Regularization of the AVO inverse problem by means of a ...

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2.3.6 Fatti’s approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Comparison of approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Effect of γ = β α on forward modeling . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 Ray Tracing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6.1 Ray tracing method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6.2 Ray tracing method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Bayesian Inversion Approach and Algorithms 26 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Prior distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 Inversion Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4.1 Parameter selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.2 The least squares method . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4.3 Weighted Least squares method . . . . . . . . . . . . . . . . . . . . . . 35 3.4.4 Iterative re-weighed least squares method . . . . . . . . . . . . . . . . 35 3.5 Target oriented AVO inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Two-term AVO Inversion 39 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Two-term AVO inversion formulation . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.1 Likelihood function of the data . . . . . . . . . . . . . . . . . . . . . . 41 4.2.2 Multivariate Gaussian prior . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.3 Univariate Cauchy prior . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.4 Bivariate Cauchy prior . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 Synthetic data examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Three-term AVO Inversion 55 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Three-term inversion formulation . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.1 Likelihood function of the data . . . . . . . . . . . . . . . . . . . . . . 57 5.2.2 Multivariate Gaussian prior . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2.3 Univariate Cauchy prior . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.4 Trivariate Cauchy prior . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3 Synthetic data examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4 Real data examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6 Conclusions 76 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Bibliography 79 Appendices A Multivariate t distribution and Regularizations 83 A.1 Multivariate t distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 A.2 Differentiation of the Regularization R(m) . . . . . . . . . . . . . . . . . . . . 84 B EM Algorithm 87 B.1 EM-Algorithm to estimate the scale matrix Ψ . . . . . . . . . . . . . . . . . . 87 C Calculation of root mean square error 89
Page 1 and 2: University of Alberta Regularizatio
Page 3 and 4: Abstract Amplitude Variation with O
Page 5: Contents 1 Introduction 1 1.1 Forwa
Page 9 and 10: List of Figures 1.1 A simple diagra
Page 11 and 12: 5.3 (a) P-wave reflectivity (RMSEA
Page 13 and 14: CHAPTER 1 Introduction The reflecti
Page 15 and 16: CHAPTER 1. INTRODUCTION 3 ! !!!!!!!
Page 17 and 18: CHAPTER 1. INTRODUCTION 5 1.4.1 Rel
Page 19 and 20: CHAPTER 1. INTRODUCTION 7 three-ter
Page 21 and 22: 2.1 Introduction CHAPTER 2 AVO Mode
Page 23 and 24: CHAPTER 2. AVO MODELING 11 (S-wave
Page 25 and 26: CHAPTER 2. AVO MODELING 13 At this
Page 27 and 28: CHAPTER 2. AVO MODELING 15 The resu
Page 29 and 30: CHAPTER 2. AVO MODELING 17 2.3.3 Sh
Page 31 and 32: CHAPTER 2. AVO MODELING 19 Assuming
Page 33 and 34: CHAPTER 2. AVO MODELING 21 Reflecti
Page 35 and 36: CHAPTER 2. AVO MODELING 23 ! !"# %$
Page 37 and 38: CHAPTER 2. AVO MODELING 25 can be r
Page 39 and 40: CHAPTER 3. BAYESIAN INVERSION APPRO
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CHAPTER 4. TWO-TERM AVO INVERSION 4
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5.1 Introduction CHAPTER 5 Three-te
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CHAPTER 5. THREE-TERM AVO INVERSION
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CHAPTER 6. CONCLUSIONS 77 two terms
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Bibliography Aki, K. and P. G. Rich
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BIBLIOGRAPHY 81 Sacchi, M.D. and T.
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APPENDIX A Multivariate t distribut
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APPENDIX A. MULTIVARIATE T DISTRIBU
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APPENDIX B EM Algorithm B.1 EM-Algo
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APPENDIX C Calculation of root mean
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