Migration in Seismic Prospecting 9061919088, 9789061919087

Table of contents :
Cover
Series Page
Half Title
Title Page
Copyright Page
Preface to English Translation
Table of Contents
Introduction
Chapter 1: The Principles of Migration
1.1 Wave Equation
1.2 Interaction of Incident and Reflected Waves
1.3 Kirchhoff’s Integral
1.4 Wavefield Continuation on the Basis of Analytical Integral Solutions of Wave Equations
1.5 Wavefield Continuation by Finite-Difference Solution of the Wave Equation
1.6 Models of the Medium and the Seismic Field; Formulation of the Problem of Migration
Chapter 2: Migration in the Space-Time Domain
2.1 Forward Wavefield Continuation
2.2 Inverse Wavefield Continuation
2.3 Features of Migration Based on Integral Transforms in the x, t-Domain
2.4 Finite-Difference Migration in the X, /-domain
Chapter 3: Migration in the Frequency Domain
3.1 Migration in Kx, -Domain
3.2 Migration in the Kx, t-Domain
3.3 Migration in x, -Domain
Chapter 4: Optimised and Non-Linear Migration
4.1 Ideal and Regularised Migration Filters
4.2 Migration in the Presence of Noise: Unordered Medium
4.3 Migration in the Presence of Noise: Ordered Medium
Chapter 5: Effect of Velocity Structure Inhomogeneity
5.1 Direct Migration
5.2 Recursive Migration
5.3 Estimation of Migration Velocities
5.4 Directivity Features of Migration
Chapter 6: Pseudo-Three-Dimensional and Three-Dimensional Migrations
6.1 Pseudo-Three-Dimensional Migration for Individual Seismic Profiles
6.2 Three-Dimensional Migration
6.3 Preparation of Input Data for 3-D Migration
Chapter 7: Some Peculiarities of Migration Results and Technology
7.1 Effect of Limited Size of Time Section
7.2 Aliasing Effects
7.3 Resolving Power of Migration
7.4 Migration Strategy
7.5 Summary
References

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E \T . Sarukhanyan: Structure and Variability o f the Antarctic Circumpolar Current V.A. Shapa (Chief editor): Biological Plant Protection A.I. Zakharova: Estimation of Seismicity Parameters Using a Computer M.A. Mardzhanishvili & L.M. Mardzhanishvili: Theoretical and Experimental Analysis of Members of Earthquake-proof Frame-panel Buildings S.G Shul’man: Seismic Pressure of Water on Hydraulic Structures Yu. A. Ibad-zade: Movement of Sediments in Open Channels I.S. Popushoi (Chief editor): Biological and Chemical Methods of Plant Protection K.V. Novozhilov (Chief editor): Microbiological Methods for Biological Control of Pests o f Agricultural Crops K.I. Rossinskii (editor): Dynamics and Thermal Regimes o f Rivers K.V. Gnedin: Operating Conditions and Hydraulics o f Horizontal Settling Tanks G.A. Zakladnoi & V.F. Ratanova: Stored-grain Pests and Their Control Ts.E. Mirtskhulava: Reliability of Hydro-reclamation Installations Ia.S. Ageikin: Off-the-road Mobility o f Automobiles A.A. Kmito & Yu.A. Sklyarov: Pyrheliometry N.S. Motsonelidze: Stability and Seismic Resistance o f Buttress Dams Ia.S. Ageikin: Off-the-road Wheeled and Combined Traction Devices Iu.N. Fadeev & K.V. Novozhilov: Integrated Plant Protection N.A. Izyumova: Parasitic Fauna of Reservoir Fishes of the USSR and Its Evolution O.A. Skarlato (Editor-in-Chief): Investigation of Monogeneans in the USSR A.I. Ivanov: Alfalfa Z.S. Bronshtein: Fresh-water Ostracoda M.G. Chukhrii: An Atlas of the Ultrastructure of Viruses o f Lepidopteran Pests of Plants E.S. Bosoi et a!.: Theory, Construction and Calculations of Agricultural Machines, Volume 1 G.A. Avsyuk (Editor-in-Chief): Data of Glaciological Studies G.A. Mchedlidze: Fossil Cetacea of the Caucasus A.M. Akramkhodzhaev: Geology and Exploration o f Oil- and Gas-bearing Ancient Deltas N.M. Berezina & D.A. Kaushanskii: Presowing Irradiation o f Plant Seeds G.U. Lindberg & Z.V. Krasyukova: Fishes of the Sea o f Japan and the Adjacent Areas of the Sea of Okhotsk and the Yellow Sea N.I. Plotnikov & 1.1. Roginets: Hydrogeology o f Ore Deposits A.V. Balushkin: Morphological Bases of the Systematics and Phylogeny o f the Nototheniid Fishes E.Z. Pozin et al.: Coal Cutting by Winning Machines S.S. ShuFman: Myxosporidia of the USSR G.N. Gogonenkov: Seismic Prospecting for Sedimentary Formations I.M. Batugina & I.M. Petukhov: Geodynamic Toning o f Mineral Deposits for Planning and Exploitation of Mines 1.1. Abramovich & I.G. Klushin: Geodynamics and Metallogeny of Folded Belts M.V. Mina: Microevolution of Fishes K.V. Konyaev: Spectral Analysis of Physical Oceanographic Data 1.1. Tseitlin & A. A. Kusainov: Role of Internal Friction in Dynamic Analysis of Structures E.A. Kozlov: Migration in Seismic Prospecting

