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

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List of Symbols Symbols Description s Second ms Millisecond m Meter kg Kilogram α P-wave velocity β S-wave velocity ρ Density σ Standard deviation A P-wave reflectivity B S-wave reflectivity C Density reflectivity rα P-wave impedance rβ S-wave impedance λ First Lamé parameter µ Bulk modulus µh Hyper-parameter γ S-wave to P-wave velocity ratio p Ray parameter h or X Offset Vrms Root mean square velocity ν Degree of freedom J Objective function mg Index for Multivariate Gaussian uc Index for Univariate Cauchy bc Index for Bivariate Cauchy tc Index for Trivariate Cauchy Q Regularization term Ψ Scale matrix Cf Covariance matrix for Fatti’s parameters Ca Covariance matrix for Aki and Richard’s parameters S/N Signal to noise ratio CDP Common depth point RMSE Root mean square error xii
CHAPTER 1 Introduction The reflection seismic method is one of the geophysical tools that is used to explore subsurface physical structures and properties. It is extensively used in exploration for hydrocarbons (i.e.., petroleum, natural gas) and to lesser extent for hydrogeology (Haeni, 1986) and mineral exploration (Eaton et al., 2003). The basic procedure in seismic exploration uses seismic sources such as explosives, vibrators or air guns to generate seismic waves and record the amplitudes to build an image of the subsurface. The recorded seismic signal shows the variation in amplitude and phase as a function of the source-receiver distance (offset) and two-way travel time. Amplitude Variation with Offset (AVO) is one of the techniques which study the change in the amplitude of a seismic signal with respect to offset. If the offset information is mapped into angle of incidence, the method is called Amplitude Variation with Angle (AVA). The amplitude and the total travel time depends on the angle of incidence, the properties and the structure of the underlying rocks. Usually, due to the limitations in acquisition and noise, the seismic data has to be processed to extract the signal. After routine processing steps, the seismic data are used to analyze and infer the nature of the subsurface. The first information that can be extracted from a given seismic data are about the geological structures of the subsurface. A direct detection of hydrocarbon from the amplitude of a seismic section was first suggested by Ostrander (1982). In other words, detailed investigation of the amplitude of reflections on a seismic section can provide im- portant information about the presence of hydrocarbons. The quality of the interpretation depends on the processing steps carried out to accurately represent the amplitudes of the recorded waveforms (Chopra and Castagna, 2007). For instance, the presence of gas or oil in a pore space of rocks lowers the velocity as compared to water. Although the change in density is small, gas and oil also lower density which in turn has a great effect in lowering the acoustic impedance. Depending on the local circumstances, the presence of the hydrocarbon 1
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: 5.3 (a) P-wave reflectivity (RMSEA
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 5
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CHAPTER 4. TWO-TERM AVO INVERSION 5
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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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