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

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APPENDIX B. EM ALGORITHM 88 From equation (B.3), the maximum likelihood estimates of Ψ satisfies and where N ωi∆i = 0 (B.4) i=1 Ψ = 1 N ωi = N i=1 ωi∆i∆ T i , (B.5) 4 1 + ∆ T i Ψ −1 ∆i . (B.6) Table B.1: EM Algorithm 1. Calculate the model parameters, m, from bore-hole data. This is an input for the algorithm. 2. Intialize Ψ (3 × 3). An identity or sample covariance matrix. 3. Set maximum number of iteration (Max) or a convergence criteria E-step, 4. For j
APPENDIX C Calculation of root mean square error In this section, the calculation of root mean square error (RMSE) which is used for the Monte Carlo Simulation is given. Let the true (t) and the inverted (i) parameters are represented by vectors x t j and xi j respectively at the jth realization. For N number of time samples, each vector has N number of elements. Then the root mean square error at the j th realization, R j x, is defined as R j x = 1 N N k=1 (x t jk − xi jk )2 . (C.1) Therefore, for n realizations, the root mean square error can be calculated using ¯Rx = 1 n n R j x. (C.2) j=1 The corresponding standard deviation for ¯ Rx is given by σx = 1 n (R n j x − ¯ R j x) 2 . (C.3) j=1 From equations (C.2) and (C.3), we can easily see that ¯Rx − σx < RMSEx < ¯ Rx + σx, (C.4) Note that this calculation is for n realizations at a specific S/N. The x represents an AVO parameter; P-wave reflectivity, S-wave reflectivity or density reflectivity. 89
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University of Alberta Regularizatio
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Abstract Amplitude Variation with O
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Contents 1 Introduction 1 1.1 Forwa
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5 Three-term AVO Inversion 55 5.1 I
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List of Figures 1.1 A simple diagra
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5.3 (a) P-wave reflectivity (RMSEA
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CHAPTER 1 Introduction The reflecti
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CHAPTER 1. INTRODUCTION 3 ! !!!!!!!
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CHAPTER 1. INTRODUCTION 5 1.4.1 Rel
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CHAPTER 1. INTRODUCTION 7 three-ter
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2.1 Introduction CHAPTER 2 AVO Mode
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CHAPTER 2. AVO MODELING 11 (S-wave
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CHAPTER 2. AVO MODELING 13 At this
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CHAPTER 2. AVO MODELING 15 The resu
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CHAPTER 2. AVO MODELING 17 2.3.3 Sh
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CHAPTER 2. AVO MODELING 19 Assuming
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CHAPTER 2. AVO MODELING 21 Reflecti
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CHAPTER 2. AVO MODELING 23 ! !"# %$
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CHAPTER 2. AVO MODELING 25 can be r
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CHAPTER 3. BAYESIAN INVERSION APPRO
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Page 49 and 50: CHAPTER 3. BAYESIAN INVERSION APPRO
Page 51 and 52: 4.1 Introduction CHAPTER 4 Two-term
Page 53 and 54: CHAPTER 4. TWO-TERM AVO INVERSION 4
Page 67 and 68: 5.1 Introduction CHAPTER 5 Three-te
Page 69 and 70: CHAPTER 5. THREE-TERM AVO INVERSION
Page 89 and 90: CHAPTER 6. CONCLUSIONS 77 two terms
Page 91 and 92: Bibliography Aki, K. and P. G. Rich
Page 93 and 94: BIBLIOGRAPHY 81 Sacchi, M.D. and T.
Page 95 and 96: APPENDIX A Multivariate t distribut
Page 97 and 98: APPENDIX A. MULTIVARIATE T DISTRIBU
Page 99: APPENDIX B EM Algorithm B.1 EM-Algo
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