# Seismic Migration

## 1st Edition

### Imaging of Acoustic Energy by Wave Field Extrapolation

**Authors:**A. J. Berkhout

**eBook ISBN:**9780444601582

**Imprint:**Elsevier

**Published Date:**1st January 1980

**Page Count:**352

## Description

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

Introduction

Introduction

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

Appendices

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

## Details

- No. of pages:
- 352

- Language:
- English

- Copyright:
- © Elsevier 1980

- Published:
- 1st January 1980

- Imprint:
- Elsevier

- eBook ISBN:
- 9780444601582

## About the Author

### A. J. Berkhout

### Affiliations and Expertise

Delft University of Technology, Delft, Netherlands