MIGRATION IN SEISMIC PROSPECTING

MIGRATION IN SEISMIC PROSPECTING

E.A. KOZLOV

RUSSIAN TRANSLATION SERIES 82

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

A BALKEMA BOOK

Published by: CRCPress/Balkema P.O. Box 447, 2300 AK Leiden, The Netherlands e-mail: [email protected] www.crcpress.com - www.taylorandfrancis.com © 1990 by Taylor & Francis Group, LLC CRC Press/Balkema is an imprint of the Taylor & Francis Group, an iriforma business

No claim to original U.S. Government works ISBN 13: 978-90-6191-908-7 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Translation of: Migratsionnie preobrazovaniya v seismorazvedke, Nedra, Moscow, 1986. The author revised and updated the manuscript of the English Edition in 1989. Translator Technical Editor Language Editor

B.B. Bhattacharya S.H. Rao Margaret Majithia

The translation was checked by the author, Dr. E.A. Kozlov.

PREFACE TO ENGLISH TRANSLATION

This book first attempts to formulate the problem of 'migration’ as a method applied in seismic prospecting and then to comprehend the unifying fundamentals of the numerous and various migration techniques now in use. Guidelines are offered for the practical implementation of this method in complex structural as well as stratigraphic environment and the shortcomings and limitations that ought to be kept in mind when accomplishing migration or interpreting the resultant data are discussed. Hopefully, the book will prove of use not only to practising professionals engaged in designing or running migration procedures, but also to those who would rather deal with the results of migration. The book is not entirely my personal creation. Significant contributions were made, either in the form of discussions on unclear aspects or by direct participation in developing certain algorithms, by A.S. Alekseev, Yu.V. Timoshin, A.K. Urupov, S.A. Vasil’ev, V.M. Glogovskii, I.A. Mushin, V.I. Meshbei, as well as N.M. Tsyplakova and E.Yu. Arkhipova. I thank them sincerely for their assistance. I also thank Dr. B.B. Bhattacharya for his fine translation of this work into English. September 1989

E.A. Kozlov

CONTENTS

INTRODUCTION

1

CHAPTER 1. THE PRINCIPLES OF MIGRATION

6

1.1 Wave Equation 1.2 Interaction of Incident and Reflected Waves 1.3 Kirchhoff’s Integral 1.4 Wavefield Continuation on the Basis of Analytical Integral Solutions of Wave Equations 1.5 Wavefield Continuation by Finite-Difference Solution of the Wave Equation 1.6 Models of the Medium and the Seismic Field; Formulation of the Problem of Migration CHAPTER 2.

