Computation, Optimization, and Machine Learning in Seismology

Preface xiii

Availability Statement xv

About the Companion Website xvii

1 Introduction to Key Concepts in Seismic Inversion and Elastic Wave Theory 1

1.1 Background 1

1.2 Seismology-A Historical Perspective 1

1.2.1 Earthquake Seismology 1

1.2.2 Exploration Seismology 2

1.3 Mathematical Foundations of Seismology 3

1.4 Seismic Inversion 3

1.4.1 The Meaning of Inversion 3

1.4.2 Seismic Problems 4

1.4.3 Operator-Based and Model-Based Inversions-The Concept of Optimization 9

1.4.4 Fundamental Concepts of the Optimization Method (Model-Based Inversion) 10

1.5 Model, Data, and Objective Spaces 11

1.6 Different Flavors of Optimization 11

1.6.1 Local (Gradient-Based) Optimization 12

1.6.2 Global Optimization 12

1.6.3 Machine-Learning-Based Optimization 14

1.6.4 Single and Multi-objective Optimization 15

1.7 Bayesian Approach to Inversion/Optimization 16

1.8 Summary and Organization of the Book 16

1.9 Exercises 17

References 18

2 Mathematical Background 23

2.1 Fourier Series and Fourier Integrals 23

2.1.1 Fourier Series 23

2.1.2 Fourier Integrals 26

2.1.3 Fourier Transforms 27

2.2 Partial Differential Equations 33

2.2.1 How Do the Simplest Partial Differential Equations Arise? 33

2.2.2 Elliptic, Hyperbolic, and Parabolic Partial Differential Equations: Theory of Characteristics 35

2.2.3 Simple Examples of the Partial Differential Equations 36

2.2.4 Adjoint Differential Forms 40

2.3 Fundamentals of Tensor Algebra and Tensor Calculus 40

2.3.1 System of Coordinates 41

2.3.2 What Are Tensors? 41

2.3.3 Basis Vectors 42

2.3.4 The Gradient Operator and the Covariant and Contravariant Basis Vectors 43

2.3.5 Concept of Tensors 44

2.3.6 The Identity Tensor 44

2.3.7 Elements of Tensor Algebra 45

2.3.8 Elements of Tensor Calculus 49

2.3.9 Useful Theorems in Tensor Calculus 55

2.4 Chapter Summary 56

2.5 Exercises 56

References 57

3 Fundamentals of the Linearized Elastic Wave Theory 59

3.1 Introduction 59

3.2 The Stress Tensor and Traction 59

3.3 Strain (Deformation) Tensor 60

3.4 Static Relation-The First Fundamental Equation in Elasticity 62

3.4.1 Orthogonal Transformation of the Elastic Stiffness Matrix 63

3.4.2 Elastic Symmetries 68

3.4.3 Geological Interpretation of the Elastic Symmetries-The Concept of an Equivalent Medium 71

3.5 Strain Energy Function and the Positive-Definite Conditions 74

3.6 Dynamic Relation-Second Fundamental Equation in Elasticity 75

3.7 Elastodynamic Equation 75

3.8 Solution of the Elastodynamic Equation in Homogeneous Elastic Medium 76

3.8.1 Solution of the Christoffel Equation for a Hexagonally Symmetric Medium 77

3.8.2 Solution of the Christoffel Equation for an Orthorhombically Symmetric Medium 80

3.9 Ray (Group) Angle and Ray (Group) Velocity 83

3.9.1 Mathematical Formulation of the Group and Phase Directions for an Elastic Medium with Arbitrary Anisotropy 83

3.9.2 Analytical Expressions for Group Velocity and Angle for Specific Symmetries 85

3.9.3 Importance of the Group and Phase-An Optimization Problem of Practical Importance 88

3.10 Radiation Patterns from Seismic Sources 92

3.10.1 The Laplacian Operator and Its Inverse 92

3.10.2 Helmholtz Representation Theorem 93

