Random Field Models in Earth Sciences (eBook)

Front Cover 1
Random Field Models in Earth Sciences 4
Copyright Page 5
Table of Contents 8
Dedication 6
Foreword 18
Preface 22
Chapter 1. Prolegomena 30
1. The Science of the Probable and the Random Field Model 30
2. The Physical Significance of the Random Field Model 35
3. The Mathematics of Random Fields 40
4. The Philosophical Theses of the Stochastic Research Program 42
5. The Practice of the Stochastic Research Program and the Spectrum of Its Applications 45
Chapter 2. The Spatial Random Field Model 50
1. Introduction 50
2. Basic Notions 51
3. Characterization of Spatial Random Fields by Means of Their Second-Order Statistical Moments—Correlation Theory 60
4. Certain Geometrical Properties of Spatial Random Fields 69
5. Spectral Characteristics of Spatial Random Fields 78
6. Auxiliary Hypotheses 84
7. Homogeneous Spatial Random Fields 86
8. Isotropic Spatial Random Fields 98
9. Scales of Spatial Correlation 105
10. Relationships between the Spatial and the Frequency Domains—The Uncertainty Principle 105
11. Spatial Random Fields with Homogeneous Increments 107
12. On the Ergodicity Hypotheses of Spatial Random Fields 127
13. Information and Entropy of Spatial Random Fields 132
Chapter 3. The Intrinsic Spatial Random Field Model 136
1. Introduction 136
2. Generalized Spatial Random Fields 137
3. Spatial Random Fields with Space Homogeneous Increments or Intrinsic Spatial Random Fields 144
4. Discrete Linear Representations of Spatial Random Fields 164
5. Stochastic Differential and Difference Equations 175
Chapter 4. The Factorable Random Field Model 182
1. Introduction 182
2. The Theory of Factorable Random Fields 183
3. Nonlinear Transformations of Factorable Random Fields 189
4. Construction of Factorable Random Fields 190
5. The Nonlinear State-Nonlinear Observation System 194
Chapter 5. The Spatiotemporal Random Field Model 197
1. Introduction 197
2. Spatiotemporal Natural Processes—A Review 198
3. Ordinary Spatiotemporal Random Fields 204
4. Generalized Spatiotemporal Random Fields 216
5. Spatiotemporal Random Fields of Order v/µ (Ordinary and Generalized) 225
6. Stochastic Partial Differential Equations 235
7. Discrete Linear Representations of Spatiotemporal Random Fields 238
Chapter 6. Space Transformations of Random Fields 244
1. Introduction 244
2. Space Transformations 245
3. Space Transformation Representations of Spatial Random Fields 252
4. Stochastic Differential Equation Models 255
5. Criteria of Permissibility 265
Chapter 7. Random Field Modeling of Natural Processes 267
1. Introduction 267
2. Descriptive Features of Natural Processes and the Basic Working Hypotheses 269
3. Duality Relations between the Natural Process and the Spatial Random Field Model—Examples from the Geosciences 274
4. Certain Practical Aspects of Spatial and Temporal Variability Characterization 284
5. Qualitative (Soft) Information 304
6. Some Final Comments about the Stochastic Research Program 321
Chapter 8. Simulation of Natural Processes 323
1. Introduction 323
2. The Physical Significance of Simulation 324
3. Simulation of Random Fields 328
4. Simulation of Spatial Random Field by Space Transformations—Examples 331
5. Techniques of One-Dimensional Simulation 345
6. Simulation of Integrated Natural Processes 350
7. Simulation of Dynamic Stochastic Systems 351
8. The Effect of Measurement Error 353
9. Simulation of Spatial Random Fields by Means of Frequency Domain Techniques 354
10. The Lower-Upper Triangular Matrix Technique 355
11. The Karhunen-Loeve Expansion Technique 357
12. Simulation of Vector Spatial Random Fields 358
13. Simulation of Non-Gaussian Spatial Random Fields 361
14. Simulation in Space-Time 364
Chapter 9. Estimation in Space and Time 366
1. Introduction 366
2. A Brief Review of Nonstochastic Estimators and the Emergence of Stochastic Estimation 368
3. Optimum Stochastic Spatial Estimation 370
4. Certain Classes of Linear Spatial Estimators 375
5. Properties and Physical Interpretations of Linear Spatial Estimators 386
6. Nonlinear Estimation 399
7. Optimal Estimation of Spatiotemporal Random Fields 408
8. A Bayesian/Maximum-Entropy View of the Estimation Problem 415
Chapter 10. Sampling Design 430
1. Introduction 430
2. About Sampling 432
3. Simple Global Approaches to Sampling Design 444
4. Optimal Linear Estimation Approaches to Global Sampling Design 448
5. Local Sampling Design Approaches 462
6. Statistical Inference Problems in Sampling Design 466
7. The Design of Spatiotemporal Sampling Networks 467
8. A Taxonomy of Site Exploration Tasks 468
9. Terminal Decision Analysis and Sampling Design 469
References 476
Index 488