MIGRATION IN THE SPACE-TIME DOMAIN

2.1 Forward Wavefield Continuation 2.2 Inverse Wavefield Continuation 2.3 Features of Migration Based on Integral Transforms in the x, /-Domain 2.4 Finite-Difference Migration in the x, /-Domain CHAPTER 3.

MIGRATION IN THE FREQUENCY DOMAIN

7 13 15 17 28 35 44 44 57 70 82 107

3.1 Migration in kx, «^-Domain 3.2 Migration in the k Xi /-Domain 3.3 Migration in x, ^ -D o m a in

107 121 125

CHAPTER 4. OPTIMISED AND NON-LINEAR MIGRATION

128

4.1 Ideal and Regularised Migration Filters 4.2 Migration in the Presence of Noise: Unordered Medium 4.3 Migration in the Presence of Noise: Ordered Medium

128 132 141

vii

viii

CHAPTER 5. 5.1 5.2 5.3 5.4

EFFECT OF VELOCITY STRUCTURE INHOMOGENEITY

Direct Migration Recursive Migration Estimation of Migration Velocities Directivity Features of Migration

CHAPTER 6.

PSEUDO-THREE-DIMENSIONAL AND THREE-DIMENSIONAL MIGRATIONS

6.1

Pseudo-Three-Dimensional Migration for Individual Seismic Profiles 6.2 Three-Dimensional Migration 6.3 Preparation of Input Data for 3-D Migration

CHAPTER 7. 7.1 7.2 7.3 7.4 7.5

SOME PECULIARITIES OF MIGRATION RESULTS AND TECHNOLOGY

Effect of Limited Size of Time Section Aliasing Effects Resolving Power of Migration Migration Strategy Summary

REFERENCES

150 150 170 186 188 197 197 209 228 235 235 239 247 252 261 269

INTRODUCTION

The term ‘migration’ is currently widely used in connection with seismic prospecting. It has been borrowed from western literature and is often used as a synonym for diffraction transform, continuation (or extrapolation) of seismic fields, seismic focussing, mapping, seismic holography, etc. However, all these terms actually depict different aspects of the physical meaning of the procedure. The term ‘migration’ best reveals the geophysical sense of this procedure. It denotes primarily seismic ‘stripping’ or, to be more precise, placing the images of reflectors and scatterers on a seismic section in their proper position. The physical principle of migration, based on the theory of propagation of waves, is utilised in the present book in-so-far as necessary to understand the formulae used in migration processes. Emphasis is mainly given to those aspects that lead to the improvement and use of algorithms of migration, namely, its computational nature (which, in fact, is reduced to two- or three-dimensional space and time filtering), noise immunity and possibilities of optimisation as well as geological effectiveness. The problem of migration is formulated and discussed in Chapters 1 and 2. Only the general concept of the method is outlined here. During interpretation, reflection seismic records are considered images of the medium which characterise the reflectivity of every point of the interior. The images (seismograms or time sections) are generally presented in .v and t co-ordinates, where x is the abscissa and r vertical time. Let each point x,y, z of the medium with average velocity v = v(x9y, z) be characterised by reflectivity (or scattering capacity) k . Let us assume that at a point xi, yi9 z\ the reflectivity is equal to /q and at the rest of the points it is zero. Accordingly, let the undistorted image of the medium in the plane x 9 t , y = y\ (where r = 2z/v) be represented by function k = k (x 9 r) which is zero everywhere except at the point x \9 n = 2zi/v where it is equal to /q, thereby serving as a true image of the scatterers. Due to this scatterer, a diffracted event is registered on the seismogram or time section along profile y — y\. The time-distance curve of this event is approximately hyperbolic, and its amplitude is approximately proportional 1

2

to K. This event may be considered a registered image of the scattered Compared to the true image (which is situated at the apex of the hyperbola, as shown in Fig. 1), the registered image is distorted. It is seen from Fig. 1 that the displacement r of every point of the diffraction hyperbola from its apex may serve as a measure of distortion inherent in the registered image. (Such displacements on seismic records or maps are often referred to as seismic drift in Soviet literature.)

Fig. 1. Seismic drift in the time section for a simple diffracting point. 0—diffracting point in the x, z plane of the medium; 0*—its image in the t section.

