Computation, Optimization, and Machine Learning in Seismology

Subhashis Mallick, University of Wyoming, USA

• Preface xiiiAvailability Statement xvAbout the Companion Website xvii1 Introduction to Key Concepts in Seismic Inversion and Elastic Wave Theory 11.1 Background 11.2 Seismology—A Historical Perspective 11.2.1 Earthquake Seismology 11.2.2 Exploration Seismology 21.3 Mathematical Foundations of Seismology 31.4 Seismic Inversion 31.4.1 The Meaning of Inversion 31.4.2 Seismic Problems 41.4.3 Operator-Based and Model-Based Inversions—The Concept of Optimization 91.4.4 Fundamental Concepts of the Optimization Method (Model-Based Inversion) 101.5 Model, Data, and Objective Spaces 111.6 Different Flavors of Optimization 111.6.1 Local (Gradient-Based) Optimization 121.6.2 Global Optimization 121.6.3 Machine-Learning-Based Optimization 141.6.4 Single and Multi-objective Optimization 151.7 Bayesian Approach to Inversion/Optimization 161.8 Summary and Organization of the Book 161.9 Exercises 17References 182 Mathematical Background 232.1 Fourier Series and Fourier Integrals 232.1.1 Fourier Series 232.1.2 Fourier Integrals 262.1.3 Fourier Transforms 272.2 Partial Differential Equations 332.2.1 How Do the Simplest Partial Differential Equations Arise? 332.2.2 Elliptic, Hyperbolic, and Parabolic Partial Differential Equations: Theory of Characteristics 352.2.3 Simple Examples of the Partial Differential Equations 362.2.4 Adjoint Differential Forms 402.3 Fundamentals of Tensor Algebra and Tensor Calculus 402.3.1 System of Coordinates 412.3.2 What Are Tensors? 412.3.3 Basis Vectors 422.3.4 The Gradient Operator and the Covariant and Contravariant Basis Vectors 432.3.5 Concept of Tensors 442.3.6 The Identity Tensor 442.3.7 Elements of Tensor Algebra 452.3.8 Elements of Tensor Calculus 492.3.9 Useful Theorems in Tensor Calculus 552.4 Chapter Summary 562.5 Exercises 56References 573 Fundamentals of the Linearized Elastic Wave Theory 593.1 Introduction 593.2 The Stress Tensor and Traction 593.3 Strain (Deformation) Tensor 603.4 Static Relation—The First Fundamental Equation in Elasticity 623.4.1 Orthogonal Transformation of the Elastic Stiffness Matrix 633.4.2 Elastic Symmetries 683.4.3 Geological Interpretation of the Elastic Symmetries—The Concept of an Equivalent Medium 713.5 Strain Energy Function and the Positive-Definite Conditions 743.6 Dynamic Relation—Second Fundamental Equation in Elasticity 753.7 Elastodynamic Equation 753.8 Solution of the Elastodynamic Equation in Homogeneous Elastic Medium 763.8.1 Solution of the Christoffel Equation for a Hexagonally Symmetric Medium 773.8.2 Solution of the Christoffel Equation for an Orthorhombically Symmetric Medium 803.9 Ray (Group) Angle and Ray (Group) Velocity 833.9.1 Mathematical Formulation of the Group and Phase Directions for an Elastic Medium with Arbitrary Anisotropy 833.9.2 Analytical Expressions for Group Velocity and Angle for Specific Symmetries 853.9.3 Importance of the Group and Phase—An Optimization Problem of Practical Importance 883.10 Radiation Patterns from Seismic Sources 923.10.1 The Laplacian Operator and Its Inverse 923.10.2 Helmholtz Representation Theorem 933.10.3 Momentum Equation for Isotropic Elastic System 943.10.4 Green’s Function for Hyperbolic Partial Differential Equations 943.10.5 Radiation Patterns from a Uniform (Explosive) Point Source 953.10.6 Radiation Patterns from a Point Double Couple or a Moment-Tensor Source 973.10.7 Radiation Patterns from a Point Force 1043.10.8 Summary of Radiation Patterns 1053.11 Chapter Summary 1063.12 Exercises 106References 1084 Computation of Synthetic Seismograms in Inhomogeneous Medium: Approximate (Partial) Solutions 1114.1 Introduction 1114.2 Fundamentals of Ray Theory and Computation of Partial Synthetic Seismic Response 1114.2.1 Snell’s Law 1124.2.2 Sign Convention for the Fourier Transforms in Seismology 1144.2.3 Ray Tracing in a Smoothly Varying Laterally Homogeneous Medium 1154.2.4 Travel Time and Distance 1204.2.5 A Practical Example—Linearly Varying Velocity with Depth 1234.2.6 Reflection and Transmission Problem 1234.2.7 Ray-Theoretical Seismogram Computations 1314.2.8 Ray Tracing in an Anisotropic Medium 1324.2.9 Other Methods for Computing Partial Seismic Response 1324.3 Amplitude-Variation-With-Angle Synthetic Seismograms 1334.4 Chapter Summary 1344.5 Exercises 134References 1355 Computation of Synthetic Seismograms in Inhomogeneous Medium: Exact Solutions 1375.1 Introduction 1375.2 Motivations Behind Computing a Complete Synthetic Seismic Response 1375.3 Analytical Computation of Exact Synthetic Seismograms for a Horizontally Stratified Earth Model 1405.3.1 Conventions and Notations 1405.3.2 The Elastic