Seismic Migration
Imaging of Acoustic Energy by Wave Field Extrapolation
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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
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
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