Full Seismic Waveform Modelling and Inversion (eBook)

Advances in Geophysical and EnvironmentalMechanics and Mathematics 2
Foreword 6
Preface 8
Acknowledgements 11
Contents 13
Chapter 1 Preliminaries 19
1.1 A Brief Historical Overview 19
1.2 The Full Waveform Tomographic Inverse Problem -- Probabilistic vs. Deterministic 21
1.3 Terminology: Full Language Confusion 22
Part I Numerical Solution of the Elastic Wave Equation 24
Chapter 2 Introduction 26
2.1 Notational Conventions 26
2.2 The Elastic Wave Equation 28
2.2.1 Governing Equations 28
2.2.2 Formulations of the Elastic Wave Equation 30
2.3 The Acoustic Wave Equation 31
2.4 Discretisation in Space 32
2.5 Discretisation in Time or Frequency 33
2.5.1 Time-Domain Modelling 33
2.5.2 Frequency-Domain Modelling 35
2.6 Summary of Numerical Methods 36
Chapter 3 Finite-Difference Methods 40
3.1 Basic Concepts in One Dimension 41
3.1.1 Finite-Difference Approximations 41
3.1.2 Discretisation of the 1D Wave Equation 47
3.1.3 von Neumann Analysis: Stability and Numerical Dispersion 51
3.2 Extension to the 3D Cartesian Case 55
3.2.1 The Staggered Grid 56
3.2.2 Anisotropy and Interpolation 60
3.2.3 Implementation of the Free Surface 62
3.3 The 3D Spherical Case 67
3.4 Point Source Implementation 70
3.5 Accuracy and Efficiency 72
Chapter 4 Spectral-Element Methods 75
4.1 Basic Concepts in One Dimension 75
4.1.1 Weak Solution of the Wave Equation 76
4.1.2 Spatial Discretisation and the Galerkin Method 76
4.2 Extension to the 3D Case 82
4.2.1 Mesh Generation 82
4.2.2 Weak Solution of the Elastic Wave Equation 86
4.2.3 Discretisation of the Equations of Motion 87
4.2.4 Point Source Implementation 92
4.3 Variants of the Spectral-Element Method 95
4.4 Accuracy and Efficiency 97
Chapter 5 Visco-elastic Dissipation 98
5.1 Memory Variables 98
5.2 Q Models 100
Chapter 6 Absorbing Boundaries 104
6.1 Absorbing Boundary Conditions 104
6.1.1 Paraxial Approximations of the Acoustic Wave Equation 105
6.1.2 Paraxial Approximations as Boundary Conditions for Acoustic Waves 107
6.1.3 High-Order Absorbing Boundary Conditions for Acoustic Waves 109
6.1.4 Generalisation to the Elastic Case 111
6.1.5 Discussion 112
6.2 Gaussian Taper Method 113
6.3 Perfectly Matched Layers (PML) 114
6.3.1 General Development 114
6.3.2 Standard PML 118
6.3.3 Convolutional PML 119
6.3.4 Other Variants of the PML Method 123
Part II Iterative Solution of the Full Waveform Inversion Problem 126
Chapter 7 Introduction to Iterative Non-linear Minimisation 128
7.1 Basic Concepts: Minima, Convexity and Non-uniqueness 129
7.1.1 Local and Global Minima 129
7.1.2 Convexity: Global Minima and (Non)Uniqueness 131
7.2 Optimality Conditions 136
7.3 Iterative Methods for Non-linear Minimisation 137
7.3.1 General Descent Methods 137
7.3.2 The Method of Steepest Descent 140
7.3.3 Newton's Method and Its Variants 141
7.3.4 The Conjugate-Gradient Method 143
7.4 Convergence 149
7.4.1 The Multi-Scale Approach 149
7.4.2 Regularisation 152
Chapter 8 The Time-Domain Continuous Adjoint Method 156
8.1 Introduction 156
8.2 General Formulation 158
8.2.1 Fréchet Kernels 160
8.2.2 Translation to the Discretised Model Space 160
8.2.3 Summary of the Adjoint Method 161
8.3 Derivatives with Respect to the Source 162
8.4 Second Derivatives 163
8.4.1 Motivation: The Role of Second Derivatives in Optimisation and Resolution Analysis 164
8.4.2 Extension of the Adjoint Method to Second Derivatives 167
8.5 Application to the Elastic Wave Equation 172
8.5.1 Derivation of the Adjoint Equations 172
8.5.2 Practical Implementation 176
Chapter 9 First and Second Derivatives with Respect to Structural and Source Parameters 177
