Statistical Methods and Modeling of Seismogenesis

The study of earthquakes is a multidisciplinary field, an amalgam of geodynamics, mathematics, engineering and more. The overriding commonality between them all is the presence of natural randomness.

Stochastic studies (probability, stochastic processes and statistics) can be of different types, for example, the black box approach (one state), the white box approach (multi-state), the simulation of different aspects, and so on. This book has the advantage of bringing together a group of international authors, known for their earthquake-specific approaches, to cover a wide array of these myriad aspects. A variety of topics are presented, including statistical nonparametric and parametric methods, a multi-state system approach, earthquake simulators, post-seismic activity models, time series Markov models with regression, scaling properties and multifractal approaches, selfcorrecting models, the linked stress release model, Markovian arrival models, Poisson-based detection techniques, change point detection techniques on seismicity models, and, finally, semi-Markov models for earthquake forecasting.

Nikolaos Limnios is Full Professor of Applied Mathematics at Université de Technologie de Compiègne, Sorbonne University, France. His research interests include stochastic processes and statistics, Markov and semi-Markov processes and random evolutions with varied applications.

Eleftheria Papadimitriou is Professor of Seismology at the Aristotle University of Thessaloniki, Greece. Her research interests are related to Earthquake Seismology and she engages in scientific exchange and collaboration with several international institutions.

George Tsaklidis is Professor of Probability and Statistics at the Aristotle University of Thessaloniki, Greece. His research interests include stochastic processes and computational statistics with applications in seismology, finance and continuum mechanics, and state-space modeling.

Preface xi
Nikolaos LIMNIOS, Eleftheria PAPADIMITRIOU and George TSAKLIDIS

Chapter 1. Kernel Density Estimation in Seismology 1
Stanisław LASOCKI

1.1. Introduction 1

1.2. Complexity of magnitude distribution 7

1.3. Kernel estimation of magnitude distribution 13

1.4. Implications for hazard assessments 14

1.5. Interval estimation of magnitude CDF and related hazard parameters 16

1.6. Transformation to equivalent dimensions 19

1.7. References 23

Chapter 2. Earthquake Simulators Development and Application 27
Rodolfo CONSOLE, Roberto CARLUCCIO

2.1. Introduction 28

2.2. Development of earthquake simulators in the seismological literature 28

2.2.1. ALLCAL 28

2.2.2. Virtual quake 29

2.2.3. RSQSim 30

2.2.4. ViscoSim 30

2.2.5. Other simulation codes 30

2.2.6. Comparisons among simulators 31

2.3. Conceptual evolution of a physics-based earthquake simulator 32

2.3.1. A physics-based earthquake simulator (2015) 33

2.3.2. Frequency-magnitude distribution of the simulated catalog (2015) 36

2.3.3. Temporal features of the synthetic catalog (2015) 38

2.3.4. Improvements in the physics-based earthquake simulator (2017–2018) 41

2.3.5. Application to the seismicity of Central Italy 42

2.3.6. Further improvements of the simulator code (2019) 46

2.4. Application of the last version of the simulator to the Nankai mega-thrust fault system 49

2.5. Appendix 1: Relations among source parameters adopted in the simulation model 54

2.6. Appendix 2: Outline of the simulation program 56

2.7. References 58

Chapter 3. Statistical Laws of Post-seismic Activity 63
Peter SHEBALIN, Sergey BARANOV

3.1. Introduction 63

3.2. Earthquake productivity 64

3.2.1. The proposed method to study productivity 65

3.2.2. Earthquake productivity at the global level 69

3.2.3. Independence of the proximity function 72

3.2.4. Earthquake productivity at the regional level 76

3.2.5. Productivity in relation to the threshold of the proximity function 78

3.2.6. Discussion 79

3.3. Time-dependent distribution of the largest aftershock magnitude 81

3.3.1. The distribution of the magnitude of the largest aftershock in relation to time 82

3.3.2. The agreement between the dynamic Båth law and observations 85

3.3.3. Discussion 86

3.4. The distribution of the hazardous period 88

3.4.1. A model for the duration of the hazardous period 89

3.4.2. Determining the model parameters 91

3.4.3. Using the early aftershocks 96

3.5. Conclusion 98

3.6. References 100

Chapter 4. Explaining Foreshock and the Båth Law Using a Generic Earthquake Clustering Model 105
Jiancang ZHUANG

