# Application of Monte Carlo Techniques for Solving Selected Seismological Inverse Problems

**Author(s):**Dębski W.

**Volume:**367

**Series:**B-34

This book is not intended to be an essay on the general inverse theory but rather a discussion of a particular technique used in the framework of the probabilistic inverse theory, including some new axiomatic elements of the theory, with advanced applications based on the Monte Carlo technique. For this reason, to present inverse problems and the techniques used to solve them, I chose the pragmatic, phenomenological approach which underlies the meaning of the various elements of inverse problems at the cost of mathematical rigors.

MONOGRAPHIC VOLUME

CONTENTS

Preface and acknowledgments, ...3

1. Introduction, ...5

2.Inverse problems, ...9

2.1 Forward and inverse problems, ...9

2.2 Inverse algorithms, ...10

2.3 Why the probabilistic approach?, ...14

3. Probabilistic inverse theory, ...17

3.1 Mathematical framework, ...17

3.2 A posteriori PDF, ...22

3.3 Solving inverse problems - probabilistic point of view, ...26

4. Monte Carlo technique, ...33

4.1 Introduction, ...33

4.2 Principles of the Monte Carlo technique, ...34

4.3 Markov chain Monte Carlo sampling, ...36

4.4 Global optimization, ...46

4.5 Sampling a posteriori PDF by MCMC, ...56

5. Location of seismic/acoustic events with imprecise receiver positioning, ...61

5.1 Monitoring ocean crust deformation, ...62

5.2 Location algorithm, ...63

5.3 Numerical examples, ...65

5.4 Discussion, ...67

6. Velocity tomography, ...69

6.1 Introduction, ...69

6.2 Western Honshu, ...72

6.3 Rudna copper mine case, ...105

7. Inversion of the source time function, ...117

7.1 Introduction, ...117

7.2 The elements of the source imaging theory, ...119

7.3 Case study: Lubin event of July 1998, ...125

8. Final remarks and conclusions, ...135

Appendix A. Comments on the probabilistic inverse theory, ...139

A.1 The inverse problem as an inference task, ...139

A.2 Models, model parameters and observable parameters, ...140

A.3 Description of experimental, a priori and theoretical information, ...142

A.4 Probability calculus and solution of real problems, ...147

A.5 Homogeneous (non-informative) PDF´s, ...148

A.6 Statistical interpretation of the likelihood function, ...151

A.7 The linear inverse problem and quadratic misfit function, ...152

A.8 A posteriori PDF - special cases, ...154

A.9 Optimization vs. probabilistic techniques, ...158

Appendix B. Selected numerical aspects of inverse problems, ...161

B.1 Robustness property of norms, ...161

B.2 Sampling in high-dimensional spaces, ...163

B.3 Stability of numerical integration, ...164

B.4 Random number generators, ...166

B.5 Inhomogeneous Markov chain - SA algorithm, ...169

Appendix C. Illustrations of probabilistic inference, ...171

C.1 The linear inverse problem, ...172

C.2 The nonlinear forward problem, ...173

C.3 Inversion with a priori information, ...174

C.4 Approximate forward modeling, ...175

C.5 Null space, ...176

C.6 Non-resolved parameters, ...177

C.7 Complex theory, ...178

C.8 No observational information, ...178

Bibliography, ...181

Summary, ...199

Streszczenie, ...201