To transform a registered image into a true one, it is necessary, provided the possible noise has been eliminated, to carry out two fundamental procedures: 1. Compression in time of the seismic impulse on all traces of the record into sharp spikes, confined to the first arrivals of wave events, and assigning to each spike an amplitude proportional to the amplitude of the event on the corresponding trace. 2. Compression in time as well as space of the resultant dynamic timedistance curve into a single spike at point xi, t \ with an amplitude proportional to *i. The first of these procedures is known as inverse filtering in time. The second procedure is known as migration. The latter procedure looks like the elimination of the displacement r, i.e., shifting (migrating) every point of the diffraction hyperbola back to its apex. In the plane x, r this procedure appears as focussing of waves at a point from which the waves originated. Note that the thinner the overburden over the diffractor, the less the

3

displacements r . This explains why migration may be considered seismic stripping: the migrated records should be identical to those that would have been obtained had the observation line been placed immediately above the level z\ at which the diffractor is situated. Focussing, as well as seismic stripping, describes the geometrical nature of migration. — >

From yet another point of view, the distortions r characteristic of the registered image are due to the diffracted wave propagation through the overburden of non-zero thickness 0 < z < zi. As such, these distortions can be predicted by solving the forward diffraction problem of the wave propagation theory. Hence, in terms of the latter, migration may be considered the inverse problem of diffraction. Our interest in migration techniques is not casual. The methods of seismic drift elimination are rather cumbersome but since the very initiation of exploratory seismology have been considered essential for the processing and interpretation of seismic data. In the past, when analogue processing was common, the effect of seismic drift was taken care of by constructing seismic horizons—lines z(x)—in seismic depth sections using aplanetic surface methods, or by transforming normal-incidence time t maps into vertical time r maps. Like procedures are still in use in computer processing of seismic data. Essentially, this consists of the transformation of parameters of seismic horizons from the x, ¿-domain to parameters in the x, t or x, z-domains. Today the term 'migration’ is mostly applied to procedures for eliminating seismic drift on seismic records, either unstacked trace gathers or stacked time sections; yet the term preserves its meaning in application to the domain of parameters. The general shift from migrating parameters to migrating records has come about for the following reasons. Firstly, seismic drift leads to distortions that rapidly increase with an increase in depth of investigation as well as in tectonic complexity. In present-day exploratory efforts, the main seismic targets are deeper horizons as well as complex regions, such as the Pre-Caspian and Dnieper-Donets basin, the Lower Mesozoics and Paleozoics of western Siberia etc. Under such conditions, the original interference of signal events on unmigrated records is undecipherable, so the determination of parameters of separate events becomes either difficult or impossible. Consequently, the elimination of drift in the parameter domain cannot be accomplished. Secondly, the main purpose of the seismic method nowadays is not only to delineate anticlinal structures, but also to locate non-structural traps or, directly, hydrocarbon-saturated reservoirs. The main object of interpretation has become the wavefield as a whole. Obviously, an interpretation of a registered wavefield heavily distorted by seismic drift cannot be meaningful.