4.1. Introduction 105

4.1.1. Issues related to foreshocks 106

4.1.2. Issues related to the Båth law 108

4.1.3. Study objectives 108

4.2. Theories related to foreshock probability and the Båth law under the assumptions of the ETAS model 109

4.2.1. Space–time ETAS model, stochastic declustering and classification of earthquakes 109

4.2.2. Master equation 110

4.2.3. Asymptotic property of F(m’) 113

4.2.4. Foreshock probabilities and their magnitude distribution in the ETAS model 117

4.2.5. Explanation of the Båth law by the ETAS model 118

4.3. Foreshock simulations based on the ETAS model 120

4.3.1. Works by Helmstetter and others 120

4.3.2. Works by Zhuang and others 120

4.3.3. Evidence of statistics between mainshocks and foreshocks 121

4.3.4. Different simulation results 121

4.4. Simulation of the Båth law based on the ETAS model 123

4.4.1. On the simulation study by Helmstetter 123

4.4.2. Observation on Båth’s law for volcanic earthquake swarms 124

4.5. Conclusion 125

4.5.1. Back to the starting point 125

4.5.2. On the comparison between foreshock probability in the ETAS model and real catalogs 125

4.5.3. Impracticality of the foreshock concept 126

4.5.4. What should we do? 126

4.6. Acknowledgments 127

4.7. References 127

Chapter 5. The Genesis of Aftershocks in Spring Slider Models 131
Eugenio LIPPIELLO, Giuseppe PETRILLO, François LANDES and Alberto ROSSO

5.1. Introduction 131

5.2. The rate-and-state equation 133

5.3. The Dieterich model 134

5.3.1. Time to instability 135

5.3.2. Initial conditions during stationary seismicity 137

5.3.3. Effect of a constant stress increase Δτ 137

5.4. The mechanics of afterslip 138

5.5. The two-block model 140

5.5.1. Synthetic catalogs 142

5.6. Conclusion 146

5.7. References 148

Chapter 6. Markov Regression Models for Time Series of Earthquake Counts 153
Dimitris KARLIS, Katerina ORFANOGIANNAKI

6.1. Introduction 153

6.2. Markov regression HMMs: definition and notation 156

6.3. Application 157

6.3.1. Data 157

6.3.2. Results 160

6.4. Conclusion 163

6.5. Acknowledgments 166

6.6. References 166

Chapter 7. Scaling Properties, Multifractality and Range of Correlations in Earthquake Time Series: Are Earthquakes Random? 171
Georgios MICHAS, Filippos VALLIANATOS

7.1. Introduction 171

7.2. The range of correlations in earthquake time series 173

7.2.1. Short-range correlations 173

7.2.2. Long-range correlations 177

7.3. Scaling properties of earthquake time series 183

7.3.1. The probability distribution function 184

7.3.2. A stochastic dynamic mechanism with memory effects 192

7.3.3. The cumulative distribution function 195

7.4. Fractal and multifractal structures 197

7.5. Discussion and conclusion 201

7.6. References 204

Chapter 8. Self-correcting Models in Seismology: Possible Coupling Among Seismic Areas 211
Ourania MANGIRA, Eleftheria PAPADIMITRIOU, Georgios VASILIADIS and George TSAKLIDIS

8.1. Introduction 211

8.2. Review of applications 212

8.3. Formulation of the models 218

8.3.1. Simple Stress Release Model 218

8.3.2. Independent Stress Release Model 220

8.3.3. Linked Stress Release Model 220

8.4. Applications 222

8.4.1. Greece and the surrounding area 222

8.4.2. Gulf of Corinth 229

8.5. Conclusion 235

8.6. References 236

Chapter 9. Markovian Arrival Processes for Earthquake Clustering Analysis 241
Polyzois BOUNTZIS, Eleftheria PAPADIMITRIOU and George TSAKLIDIS

9.1. Introduction 241

9.2. State of the art 243

9.2.1. Earthquake clustering methods and applications 243

9.2.2. Hidden Markov models and applications in seismology 244

9.3. Markovian Arrival Process 247

9.3.1. Definition and basic results 248

9.3.2. Parameter fitting 250

9.3.3. Inference of the latent states 252

9.4. Methodology and results 254

9.4.1. Motivation 254

9.4.2. Clustering detection procedure 254

9.5. Conclusion 264

9.6. References 265

Chapter 10. Change Point Detection Techniques on Seismicity Models 271
Rodi LYKOU, George TSAKLIDIS

10.1. Introduction 271

10.2. The change point framework 272

10.3. Changes in a Poisson process 276

10.4. Changes in the Epidemic Type Aftershock Sequence model 279

10.5. Changes in the Gutenberg–Richter law 282

10.6. ZMAP 286

10.7. Other statistical tests 287

10.8. Detection of changes without hypothesis testing 289

10.9. Discussion and conclusion 290

10.10. References 291

Chapter 11. Semi-Markov Processes for Earthquake Forecast 299
Vlad Stefan BARBU, Alex KARAGRIGORIOU and Andreas MAKRIDES

11.1. Introduction 299

11.2. Semi-Markov processes – preliminaries 300

11.2.1. Special class of distributions 303

11.3. Transition probabilities and earthquake occurrence 304

11.3.1. Likelihood and estimation 304

11.4. Semi-Markov transition matrix 305

11.5. Illustrative example 307

11.6. References 308

List of Authors 309

Index 311

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