3.10.3 Momentum Equation for Isotropic Elastic System 94

3.10.4 Green's Function for Hyperbolic Partial Differential Equations 94

3.10.5 Radiation Patterns from a Uniform (Explosive) Point Source 95

3.10.6 Radiation Patterns from a Point Double Couple or a Moment-Tensor Source 97

3.10.7 Radiation Patterns from a Point Force 104

3.10.8 Summary of Radiation Patterns 105

3.11 Chapter Summary 106

3.12 Exercises 106

References 108

4 Computation of Synthetic Seismograms in Inhomogeneous Medium: Approximate (Partial) Solutions 111

4.1 Introduction 111

4.2 Fundamentals of Ray Theory and Computation of Partial Synthetic Seismic Response 111

4.2.1 Snell's Law 112

4.2.2 Sign Convention for the Fourier Transforms in Seismology 114

4.2.3 Ray Tracing in a Smoothly Varying Laterally Homogeneous Medium 115

4.2.4 Travel Time and Distance 120

4.2.5 A Practical Example-Linearly Varying Velocity with Depth 123

4.2.6 Reflection and Transmission Problem 123

4.2.7 Ray-Theoretical Seismogram Computations 131

4.2.8 Ray Tracing in an Anisotropic Medium 132

4.2.9 Other Methods for Computing Partial Seismic Response 132

4.3 Amplitude-Variation-With-Angle Synthetic Seismograms 133

4.4 Chapter Summary 134

4.5 Exercises 134

References 135

5 Computation of Synthetic Seismograms in Inhomogeneous Medium: Exact Solutions 137

5.1 Introduction 137

5.2 Motivations Behind Computing a Complete Synthetic Seismic Response 137

5.3 Analytical Computation of Exact Synthetic Seismograms for a Horizontally Stratified Earth Model 140

5.3.1 Conventions and Notations 140

5.3.2 The Elastic System in 1D 141

5.3.3 Solution of the Elastic System: A Homogeneous Region 142

5.3.4 Reflection and Transmission 144

5.3.5 The Eigenvalue and Eigenvector Matrices and the Inverse of the Eigenvector Matrix 148

5.3.6 Reflection and Transmission in a Homogeneous Medium 151

5.3.7 Reflection and Transmission in a Stack of Layers 152

5.3.8 Reflection and Transmission in a Homogeneous Layer and an Interface 153

5.3.9 Iteration Equations 153

5.3.10 The Source Term 156

5.3.11 Computation of the Source Wavefield 159

5.3.12 Computation of the Receiver Wavefield 160

5.3.13 Response Computation in Different Domains 161

5.3.14 Inelastic Attenuation 162

5.4 Synthetic Seismograms for Vertically and Laterally Varying Media 165

5.4.1 Governing Equations 165

5.4.2 Spatial Discretization 167

5.4.3 Temporal Discretization 168

5.4.4 Overview of Different Numerical Methods 169

5.4.5 Boundary Conditions 170

5.4.6 Summary of Different Methods for Computing Synthetic Seismic Responses for Heterogeneous Media 171

5.5 Chapter Summary 171

5.6 Exercises 171

References 173

6 Optimization of Functions 179

6.1 Introduction 179

6.2 One-Dimensional Optimization 179

6.2.1 Golden Section Search in One Dimension 180

6.2.2 Inverse Parabolic Interpolation and Brent's Method in One Dimension 183

6.2.3 Van Wijngaarden-Dekker-Brent Method 184

6.2.4 One-Dimensional Optimization Using First Derivatives 186

6.2.5 Practical Examples of One-Dimensional Optimization 187

6.3 Multidimensional Optimization 192

6.3.1 Fundamental Concepts 192

6.3.2 Conjugate Directions 194

6.3.3 Steepest Descent (Gradient Descent) Method 195

6.3.4 Conjugate Gradient Method 196

