Computation, Optimization, and Machine Learning in Seismology The goal of computational seismology is to digitally simulate seismic waves, create subsurface models, and match these models with observations to identify subsurface rock properties. With recent advances in computing technology, including machine learning, it is now possible to automate matching procedures and use waveform inversion or optimization to create large-scale models. Computation, Optimization, and Machine Learning in Seismology provides students with a detailed understanding of seismic wave theory, optimization theory, and how to use machine learning to interpret seismic data. Volume highlights include: Mathematical foundations and key equations for computational seismology Essential theories, including wave propagation and elastic wave theory Processing, mapping, and interpretation of prestack data Model-based optimization and artificial intelligence methods Applications for earthquakes, exploration seismology, depth imaging, and multi-objective geophysics problems Exercises applying the main concepts of each chapter

Table of Contents:
CHAPTER 1 INTRODUCTION TO SEISMOLOGY, OPTIMIZATION, AND MACHINE LEARNING 1.1 Seismology: a historical perspective 1.2 Earthquake Seismology 1.2.1 Finding the structure of earth 1.2.2 Developments of the mathematical foundations 1.2.3 Development of the concepts of plate-tectonics and earthquake seismology 1.3 Exploration Seismology 1.3.1 Historical perspective 1.3.2 Refraction seismology 1.3.2.1 Fundamentals 1.3.2.2 Interpretation 1.3.3 Reflection seismology 1.3.3.1 Fundamentals 1.3.3.2 Data processing 1.3.3.3 Interpretation 1.4 Seismic inversion or optimization 1.4.1 Overview 1.4.2 Traveltime inversion 1.4.3 Amplitude-variation-with-angle analysis and inversion 1.4.4 Waveform inversion 1.5 General concepts of machine-learning 1.6 Description of the different Chapters of the book 1.7 References CHAPTER 2 MATHEMATICAL BACKGROUND TO UNDERSTAND GEOLOGIC AND GEOPHYSICAL SOURCES 2.1 Fourier Series and integrals 2.1.1 Fourier Series 2.1.2 Fourier Integrals 2.1.3 Fourier Transforms 2.2 Partial Differential Equations 2.2.1 Parabolic Partial Differential Equations 2.2.2 Elliptic Partial Differential Equations 2.2.3 Hyperbolic Partial Differential Equations 2.2.4 Green’s Theorem and Seismic Response from Point Sources 2.3 Fundamentals of Tensor Algebra 2.3.1 Scalars, Vectors, and Tensors 2.3.2 Basis Vectors 2.3.3 Covariant and contravariant components of a Tensor 2.3.4 Identity Tensor 2.3.5 Representation of Tensors in Different Coordinate Systems 2.3.6 Elements of Tensor Algebra and Tensor Calculus 2.4 Exercises 2.5 References CHAPTER 3 FUNDAMENTALS OF THE LINEARIZED ELASTIC WAVE THEORY WITH APPLICATIONS TO GEOLOGIC SURFACES 3.1 Strain (deformation) Tensor 3.2 Stress Tensor 3.3 Linearized Theory of Elasticity (Hooke’s Law) 3.3.1 Elastic Stiffness Matrix 3.3.2 Orthogonal Transformations and Different Anisotropic Symmetries 3.4 Equivalent Medium 3.4.1 Layered System 3.4.2 Effect of Fractures and Cracks 3.5 Momentum Equation 3.5.1 Dynamic Relation of Elasticity 3.6 Elastodynamic Equation 3.6.1 Combination of the Static and Dynamic Relation 3.6.2 Derivation of Elastodynamic Equation 3.7 Elastic Waves in a Homogeneous Medium 3.7.1 Christoffel Equation 3.7.2 Slowness (Inverse Velocity) Surface 3.7.2 Phase Angle and