Preface ........................................................ ix
1 Introduction to the Wave Theory .............................. 1
1.1 Wave Motion in Continuous Media ......................... 2
1.2 Vibration ............................................... 5
1.3 Propagation and Diffusion ............................... 6
1.4 Acoustic Wave Equation .................................. 8
1.5 Acoustic Wave Equation with Complex Coefficients ....... 11
1.5.1 Complex Elastic Modulus and the Complex Wave
Velocity ........................................ 11
1.5.2 Damping Wave Equations in Viscoelastic Media .... 13
1.5.3 Viscoelastic Models ............................. 14
1.6 Acoustic Wave Equation with Valiant Density or
Velocity ............................................... 16
1.7 Summary ................................................ 18
2 Elastic Waves in a Perfect Elastic Solid .................... 19
2.1 Stress Tensor and Strain Tensor ........................ 20
2.2 Vector Wave Equation in Fully Elastic Media ............ 23
2.3 Scalar Wave Equations in Fully Elastic Media ........... 27
2.4 Elastic Wave Equation in Two-Dimensional Media ......... 30
2.5 Elastic Wave Equations in Anisotropic Media ............ 31
2.6 Boundary Conditions for Elastic Wave Equations ......... 34
2.7 Elastic Wave Velocities of Rocks ....................... 37
3 From Elastic Waves to Seismic Waves ......................... 47
3.1 On Acoustic Wave Equations with Variant Coefficients ... 48
3.2 Seismic Reflection Records and Corresponding
Equations .............................................. 54
3.2.1 Wave Equations for Marine Reflection Records .... 54
3.2.2 Wave Equations for Land Single-Component
Records ......................................... 55
3.2.3 Wave Equations for Land Three-Component
Records ......................................... 55
3.3 Elastic Waves in Horizontally Multilayered Media ....... 58
3.3.1 Elastic Wave Equations in a Cylindrical
Coordinate System ............................... 58
3.3.2 Boundary Conditions ............................. 63
3.3.3 Acoustic Wave Propagation in Layered Half
Space ........................................... 64
3.4 Elastic Waves in Fluid-Saturated Solid (I):
Gassmann's Model ....................................... 66
3.4.1 The Gassmann Model .............................. 66
3.4.2 The Generalized Gassmann Model .................. 69
3.5 Elastic Waves in Fluid-Saturated Solid (II): Biot's
Theory ................................................. 71
3.5.1 Low-Frequency Elastic Waves in a Fluid-
Saturated Porous Solid .......................... 72
3.5.2 All Frequency Elastic Waves in a Fluid-
Saturated Porous Solid .......................... 76
3.6 Tracking Reservoirs with the Gassmann Model ............ 77
4 Wave Equation Reduction with Reflection Seismic Data
Processing .................................................. 83
4.1 The Statics of Land Seismic Data ....................... 84
4.2 Muting and Deghost Filtering ........................... 87
4.3 Shear Wave Decoupling Process .......................... 88
4.4 Suppression of Multiples Generated by the Ocean
Bottom ................................................. 89
4.5 CMP Stacking ........................................... 91
4.6 The One-Way Wave Equation and the Wave Migration
Equations .............................................. 95
4.7 Reflectors ............................................. 99
4.8 Summary ............................................... 105
5 Integral Solutions of the Wave Equation with Boundary
and Initial Value Conditions ............................... 107
5.1 Integral Solutions for Mixed Cauchy Boundary Value
Problems .............................................. 109
5.2 The Kirchhoff Integral Formula for the Boundary
Value Wave Equation Problems .......................... 112
5.3 The Green's Function of Boundary Value Problems for
Wave Motion ........................................... 117
5.3.1 The Green's Function Method .................... 117
5.3.2 Green's Function for the Wave Equation with
Zero Initial Value Problems .................... 119
5.3.3 Green's Function of the Wave Equation in Half
Space with a Point Source ...................... 122
5.4 The Green's Function in Medium with Linear Velocity ... 124
5.5 The Eikonal Equation and the Transport Equations ...... 127
5.6 The Second-Type Green's Function with Nonhomogeneous
Boundary Conditions ................................... 131
5.7 Summary ............................................... 133
6 Appe equal Refe Inde: Decomposition and Continuation of
Seismic Wave Field ......................................... 135
6.1 The Equations of Acoustic Upgoing and Downgoing
Waves ................................................. 137
6.2 Kirchhoff Migration of the Prestack Seismic Data ...... 140
6.3 Downward Continuation of the Reflective Seismic Wave
Field in Homogenous Media ............................. 143
6.4 Downward Continuation of Seismic Wave Field in
Vertically Inhomogeneous Media ........................ 148
6.5 The Pseudo-Differential Operator and Fourier
Integral Operator ..................................... 151
6.5.1 Analysis of the Boundary Value Problem of
Wave Equation with Variant Coefficients ........ 151
6.5.2 The Oscillatory Integral ....................... 153
6.5.3 The Fourier Integral Operator .................. 156
6.5.4 Decomposition of Fourier Integral Operator ..... 159
6.6 Downward Continuation of the Seismic Wave Field in
Inhomogeneous Medium .................................. 161
6.7 Decomposition of Body Waves in Reflection Seismic
Wave Field ............................................ 165
6.8 Brief Summary ......................................... 169
7 Seismic Inversion .......................................... 171
7.1 Introduction to Inverse Problems in Seismology ........ 172
7.1.1 Inverse Problems in Seismic Exploration ........ 172
7.1.2 The Generalized Solutions ...................... 174
7.1.3 Linearized Iterative Seismic Inversion ......... 177
7.1.4 Nonlinear Stochastic Inversions ................ 178
7.2 Born Approximation Inversion by Inverse Scattering .... 180
7.3 Acoustic Wave Equation Inversion in Vertically
Inhomogeneous Background Media ........................ 187
7.4 Acoustic Inverse Scattering Problems in Variant
Velocity Media ........................................ 190
7.4.1 Acoustic Generalized Radon Transformation ...... 190
7.4.2 The Inverse Acoustic Generalized Radon
Transformation ................................. 192
7.4.3 Some Supplements about Inverse Scattering
Procedures ..................................... 196
7.5 Linearized Iterative Inversion of Seismic Reflection
Data .................................................. 198
7.6 The Maximum Entropy Inversion and Inversion for
Reservoir Parameters .................................. 207
7.6.1 Bayes' Theorem and Maximum Entropy Inversion ... 207
7.6.2 Probability Density Inversion Based on
Statistical Estimation of Rock Physical
Properties ..................................... 210
7.7 Summary ............................................... 212
Appendix: Finite difference method for solving the acoustic
wave equation with velocity and density variant media ......... 215
References .................................................... 245
Index ......................................................... 249
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