Seismic Applications of Acoustic Reciprocity (eBook)

P.M. van den Berg, J.T. Fokkema (Autoren)

eBook Download: PDF

2013 | 1. Auflage
364 Seiten
Elsevier Science (Verlag)
978-1-4832-9120-8 (ISBN)

Infos

The seismic applications of the reciprocity theorem developed in this book are partly based on lecture notes and publications from Professor de Hoop. Every student Professor de Hoop has taught knows the egg-shaped figure (affectionately known as 'de Hoop's egg') that plays such an important role in his theoretical description of acoustic, electromagnetic and elastodynamic wave phenomena.On the one hand this figure represents the domain for the application of a reciprocity theorem in the analysis of a wavefield and on the other hand it symbolizes the power of a consistent wavefield description of this theorem.The roots of the reciprocity theorem lie in Green's theorem for Laplace's equation and Helmholtz's extension to the wave equation. In 1894, J.W. Strutt, who later became Lord Rayleigh, introduced in his book The Theory of Sound this extension under the name of Helmholtz's theorem. Nowadays it is known as Rayleigh's reciprocity theorem.Progress in seismic data processing requires the knowledge of all the theoretical aspects of the acoustic wave theory. The reciprocity theorem was chosen as the central theme of this book as it constitutes the fundaments of the seismic wave theory. In essence, two states are distinguished in this theorem. These can be completely different, although sharing the same time-invariant domain of application, and they are related via an interaction quantity. The particular choice of the two states determines the acoustic application, in turn making it possible to formulate the seismic experiment in terms of a geological system response to a known source function.In linear system theory, it is well known that the response to a known input function can be written as an integral representation where the impulse response acts as a kernel and operates on the input function. Due to the temporal invariance of the system, this integral representation is of the convolution type. In seismics, the temporal behaviour of the system is dealt with in a similar fashion; however the spatial interaction needs a different approach. The reciprocity theorem handles this interaction by identifying one state with the spatial impulse function, also known as the Green's function, while the other state is connected with the actual source distribution. In general, the resulting integral representation is not a spatial convolution. Moreover, the systematic use of the reciprocity theorem leads to a hierarchical description of the seismic experiment in terms of increasing complexity. Also from an educational point of view this approach provides a hierarchy and the student learns to break down the seismic problem into constituent partial solutions.This book should contribute to the understanding that the reciprocity theorem is a powerful tool in the analysis of the seismic experiment.

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The seismic applications of the reciprocity theorem developed in this book are partly based on lecture notes and publications from Professor de Hoop. Every student Professor de Hoop has taught knows the egg-shaped figure (affectionately known as "e;de Hoop's egg"e;) that plays such an important role in his theoretical description of acoustic, electromagnetic and elastodynamic wave phenomena.On the one hand this figure represents the domain for the application of a reciprocity theorem in the analysis of a wavefield and on the other hand it symbolizes the power of a consistent wavefield description of this theorem.The roots of the reciprocity theorem lie in Green's theorem for Laplace's equation and Helmholtz's extension to the wave equation. In 1894, J.W. Strutt, who later became Lord Rayleigh, introduced in his book The Theory of Sound this extension under the name of Helmholtz's theorem. Nowadays it is known as Rayleigh's reciprocity theorem.Progress in seismic data processing requires the knowledge of all the theoretical aspects of the acoustic wave theory. The reciprocity theorem was chosen as the central theme of this book as it constitutes the fundaments of the seismic wave theory. In essence, two states are distinguished in this theorem. These can be completely different, although sharing the same time-invariant domain of application, and they are related via an interaction quantity. The particular choice of the two states determines the acoustic application, in turn making it possible to formulate the seismic experiment in terms of a geological system response to a known source function.In linear system theory, it is well known that the response to a known input function can be written as an integral representation where the impulse response acts as a kernel and operates on the input function. Due to the temporal invariance of the system, this integral representation is of the convolution type. In seismics, the temporal behaviour of the system is dealt with in a similar fashion; however the spatial interaction needs a different approach. The reciprocity theorem handles this interaction by identifying one state with the spatial impulse function, also known as the Green's function, while the other state is connected with the actual source distribution. In general, the resulting integral representation is not a spatial convolution. Moreover, the systematic use of the reciprocity theorem leads to a hierarchical description of the seismic experiment in terms of increasing complexity. Also from an educational point of view this approach provides a hierarchy and the student learns to break down the seismic problem into constituent partial solutions.This book should contribute to the understanding that the reciprocity theorem is a powerful tool in the analysis of the seismic experiment.