6.3.5 Variable Metric Method 198

6.3.6 Other Popular Methods 201

6.3.7 A Final Note to Multidimensional Optimization Problems 202

6.4 Chapter Summary 202

6.5 Exercises 202

References 203

7 Local Optimization Methods in Seismology 205

7.1 Introduction 205

7.2 Fréchet Derivative (Jacobi) Matrix and the Computation of the Gradient of the Objective Function 206

7.3 Regularization of the Objective 207

7.3.1 Variance of the Model Parameter Estimates 208

7.3.2 Variance and Prediction Error of the Least-Squares Solutions 208

7.3.3 Data Resolution Matrix 209

7.3.4 Model Resolution Matrix 209

7.3.5 Objective Regularization 210

7.4 Implementation of Local Optimization Methods 210

7.4.1 Steepest Descent (Gradient Descent) and Conjugate-Gradient Methods 211

7.4.2 Gauss-Newton Method 215

7.4.3 Other Methods 216

7.5 Computation of the Jacobi (Fréchet Derivative) Matrix 217

7.5.1 Amplitude-Variation-With-Angle Inversion 217

7.5.2 Full Waveform Inversion 218

7.6 Examples 228

7.6.1 Poststack Inversion 228

7.6.2 Prestack Inversion 229

7.7 Chapter Summary 238

7.8 Exercises 238

References 239

8 Global Optimization Methods in Seismology 243

8.1 Introduction 243

8.2 Bayesian Approach to Optimization Problems 246

8.2.1 Simple Monte Carlo Integration 251

8.3 Global Optimization Methods 253

8.3.1 Markov Chain Monte Carlo Optimization 255

8.3.2 Simulated Annealing Optimization 261

8.3.3 Genetic Algorithm Optimization 264

8.4 Multi-Objective Optimization 287

8.4.1 The Concepts of Pareto-Optimality, Pareto-Optimal Solution Sets, and Dominance 287

8.4.2 Why Multi-Objective Methods Are Necessary? 289

8.4.3 Multi-Objective Optimization: A General Overview 290

8.4.4 Geophysical Applications of Multi-Objective Optimization 292

8.5 Examples 295

8.6 Chapter Summary 302

8.7 Exercises 302

References 304

9 Artificial Intelligence for Seismic Inverse Problems 311

9.1 Introduction 311

9.2 Artificial Neural Network 311

9.2.1 Anatomy of a Biological Neuron 312

9.2.2 An Equivalent Artificial Neuron 312

9.2.3 The Activation and Bias 312

9.2.4 From a Single Neuron to Multiple Neurons: A Simple Neural Network 316

9.3 Deep Neural Networks 323

9.3.1 Number of Hidden Layers 324

9.3.2 Number of Neurons in Each Hidden Layer 324

9.3.3 Stopping Criteria for Training 325

9.3.4 Optimum Number of Training (and Validation) Data 326

9.3.5 Revisiting the Network Design 326

9.3.6 Different Flavors of DNNs 332

9.4 Other Machine-Learning Methods 344

9.4.1 Support Vector Machine 344

9.4.2 Gradient Boosting 353

9.5 Physics-Informed Machine Learning 354

9.6 Multi-task Learning 355

9.7 Machine Learning in a Bayesian Framework 355

9.8 Examples 355

9.9 Chapter Summary 356

9.10 Exercises 356

References 358

10 The Future of Seismic Inversion and Machine Learning 367

10.1 Introduction 367

10.2 The Road Ahead 367

10.2.1 Carbon Capture, Utilization, and Storage 368

10.2.2 Hydrogen Storage Systems 373

10.2.3 Geohazards and Related Environmental Impacts 373

10.2.4 Use of Seismology for Weather Prediction, Climate Modeling, and Marine Biology Research 374

10.2.5 The Overall Picture for Future Developments and Applications 377

10.3 Few Aspects of Practical Importance 378

10.4 Conclusions 383

References 384

Index 389