 Abstract
 Affiliation
 All
 Authors
 Book Series
 DOI
 EISBN
 EISSN
 Full Text
 GeoRef ID
 ISBN
 ISSN
 Issue
 Keyword (GeoRef Descriptor)
 Meeting Information
 Report #
 Title
 Volume
 Abstract
 Affiliation
 All
 Authors
 Book Series
 DOI
 EISBN
 EISSN
 Full Text
 GeoRef ID
 ISBN
 ISSN
 Issue
 Keyword (GeoRef Descriptor)
 Meeting Information
 Report #
 Title
 Volume
 Abstract
 Affiliation
 All
 Authors
 Book Series
 DOI
 EISBN
 EISSN
 Full Text
 GeoRef ID
 ISBN
 ISSN
 Issue
 Keyword (GeoRef Descriptor)
 Meeting Information
 Report #
 Title
 Volume
 Abstract
 Affiliation
 All
 Authors
 Book Series
 DOI
 EISBN
 EISSN
 Full Text
 GeoRef ID
 ISBN
 ISSN
 Issue
 Keyword (GeoRef Descriptor)
 Meeting Information
 Report #
 Title
 Volume
 Abstract
 Affiliation
 All
 Authors
 Book Series
 DOI
 EISBN
 EISSN
 Full Text
 GeoRef ID
 ISBN
 ISSN
 Issue
 Keyword (GeoRef Descriptor)
 Meeting Information
 Report #
 Title
 Volume
 Abstract
 Affiliation
 All
 Authors
 Book Series
 DOI
 EISBN
 EISSN
 Full Text
 GeoRef ID
 ISBN
 ISSN
 Issue
 Keyword (GeoRef Descriptor)
 Meeting Information
 Report #
 Title
 Volume
NARROW
GeoRef Subject

Primary terms

data processing (8)

geophysical methods (37)

seismology (1)

3D trueamplitude finiteoffset migration
Abstract Compressional primary nonzero offset reflections can be imaged into threedimensional (3D) time or depthmigrated reflections so that the migrated wavefield amplitudes are a measure of angledependent reflection coefficients. Various migration/inversion algorithms involving weighted diffraction stacks recently proposed are based on Born or Kirchhoff approximations Here a 3D Kirchhofftype prestack migration approach is proposed where the primary reflections of the wavefields to be imaged are a priori described by the zeroorder ray approximation As a result, the principal issue in the attempt to recover angledependent reflection coefficients becomes the removal of the geometrical spreading factor of the primary reflections The weight function that achieves this aim is independent of the unknown reflector and correctly accounts for the recovery of the source pulse in the migrated image irrespective of the sourcereceiver configurations employed and the caustics occurring in the wavefield Our weight function which is computed using paraxial ray theory is compared with the one of the inversion integral based on the Beylkin determinant It differs by a factor that can be easily explained
From Volume Editor’s Preface by Robert H. Stolt Seismic TrueAmplitude Imaging is a raytheoretical exposition of seismic imaging processes, unapologetic in that it bristles at any suggestion that ray theory is not wave theory. This attitude is justified in part by being right – ray theory does come from an asymptotic approximation to the wave equation – and by the fact that similar asymptotic assumptions tend to sneak like viruses into socalled fullwavetheoretical processes as well. On other issues, the book is more accommodating. After noting that the original Kirchhoff diffraction integral was devised only for forwardwave propagation, the authors concede that the term Kirchhoff migration has passed into common usage, and they learn to live with it. Similarly, although the term true amplitude is a lexicological minefield, it is widely applied to processes that seek to faithfully preserve amplitude information. The authors are comfortable with that, although they apply a very specific definition of the term in their text. This book uses the term imaging in the widest possible sense, which was an eyeopener for me. I have tended to use imaging as a synonym for migration, but I can’t do that anymore. The authors note that seismic data in any stage of processing is likely to contain discrete events or images, perhaps geometrically distorted but nevertheless pictures or images of the earth’s interior. This observation is facilitated by the raytheoretical point of view, because it is largely in asymptopia that such images form. In consequence, any operation that affects or changes the images in the data can be considered an imaging process. This brings under the imaging umbrella a variety of processes, including partial and residual migration, conversion of one experimental configuration into another, and so on. Strictly, I suppose, Kirchhoff imaging employs a raytheoretical model for propagation and a farfield diffractive model for reflections. It is best suited for a medium composed of regions where earth properties change slowly, divided by reflecting surfaces where properties change abruptly. Where those conditions are met imperfectly, the method may tend to impose this model, because this is how the method views the world. That is not necessarily bad – geophysicists tend to view the world that way too. Kirchhoff depth migration has enjoyed a long period of preeminence. Advances in computer power in recent years have allowed contemplation of fullwaveequation depthmigration algorithms, but the authors of Seismic TrueAmplitude Imaging show no deep concern for the future of raytheoretical imaging. In that, they are probably justified, for several reasons. First, migration methods, once established, never die. The inverse seismic problem is so tough and complex that no single technique, however powerful, can be universally successful. New techniques may be added to the arsenal but are not likely to completely displace proven methodologies. Second, the asymptotic approximation is