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`3D Seismic Imaging
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`Biondo L. Biondi
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`Investigations in Geophysics Series No. 14
`
`Michael R. Cooper, series editor
`
`Gerry Gardner, volume editor
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`
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`SOCIETY OF EXPLORATION GEOPHYSICISTS
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`Tulsa, Oklahoma, USA.
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`ISBN 0-931830—46—X
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`ISBN 1-56080-137—9
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`Society of Exploration Geophysicists
`P. O. Box 702740
`
`Tulsa, OK 74170—2740
`
`@2006 by Society of Exploration Geophysicists
`All rights reserved. This book or parts hereof may not be reproduced in any form without written permission from
`
`the publisher.
`
`Published 2006
`
`Printed in the United States of America
`
`Library of Congress Cataloging-in—Publication Data
`Biondi, Biondo, 1959-
`3D seismic imaging / Biondo L. Biondi.
`p. cm. -— (Investigations in geophysics ; no. 14)
`Includes bibliographical references and index.
`ISBN 0—931830—46-X -- ISBN 1-56080-137—9
`1. Three-dimensional imaging. 2. Seismic reflection method. I. Title. II.Tit1e: Three dimensional seismic
`imaging.
`QE538.5.B56
`551.0285’6693-—d022
`
`2006
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`_
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`2006050641
`CIP
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` Chapter 9
`
`Introduction
`
`In Chapter 8, we analyzed how the spatial sampling
`rate influences image quality. If data sampling is not suffi-
`ciently dense, the seismic image may lose resolution and/or
`it may be affected by artifacts.
`Unfortunately, however, density of spatial sampling is
`not the only problem encountered with realistic 3D acquisi-
`tion geometries. An even more common problem is irreg-
`ularity of the spatial sampling. Often, irregular sampling in
`space is a product of practical constraints, examples of which
`include cable feathering in marine acquisition and surface
`obstacles in land acquisition. In other cases (e.g., with but—
`ton-patch geometries), irregular sampling geometry might
`be inherent in the survey design.
`The main effect of irregular sampling geometries is
`either uneven illumination or incomplete illumination of the
`subsurface. Such partial illumination causes distortions in
`the image. In milder cases, distortions are limited to the
`image amplitudes, and they are clearly visible in depth or
`time slices. Those distortions often are called acquisition
`footprint. Figure 1 shows an example of acquisition foot-
`prints in a migrated depth slice taken from a marine data set.
`On the right-hand side, horizontalstriping is clearly visible,
`superimposed over the image of a complex turbidite sys-
`tem with crossing channels. The horizontal striping is not
`linked to geology; it is along the direction of the sailing
`lines of the recording vessel.
`When subsurface illumination is not only uneven but is
`also incomplete, the phase of the image is distorted, and
`strong artifacts are created. At the limit, when the acquisi—
`tion geometry has holes, the data are aliased, at least 10-
`cally. In such cases, a distinction between the effects of
`coarse sampling (which we called aliasing in Chapter 8)
`and the effects of irregular geometries obviously is artifi-
`cial. However, it helps to analyze such effects separately
`and to develop independent methods for alleviating the
`problems.
`Either uneven or incomplete illumination can be caused
`by complexity of the velocity function in the overburden,
`as well as by irregular acquisition geometries. Imaging under
`
`Imaging and Partial Subsurfacevlllumination
`
`salt edges is an example Of an important task that suffers
`from partial illumination of the reflectors. The problem
`often is caused by sharp velocity—model variations that pre—
`vent the seismic energy either from reaching the reflectors
`or from propagating back to the surface. Although the im—
`mediate causes of partial illumination differ in the two
`cases — irregular acquisition geometry versus complex over—
`burden —- the final manifestation is the same: The wave-
`field is not sampled sufficiently at depth for migration to
`image the reflectors without artifacts. The concepts and
`methods used to address the uneven—illumination problem
`are similar, regardless of its origin, and consequently I pres-
`ent them in a unified manner.
`
`When illumination is uneven but without gaps, the
`image can be improved substantially by a simple normal—
`ization of the imaging operator or, as it often is called, by
`an operator equalization. In this chapter, we introduce the
`basic concepts of operator equalizations, using a simple
`imaging operator — interpolation followed by partial stack-
`ing — as a proxy for more complex imaging operators. In
`cases when uneven illumination of the reflectors relates
`mostly to irregular acquisition geometry and the velocity in
`the overburden is fairly simple, the DMO or AMO opera-
`tors (Chapter 3) are normalized (Beasley and Mobley,
`1988; Canning and Gardner, 1998; Chemingui, 1999). In
`more complex situations, in which the velocity in the over—
`burden is sufficiently Complex to distort the wavefield or
`even to cause illumination gaps, normalization should be
`applied in the image domain after full prestack migration
`(Bloor et a1., 1999; Rickett, 2003).
