Seismic Migration

Seismic Migration

Imaging of Acoustic Energy by Wave Field Extrapolation

1st Edition - January 1, 1980

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  • Author: A. J. Berkhout
  • eBook ISBN: 9780444601582

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Seismic Migration: Imaging of Acoustic Energy by Wave Field Extrapolation derives the migration theory from first principles. This book also obtains a formulated forward modeling and migration theory by introducing the propagation matrices and the scattering matrix. The book starts by presenting the basic results from vector analysis, such as the scalar product, gradient, curl, and divergence. It also describes the theorem of Stokes, theorem of Gause and the Green’s theorem. The book also deals with discrete spectral analysis, two-dimensional Fourier theory and plane wave analysis. It also describes the wave theory, including the plane waves and k-f diagram, spherical waves, and cylindrical waves. This book explores the forward problem and the inward problem of the wave field extrapolation, as well as the modeling by wave field extrapolation. Furthermore, the book explains the migration in the wave number-frequency domain. It also includes the summation approach and finite-difference approach to migration, as well as a comparison between the different approaches to migration. Finally, the book offers the limits of lateral resolution as the last chapter.

Table of Contents

  • Preface



    Chapter 1. Basic Results from Vector Analysis

    1.1. Introduction

    1.2. Scalar product, gradient, curl and divergence

    1.3. Theorem of Stokes, theorem of Gauss and Green's theorems

    1.4. References

    Chapter 2. Discrete Spectral Analysis

    2.1. Introduction

    2.2. The delta pulse and discrete functions

    2.3. Fourier series of periodic time functions

    2.4. Fourier integral of transients

    2.5. Relationship between the discrete property and periodicity

    2.6. Sampling and aliasing in time and frequency

    2.7. References

    Chapter 3. Two-Dimensional Fourier Transforms

    3.1. Introduction

    3.2. Basic theory

    3.3. Spatial aliasing

    3.4. Two-dimensional Fourier theory and plane wave analysis

    3.5. References

    Chapter 4. Wave Theory

    4.1. Introduction

    4.2. Derivation of the wave equation

    4.3. Plane waves and k-f diagrams

    4.4. Spherical waves and directivity patterns

    4.5. Cylindrical waves

    4.6. Angle dependence of reflection coefficients

    4.7. References

    Chapter 5. Wave Field Extrapolation: The Forward Problem

    5.1. Introduction

    5.2. Derivation of the Kirchhoff integral

    5.3. The Rayleigh integral I

    5.4. The Rayleigh integral II

    5.5. Forward extrapolation scheme in the space-time domain

    5.6. Forward extrapolation scheme in the space-frequency domain

    5.7. Forward extrapolation scheme in the wavenumber-frequency domain

    5.8. References

    Chapter 6 . Modeling by Wave Field Extrapolation

    6.1. Introduction

    6.2. Modeling of one physical experiment

    6.3. Focussing of one physical experiment

    6.4. Modeling of a plane wave response

    6.5. Modeling with the two-way propagation matrix

    6.6. Modeling of multi-record datasets

    6.7. References

    Chapter 7. Wave Field Extrapolation: The Inverse Problem

    7.1. Introduction

    7.2. Upward extrapolation of multi-record datasets in terms of spatial convolution

    7.3. Downward extrapolation of multi-record data sets in terms of spatial inverse filtering

    7.4. Kirchhoff-summation approach and matched filtering

    7.5. Downward extrapolation in the presence of noise

    7.6. Least-squares downward extrapolation in two dimensions

    7.7. Downward extrapolation of one source gather by inversion of the two-way propagation matrix

    7.8. Downward extrapolation of one detector gather by inversion of the two-way propagation matrix

    7.9. Downward extrapolation of one source - or receiver gather by combined forward and inverse extrapolation

    7.10. Downward extrapolation of plane wave data

    7.11. Downward extrapolation of zero-offset data

    7.12. Imaging

    7.13. References

    Chapter 8. Migration in the Wavenumber-Frequency Domain

    8.1. Introduction

    8.2. Migration as a mapping procedure to the kx-kz domain

    8.3. Recursive migration in the kx-k domain

    8.4. Migration of plane-wave data in the kx-k domain

    8.5. Migration of zero-offset data in the kx-k domain

    8.6. References

    Chapter 9. Summation Approach to Migration

    9.1. Introduction

    9.2. Summation method in the space-frequency domain

    9.3. Summation method in the space-time domain

    9.4. Summation method for plane-wave zero-offset data

    9.5. Practical summation schemes for recursive migration

    9.6. Multi-level extrapolation

    9.7. References

    Chapter 10. Finite-Difference Approach to Migration

    10.1. Introduction

    10.2. Wave field extrapolation with the Taylor series

    10.3. Floating reference

    10.4. Approximate expressions for the spatial derivatives with respect to z

    10.5. Approximations of the wave equation for delayed pressure

    10.6. Finite-difference migration in the space-frequency domain

    10.7. Errors in finite-difference migration

    10.8. Finite-difference schemes in three dimensions

    10.9. References

    Chapter 11. Comparison Between the Different Approaches to Migration

    11.1. Introduction

    11.2. Review of the seismic model

    11.3. Review of the inversion philosophy

    11.4. Taylor series and wave equation

    11.5. Extrapolation by means of multiplication

    11.6. Replacement of the multiplication procedure by one-dimensional convolution

    11.7. Replacement of the multiplication procedure by two-dimensional convolution

    11.8. Series expansion of the convolution operators

    11.9. Summary of extrapolation methods

    11.10. Summary of imaging methods

    11.11. Possibilities and limitations in practical situations

    11.12. Some concluding remarks

    11.13. References

    Chapter 12. Limits of Lateral Resolution

    12.1. Introduction

    12.2. Ultimate limits of lateral resolution

    12.3. Lateral resolution in practical situations

    12.4. Influence of finite apertures

    12.5. References

    Subject Index


    A. Hooke’s Law for Fluids and Solids

    B. Linear Equations for Compressional Waves in Homogeneous Solids

    C. Wave Equation for Inhomogeneous Fluids

    D. Spatial Fourier Transforms of Green’s Functions in the Rayleigh Integrals

    E. Summation Operator for Small Extrapolation Steps

    F. Differentiation in Terms of Convolution

Product details

  • No. of pages: 352
  • Language: English
  • Copyright: © Elsevier 1980
  • Published: January 1, 1980
  • Imprint: Elsevier
  • eBook ISBN: 9780444601582

About the Author

A. J. Berkhout

Affiliations and Expertise

Delft University of Technology, Delft, Netherlands

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