very well met under most circumstances by seismic data, and even when that is not the case, raytheoretical techniques often are able to extract useful information. Third, ray theory is well suited to trueamplitude processing because amplitudes can be computed explicitly at every point and related back to the underlying earth properties. Fourth, Kirchhoffbased imaging is uniquely suited to composite operations (e.g., demigration using one velocity structure followed by migration using another velocity structure) because, in asymptopia, a composite operation can be condensed into a single operation. For extremely complex geology, ray theory might have difficulty providing a complete description of wave propagation, even where it is technically possible to do so. A fullwaveequation method, in contrast, may provide a complete description without extra effort. That might give some advantage to a fullwave method, but the blessings are mixed. In ray theory, one knows exactly which waves are where. If some portion of the complete waveform (perhaps a multiple or a converted wave) does not contribute positively to the desired image, a raytheoretical method might eliminate it. Even where a fullwave method can produce the better image, one would likely want to run a Kirchhoff algorithm concurrently to aid in analysis. Seismic TrueAmplitude Imaging provides a clear, readable, and reasonably complete presentation of Kirchhoff imaging theory. Although subjects such as beam forming and multipath imaging are not presented in detail, the tools to deal with them are present. For those of us not steeped in ray theory, the book provides a good introduction and tutorial, then digs deeply and profoundly into a theory of generalized imaging.
The front matter contains the title page, copyright page, dedication, table of contents, about the authors, foreword, volume editor’s preface, authors’ preface, acknowledgments, and list of symbols and abbreviations.
In this book, we present a unified theory of 3D seismic trueamplitude imaging that can be applied to seismic records under general measurement configurations. The precise formulation of the trueamplitude concept, which depends on the specific imaging task under consideration, will be given below. The theory relies on the raytheoretical description of the seismic wave propagation involved and assumes an a priori macrovelocity model. This is an initial, or reference, velocity model that incorporates the basic information we have from the subsurface to be imaged. The imaging theory consists of a weighted trueamplitude diffraction stack to migrate the seismicreflection data from the timetrace domain into the depth domain, and a weighted trueamplitude isochron stack to demigrate the migrated seismic image from the depth domain back into the timetrace domain. The diffraction and isochron stacks are explained in connection with trueamplitude migration and demigration. The stacking operations can be cascaded, or chained, for different measurement configurations, velocity models, or elementary waves to permit a variety of trueamplitude image transformations. Many of the described ideas and results are contained in research articles that we have published over the last few years. Our goal is to provide an updated, didactic tutorial of the subject that is accessible to a broader audience that wishes to understand it and, above all, use it.
In this chapter, we discuss in more detail and from a mainly geometric point of view the principles on which the two fundamental seismic processes of trueamplitude migration and demigration are based. We show how they form the basis for a unified theory of Kirchhofftype seismicreflection imaging. By applying these two operations in sequence (i.e., chaining them), a wide class of seismicimaging problems can be solved. These include: 1) Transformation of a seismicdata section in the timetrace domain that was recorded with a given measurement configuration into a section that is as if it were recorded with another configuration, except for the reflection and transmission coefficients. This imaging process generally is referred to as a configuration transform (CT). As particular CTs, we can cite dip moveout (DMO), azimuth moveout (AMO), migrationtozero offset (MZO), shot or offset continuation, etc. In this chapter, we concentrate on the MZO operation, i.e., the transformation of a commonoffset (CO) section into a zerooffset (ZO) section. 2) Transformation of a 3D migrated image in the depth domain into another one for a different (improved) macrovelocity model. This imaging process is referred to as remigration. Other possible image transforms that can be solved by chaining the migration and demigration operations include redatuming, wavemode transformation, transformation of surface data into vertical seismic profiling (VSP) data. This book does not discuss these possible applications in detail. Note that in this context, imaging implies not only going from the timetrace domain to the depth domain or vice versa, but also, in the framework of the unified approach, going from one timetrace domain to another, or from one depth domain to another. We start by briefly describing the earth and macrovelocity models to be considered and the seismic measurement configurations that commonly are used. Then we summarize the basic (mainly kinematic) aspects of the theory to provide a good geometric understanding of all imaging operations involved.
