1 Preliminaries ................................................ 1
1.1 A Brief Historical Overview ............................. 1
1.2 The Full Waveform Tomographic Inverse Problem -
Probabilistic vs. Deterministic ......................... 3
1.3 Terminology: Full Language Confusion .................... 4
Part I Numerical Solution of the Elastic Wave Equation
2 Introduction ................................................. 9
2.1 Notational Conventions .................................. 9
2.2 The Elastic Wave Equation .............................. 11
2.2.1 Governing Equations ............................. 11
2.2.2 Formulations of the Elastic Wave Equation ....... 13
2.3 The Acoustic Wave Equation ............................. 14
2.4 Discretisation in Space ................................ 15
2.5 Discretisation in Time or Frequency .................... 16
2.5.1 Time-Domain Modelling ........................... 16
2.5.2 Frequency-Domain Modelling ...................... 18
2.6 Summary of Numerical Methods ........................... 19
3 Finite-Difference Methods ................................... 23
3.1 Basic Concepts in One Dimension ........................ 24
3.1.1 Finite-Difference Approximations ................ 24
3.1.2 Discretisation of the ID Wave Equation .......... 30
3.1.3 von Neumann Analysis: Stability and Numerical
Dispersion ...................................... 34
3.2 Extension to the 3D Cartesian Case ..................... 38
3.2.1 The Staggered Grid .............................. 39
3.2.2 Anisotropy and Interpolation .................... 43
3.2.3 Implementation of the Free Surface .............. 45
3.3 The 3D Spherical Case .................................. 50
3.4 Point Source Implementation ............................ 53
3.5 Accuracy and Efficiency ................................ 55
4 Spectral-Element Methods .................................... 59
4.1 Basic Concepts in One Dimension ........................ 59
4.1.1 Weak Solution of the Wave Equation .............. 60
4.1.2 Spatial Discretisation and the Galerkin
Method .......................................... 60
4.2 Extension to the 3D Case ............................... 66
4.2.1 Mesh Generation ................................. 66
4.2.2 Weak Solution of the Elastic Wave Equation ...... 70
4.2.3 Discretisation of the Equations of Motion ....... 71
4.2.4 Point Source Implementation ..................... 76
4.3 Variants of the Spectral-Element Method ................ 79
4.4 Accuracy and Efficiency ................................ 81
5 Visco-elastic Dissipation ................................... 83
5.1 Memory Variables ....................................... 83
5.2 Q Models ............................................... 85
6 Absorbing Boundaries ........................................ 89
6.1 Absorbing Boundary Conditions .......................... 89
6.1.1 Paraxial Approximations of the Acoustic Wave
Equation ........................................ 90
6.1.2 Paraxial Approximations as Boundary Conditions
for Acoustic Waves .............................. 92
6.1.3 High-Order Absorbing Boundary Conditions for
Acoustic Waves .................................. 94
6.1.4 Generalisation to the Elastic Case .............. 96
6.1.5 Discussion ...................................... 97
6.2 Gaussian Taper Method .................................. 98
6.3 Perfectly Matched Layers (PML) ......................... 99
6.3.1 General Development ............................. 99
6.3.2 Standard PML ................................... 103
6.3.3 Convolutional PML .............................. 104
6.3.4 Other Variants of the PML Method ............... 108
Part II Iterative Solution of the Full Waveform Inversion
Problem
7 Introduction to Iterative Non-linear Minimisation .......... 113
7.1 Basic Concepts: Minima, Convexity and
Non-uniqueness ........................................ 114
7.1.1 Local and Global Minima ........................ 114
7.1.2 Convexity: Global Minima and (Non)Uniqueness ... 116
7.2 Optimally Conditions .................................. 121
7.3 Iterative Methods for Non-linear Minimisation ......... 122
7.3.1 General Descent Methods ........................ 122
7.3.2 The Method of Steepest Descent ................. 125
7.3.3 Newton's Method and Its Variants ............... 126
7.3.4 The Conjugate-Gradient Method .................. 128
7.4 Convergence ........................................... 134
7.4.1 The Multi-Scale Approach ....................... 134
7.4.2 Regularisation ................................. 137
8 The Time-Domain Continuous Adjoint Method .................. 141
8.1 Introduction .......................................... 141
8.2 General Formulation ................................... 143
8.2.1 Frechet Kernels ................................ 145
8.2.2 Translation to the Discretised Model Space ..... 145
8.2.3 Summary of the Adjoint Method .................. 146
8.3 Derivatives with Respect to the Source ................ 147
8.4 Second Derivatives .................................... 148
8.4.1 Motivation: The Role of Second Derivatives in
Optimisation and Resolution Analysis ........... 149
8.4.2 Extension of the Adjoint Method to Second
Derivatives .................................... 152
8.5 Application to the Elastic Wave Equation .............. 157
8.5.1 Derivation of the Adjoint Equations ............ 157
8.5.2 Practical Implementation ....................... 161
9 First and Second Derivatives with Respect to Structural
and Source Parameters ...................................... 163
9.1 First Derivatives with Respect to Selected
Structural Parameters ................................. 163
9.1.1 Perfectly Elastic and Isotropic Medium ......... 165
9.1.2 Perfectly Elastic Medium with Radial
Anisotropy ..................................... 167
9.1.3 Isotropic Visco-Elastic Medium: Qμ and Qk ...... 170