Phase Velocity 3.7.3 Group (ray) Angle and Velocity 3.7.4 Solution of the Christoffel Equation 3.8 Exercises 3.9 References CHAPTER 4 SEISMIC WAVES IN AN INHOMOGENEOUS GEOLOGIC MEDIUM 4.1 Introduction 4.2 Seismic Waves in a Horizontally Stratified Medium 4.2.1 Derivation of the Elastic Systems 4.2.2 Solution of the Elastic System 4.2.3 Propagators and Wave Propagators 4.2.4 Reflection and Transmission Coefficients at an Interface 4.2.5 Reflection and Transmission Matrices in a Layered System 4.2.6 Point Source in a Layered System 4.2.7 Computation of the Source Wavefield 4.2.8 Computation of the Receiver Wavefield 4.2.9 Computation of Seismic Response in Different Domains 4.2.10 Implementations in High-performance Parallel Computing Environments 4.3 Seismic Waves in a Three-dimensionally Inhomogeneous Medium 4.3.1 Formulation of the Problem from Elastodynamic Equation 4.3.2 Finite-difference Method 4.3.2 Finite-element Method 4.3.3 Implementation in High-performance Parallel Computing Environments 4.4 Amplitude-Variation-With-Angle (AVA) Methods 4.4.1 Introduction 4.4.2 AVA Formula for Isotropic and Anisotropic Medium 4.4.3 Applications of AVA 4.4.3 Fundamental Assumptions of AVA with Implications 4.4.5 Examples 4.5 Exercises 4.6 References CHAPTER 5 FUNDAMENTALS OF OPTIMIZATION IN GEOLOGICAL MODELS 5.1 What is Optimization? 5.1.1 A Simple Example- Person Kicking a Soccer Ball 5.1.2 Geophysical Analogies 5.1.2.1 Post-stack Seismic Inversion 5.1.2.2 AVA Inversion 5.1.3 Definitions of Forward and Inverse Problems 5.2 Operator-based and Model-based Inversion/Optimization 5.2.1 Operator-based Optimization 5.2.2 Model-based Optimization 5.2.3 Demonstration of the Need for Model-based Optimization 5.2.4 Soccer Ball Example Revisited and Formulation of Model-based Optimization 5.2.5 Definition of the Objective 5.3 Fundamental Definitions 5.3.1 Data Space 5.3.2 Model (Decision) Space 5.3.3 Objective Space 5.4 Different Flavors of Model-based Optimization 5.4.1 Local Optimization 5.4.2 Global Optimization 5.4.3 Single and Multi-objective Optimization 5.5 Optimization Cast in a Bayesian Framework 5.5.1 Fundamental Concept 5.5.2 Bayes’ Theorem 5.5.3 Uncertainty Quantification 5.6 Exercises 5.7 References CHAPTER 6 OPTIMIZATION OF FUNCTIONS IN GEOPHYSICAL REFERENCE 6.1 Introduction 6.2 Optimization in One Dimension 6.2.1 Golden Section Search 6.2.2 Inverse Hyperbolic Interpolation 6.3.2 Brent’s Root Finding Method 6.4 Geophysical Examples of One-Dimensional Optimization 6.4.1 Raytracing in an Isotropic Medium 6.4.2 Phase and Group Velocities and Angles in An Anisotropic Medium 6.4.3 Raytracing in an Anisotropic Medium 6.4 Optimization in Multiple Dimensions 6.4.1 Fundamental Concepts 6.4.2 Concept of Conjugate Directions 6.4.3 Conjugate Gradient Methods 6.4.4 Variable-Metric or Quasi-Newton Methods 6.5 Numerical Implementations of Different Optimization Methods 6.6 Exercises 6.7 References CHAPTER 7 LOCAL OPTIMIZATION METHODS IN GEOPHYSICS 7.1 Introduction 7.2 Formulation of the Problem 7.3.1 Model (Decision), Data, and Objective Spaces 7.3.2 Model to Data Mapping 7.3.1 The Jacobian of Fréchet Derivative Matrix 7.4 Data and Model Covariance Matrices 7.5 Measurement of the Goodness of Fit 7.5.1 Data Resolution Matrix 7.5.2 Model Resolution Matrix 7.6 Regularization of the Objective 7.6.1 Fundamental Concepts 7.6.2 Formulation