In this chapter, we briefly introduce highfrequency wave propagation in isotropic, inhomogeneous, layered 3D media as is described by zeroorder ray theory. We will cover as much theory as is needed for the development and understanding of the imaging theory that is described in later chapters of this book. The main purpose of this chapter therefore is twofold. One aim is to formulate raytheoretical expressions for the elementary seismic waves by which seismic reflections are described in seismic records (as, e.g., commonshot, commonreceiver, commonmidpoint, or commonoffset gathers). It is these elementary waves from which the seismic images (e.g., depthmigrated images) are to be constructed by the imaging processes that are studied below. The other aim is to provide raytheoretical expressions for all quantities that will appear in the imaging theory to be developed. For a more detailed treatment of most of the topics discussed in this chapter, see Červený (1985, 1987, 1995, 2001). If your main interests involve trueamplitude imaging as presented in Chapter 2, without a need for the underlying details of forward wave propagation, we suggest that you continue your reading of this book with Chapter 7.
SurfacetoSurface Paraxial Ray Theory
This chapter contains paraxialraytheory foundations for understanding most of the derivations and analyses presented later in connection with seismic trueamplitude imaging. It also provides the links between the factors that appear later in the imaging formulas and dynamic ray tracing, and thus indicates how to practically compute the quantities involved in the Kirchhofftype imaging methods. Wherever possible, we chose definitions of quantities and variables that agreed with the conventions previously used by us and by other authors. However, some notational differences from previous publications were unavoidable.
We mentioned in Chapter 1 that all map and image transformation procedures discussed in this book rely on two basic geometric concepts: the Huygens surface (also called the diffractiontraveltime surface or the maximumconvexity surface) and the isochron (also called the aplanat, aplanatic, or equaltraveltime surface). In this chapter, we elaborate further on the common properties and the mutual relationship of these two fundamental surfaces, as well as on their relationship to the 3D reflectiontraveltime surface and the 3D target reflector. We always assume a fixed measurement configuration as discussed in the “Measurement configurations” section of Chapter 2. Thus, all traveltime surfaces considered here are functions of a 2D vector parameter ?, rather than of the complete set of source and receiver coordinates xS and xG.
In this chapter, we take a closer look at the highfrequency approximation of the traditional Kirchhoff integral. It provides an integral representation of the seismicreflection response at a receiver, given the locations of a sourcereceiver pair, a laterally inhomogeneous velocity model, and a reflector. Use of the KirchhoffHelmholtz approximation for the elementary wave after reflection at the reflector results in an approximate forwardmodeling integral that we more appropriately call the KirchhoffHelmholtz integral (KHI).
TrueAmplitude Kirchhoff Migration
In this chapter, we address the inverse problem, i.e., recovery of the reflector image and determination of reflection coefficients from the recorded reflected field. This will lead us to the trueamplitude diffractionstack migration operation that is the first of two building blocks for the unified approach to seismicreflection imaging. The second building block and the complete theory of trueamplitude seismic imaging are presented in Chapter 9.
Further Aspects of Kirchhoff Migration
In Chapter 7, we introduced the trueamplitude Kirchhoff migration integral, which is the first of the two building blocks of the unified approach to seismicreflection imaging. The second building block is the trueamplitude Kirchhoff demigration integral, which is introduced in Chapter 9. In this chapter, we further elaborate on the properties of Kirchhoff migration. However, all aspects of Kirchhoff migration that are needed to understand the general imaging theory were discussed in Chapter 7. Thus, readers whose main interest is with the unified approach to seismicreflection imaging may go directly to Chapter 9.
