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

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CHAPTER 2. AVO MODELING 22 ! $%! $! !#! !&! !"# $&! $#! '#! '&! '%! (#! (&! Figure 2.4: The source-receiver geometry for CDP gathers assuming straight line ray path from a source to a reflection point and from reflection point to the receiver. The symbols S1, S2, S3 are sources and R1, R2, and R3 are receivers. The depths of each layer represented by Z1 and Z2 with corresponding interval velocity α1 and α2. This geometry is used for ray tracing method 1. where θn the angle of incidence on that layer and αn is the interval velocity of the layer. For zero offset two-way travel time, τn = n∆t , and offset, h. According to the geometry shown in Figure 2.4, the approximate two-way travel time, tn, has the form t 2 n = τ 2 h n + tn = τ 2 n + V n rms h V n rms 2 , (2.38) 2 , (2.39) where V n rms is the root mean square velocity down to the n th layer. The ray parameter can also be derived from the two-way travel as p = ∂tn ∂h = h tn(V n . (2.40) 2 vrms) Combing equations (2.37) and (2.40), the angle of incidence at the n th layer takes the form sin θ(tn, h) = 2.6.2 Ray tracing method 2 hαn tn(V n . (2.41) 2 vrms) Unlike the first method, this method tracks the ray path from the source to the receiver according to the layer velocity to determine the angle of incidence and calculate the total
CHAPTER 2. AVO MODELING 23 ! !"# %$! %&! %#! $! '#! '&! '$! !!# $! %! !& ! %! Figure 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. travel time for each path (Dahl and Ursin, 1991). I prefer to work by choosing an emerging angle instead of a ray parameter. A simplified geometry of the ray paths for 3 source-receiver and two-layered model is shown in Figure 2.5. The first step in this algorithm is to determine the angle by which the ray emerges from the source or the ray parameter for a given ray path that matches the given offset. For a given ray path, let the angle of incidences at each layer, down to a given layer (n th layer ) be θ1, θ2, ..., θi, ...θN. Using snell’s law, the angle of incidence and the velocity model are related as sin θ1 p = = sin θ2 α1 α2 sin θi sin θn = ... = = ... = . (2.42) Choosing an emerging angle, θe, and using the relationship given by equation (2.42), we have sin θi = αi α1 αi αn !(# $! %! !(& $! sin θe. (2.43) Using the chosen emerging angle, the sum of the horizontal distance traveled by the ray in each layer should give the offset for that ray path, X(τn, θe) = n α 2 sin θe, i ∆t α2 1 − α2 i sin θ2 , (2.44) e i=1 where θe is the emerging angle, αi velocity of the i th layer and X is the offset for the chosen emerging angle. To find the offset that approximate the actual recorded offset, h, we can %!
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 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: CHAPTER 2. AVO MODELING 21 Reflecti
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
Page 85 and 86:
CHAPTER 5. THREE-TERM AVO INVERSION
Page 87 and 88:
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 and 100:
APPENDIX B EM Algorithm B.1 EM-Algo
Page 101:
APPENDIX C Calculation of root mean
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