Fichtner Full Seismic Waveform Modelling and Inversion

1. Auflage 2010
ISBN: 978-3-642-15807-0
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 343 Seiten

Reihe: Advances in Geophysical and Environmental Mechanics and Mathematics

Recent progress in numerical methods and computer science allows us today to simulate the propagation of seismic waves through realistically heterogeneous Earth models with unprecedented accuracy. Full waveform tomography is a tomographic technique that takes advantage of numerical solutions of the elastic wave equation. The accuracy of the numerical solutions and the exploitation of complete waveform information result in tomographic images that are both more realistic and better resolved. This book develops and describes state of the art methodologies covering all aspects of full waveform tomography including methods for the numerical solution of the elastic wave equation, the adjoint method, the design of objective functionals and optimisation schemes. It provides a variety of case studies on all scales from local to global based on a large number of examples involving real data. It is a comprehensive reference on full waveform tomography for advanced students, researchers and professionals.

Autoren/Hrsg.

Fichtner, Andreas

Inhaltsverzeichnis

1;Advances in Geophysical and EnvironmentalMechanics and Mathematics;2
2;Foreword;6
3;Preface;8
4;Acknowledgements;11
5;Contents;13
6;Chapter 1 Preliminaries ;19
6.1;1.1 A Brief Historical Overview;19
6.2;1.2 The Full Waveform Tomographic Inverse Problem -- Probabilistic vs. Deterministic;21
6.2.1;1.3 Terminology: Full Language Confusion;22
7;Part I Numerical Solution of the Elastic Wave Equation;24
7.1;Chapter 2 Introduction ;26
7.1.1;2.1 Notational Conventions;26
7.1.2;2.2 The Elastic Wave Equation;28
7.1.2.1;2.2.1 Governing Equations;28
7.1.2.2;2.2.2 Formulations of the Elastic Wave Equation;30
7.1.3;2.3 The Acoustic Wave Equation;31
7.1.4;2.4 Discretisation in Space;32
7.1.5;2.5 Discretisation in Time or Frequency;33
7.1.5.1;2.5.1 Time-Domain Modelling;33
7.1.5.2;2.5.2 Frequency-Domain Modelling;35
7.1.6;2.6 Summary of Numerical Methods;36
7.2;Chapter 3 Finite-Difference Methods ;40
7.2.1;3.1 Basic Concepts in One Dimension;41
7.2.1.1;3.1.1 Finite-Difference Approximations;41
7.2.1.2;3.1.2 Discretisation of the 1D Wave Equation;47
7.2.1.3;3.1.3 von Neumann Analysis: Stability and Numerical Dispersion;51
7.2.2;3.2 Extension to the 3D Cartesian Case;55
7.2.2.1;3.2.1 The Staggered Grid;56
7.2.2.2;3.2.2 Anisotropy and Interpolation;60
7.2.2.3;3.2.3 Implementation of the Free Surface;62
7.2.3;3.3 The 3D Spherical Case;67
7.2.4;3.4 Point Source Implementation;70
7.2.5;3.5 Accuracy and Efficiency;72
7.3;Chapter 4 Spectral-Element Methods ;75
7.3.1;4.1 Basic Concepts in One Dimension;75
7.3.1.1;4.1.1 Weak Solution of the Wave Equation;76
7.3.1.2;4.1.2 Spatial Discretisation and the Galerkin Method;76
7.3.2;4.2 Extension to the 3D Case;82
7.3.2.1;4.2.1 Mesh Generation;82
7.3.2.2;4.2.2 Weak Solution of the Elastic Wave Equation;86
7.3.2.3;4.2.3 Discretisation of the Equations of Motion;87
7.3.2.4;4.2.4 Point Source Implementation;92
7.3.3;4.3 Variants of the Spectral-Element Method;95
7.3.4;4.4 Accuracy and Efficiency;97