4

Elimination of seismic drift in time or depth sections localises the signal waves at those points of imaged sub-surface where they actually originated and where they characterise the reflectivity (or scattering capacity) of the medium. Such localisation signifies higher seismic resolution along the horizontal direction whereas the inverse time filtering of time sections increases the resolution along the vertical. Thus, migration is a necessary procedure not only in regions of complex structure, but also in such regions where significant lateral changes in reflectivity exist due to tectonic ruptures, pinch outs, lithological changes etc. The idea of diffraction (here migration) transform of seismic records was clearly formulated for the first time by Yu.V. Timoshin in 1960, long before similar publications appeared abroad. He introduced the concept of superposition of diffracted waves in the seismic method and of the medium as a combination of diffraction points. The principles of the theory of diffraction transform, development of its modifications—D-, O-, M-transforms etc. (Timoshin, 1972), and analogous devices for their automatic implementation were proposed together with V.D. Zav’yalnov. Side by side with these, in the early 60s, Yu.V. Tarasov formulated, with experimental support, the principles of focussed mapping (Tarasov and Ryabinkin, 1974). Development of the theory of migration was considerably advanced by the contributions of A.S. Alekseev in two- and three-dimensional inverse dynamic seismic problems (in short, seismic inversion). The studies initiated in the early 60s are still of interest today (Alekseev, 1981). The first attempts to apply early diffraction transforms and focussing algorithms to field records were not very successful. This stimulated renewed efforts in theoretical substantiation of these transforms in the 70s (VasiTev, 1973; Vasil’ev etal., 1974; Petrashen’ and Nakhamkin, 1973). The relationship between these mappings and optical holography was also established (Timoshin, 1978). G.I. Petrashen’ and S.A. Nakhamkin introduced the concept of continuation of the seismic field as an instrument of reconstruction of the seismic field in the medium at the moments of its origination at the points of diffraction. The role of the fundamental theory of propagation of waves, especially of the wave equation and Kirchhoff’s integral as the basis for wavefield continuation, was revealed. Yu. V. Timoshin as well as S.A. Vasil’ev and A.K. Urupov (Vasil’ev et aL, 1974) showed that only three-dimensional prospecting can lead to the correct reconstruction of a scattered (i.e., reflected and/or diffracted from the inhomogeneities of the medium) field at the points of its origin. They proposed an effective method of areal aperture synthesis and developed computer algorithms for diffraction transforms using the Kirchhoff integral. During this period the first significant theoretical and algorithmic developments in migration were made abroad. J. Claerbout (1971, 1976)

5

suggested the principles of wavefield continuation by finite-difference solution of the simplified wave equation in a moving co-ordinate system facilitating numerical applicability of the algorithm. Only in the late 70s did American geophysicists realise the role of Kirchhoff’s transformation as an effective tool for migration. It was during this period that the basics of migration transforms in the time and space frequency domain were outlined (Gasdag, 1981; Stolt, 1978). By the end of the 70s and in the early 80s, migration transforms had become a standard technique in seismic exploration of complex regions. The notable geological effectiveness of such transforms was established. Fast, noise immune, and stable (even in the presence of lateral velocity changes) algorithms of migration were realised in the form of space-time filtering (Meshbei, 1980; Kozlov et al., 1982; Bratchik et al., 1983; Berkhout, 1982; Castle, 1982; Hatton et al., 1981). Experience was gained in three-dimensional (3-D) migration in the form of a two-stage wavefield continuation in two orthogonal directions (Gibson et ah, 1983; Jakubowica and Levin, 1983; Ristow, 1980). Attempts were made to use migration in refraction seismology (Clayton and McMechan, 1981), in designing a two- and threedimensional analogue of pseudo-acoustic logging (Clayton and Stolt, 1981; Loewenthal et a l , 1976), and in dealing with side-swipe1 reflections while processing conventional seismic reflection profiling data supplemented by measurements of cross-dips (Kozlov, 1982; Marchuk, 1977). The unifying fundamentals of the numerous and various migration techniques now in use are reviewed in this book as also their shortcomings and limitations, which ought to be kept in mind when accomplishing migration or interpreting the resultant data.

Hhis term refers to the fact that the reflection points are situated outside the vertical plane through the seismic line.

1

THE PRINCIPLES OF MIGRATION

The process of acquisition of seismic signals can be modelled (Berkhout, 1982) as a system which transforms the input information by a succession of operators S, Ws r , R, Wr d , D in the presence of noise contributions N\ and Ni; the former are related to the source, while the latter are not (Fig. 2). Here the operator S shows the effect of source, Wsr the propagation of the wavefield from the source to a scattering inhomogeneity of the Earth’s interior, R the transformation of the field by this inhomogeneity, Wr d the propagation of the field from the inhomogeneity to the detector, and D the transformation of the field by the detector.

Fig. 2. Ray path diagram (a) and block diagram (b) for reflection seismic model.

