Computation, Optimization, and Machine Learning in Seismology
Description
A textbook applying fundamental seismology theories to the latest computational tools
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
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Other editions
Additional editions
Person
Subhashis Mallick, University of Wyoming, USA
Content
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
1
Introduction to Key Concepts in Seismic Inversion and Elastic Wave Theory
1.1 Background
Geophysical inversion is a well-established technique that yields quantitative descriptions of the earth's subsurface. Inversion of measurements of earthquake waves, for example magnetic, electrical, and gravity fields, has long been used to understand the macroscopic structure of the earth. Seismic inversion is routinely used in the oil and gas industry for exploration risk assessment and for reservoir characterization. The purpose of this book is to familiarize readers with the theory behind various approaches to inversion and their common application. Our focus is on seismic applications of inversion, including methods that utilize machine learning (ML) and artificial intelligence (AI) that are now being increasingly applied in all fields of geophysics.
1.2 Seismology-A Historical Perspective
Broadly speaking, seismology is the study of the mechanical vibrations of the earth. These vibrations are generated either from natural sources such as earthquakes or from the artificial (human-generated) sources such as from the use of explosives or vibrators. Elastic waves associated with these vibrations propagate within the earth and are recorded as seismograms on various measuring devices, the most common of which are "seismometers" or "geophones." Analyzing these seismograms to estimate the elastic properties of the earth, and relating these to subsurface lithology, fluids, in situ stress fields, etc., is the primary objective of seismology.
Based on the source generating the mechanical vibrations, seismology is broadly classified into (1) earthquake or passive-source seismology and (2) exploration or active-source seismology. Irrespective of a passive or an active source, the fundamental mathematical theory for both is the propagation of the elastic waves. This book focuses on elastic wave theory and how it is used to decipher the earth's internal structure and physical properties from recorded seismograms. From a historical perspective, however, it is important to briefly introduce the above two broad categories of seismology and their respective applications.
1.2.1 Earthquake Seismology
The study of earthquakes dates back thousands of years, a comprehensive review of which can be found in Ben-Menahem (1995). Earthquakes are caused by a sudden release of energy within the subsurface, primarily due to the release of elastic strain. Analysis of the seismograms generated from earthquakes that are recorded by permanent seismometers deployed around the globe enables the determination of their location, magnitude, focal mechanism, and the subsurface structure of the earth. Additionally, earthquake seismology plays a major role in developing our understanding of plate tectonics. In earthquake seismology, the relative distances between the permanent seismic sensors are large (on the order of hundreds of kilometers) and the temporal frequencies of the elastic waves generated from the earthquakes are low (typically less than 1 Hz). Consequently, the vertical and lateral resolution of the earth structure obtained from the analysis of these seismograms is low (tens to hundreds of kilometers). The depth of penetration of these waves is, however, high, and therefore earthquake seismology is useful for studying the structure of earth's deep interior. A very good account of the science of earthquake seismology can be found in Bath (1979), Stein and Wysession (2002), Shearer (2009), among others.
Finding the structure of the earth's deep interior, developing the concepts of plate tectonics, etc., are the fundamental focus of earthquake seismology. In the past few decades, the oil and gas industry has made an interesting adaptation of the concepts of earthquake seismology in a technique known as "microseismic monitoring." Where hydrocarbon-bearing reservoirs are impermeable, the reservoir fluids are squeezed out by fracturing under great pressure. In microseismic monitoring, the microearthquakes (microseisms) resulting from such fracturing are recorded by borehole seismic sensors and analyzed to characterize the distribution, extent, size, and nature of the induced fractures. This helps to optimize well placement and design for efficient drainage of the reservoir hydrocarbons (Grechka and Heigl, 2017). Microseismic monitoring is also used in carbon dioxide sequestration and steam injection related applications.
1.2.2 Exploration Seismology
Exploration seismology evolved from earthquake seismology. Here, the source generating elastic waves is artificially excited. Compared with the earthquake seismology, in exploration seismology the seismic sensors (usually geophones on land and hydrophones in marine environments) are placed at relatively short distances apart (50 m or even less), and the temporal frequencies of the elastic waves are high with a dominant frequency between 20 and 50 Hz. The lateral and vertical resolution of the subsurface structures typically resolved from exploration seismology is therefore higher than earthquake seismology. However, because high frequencies rapidly attenuate with depth, the depth of penetration in exploration seismology is much lower than for earthquake seismology, and its use is limited to finding detailed structures at shallow depths, usually less than 10 km.
In exploration seismology, artificial sources are placed at or the near surface of the earth or within a borehole. On land, the seismic sensors (usually geophones) are placed either at the surface or within a borehole. In marine environments on the other hand, these sensors (usually hydrophones) are placed either near the water surface or at the water bottom, or within a borehole below the ocean floor. Irrespective of the land or marine environment, the elastic or acoustic wavefields, excited at the source locations propagate within the earth layers. The propagating wavefield, characterized by a temporal seismic wavelet, is reflected at boundaries between subsurface layers of contrasting elastic or acoustic impedance. The sequence of reflections is recorded as seismograms at the receiver (sensor) locations. Analyzing these recorded seismograms to interpret subsurface structure and rock properties is the focus of exploration seismology (Dobrin and Savit, 1988; Telford et al., 1990). Of all the different types of geophysical exploration methods, exploration seismology is by far the most commonly used and the most successful in terms of accuracy, depth of penetration, and subsurface resolution (Selley and Sonnenberg, 2014).
1.3 Mathematical Foundations of Seismology
Following the discovery of Hooke's law of linearized elasticity in 1660, there have been many advances in the mathematics of elastic wave theory, a comprehensive account of which can be found in the classic textbook written by Love (1892, reprinted 1944), and in more recently published textbooks by Aki and Richards (2002) and Chapman (2004).
Before the digital era, early applications of seismology were limited to the analysis of wave arrival travel times in analog records of seismic data. With the advent of computers these travel-time-based analyses gradually evolved into more quantitative analyses of the recorded seismic waveforms (Chapman, 2004).
With current advances in high-performance computing, seismology has developed into a more mature science. Complex partial differential equations that describe the propagation of seismic wave fields can now be numerically solved in reasonable timeframes. This, in turn, allowed going beyond the travel-time-based qualitative analyses into quantitative analyses using sophisticated optimization techniques. More recently, the use of ML methods, a subset of AI, has been introduced in all aspects of seismology for these analyses. The primary focus of this book is to provide an overview of these quantitative methods.
1.4 Seismic Inversion
Seismic optimization is a subset of a broad category, called seismic inversion. Therefore, before understanding the meaning of optimization, it is first necessary to understand the meaning of inversion by explaining some fundamental concepts.
1.4.1 The Meaning of Inversion
Many applications in science, engineering, economics, etc., require finding an optimal model that satisfies a given set or sets of observations. Finding such an optimal model from a set or sets of observations is called inversion. To understand what is meant by inversion, consider the scenario shown in Figure 1.1.
Figure 1.1 Example of a person kicking a soccer ball.
As shown in Figure 1.1, if a person kicks a soccer ball so that it travels with a speed v, the distance d the ball will travel in a time t is given by
(1.1)To obtain the speed v with which the ball travels, we can measure the distance d that it travels. Additionally, we can also use a stopwatch and measure the time t that the ball takes to travel the distance d. Thus, given the set of observation {d, t}, it is straightforward to find the speed v...
