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

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CHAPTER 2. AVO MODELING 14 Employing equations (2.5) in equation (2.20) and applying differentiation, we have 2ρ1β 2 ω 1p cos θi A0 − 2ρ1β α1 2 1p cos θi α1 ω A1 − ρ1β1(1 − 2β 2 1p 2 ) ω B1 β1 = ρ2β 2 ω 2p cos ϕt A2 + ρ2β2(1 − 2β 2 2p 2 ) ω B2. (2.21) β2 The four equations (2.8) ,(2.10), (2.17), and (2.21) are the final result after imposing the boundary conditions. But they are expressed in terms of the potential amplitudes. As the reflection and transmission coefficients are defined as ratios of the displacement amplitude, we can rewrite the four equations as where −αpRpp − cos ϕrRps + α2pTpp + cos ϕtTps = αp, cos θiRpp − β1pRps + cos θtTpp − β2pTps = cos θi, −ρ1α1(1 − 2β 2 1p 2 )Rpp − 2ρ1β 2 1p cos ϕrRps + ρ2α2(1 − 2β 2 2p 2 )Tpp −2ρ2β 2 2p cos ϕtTpp = ρ1α1(1 − 2β 2 1p 2 ), 2ρ1β 2 1p cos θiRpp + ρ1β1(1 − 2β 2 1p 2 )Rps + ρ2β 2 2p cos ϕtTpp Rpp = A1 A0 +ρ2β2(1 − 2β 2 2p 2 )Tps = 2ρ1β 2 1p cos θi, (2.22) , Rps = α1 B1 β1 A0 , Tpp = α1 A2 α2 A0 , Tps = α1 B2 β2 A0 which are the reflection and transmission coefficients. The elegant way of expressing the Zoeppritz equation is in matrix form. In matrix form, equation (2.22) takes the form MX = Y, (2.23) where ⎛ ⎜ M = ⎜ ⎝ −αp cos θi −ρ1α1(1 − 2β − cos ϕr −β1p α2p cos θt cos ϕt −β2p 2 1p2 ) −2ρ1β2 1p cos ϕr ρ2α2(1 − 2β2 2p2 ) −2ρ2β2 2p cos ϕt 2ρ1β2 1p cos θi ρ1β1(1 − 2β2 1p2 ) ρ2β2 2p cos ϕt 2ρ2β2 ⎞ ⎟ ⎠ 2p cos ϕt , ⎛ ⎜ X = ⎜ ⎝ Rpp Rps Tpp Tps ⎞ ⎟ ⎠ ⎛ ⎜ , Y = ⎜ ⎝ αp cos θi ρ1α1(1 − 2β 2 1p 2 ) 2ρ1β 2 1p cos θi ⎞ , α2 ⎟ ⎠ .
CHAPTER 2. AVO MODELING 15 The resulting equation is a specific case in which we have a down-going P-wave (incident), reflected P-wave, reflected S-wave, transmitted P-wave and transmitted S-wave. Following a similar procedure given above, it is also possible to generalize for all possible incidence, reflected and transmitted waves. For AVO analysis, the Rpp and Rps are the most exploited reflection coefficients i.e in typical seismic experiment using compressional wave sources , P-waves, and receiving the P-wave component and/ or S-wave component by receivers. In this thesis, only the Rpp of equation (2.23) part is used. 2.3 Approximation of the Zoeppritz Equations Solving for Rpp from Zoeppritz equation, the exact P P wave reflection coefficients in terms of the ray parameter p, velocity and density of the media can be expressed as where Rpp = (b cos θi α1 cos θt cos θi cos θt − c )F − (a + d α2 α1 α2 )Hp2 D a = ρ2(1 − 2β2p 2 ) − ρ1(1 − 2β1p 2 ), b = ρ2(1 − 2β2p 2 ) + 2ρ1β1p 2 , c = ρ1(1 − 2β1p 2 ) + 2ρ2β2p 2 , d = 2(ρ2β2 − ρ1β 2 1), cos ϕr cos ϕt F = b + c , β1 β2 (2.24) H = cos θt cos ϕr a − d , α2 β1 D = det M . (2.25) α1α2β1β2 Although the above equation shows the exact transformation of the reflection coefficient, it is non-linear and complicated with respect to the the model parameters of the medium. For this reason, under the assumption of small property contrasts, expanding and retaining only linear terms, a number of papers has been published on linearized approximations which can be directly used for AVO inversion. The next section is devoted in revisiting the approximations
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 and 8: 5 Three-term AVO Inversion 55 5.1 I
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: CHAPTER 2. AVO MODELING 13 At this
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
Page 67 and 68: 5.1 Introduction CHAPTER 5 Three-te
Page 69 and 70: CHAPTER 5. THREE-TERM AVO INVERSION
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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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