9.1 First Derivatives with Respect to Selected Structural Parameters 177
9.1.1 Perfectly Elastic and Isotropic Medium 179
9.1.2 Perfectly Elastic Medium with Radial Anisotropy 181
9.1.3 Isotropic Visco-Elastic Medium: Q and Q 184
9.2 First Derivatives with Respect to Selected Source Parameters 186
9.2.1 Distributed Sources and the Relation to Time-Reversal Imaging 186
9.2.2 Moment Tensor Point Source 186
9.3 Second Derivatives with Respect to Selected Structural Parameters 187
9.3.1 Physical Interpretation and Structure of the Hessian 187
9.3.2 Practical Resolution of the Secondary Adjoint Equation 192
9.3.3 Hessian Recipe 193
9.3.4 Perfectly Elastic and Isotropic Medium 195
9.3.5 Perfectly Elastic Medium with Radial Anisotropy 197
9.3.6 Isotropic Visco-Elastic Medium 199
Chapter 10 The Frequency-Domain Discrete Adjoint Method 202
10.1 General Formulation 202
10.2 Second Derivatives 204
Chapter 11 Misfit Functionals and Adjoint Sources 206
11.1 Derivative of the Pure Wave Field and the Adjoint Greens Function 207
11.2 L2 Waveform Difference 208
11.3 Cross-Correlation Time Shifts 210
11.4 L2 Amplitudes 213
11.5 Time-Frequency Misfits 214
11.5.1 Definition of Phase and Envelope Misfits 215
11.5.2 Practical Implementation of Phase Difference Measurements 216
11.5.3 An Example 218
11.5.4 Adjoint Sources 220
Chapter 12 Fréchet and Hessian Kernel Gallery 224
12.1 Body Waves 225
12.1.1 Cross-Correlation Time Shifts 226
12.1.2 L2 Amplitudes 232
12.2 Surface Waves 234
12.2.1 Isotropic Earth Models 234
12.2.2 Radial Anisotropy 237
12.3 Hessian Kernels: Towards Quantitative Trade-Off and Resolution Analysis 238
12.4 Accuracy-Adaptive Time Integration 242
Part III Applications 244
Chapter 13 Full Waveform Tomography on Continental Scales 246
13.1 Motivation 246
13.2 Solution of the Forward Problem 248
13.2.1 SpectralElements in Natural Spherical Coordinates 248
13.2.2 Implementation of Long-Wavelength Equivalent Crustal Models 251
13.3 Quantification of Waveform Differences 259
13.4 Application to the Australasian Upper Mantle 262
13.4.1 Data Selection and Processing 264
13.4.2 Initial Model 266
13.4.3 Model Parameterisation 268
13.4.4 Tomographic Images and Waveform Fits 269
13.4.5 Resolution Analysis 273
13.5 Discussion 274
13.5.1 Forward Problem Solution 275
13.5.2 The Crust 275
13.5.3 Time--Frequency Misfits 275
13.5.4 Dependence on the Initial Model 276
13.5.5 Anisotropy 276
13.5.6 Resolution 277
Chapter 14 Application of Full Waveform Tomography to Active-Source Surface-Seismic Data 279
14.1 Introduction 279
14.2 Data 280
14.3 Data Pre-conditioning and Weighting 283
14.4 Misfit Functional 284
14.5 Initial Model 284
14.6 Inversion and Results 286
14.7 Data Fit 288
14.8 Discussion 290
Chapter 15 Source Stacking Data Reduction for Full Waveform Tomography at the Global Scale 293
15.1 Introduction 293
15.2 Data Reduction 294
15.3 The Source Stacked Inverse Problem 295
15.4 Validation Tests 296
15.4.1 Parameterisation 297
15.4.2 Experiment Setup and Input Models 297
15.4.3 Test in a Simple Two-Parameter Model 299
15.4.4 Tests in a Realistic Degree-6 Global Model 301
15.5 Towards Real Cases: Dealing with Missing Data 306
15.6 Discussion and Conclusions 310
Appendix A Mathematical Background for the Spectral-Element Method 312
A.1 Orthogonal Polynomials 312
A.2 Function Interpolation 313
A.2.1 Interpolating Polynomial 313
A.2.2 Lagrange Interpolation 314
A.2.3 Lobatto Interpolation 316
A.2.4 Fekete Points 320
A.2.5 Interpolation Error 321
A.3 Numerical Integration 323
A.3.1 Exact Numerical Integration and the Gauss Quadrature 323
A.3.2 Gauss--Legendre--Lobatto Quadrature 325
Appendix B Time--Frequency Transformations 327
References 331
Index 348