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B-19 (293) W. Dębski, The Probabilistic Formulation of the Inverse Theory with Application to Selected Seismological Problems

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Abstract, 0

Preface and acknowledgements, 0

Introduction, 1

1. Inverse theory, 4

1.1 Introduction, 4

1.2 States of information, 5

1.3 Inverse problems, 21

1.4 Solving inverse problems, 31

2. Location of seismic events, 36

2.1 Introduction, 36

2.2 The resolution of hypocentre coordinates, 38

2.3 Location uncertainties, 52

2.4 Location of natural seismic earthquakes, 62

3. Inversion of amplitude variations, 100

3.1 Introduction, 100

3.2 Parameters describing and elastic earth, 101

3.3 The synthetic experiment, 102

3.4 The inversion, 104

3.5 The a priori information, 105

3.6 The a posteriori information, 111

3.7 Discussion, 117

4. Amplitude tomography, 121

4.1 Introduction, 121

4.2 Tomography imaging in mines, 123

4.3 The numerical experiment, 124

4.4 Results, 132

4.5 Conclusions, 142

Appendix A. Norms and probability densities, 144

A.1 The least squares norm ℓ2, 145

A.2 The least-absolute-values norm ℓ1, 149

A.3 The generalized Gaussian norm ℓp, 151

A.4 The minimax norm ℓ∞, 151

A.5 The Cauchy norm ℓC, 153

A.6 The hyperbolic secant norm ℓS, 154

References, 158

Streszczenie, 164

List of symbols, 168

Contents, 171


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