7.4;Chapter 5 Visco-elastic Dissipation ;98
7.4.1;5.1 Memory Variables;98
7.4.2;5.2 Q Models;100
7.5;Chapter 6 Absorbing Boundaries ;104
7.5.1;6.1 Absorbing Boundary Conditions;104
7.5.1.1;6.1.1 Paraxial Approximations of the Acoustic Wave Equation;105
7.5.1.2;6.1.2 Paraxial Approximations as Boundary Conditions for Acoustic Waves;107
7.5.1.3;6.1.3 High-Order Absorbing Boundary Conditions for Acoustic Waves;109
7.5.1.4;6.1.4 Generalisation to the Elastic Case;111
7.5.1.5;6.1.5 Discussion;112
7.5.2;6.2 Gaussian Taper Method;113
7.5.3;6.3 Perfectly Matched Layers (PML);114
7.5.3.1;6.3.1 General Development;114
7.5.3.2;6.3.2 Standard PML;118
7.5.3.3;6.3.3 Convolutional PML;119
7.5.3.4;6.3.4 Other Variants of the PML Method;123
8;Part II Iterative Solution of the Full Waveform Inversion Problem;126
8.1;Chapter 7 Introduction to Iterative Non-linear Minimisation ;128
8.1.1;7.1 Basic Concepts: Minima, Convexity and Non-uniqueness;129
8.1.1.1;7.1.1 Local and Global Minima;129
8.1.1.2;7.1.2 Convexity: Global Minima and (Non)Uniqueness;131
8.1.2;7.2 Optimality Conditions;136
8.1.3;7.3 Iterative Methods for Non-linear Minimisation;137
8.1.3.1;7.3.1 General Descent Methods;137
8.1.3.2;7.3.2 The Method of Steepest Descent;140
8.1.3.3;7.3.3 Newton's Method and Its Variants;141
8.1.3.4;7.3.4 The Conjugate-Gradient Method;143
8.1.4;7.4 Convergence;149
8.1.4.1;7.4.1 The Multi-Scale Approach;149
8.1.4.2;7.4.2 Regularisation;152
8.2;Chapter 8 The Time-Domain Continuous Adjoint Method ;156
8.2.1;8.1 Introduction;156
8.2.2;8.2 General Formulation;158
8.2.2.1;8.2.1 Fréchet Kernels;160
8.2.2.2;8.2.2 Translation to the Discretised Model Space;160
8.2.2.3;8.2.3 Summary of the Adjoint Method;161
8.2.3;8.3 Derivatives with Respect to the Source;162
8.2.4;8.4 Second Derivatives;163
8.2.4.1;8.4.1 Motivation: The Role of Second Derivatives in Optimisation and Resolution Analysis;164
8.2.4.2;8.4.2 Extension of the Adjoint Method to Second Derivatives;167
8.2.5;8.5 Application to the Elastic Wave Equation;172
8.2.5.1;8.5.1 Derivation of the Adjoint Equations;172
8.2.5.2;8.5.2 Practical Implementation;176
8.3;Chapter 9 First and Second Derivatives with Respect to Structural and Source Parameters ;177
8.3.1;9.1 First Derivatives with Respect to Selected Structural Parameters;177
8.3.1.1;9.1.1 Perfectly Elastic and Isotropic Medium;179
8.3.1.2;9.1.2 Perfectly Elastic Medium with Radial Anisotropy;181
8.3.1.3;9.1.3 Isotropic Visco-Elastic Medium: Q and Q;184
8.3.2;9.2 First Derivatives with Respect to Selected Source Parameters;186
8.3.2.1;9.2.1 Distributed Sources and the Relation to Time-Reversal Imaging;186
8.3.2.2;9.2.2 Moment Tensor Point Source;186
8.3.3;9.3 Second Derivatives with Respect to Selected Structural Parameters;187
8.3.3.1;9.3.1 Physical Interpretation and Structure of the Hessian;187
8.3.3.2;9.3.2 Practical Resolution of the Secondary Adjoint Equation;192
8.3.3.3;9.3.3 Hessian Recipe;193
8.3.3.4;9.3.4 Perfectly Elastic and Isotropic Medium;195
8.3.3.5;9.3.5 Perfectly Elastic Medium with Radial Anisotropy;197
8.3.3.6;9.3.6 Isotropic Visco-Elastic Medium;199
8.4;Chapter 10 The Frequency-Domain Discrete Adjoint Method ;202
8.4.1;10.1 General Formulation;202
8.4.2;10.2 Second Derivatives;204