In this chapter, we describe quantitatively the actual asymptotic inverse process for diffractionstack migration – namely, isochronstack demigration. We show that the diffractionstack integral (DSI) and the isochronstack integral (ISI) constitute an asymptotic transform pair that is interlinked well by the duality theorems that were derived in Chapter 5. This transform pair can be used to solve a multitude of trueamplitude targetoriented seismicimaging (or imagetransformation) problems, including, e.g., the dynamic counterparts of the kinematic maptransformation examples that Chapter 2 discussed qualitatively. All imagetransformation problems can be addressed by applying the two stacking integrals in sequence, whereby the macrovelocity model, the measurement configuration, or the ray code of the considered elementary reflections may change from step to step. Alternatively, the two stacking procedures can be combined mathematically into a single process, which leads to weighted (Kirchhofftype) summations along certain stacking surfaces (or inplanats). This chapter provides the general formulas for the stacking surfaces and the trueamplitude weights of these new Kirchhoff processes. To demonstrate the value of the proposed imaging theory, which is based on analytically chaining the two stacking integrals, we solve the trueamplitude configurationtransform and remigration problems for the case of a 3D isotropic, laterally inhomogeneous medium. In this important chapter of our book, we present the theoretical background for the geometrically motivated mapping and imaging concepts that were discussed in Chapter 2. The diffractionstack theory, as presented in Schleicher et al. (1993a) and discussed in detail in Chapter 7 of this book, serves as our point of departure for presenting the theory of trueamplitude isochronstack demigration along very similar lines. These Kirchhofftype migration and demigration operations provide the basis for the unified approach to seismicreflection imaging (Hubral et al., 1996a; Tygel et al., 1996) that is the central subject of this book. In the last section of this chapter, we show how to chain the DSI and the ISI to solve various seismicimagetransformation problems. As in the papers just cited, we present this unified theory here in the time domain. Bleistein and Jaramillo (2000) and Bleistein et al. (2001) discuss an equivalent frequencydomain theory.
Reflection and Transmission Coefficients
In this appendix, we provide the formulas for the amplitudenormalized planewave reflection and transmission coefficients, as derived in Červený et al. (1977) and Červený (2001) on the basis of the boundary conditions of Zoeppritz (1919). We assume the incidence angle to be between 0° and 90° or between 90° and 180°, depending on the direction that is chosen for the normal vector of the interface. In this appendix, we denote the incidence angle of a Pwave by θ and the incidence angle of an Swave by φ. We also state several linearized expressions for the reflection coefficients because generally, contrasts at seismic reflectors are rather small. The linearized formulas have proved to be very useful when we are inverting the reflection coefficients for medium parameters.
In Chapter 3, we derived formulas that describe how the scalar amplitude of particle displacement changes along a ray. However, in a seismic survey, this scalar amplitude is not directly recorded because the geophones or hydrophones are at a free surface. This appendix addresses how the described scalar amplitude of particle displacement can be computed from landseismic threecomponent freesurface recordings. It also addresses how pressure is described in a seaseismic survey. The formulas in this appendix are based on the conversion coefficients given by Červený et al. (1977).
In this appendix, we derive the relationship between the Hessian matrix of second derivatives of a given surface ∑ in arbitrary Cartesian coordinates and the curvature matrix of that surface ∑.
Relationship to Beylkin’s Determinant
In this appendix, we derive the relationship (equation 77 of Chapter 5) between the determinant of ??(r) as defined in equation 11 of Chapter 5 and the Beylkin determinant hB as defined in equation 76 of Chapter 5.
The Scalar Elastic KirchhoffHelmholtz Integral
In this appendix, we derive the scalar KirchhoffHelmholtz integral (KHI), equation 14 of Chapter 6, for isotropic, elastic elementary waves, from the general anisotropic representation theorem (Aki and Richards, 1980). Because the concepts are similar or identical to those discussed in Appendix F: “Derivation of the scalar elastic Kirchhoff integral,” we do not repeat them here in detail. We use Einstein’s summation convention.
Derivation of the Scalar Elastic Kirchhoff Integral
In this appendix, we derive the scalar version of the elastic Kirchhoff integral for direct, transmitted, and primary reflected elementary waves in isotropic media. This scalar integral is useful only for deriving the scalar KirchhoffHelmholtz integral (KHI) in Chapter 6. For any other purposes, the general isotropic Kirchhoff integral in Appendix E, “The scalar elastic KirchhoffHelmholtz integral,” should be used.
KirchhoffHelmholtz Approximation
In this appendix, we explain the ansatz used in the KirchhoffHelmholtz approximation. First, let us consider the simple cases of transmission and reflection of a plane wave at a planar interface between two homogeneous halfspaces.
In this appendix, we use the stationaryphase method to evaluate certain stacking integrals that appear when we chain the diffractionstack integral (DSI) and the isochronstack integral (ISI), i.e., when we insert them into each other to solve the configurationtransform (CT) and remigration (RM) problems.