# Seismic Migration: Imaging of Acoustic Energy by Wave Field Extrapolation..

## 2nd Edition

### Imaging of Acoustic Energy by Wave Field Extrapolation

**Authors:**A. J. Berkhout

**eBook ISBN:**9780444602008

**Imprint:**Elsevier

**Published Date:**1st January 1984

**Page Count:**366

## Description

Seismic Migration: Imaging of Acoustic Energy by Wave Field Extrapolation, Second Edition, Volume A: Theoretical Aspects covers the theoretical aspects of seismic migration techniques. This volume is divided into 11 chapters that consider the concept of propagation and scattering matrices. This book begins with a presentation of a selection of concepts and properties of seismic migration from vector analysis. These topics are followed by considerable chapters on the mathematical aspects of migration, including discrete spectral analysis, two-dimensional Fourier transforms, and wave theory. The subsequent chapters describe the derivation of the Kirchhoff integral for upward traveling wave field and wave field extrapolation for downward traveling source waves and upward traveling reflected waves. These chapters also propose a matrix formulation to represent single seismic record and multi-record data sets, along with different modeling algorithms. A chapter examines inverse wave field extrapolation, in which the medium must be horizontally layered, the layers being homogeneous. The book ends with a summary and comparison of different approaches to seismic migration.

## Table of Contents

Preface

Introduction

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

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

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

4 Wave Theory

4.1 Introduction

4.2 Derivation of the Wave Equation

4.3 Plane Waves and K-F Diagrams

(Solution of the One-Dimensional Wave Equation)

4.4 Spherical Waves and Directivity Patterns

(Solution of the Three-Dimensional Wave Equation)

4.5 Cylindrical Waves

(Solution of the Two-Dimensional Wave Equation)

4.6 Angle Dependence of Reflection Coefficients

4.7 Use of Discretized Reflectivity Models

4.8 References

5 Properties of the Kirchhoff Integral

5.1 Introduction

5.2 Derivation of the Kirchoff Integral for Homogeneous Media

5.3 The Rayleigh Integral I

5.4 The Rayleigh Integral II

5.5 Principle of Forward Extrapolation in the Space-Time Domain

5.6 Principle of Forward Extrapolation in the Space-Frequency Domain

5.7 Principle of Forward Extrapolation in the Wavenumber-Frequency Domain

5.8 Kirchhoff Integral for Inhomogeneous Media

5.9 References

6 Wave Field Extrapolation: The Forward Problem

6.1 Introduction

6.2 Modeling of One Seismic Experiment

6.3 Design of Source and Detector Arrays If Focussing Is Required

6.4 Modeling of Data From Plane-Wave Sources

6.5 Modeling with the Two-Way Propagation Matrix

6.6 Matrix Presentation of Multi-Record Data Sets

6.7 Modeling of Common-Offset Data

6.8 Modeling of Zero-Offset Data with the 'Half-Velocity-Substitution'

6.9 Recursive Modeling Scheme For Primary And Multiple Energy

6.10 Iterative Modeling of Primary and Multiple Energy by Using a Homogeneous Reference Medium

6.11 Simultaneous Forward Extrapolation of Pressure and Particle Velocity

6.12 References

7 Wave Field Extrapolation: The Inverse Problem

7.1 Introduction

7.2 Principle of Inverse Wave Field Extrapolation

7.3 Inverse Extrapolation of Multi-Record Data Sets

7.4 Imaging Principle

7.5 A Recursive Pre-Stack Migration Scheme for Primary Data

7.6 Multiple Elimination as Part of he Pre-Stack Migration Scheme

7.7 Migration of One Seismic Experiment

7.8 Migration of Common-Offset Data

7.9 Migration of Zero-Offset Data with the 'Half-Velocity-Substitution'

7.10 Stacking by Inverse Wave Field Extrapolation

7.11 Simultaneous Inverse Extrapolation of Pressure and Particle Velocity

7.12 References

8 Migration in the Wavenumber-Frequency Domain ('Plane Wave Method')

8.1 Introduction

8.2 Recursive Migration of Multi-Record Data Sets in the Kx-K Domain

8.3 Recursive Migration of Single Seismic Records in the Kx-K Domain

8.4 Recursive Migration of Zero-Offset Data in the Kx-K Domain, Using the 'Half-Velocity-Substitution'

8.5 Nonrecursive Migration by Mapping to the Kx-Kz Domain

8.6 Time-To-Distance Conversion Prior to Migration in the Kx-K Domain

8.7 The Shift and Stretch Process

8.8 Influence of Absorption

8.9 References

9 Summation Approach to Migration

9.1 Introduction

9.2 Recursive Summation Method in the Space-Frequency Domain

9.3 Nonrecursive Summation Method in the Space-Time Domain

9.4 A Mixed Recursive-Nonrecursive Summation Process

9.5 Three-Dimensional Migration with One-Dimensional Summation Operators

9.6 Velocity Replacement with Summation Operators

9.7 References

10 Finite-Difference Approach to Migration

10.1 Introduction

10.2 Floating Time Reference

10.3 Approximate Expressions for ∂np'/∂zn

10.4 Principle of Finite-Difference Extrapolation

10.5 Errors in Finite-Difference Extrapolation

10.6 Finite-Difference Extrapolation by Velocity Replacement

10.7 Simultaneous Extrapolation of Pressure and Particle Velocity

10.8 References

11 A 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 Wave-Number Mapping, Summation and Finite Difference: A Theoretical Comparison

11.5 References

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 Series Expansion of the Kirchhoff-Summation Operator

F Differentiation in Terms of Convolution

Subject Index

## Details

- No. of pages:
- 366

- Language:
- English

- Copyright:
- © Elsevier 1984

- Published:
- 1st January 1984

- Imprint:
- Elsevier

- eBook ISBN:
- 9780444602008

## About the Author

### A. J. Berkhout

### Affiliations and Expertise

Delft University of Technology, Delft, Netherlands