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

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List of Tables 2.1 Two layered models which are used for comparison of linear approximations to Zoeppritz equation (Rpp ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 LS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 WLS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 IRLS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Conjugate Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Target Oriented AVO inversion Via WLS/IRLS . . . . . . . . . . . . . . . . . 38 4.1 The calculated root mean square errors; RMSErα and RMSErβ for P and S-wave impedances respectively. See Appendix C for RMSE calculation. . . . 47 5.1 The root mean square errors (RMSEA) and their corresponding standard deviations (σA) for P-wave reflectivity using the three prior distributions. . . 71 5.2 The root mean square errors (RMSEB) and their corresponding standard deviations (σB) for S-wave reflectivity using the three prior distributions. . . 71 5.3 The root mean square errors (RMSEC) and their corresponding standard deviations (σC) for density reflectivity using the three prior distributions. . . 71 B.1 EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
List of Figures 1.1 A simple diagram which depicts the relationtionship between the model and data both for forward and inverse problem. . . . . . . . . . . . . . . . . . . . 3 2.1 A two layer model in welded contact shows the transformation of an incident P-wave with initial amplitude A0 into P and S-waves with their corresponding amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Comparison of approximations using the two models given in Table 2.1. (a) using model 1, and (b) using model 2. . . . . . . . . . . . . . . . . . . . . . . 20 2.3 The effect of γ on forward modeling using the two models given in Table 2.1. (a) using model 1 for (γ = 0.6129, γ1 = 0.4129, γ2 = 0.5129, and γ3 = 0.7129), and (b) using model 2 for γ = 0.6143, γ1 = 0.4143, γ2 = 0.5143, and γ3 = 0.7143. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 The source-receiver geometry for CDP gathers assuming straight line ray path from a source to a reflection point and from reflection point to the receiver. The symbols S1, S2, S3 are sources and R1, R2, and R3 are receivers. The depths of each layer represented by Z1 and Z2 with corresponding interval velocity α1 and α2. This geometry is used for ray tracing method 1. . . . . . 22 2.5 The exact ray tracing for CDP gathers. It has similar source-reciever geometry as Figure 2.4 except the ray paths which depend on change in the interval velocity of the layers. This geometry is used for the ray tracing method 2. . . 23 3.1 Probability density of the Univariate Gaussian (red), Univariate Cauchy (blue) for ν = 1, and Univariate t for ν = 2 and ν = 5 . . . . . . . . . . . . . . . . . 29 3.2 The peak of the (a) the Bivariate Cauchy Probability distribution (ν = 1), and the Bivariate t distributions for (b) ν = 2, c) ν = 5, and d) ν = 10. . . . 30 3.3 The tail of a) the Bivariate Cauchy distribution (ν = 1), and the Bivariate t distributions for b) ν = 2, c) ν = 5, and d) ν = 10. . . . . . . . . . . . . . . . 31 3.4 (a) The Chi-square test and (b) the trade-off curve for N observation data points and M number of µh trials . . . . . . . . . . . . . . . . . . . . . . . . . 34
Page 1 and 2: University of Alberta Regularizatio
Page 3 and 4: Abstract Amplitude Variation with O
Page 5 and 6: Contents 1 Introduction 1 1.1 Forwa
Page 7: 5 Three-term AVO Inversion 55 5.1 I
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
Page 51 and 52: 4.1 Introduction CHAPTER 4 Two-term
Page 53 and 54: CHAPTER 4. TWO-TERM AVO INVERSION 4
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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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