System in 1D 1415.3.3 Solution of the Elastic System: A Homogeneous Region 1425.3.4 Reflection and Transmission 1445.3.5 The Eigenvalue and Eigenvector Matrices and the Inverse of the Eigenvector Matrix 1485.3.6 Reflection and Transmission in a Homogeneous Medium 1515.3.7 Reflection and Transmission in a Stack of Layers 1525.3.8 Reflection and Transmission in a Homogeneous Layer and an Interface 1535.3.9 Iteration Equations 1535.3.10 The Source Term 1565.3.11 Computation of the Source Wavefield 1595.3.12 Computation of the Receiver Wavefield 1605.3.13 Response Computation in Different Domains 1615.3.14 Inelastic Attenuation 1625.4 Synthetic Seismograms for Vertically and Laterally Varying Media 1655.4.1 Governing Equations 1655.4.2 Spatial Discretization 1675.4.3 Temporal Discretization 1685.4.4 Overview of Different Numerical Methods 1695.4.5 Boundary Conditions 1705.4.6 Summary of Different Methods for Computing Synthetic Seismic Responses for Heterogeneous Media 1715.5 Chapter Summary 1715.6 Exercises 171References 1736 Optimization of Functions 1796.1 Introduction 1796.2 One-Dimensional Optimization 1796.2.1 Golden Section Search in One Dimension 1806.2.2 Inverse Parabolic Interpolation and Brent’s Method in One Dimension 1836.2.3 Van Wijngaarden–Dekker–Brent Method 1846.2.4 One-Dimensional Optimization Using First Derivatives 1866.2.5 Practical Examples of One-Dimensional Optimization 1876.3 Multidimensional Optimization 1926.3.1 Fundamental Concepts 1926.3.2 Conjugate Directions 1946.3.3 Steepest Descent (Gradient Descent) Method 1956.3.4 Conjugate Gradient Method 1966.3.5 Variable Metric Method 1986.3.6 Other Popular Methods 2016.3.7 A Final Note to Multidimensional Optimization Problems 2026.4 Chapter Summary 2026.5 Exercises 202References 2037 Local Optimization Methods in Seismology 2057.1 Introduction 2057.2 Fréchet Derivative (Jacobi) Matrix and the Computation of the Gradient of the Objective Function 2067.3 Regularization of the Objective 2077.3.1 Variance of the Model Parameter Estimates 2087.3.2 Variance and Prediction Error of the Least-Squares Solutions 2087.3.3 Data Resolution Matrix 2097.3.4 Model Resolution Matrix 2097.3.5 Objective Regularization 2107.4 Implementation of Local Optimization Methods 2107.4.1 Steepest Descent (Gradient Descent) and Conjugate-Gradient Methods 2117.4.2 Gauss–Newton Method 2157.4.3 Other Methods 2167.5 Computation of the Jacobi (Fréchet Derivative) Matrix 2177.5.1 Amplitude-Variation-With-Angle Inversion 2177.5.2 Full Waveform Inversion 2187.6 Examples 2287.6.1 Poststack Inversion 2287.6.2 Prestack Inversion 2297.7 Chapter Summary 2387.8 Exercises 238References 2398 Global Optimization Methods in Seismology 2438.1 Introduction 2438.2 Bayesian Approach to Optimization Problems 2468.2.1 Simple Monte Carlo Integration 2518.3 Global Optimization Methods 2538.3.1 Markov Chain Monte Carlo Optimization 2558.3.2 Simulated Annealing Optimization 2618.3.3 Genetic Algorithm Optimization 2648.4 Multi-Objective Optimization 2878.4.1 The Concepts of Pareto-Optimality, Pareto-Optimal Solution Sets, and Dominance 2878.4.2 Why Multi-Objective Methods Are Necessary? 2898.4.3 Multi-Objective Optimization: A General Overview 2908.4.4 Geophysical Applications of Multi-Objective Optimization 2928.5 Examples 2958.6 Chapter Summary 3028.7 Exercises 302References 3049 Artificial Intelligence for Seismic Inverse Problems 3119.1 Introduction 3119.2 Artificial Neural Network 3119.2.1 Anatomy of a Biological Neuron 3129.2.2 An Equivalent Artificial Neuron 3129.2.3 The Activation and Bias 3129.2.4 From a Single Neuron to Multiple Neurons: A Simple Neural Network 3169.3 Deep Neural Networks 3239.3.1 Number of Hidden Layers 3249.3.2 Number of Neurons in Each Hidden Layer 3249.3.3 Stopping Criteria for Training 3259.3.4 Optimum Number of Training (and Validation) Data 3269.3.5 Revisiting the Network Design 3269.3.6 Different Flavors of DNNs 3329.4 Other Machine-Learning Methods 3449.4.1 Support Vector Machine 3449.4.2 Gradient Boosting 3539.5 Physics-Informed Machine Learning 3549.6 Multi-task Learning 3559.7 Machine Learning in a Bayesian Framework 3559.8 Examples 3559.9 Chapter Summary 3569.10 Exercises 356References 35810 The Future of Seismic Inversion and Machine Learning 36710.1 Introduction 36710.2 The Road Ahead 36710.2.1 Carbon Capture, Utilization, and Storage 36810.2.2 Hydrogen Storage Systems 37310.2.3 Geohazards and Related Environmental Impacts 37310.2.4 Use of Seismology for Weather Prediction, Climate Modeling, and Marine Biology Research 37410.2.5 The Overall Picture for Future Developments and Applications 37710.3 Few Aspects of Practical Importance 37810.4 Conclusions 383References 384Index 389