Inhalt

Front Cover 1
Seismic Applications of Acoustic Reciprocity 4
Copyright Page 5
Table of Contents 10
Preface 8
Introduction 16
Chapter 1. Integral Transformations 28
1.1. Cartesian vectors 29
1.2. Integral-transformation methods 31
1.3. Discrete Fourier-transformation methods 45
Chapter 2. Iterative Solution of Integral Equations 56
2.1. The integral equation 57
2.2. Direct minimization of the error 59
2.3. Recursive minimization of the error 60
2.4. Selfadjoint operator LT 65
2.5. The Neumann expansion 69
2.6. Special choices of the operator T 71
2.7. Operators of convolution type 74
Chapter 3. Basic Equations in Acoustics 78
3.1. The acoustic wave equations 78
3.2. The acoustic equations in the Laplace-transform domain 85
Chapter 4. Radiation in an Unbounded, Homogeneous Medium 90
4.1. Source representations in the spectral domain 90
4.2. Source representations in the s-domain 93
4.3. Far-field radiation characteristics in the s-domain 96
4.4. Source representations in the time domain 98
4.5. Far-field characteristics in the time domain 99
4.6. The Cagniard-de Hoop method 101
4.7. The acoustic wavefield of point sources 106
Chapter 5. Reciprocity Theorems 110
5.1. The s-domain field reciprocity theorem 111
5.2. The time-domain reciprocity theorem of convolution type 114
5.3. The s-domain power reciprocity theorem 116
5.4. The time-domain reciprocity theorem of correlation type 118
Chapter 6. Field Reciprocity between Transmitter and Receiver 122
6.1. Point-transducer description 122
6.2. Volume-transducer description 126
6.3. Surface-transducer description 128
Chapter 7. Radiation in an Unbounded, Inhomogeneous Medium 132
7.1. The volume-source problem 132
7.2. The surface-source problem 144
Chapter 8. Scattering by a Bounded Contrasting Domain 156
8.1. The domain-integral equation formulation 156
8.2. The boundary-integral equation formulation 165
Chapter 9. Scattering by a Disk 176
9.1. Scattering by a planar object of vanishing thickness 176
9.2. Disk in a homogeneous embedding 180
9.3. Analytic solution for a pressure-free plane 184
9.4. Disk in a homogeneous halfspace 187
9.5. Two-dimensional scattering by a strip 193
Chapter 10. Wavefield Decomposition 214
10.1. Decomposition based on field reciprocity 214
10.2. Decomposition based on power reciprocity 219
10.3. Redatuming of seismic data 225
Chapter 11. Deghosting 230
11.1. Decomposition based on field reciprocity 232
11.2. Decomposition based on power reciprocity 233
11.3. The surface-related vertical particle velocity 235
11.4. Receiver deghosting 237
11.5. Source deghosting 239
11.6. Deghosting in the strip configuration 241
Chapter 12. Removal of Surface Related Wave Phenomena 248
12.1. Reciprocity between the actual and desired state 248
12.2. Auxiliary reciprocity relations 252
12.3. Alignment of vertical positions 259
12.4. Actual multiple-removal procedure 262
12.5. Multiple removal in the strip configuration 265
Chapter 13. Boundary Imaging 278
13.1. The boundary-integral representation 279
13.2. Fourier transform with respect to source coordinates 283
13.3. Fourier transform with respect to receiver coordinates 287
13.4. Angular-frequency-domain analysis 293
13.5. Imaging 295
13.6. Analytic solution for a planar reflector 298
13.7. Imaging of a disk 300
13.8. Two-dimensional case 305
13.9. Imaging of the strip configuration 311
Chapter 14. Domain Imaging 316
14.1. The domain-integral representation 316
14.2. Fourier transform of source and receiver coordinates 319
14.3. Angular-frequency-domain analysis 320
14.4. Imaging 321
14.5. Two-dimensional case 324
14.6. Two-dimensional scattering by a circular cylinder 329
14.7. Imaging of the circular cylinder 335
Chapter 15. Seismic Inversion 340
15.1. The domain-integral representation 341
15.2. Simultaneous minimization 344
15.3. Inversion algorithm 346
Bibliography 354
Index 360

Produkt-Details

• Erscheint lt. Verlag: 22.10.2013
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geografie / Kartografie
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geologie
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik ► Bauwesen
• ISBN-10: 1-4832-9120-0 / 1483291200
• ISBN-13: 978-1-4832-9120-8 / 9781483291208