Propagation of Transient Elastic Waves in Stratified Anisotropic Media

1. Auflage 2016
ISBN: 978-1-4832-9590-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 32, 298 Seiten, Web PDF

Reihe: North-Holland Series in Applied Mathematics and Mechanics

Seismic waves are one of the standard diagnostic tools used to determine the mechanical parameters (volume density of mass, compressibility, elastic stiffness) in the interior of the earth and the geometry of subsurface structures. There is increasing evidence that in the interpretation of seismic data - especially shear-wave data - the influence of anisotropy must be taken into account.This volume presents a method to compute the seismic waves that are generated by an impulsive source in a stratified anisotropic medium. Although written with the seismic applications in mind, the method that is developed is not limited to solid-earth geophysics. In fact, the methods discussed in this monograph are applicable wherever waves propagate in stratified, anisotropic media. The standard approach to this problem is to employ Fourier transformations with respect to time and with respect to the horizontal spatial coordinates. To obtain numerical results, the relevant inverse transformations then have to be evaluated numerically. In this monograph the problem is, in contrast to the standard approach, solved by applying the Cagniard-de Hoop method and by representing the wave field as a sum of generalized rays. With this method, the computational results can be obtained relatively easily with any degree of accuracy, and with considerably less computation time. For completeness, analysis of acoustic waves in stratified isotropic media is included. Furthermore, for large horizontal or vertical source-receiver separations very efficient approximations are derived. Several examples and applications are given.

Inhaltsverzeichnis

1;Front Cover;1
2;Propagation of Transient Elastic Waves in Stratified Anisotropic Media;4
3;Copyright Page;5
4;Table of Contents;6
5;CHAPTER 1. INTRODUCTION;12
5.1;1.1 Statement of the problem;12
5.2;1.2 The method of solution employed;14
5.3;1.3 Numerical considerations;17
6;CHAPTER 2. BASIC RELATIONS FOR ELASTIC WAVES IN STRATIFIED, PIECEWISE HOMOGENEOUS, ANISOTROPIC MEDIA;20
6.1;2.1 Introduction;21
6.2;2.2 Description of the configuration and formulation of the problem;21
6.3;2.3 Basic equations for the elastic wave motion;24
6.4;2.4 The transform-domain equations;27
6.5;2.5 The motion-stress vector in a homogeneous subdomain;29
6.6;2.6 The transform-domain wave vector in a source-free domain;32
6.7;2.7 The transform-domain wave vector and the motion-stress vector generated by a localized source in a homogeneous subdomain;36
6.8;2.8 The transform-domain wave vector and the motion-stress vector generated by a localized source in a stratified medium;38
6.9;2.9 The transform-domain generalized-ray wave constituents;43
6.10;2.10 Transformation of the solution back to the space-time domain;47
7;CHAPTER 3. BASIC RELATIONS FOR ELASTIC WAVES IN STRATIFIED, PIECEWISE HOMOGENEOUS, ISOTROPIC MEDIA;50
7.1;3.1 Introduction;51
7.2;3.2 Description of the configuration and formulation of the problem;51
7.3;3.3 Basic equations for the elastic wave motion;53
7.4;3.4 The transform-domain equations;56
7.5;3.5 The motion-stress vector in a homogeneous subdomain;56
7.6;3.6 The transform-domain wave vector in a source-free domain;59
7.7;3.7 The transform-domain wave vector and the motion-stress vector generated by a localized source in a homogeneous subdomain;67
7.8;3.8 The transform-domain wave vector and the motion-stress vector generated by a localized source in a stratified medium;70
7.9;3.9 The transform-domain generalized-ray wave constituents;72
7.10;3.10 Transformation of the solution back to the space-time domain;72
7.11;3.11 Basic relations for acoustic waves in a fluid;73
8;CHAPTER 4. RADIATION FROM AN IMPULSIVE SOURCE IN AN UNBOUNDED HOMOGENEOUS ISOTROPIC SOLID;76
8.1;4.1 Introduction;77
8.2;4.2 Transformation of the solution back to the space-time domain;77
8.3;4.3 The behavior of sp,s3 in the complex s plane;79
8.4;4.4 Cagniard-de Hoop contours in the complex s plane;80
8.5;4.5 Space-time expression for the motion-stress vector;82
8.6;4.6 Alternative implementation of the Cagniard-de Hoop method;85
8.7;4.7 Approximations and derived results;100
8.8;4.8 Numerical results;108
8.9;4.9 Conclusion;116
9;CHAPTER 5. RADIATION FROM AN IMPULSIVE SOURCE IN A STRATIFIED ISOTROPIC MEDIUM;118
9.1;5.1 Introduction;119
9.2;5.2 Transformation of the solution back to the space-time domain;119
9.3;5.3 The two-dimensional problem;121
9.4;5.4 The three-dimensional problem;130
9.5;5.5 Alternative implementation of the Cagniard-de Hoop method;144
9.6;5.6 Approximations and derived results;159
9.7;5.7 Numerical results;167
9.8;5.8 Conclusion;177
10;CHAPTER 6. RADIATION FROM AN IMPULSIVE SOURCE IN AN UNBOUNDED HOMOGENEOUS ANISOTROPIC SOLID;178
10.1;6.1 Introduction;179
10.2;6.2 Transformation of the solution back to the space-time domain;180
10.3;6.3 The behavior of s±n3 in the complex s plane;184
10.4;6.4 Cagniard-de Hoop contours in the complex s plane;195
10.5;6.5 Space-time expression for the motion-stress vector;207
10.6;6.6 Approximations and derived results;212
10.7;6.7 Numerical aspects of the computation of Cagniard-de Hoop contours in anisotropic media;222
10.8;6.8 Numerical results;225
10.9;6.9 Conclusion;242
11;CHAPTER 7. RADIATION FROM AN IMPULSIVE SOURCE IN A STRATIFIED ANISOTROPIC MEDIUM;244
11.1;7.1 Introduction;245
11.2;7.2 Transformation of the solution back to the space-time domain;245
11.3;7.3 The behavior of s±n3;m in the complex s plane;248
11.4;7.4 Cagniard-de Hoop contours in the complex s plane;249
11.5;7.5 Space-time expression for the motion-stress vector;255
11.6;7.6 Approximations and derived results;258
11.7;7.7 Numerical results;264
11.8;7.8 Conclusion;273
12;APPENDICES;274
13;A The slowness surface of a homogeneous anisotropic medium;274
14;B Up- and down-going waves in a homoge-neous anisotropic medium;279
15;C The values of s±n3 for large |s|;283
16;D The transversely isotropic medium and its stiffness tensor;283
17;E Simplified formulation for unbounded medium in the two-dimensional problem;286
18;REFERENCES;288
19;INDEX;294