`Simple normalization of the imaging operators is not
`sufficient to remove imaging artifacts when illumination
`gaps are large. In such conditions, the data-modeling op-
`erator —— which usually is defined as the adjoint of the im-
`aging operator —— should be inverted by a regularized inver-
`sion methodology. As is true for operator equalization, the
`methods proposed in the literature for inverting imaging
`operators can be divided‘into algorithms based on partial
`prestack migration (Ronen, 1987; Ronen and Liner, 2000;
`Chemingui and Biondi, 2002) and those based on full
`prestack migration. The methods use either a Kirchhoff
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`'l 24
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`3D Seismic Imaging
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`operator (Nemeth et al., 1999; Duquet et al., 2000) or a
`wavefield~continuation operator (Prucha and Biondi, 2002;
`Kuehl and Sacchi, 2002).
`
`Iterative inversion is expensive, especially when a full
`prestack~migration operator is inverted. In this chapter, I
`present a noniterative method for regularizing the model
`space. It improves the quality of the reconstructed data
`without the computational cost of an iterative inversion.
`However, when there are large acquisition gaps or when the
`complexity of the overburden is responsible for incomplete
`illumination of the reflectors, expensive iterative regular-
`ized inversion is unavoidable. At the end of this chapter, we
`discuss some potential applications of iterative inversion.
`
`Equalization of imaging operators
`
`To explore the methods used to equalize imaging op-
`erators, I employ interpolation followed by partial stacking
`as a proxy of more complex imaging operators. As a proxy,
`interpolation has the advantage of being simple, easy to
`understand, and easy to manipulate analytically. Its analy-
`sis will lead us to discuss fundamental issues regarding
`spatial interpolation of seismic traces and normalization, or
`equalization, of imaging operators. The lessons we learn by
`using interpolation are applicable to the equalization of
`several imaging operators.
`Stacking is the operation of averaging seismic traces
`by summation. It is an effective way to reduce the size of
`data sets and to enhance reflections while attenuating
`noise. To avoid attenuating the signal along with the noise,
`the reflections need to be coherent among the traces that
`are being stacked. To increase trace coherency, we can
`
`4000
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`6000
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`Inline midpoint (m)
`8000
`10000
`
`12000
`
`NOOO
`
`(m)
`Crosslinemidpoint
`
`
`4000
`
`apply simple normal moveout (NMO) before stacking, or a
`partial—prestack—migration operator such as DMO or AMO
`(Chapter 3).
`Global stacking of all the traces recorded at the same
`midpoint location, regardless of their offset and azimuth, is
`the most common type of stacking. Partial stacking aver-
`ages only those traces with their offset and azimuth within
`
`a given range. Partial stacking is useful if we want to pre-
`serve differences among traces when those differences are
`functions of the trace offset and azimuth and thus we must
`
`avoid global averaging. AVO studies are a useful applica-
`tion of partial stacking. Partial stacking also is useful when
`simple transformations, such as NMO, are not sufficient to
`correct for the differences in time delays among traces with
`very different offsets and azimuths. Such a situation is com-
`
`mon when velocity variations cause nonhyperbolic move—
`outs in the data. Because data redundancy is low in partial
`stacking, the results of partial stacking are more likely to be
`affected by artifacts related to irregular acquisition geom-
`etries than are the results of global stacking. Thus, in this
`section, I will focus my analysis on partial stacking, but the
`methods I present here obviously can be applied to global
`stacking operators too.
`To start our analysis, I define a simple linear model
`that links the recorded traces (at arbitrary midpoint loca-
`tions) to the stacked volume (defined on a regular grid).
`Each data trace is the result of interpolating the stacked
`traces and is equal to the weighted sum of the neighboring
`stacked traces. The interpolation weights are functions of
`the distance between the midpoint location of the model
`trace and the midpoint location of the data trace. The sum
`of all the weights corresponding to one data trace usually is
`equal to one. Because the weights are independent of time
`along the seismic traces, for notational simplicity, we col-
`lapse the time axis and consider each element d,- of the data
`space (recorded data) (1 and each element mj of the model
`space In (stacked volume) as representing a whole trace.
`The relationship between data and model is linear and can
`be expressed as
`
`d, = 21- 1,]- m], subject to the constraint Zj lij = 1.
`
`(9.1)
`
`In matrix notation, equation 9.1 becomes
`
`d = Lm.
`
`(9.2)
`
`Figure 1. Example of acquisition footprint in a migrated
`depth slice. The horizontal stripes are related to the acquisi-
`tion sail lines. Notice that the stripes bend when the reflectors
`start to dip in the vicinity of the salt (xm z 5500 m).
`
`The simplest and crudest spatial interpolation is a near—
`est—neighborhood interpolation. For example, if we have
`three model traces and four data traces and we use a simple
`nearest-neighborhood interpolator, equation 9.2 becomes
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