The medium here is characterised by the operator R and the seismic problem is formulated as the study of the medium by deciphering the information that the operator R introduces into the wavefield. Seismic drift appears as the consequence of the effects introduced by operators W s r and Wr d • The migration problem in the framework of the given model is elimination of these effects. The propagation of the wave in the medium, represented by operators W s r and Wr d , is assumed to be described by equations of the theory of elasticity or, approximately, by wave equations. These equations form the basis of migration. 6

7 1.1 Wave Equation

The equation of elastic wave propagation in the simplest medium—a homogeneous fluid in the absence of the source—is obtained directly from Newton’s second law of motion AF — Am

Cv dt

(l.D

and Hooke’s law for a fluid without heat losses dV dp = - K AF

( 1. 2)

Equation (1.1) states that for the given acceleration dv/dt of the elementary volume of the fluid of mass Am, such external pressure must be applied as to generate the forceAF. According to equation (1.2), the increment dp in the pressure p of the fluid leads to the increment dV of the elementary volume of the fluid AF, proportional to the bulk modulus K. In the Cartesian co-ordinate system, the component AFx of the force AF is proportional to (with the opposite sign) the increment Ap = (dp/dx)Ax of the pressure p on the elementary area AS — AyAz: AF*

(1.3)

(AF = AxAj;Az

Similarly, (1.4)

AF, = - ^ A F.

(1.5)

Equations (1.3)-( 1.5) permit us to express AFin terms of the gradient aF

== (d ? ly ’

sca^ar Pressure fold P = F(*> y> z):

AF = — (A V)vp . (1.6) As Am = PAF, where p is the fluid density, on substituting equation (1.6) in (1.1) one obtains the equation 1

’o, zo, then in place of (1.11) and (1.14), equations *2P ~

“ 2

P(t ~ ~-47t $(* — xo) &(y — yo) S(z — zo),

(1.17)

11

V2P +

0)2

P — —4wS(x — xo) 8(j — jo) 3(z — zo)

(1.18)

hold true. Let us recall the properties of the S-function 3(u), u = x, j , z, . • • ( co, u = 0 , (1.19) Ku) = ] {. 0, u # 0;

J /(« )

(1.20)

3(« — wo) == | f(u) 3(w0 — m) du = / ( mo) ;

1

'^ll^ cti

^ Uo ~

=

J / ( mo

“ u)

=

J

/(«)

Ku

— mo)

=

[S(m)] du = — J“ / ( m) 1«=««;

F,,[S(„)] = 1; FuiKu —

mo)]

= exp( —ik„uo).

(1.20') (1.21) (1.21')

Here and below Fu[X{u)\ denotes the Fourier transform of function X(u) with variable u; the symbol F “1 denotes the inverse Fourier transform and ku is the frequency in the a co-ordinate. Equations (1.11)—(1.13) are derived for a perfectly elastic homogeneous fluid, which is far too idealised for real media. Nevertheless, precisely these equations serve as the basis for developing all migration algorithms used in practise today. There are, however, several experimental migration algorithms available which use the elastic wave equations. Though greatly idealised per se, these algorithms supposedly better describe the wave propagation in real solid media than the scalar wave equations (1.11)—(1.13). Leaving these algorithms aside, we shall outline here the main limitations which, either explicitly or implicitly, are imposed on the model of the medium when simple wave equations (1.11)—(1.13) are used instead of elastic ones. Consider the three-dimensional elastic medium whose density p and Lamé constants A and n and, consequently, whose longitudinal and transverse wave velocities v = V (A + 2p)/P and Vs = V plP are continuous functions o f the three co-ordinates x, y 9 z. The equations of motion for the components p X9 py, pz o f the displacement vector p — p(N9 M , /), where N = (xN, y # 9 zN) is the point o f excitation, and M = ( x m 9 yM, z m ) is an arbitrary point in the medium, in accordance with the dynamic theory of propagation of seismic waves, may be written in the form (A + /*) ¿ = P

d2Px, 8t2 ’

W + l ^ 2Px¡ + | | VP + VP ( f j + VPx¡) ( 1. 22)

12

Here x\ = x 9 xi = y, x 3 = z. Since in seismic prospecting generally only the component pz is measured, we will consider only the equation with Xi =

X3 =

Z.