8.5;Chapter 11 Misfit Functionals and Adjoint Sources ;206
8.5.1;11.1 Derivative of the Pure Wave Field and the Adjoint Greens Function;207
8.5.2;11.2 L2 Waveform Difference;208
8.5.3;11.3 Cross-Correlation Time Shifts;210
8.5.4;11.4 L2 Amplitudes;213
8.5.5;11.5 Time-Frequency Misfits;214
8.5.5.1;11.5.1 Definition of Phase and Envelope Misfits;215
8.5.5.2;11.5.2 Practical Implementation of Phase Difference Measurements;216
8.5.5.3;11.5.3 An Example;218
8.5.5.4;11.5.4 Adjoint Sources;220
8.6;Chapter 12 Fréchet and Hessian Kernel Gallery ;224
8.6.1;12.1 Body Waves;225
8.6.1.1;12.1.1 Cross-Correlation Time Shifts;226
8.6.1.2;12.1.2 L2 Amplitudes;232
8.6.2;12.2 Surface Waves;234
8.6.2.1;12.2.1 Isotropic Earth Models;234
8.6.2.2;12.2.2 Radial Anisotropy;237
8.6.3;12.3 Hessian Kernels: Towards Quantitative Trade-Off and Resolution Analysis;238
8.6.4;12.4 Accuracy-Adaptive Time Integration;242
9;Part III Applications;244
9.1;Chapter 13 Full Waveform Tomography on Continental Scales ;246
9.1.1;13.1 Motivation;246
9.1.2;13.2 Solution of the Forward Problem;248
9.1.2.1;13.2.1 SpectralElements in Natural Spherical Coordinates;248
9.1.2.2;13.2.2 Implementation of Long-Wavelength Equivalent Crustal Models;251
9.1.3;13.3 Quantification of Waveform Differences;259
9.1.4;13.4 Application to the Australasian Upper Mantle;262
9.1.4.1;13.4.1 Data Selection and Processing;264
9.1.4.2;13.4.2 Initial Model;266
9.1.4.3;13.4.3 Model Parameterisation;268
9.1.4.4;13.4.4 Tomographic Images and Waveform Fits;269
9.1.4.5;13.4.5 Resolution Analysis;273
9.1.5;13.5 Discussion;274
9.1.5.1;13.5.1 Forward Problem Solution;275
9.1.5.2;13.5.2 The Crust;275
9.1.5.3;13.5.3 Time--Frequency Misfits;275
9.1.5.4;13.5.4 Dependence on the Initial Model;276
9.1.5.5;13.5.5 Anisotropy;276
9.1.5.6;13.5.6 Resolution;277
9.2;Chapter 14 Application of Full Waveform Tomography to Active-Source Surface-Seismic Data ;279
9.2.1;14.1 Introduction;279
9.2.2;14.2 Data;280
9.2.3;14.3 Data Pre-conditioning and Weighting;283
9.2.4;14.4 Misfit Functional;284
9.2.5;14.5 Initial Model;284
9.2.6;14.6 Inversion and Results;286
9.2.7;14.7 Data Fit;288
9.2.8;14.8 Discussion;290
9.3;Chapter 15 Source Stacking Data Reduction for Full Waveform Tomography at the Global Scale ;293
9.3.1;15.1 Introduction;293
9.3.2;15.2 Data Reduction;294
9.3.3;15.3 The Source Stacked Inverse Problem;295
9.3.4;15.4 Validation Tests;296
9.3.4.1;15.4.1 Parameterisation;297
9.3.4.2;15.4.2 Experiment Setup and Input Models;297
9.3.4.3;15.4.3 Test in a Simple Two-Parameter Model;299
9.3.4.4;15.4.4 Tests in a Realistic Degree-6 Global Model;301
9.3.5;15.5 Towards Real Cases: Dealing with Missing Data;306
9.3.6;15.6 Discussion and Conclusions;310
10; Appendix A Mathematical Background for the Spectral-Element Method ;312
10.1;A.1 Orthogonal Polynomials;312
10.2;A.2 Function Interpolation;313
10.2.1;A.2.1 Interpolating Polynomial;313
10.2.2;A.2.2 Lagrange Interpolation;314
10.2.3;A.2.3 Lobatto Interpolation;316
10.2.4;A.2.4 Fekete Points;320
10.2.5;A.2.5 Interpolation Error;321
10.3;A.3 Numerical Integration;323
10.3.1;A.3.1 Exact Numerical Integration and the Gauss Quadrature;323
10.3.2;A.3.2 Gauss--Legendre--Lobatto Quadrature;325
11; Appendix B Time--Frequency Transformations ;327
12;References;331
13;Index;348