of the Regularized Objective 7.6.1 Derivation of the Jacobian for Regularized Objective 7.7 Computer Implementations of Different Methods 7.7.1 Conjugate Gradient Method 7.7.2 Steepest Descent Method 7.7.3 Quasi-Newton Method 7.8 Computational Issues 7.8.1 Computation of Jacobian 7.8.2 Numerical Computation and Computational Challenges 7.8.3 Efficient Computation of the Jacobian for Practical Optimization Problems 7.9 Applications of Local Optimization in Geophysics 7.10 Exercises 7.11 References CHAPTER 8 GLOBAL OPTIMIZATION METHODS IN GEOPHYSICS 8.1 Introduction 8.1.1 Non-uniqueness of Geophysical Optimization 8.1.2 Local Versus Global Optimization 8.1.3 Different Flavors of Global Optimization 8.2 Simulated Annealing 8.2.1 Theory 8.2.2 Computer Implementation 8.2.3 Geophysical Applications 8.3 Genetic Algorithm 8.3.1 Theory 8.3.2 Computer Implementation 8.3.3 Geophysical Applications 8.4 Overview of Other Global Methods 8.4.1 Markov-Chain Monte-Carlo (MCMC) Method 8.4.2 Particle Swarm Optimization (PSO) 8.4.3 Differential Evolution 8.4.4 Hybrid Methods 8.5 Case Studies 8.6 Exercises 8.7 References CHAPTER 9 MULTI-OBJECTIVE METHODS IN GEOPHYSICS 9.1 Introduction 9.1.1 Why Multi-objective? 9.1.2 Traditional Way of Handling Multi-objective Problems 9.1.3 Need for Multi-objective Optimization 9.2 Pareto-Optimality 9.3 Dominance 9.4 Diversity Preservation 9.5 Definition of Model (Decision) and Objective Spaces for Multi-objective Optimization 9.6 Different Implementations of Multi-objective Optimization 9.6.1 Rank-based Implementation 9.6.2 Strength-based Implementation 9.7 Geophysical Applications 9.7.1 Estimation of the Transversely Isotropic Earth Properties 9.7.2 Estimation of the Orthorhombic Earth Properties 9.8 Computational Issues and Implementations in High-Performance Parallel Computing Environments 9.9 Exercises 9.10 References CHAPTER 10 MACHINE LEARNING IN GEOPHYSICS 10.1 Introduction 10.1.1 What is Machine Learning? 10.1.2 Historical Perspective from Geophysical Point of View 10.1.2.1 Automatic First-break Picking 10.1.2.2 Estimation of Acoustic Impedance 10.2 A Simple Example of Machine-Learning 10.2.1 Writing "Hello World" 10.3 Artificial Neural Networks 10.3.1 Fundamental Architecture 10.3.2 Classifier Systems 10.3.3 The Delta-rule as the Fundamental Basis of Learning Algorithms 10.3.4 The Momentum and Learning Rate 10.3.5 Statistical Indices for Learning Rate Quantification 10.4 The Universal Approximation Theorem 10.5 Feed-forward Back-propagation (FFBP) Neural Networks 10.6 Optimization Procedure 10.6.1 Number of Hidden Layers 10.6.2 Number of Neurons Per Layer 10.7 Optimal Network Training 10.7.1 Training and Validation Errors 10.7.2 Number of Training Data 10.7.3 Over and Under Fitting Issues 10.8 Different Flavors of Artificial Neural Network 10.8.1 Competitive Network 10.8.2 Hopfield Neural Network 10.8.3 Generalized Regression Neural Network 10.8.4 Radial Basis Function Neural Network 10.8.5 Modular Neural Network 10.8.6 Convolutional Neural Network 10.9 Other Machine Learning Method 10.9.1 Extreme Gradient Boosting 10.9.2 Support Vector Machines 10.10 Geophysical Applications of Machine Learning 10.11 Future of Machine Learning in Geophysics 10.11.1 Full Waveform Inversion 10.11.2 Multi-objective, Multi-Physics, and Multi-objective Optimization Problems 10.12 Exercises 10.13 References