Let us assume that the rays are nearly vertical and the angles of incidence at the reflecting interfaces (of second order) are not large. For such a situation the differences dpx/dz — dpz/8x, dpy/dz — dp2/dy are negligible compared to the derivatives dpx/dz9 dp2/dx, dpy/dz, dp2/dy. Further, Bp* Bx

Spy Bpz Bz •> By

Bpz Bz

These relations show in particular the absence of the effect of conversion of waves from one mode to the other. Thus, 8 _ 8 idpx dJ l \ By + Bz ) d z d z \ d x + ¥ 8X

Bz VP

v 2Pi ;

sx spz

+ v“-) ~ 2{rx

Spz

dp tyz , d/z d p \ + By By + dz dz )

Substituting these expressions in (1.22) for Xi = z and assuming vs/v = V = const, one obtains the relation V Pz

v2 Si2

-Cp S- ln (Pi'2) - 2Y2 dz oz

x [ I f é ln(p"2) + tr | r ln(H The right side here characterises the transformation of the field at the reflecting interfaces (in the absence of the conversion phenomena). For subhorizontal layering of the medium (gentle dips) dpz In (Pv2) dz dz

I2y2 \s x 8x ln {pv2) + By By

In (Pi2)

and therefore the next simplifying step is the transition to the relation V-Pz

_L^2pz v2 dt2

Bpz 8

~Tz rz Xn ipv) ■

This equation for a vertically inhomogeneous medium is valid for the propagation of waves whose ray paths are close to vertical. In the domain of x, y, z, co, this equation takes the form \ 2Pz

+ k 2Pz

(1.23)

13

or with substitution k = 2^/A, T f rz

v v , - - *! [ f t + If A ^

('»■)]•

=

(1.24)

where i = \ / — 1 , or ± i V k 2 - (k2 x + k 2)-P - 0 .

(1.24')

Transferring from equation (1.24) again to the spatial co-ordinates x and y yields +

Here the factor

^

+ W ' P ~°-

+ - & + $

), p0 = P{kx, ky, zo, ^); zo and zi are respectively the levels at which the field is known and at which it has to be determined (Fig. 4). It can be shown that the solution (1.39) directly follows from the formulation of the problem of continuing the wavefield from one level to another. Let us express P as the Taylor series expansion of Po: P

Po ±

dz 1! +

8z2 2! ± ■■' +

u dzn n\ ^ Substituting in equation (1.40) the relation f)nP

= (ik,YPo,

(1.40)

18

Fig. 4. Forward continuation of (a) downgoing and (b) upgoing waves from level z0 to zi.

obtained from the Table (see section 1.1), we get P = Po 1 ±

ffczAz . (//czAz)2 1!

+

2!

+ (-0 *

(/ArzAz)" + . Ml

(1.41) It is immediately seen that the expression within brackets is the series expansion of exp (±/fczAz), which reduces equation (1.40) to (1.39). Solution (1.39) of wave equation (1.13) is the basis for algorithms of migration in the domain of spatial and temporal frequencies or, in short, in the domain kx, ky, z, . The operator W (see Fig. 2), which describes the propagation of the field from one level to another, is represented here by exp ( ± ik2k z ) . The direction of continuation determines the choice of the sign of the quantities Az and kz = db \ / k 2 — (kl + k j) • If the field propagates in the direction of axis z, then the quantity kz takes a minus sign. For z\ > zo, the difference A* > 0 and, consequently, the power of the exponents is negative imaginary (for k2 > kl + k 2y) or negative real (for k,1 < kl + k 2v).

19

sign. If the field propagates in the opposite direction, then kz takes the For zi < zo, Az < 0, the power of the exponent stays negative whether imaginary or real. In other words, when the direction of continuation coincides with the direction of wave propagation, then the sign of the exponent power in equation (1.39) is negative. The coincidence of the directions of wave continuation and propagation is characteristic of the forward wavefield continuation (Petrashen’ and Nakhamkin, 1973) and therefore the operator of forward continuation (denoted here by W) may be written in the form W = W (kx, ky, z, oj) = exp ( - i k z \ & z \ ) .

(1.42)

This relation would play a major role in what follows. We shall use it to describe the operators of wave propagation for our initial model (see Fig. 2), assuming Ws r — W r d

=

W .

