972), our insight into wave equation migration has increased significantly. New methods have been added such as the Κ i r c h ο f f s u m m a t i on a p p r o a ch (French, 1975; Schneider, 1978) and k - f m i g r a t i on (Stolt, 1978; Gazdag, 1978). Berkhout and Van Wulfften Pal the (1979) introduced migration as a spatial d e c o n v ol u t i on process in the space-frequency domain. Recently, much attention is being paid to the relationship between migration and inverse scattering (Stone, 1981; Cohen and Bleistein, 1981). MIGRATION - A SYNTHETIC FOCUSSING TECHNIQUE - One of the most important objectives of seismic migration can be formulated as follows: 'to bring seismic waves into focus*. In a physical focus point the contribution of different travel paths arrive in phase. Focussing can be described by a summation procedure along wave fronts. Applications can be subdivided in three categories: 1. Fixed focussing techniques (real-time) 2. Dynamic focussing techniques (real-time/nonreal-time) 3. Synthetic focussing techniques (nonreal-time) In special situations, where the wave field of one source with a f i x ed position has to be focussed, the detectors can be positioned on a fixed wave front surface or an acoustic lens could be used (see fig. la). However, in echo-techniques the wave field to be focussed is generated by many (secondary) sources with different (unknown) positions and, therefore, d y n a m ic focussing should be applied. This means that, prior to summation, time corrections should be applied which are changing with the focussing depth. With dynamic focussing, detectors are normally situated on a plane surface (see fig. lb). We will see with the aid of the wave field extrapolation theory that for optimum focussing a frequency-dependent weighting function should be used in addition to time correction.
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