9.2 First Derivatives with Respect to Selected Source
Parameters ............................................ 172
9.2.1 Distributed Sources and the Relation to Time-
Reversal Imaging ............................... 172
9.2.2 Moment Tensor Point Source ..................... 172
9.3 Second Derivatives with Respect to Selected
Structural Parameters ................................. 173
9.3.1 Physical Interpretation and Structure of the
Hessian ........................................ 173
9.3.2 Practical Resolution of the Secondary Adjoint
Equation ....................................... 178
9.3.3 Hessian Recipe ................................. 179
9.3.4 Perfectly Elastic and Isotropic Medium ......... 181
9.3.5 Perfectly Elastic Medium with Radial
Anisotropy ..................................... 183
9.3.6 Isotropic Visco-Elastic Medium ................. 185
10 The Frequency-Domain Discrete Adjoint Method ............... 189
10.1 General Formulation ................................... 189
10.2 Second Derivatives .................................... 191
11 Misfit Functional and Adjoint Sources ...................... 193
11.1 Derivative of the Pure Wave Field and the Adjoint
Greens Function ....................................... 194
11.2 L2 Waveform Difference ................................ 195
11.3 Cross-Correlation Time Shifts ......................... 197
11.4 L2 Amplitudes ......................................... 200
11.5 Time-Frequency Misfits ................................ 201
11.5.1 Definition of Phase and Envelope Misfits ....... 202
11.5.2 Practical Implementation of Phase Difference
Measurements ................................... 203
11.5.3 An Example ..................................... 205
11.5.4 Adjoint Sources ................................ 207
12 Frécnet and Hessian Kernel Gallery ......................... 211
12.1 Body Waves ............................................ 212
12.1.1 Cross-Correlation Time Shifts .................. 213
12.1.2 L2 Amplitudes .................................. 219
12.2 Surface Waves ......................................... 221
12.2.1 Isotropic Earth Models ......................... 221
12.2.2 Radial Anisotropy .............................. 224
12.3 Hessian Kernels: Towards Quantitative Trade-Off and
Resolution Analysis ................................... 225
12.4 Accuracy-Adaptive Time Integration .................... 229
Part III Applications
13 Full Waveform Tomography on Continental Scales ............. 233
13.1 Motivation ............................................ 233
13.2 Solution of the Forward Problem ....................... 235
13.2.1 Spectral Elements in Natural Spherical
Coordinates .................................... 235
13.2.2 Implementation of Long-Wavelength Equivalent
Crustal Models ................................. 238
13.3 Quantification of Waveform Differences ................ 246
13.4 Application to the Australasian Upper Mantle .......... 249
13.4.1 Data Selection and Processing .................. 251
13.4.2 Initial Model .................................. 253
13.4.3 Model Parameterisation ......................... 255
13.4.4 Tomographic Images and Waveform Fits ........... 256
13.4.5 Resolution Analysis ............................ 260
13.5 Discussion ............................................ 261
13.5.1 Forward Problem Solution ....................... 262
13.5.2 The Crust ...................................... 262
13.5.3 Time-Frequency Misfits ......................... 262
13.5.4 Dependence on the Initial Model ................ 263
13.5.5 Anisotropy ..................................... 263
13.5.6 Resolution ..................................... 264
14 Application of Full Waveform Tomography to Active-Source
Surface-Seismic Data ....................................... 267
14.1 Introduction .......................................... 267
14.2 Data .................................................. 268
14.3 Data Pre-conditioning and Weighting ................... 271
14.4 Misfit Functional ..................................... 272
14.5 Initial Model ......................................... 272
14.6 Inversion and Results ................................. 274
14.7 Data Fit .............................................. 276
14.8 Discussion ............................................ 278
15 Source Stacking Data Reduction for Full Waveform
Tomography at the Global Scale ............................. 281
15.1 Introduction .......................................... 281
15.2 Data Reduction ........................................ 282
15.3 The Source Stacked Inverse Problem .................... 283
15.4 Validation Tests ...................................... 284
15.4.1 Parameterisation ............................... 285
15.4.2 Experiment Setup and Input Models .............. 285
15.4.3 Test in a Simple Two-Parameter Model ........... 287
15.4.4 Tests in a Realistic Degree-6 Global Model ..... 289
15.5 Towards Real Cases: Dealing with Missing Data ......... 294
15.6 Discussion and Conclusions ............................ 298
Appendix A Mathematical Background for the Spectral-
Element Method ............................................. 301
A.l Orthogonal Polynomials ................................ 301
A.2 Function Interpolation ................................ 302
A.2.1 Interpolating Polynomial ....................... 302
A.2.2 Lagrange Interpolation ......................... 303
A.2.3 Lobatto Interpolation .......................... 305
A.2.4 Fekete Points .................................. 309
A.2.5 Interpolation Error ............................ 310
A.3 Numerical Integration ................................. 312
A.3.1 Exact Numerical Integration and the Gauss
Quadrature ..................................... 312
A.3.2 Gauss-Legendre-Lobatto Quadrature .............. 314
Appendix В Time-Frequency Transformations .................... 317
|