Recall that migration is intended to compensate the effects engendered by these operators. Forward continuation leads to an increment in event travel time. The positive sign for the index of the exponent in equation (1.39) is for that case in which the directions of wave continuation and wave propagation are opposed. In such a case, the procedure is known as inverse, or downwards continuation. The operator of the inverse continuation is H = H (ikx, ky9Az, ) = exp (ikz | A* |) .

(1.43)

Inverse continuation is carried out in decreasing time; the travel times of events decrease. From equations (1.42) and (1.44) it follows that H = 1/IF .

(1.44)

Moreover, in the region (co/zi)2 < k l + k 2v H = W* , 1— »

(1.440 ’— >

where W* is the complex conjugate of IF. An important feature of the wave continuation operator is that, in the domain kX9 ky>z, , it is a multiplicative: The resultant wavefield P is calculated as the product of the initial wavefield Po and operator IF [see equation (1.39)]. Thus, the procedure of wavefield continuation is a filtering process and IF acts in (1.39) as the transfer function of the filter, which could be termed either a forward continuation filter or a forward migration

20

filter. It is obvious that in the space and time domain or in the x, y, z, w-domain, the procedure of wavefield continuation must correspond to the three-dimensional and two-dimensional convolution respectively: pi = P0{x)(y)iD W, Pi = Poix){y) W . Here the symbols (*),(*>,(*) denote the convolution along x, y and t respectively; W = W (x, y, z, ui) and w = w (x, y, z, t) act as impulse responses of the forward migration filter, related to Why two- and three-dimensional Fourier transforms. Let us now find the explicit expression for the operator W = W(x, y, z, a>) = F ^ F ^ [ W ] . For this, following Brekhovskikh (1973), we shall consider the forward two-dimensional Fourier transform P = P (kx, ky, z, a>) of field P = P (x9 y5 z, w) of a point source situated at the origin of the co-ordinate system. The latter, as is known, can be expressed as P = exp (— ikr)/r ,

(1.45)

where r — V x2 + y2 + z2 . Thus, one can write: P = ^

| J CXP -■y

j

x exp [— i (kx x + k y y)] dx dy .

(1.45')

—00

To begin with, consider the field at the plane z = 0. Denoting P\z=o = Fo, P \z~o = P°, \ / x 2 + y2 = ro, we obtain instead of (1.45): Po =

I I exp ^ ro lkr^

X

exp [— i{kxX + ky y)] dx dy .

—00

Function P describes plane waves while function P refers to spherical waves. For the sake of convenience, we shall transfer to the polar co-ordinates. Denoting *

=

Vkl

+

kl ,

kx — q cos i[>,ky = q sin >p , x = ro cos 9 ,y — ro sin p, dx dy = ro dro dp , we get 2t t

oo

* - & ) >o /o f

C

exp — /ro [k + g cos (^ - dro .

21

Integrating (Brekhovskikh, 1973) yields Po = — il2tt \ / k2 — q2 . But V k 2 - q2 = V k 2 - (k2 x + k 2) = kx ; hence Po = — ijl^kz = ( 1/2tt) ( 1//Æ2) .

Now the field due to a point source Po can be expressed via its plane wave expansion, using inverse Fourier transforms: «

Po =

II

Po exp

[¡(fcx +

k,y)]dkx dky ,

or 00

= -i- f f 7^- x exp [i(kxx + kyy)] dkx dky .

ro

( 1.46)

2*™ J J ikz — 00

The field Po, initially determined at the plane z = 0, needs to be continued at some level | Az| ^ 0. To do that, the Fourier transform Po of the field Po at the right side of ( 1.46) must be multiplied by transfer function ( 1.42) of the forward wavefield continuation, and the field Po^P(ro) at the left of ( 1.46) must be replaced by P^P(r) = exp (— ikr)\r. This gives

'*» Ar

(1.54)

is fulfilled, so that 0.54')

G' = — exp (— /&Ar')/Ar' . In that case 8 ,

)=

1 + ik&r' , , (Ar;) ~ COS 9 CXP

.» a /\ lkAr ) =

1 + ik&r cos