Sirgue, Laurent, 2003. Inversion de la forme d'onde dans le ...

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$Fayek & Kyser 2000 uraninite fractionations.pdf - Queen's University$

10 CHAPTER 1. INTRODUCTION 1.8 Summary of the thesis This thesis begins by reviewing the basics of inverse theory in Chapter 2. The resolution of linear inverse problem using the singular value decomposition (SVD) method is first described. This is followed by the definition of local methods using Gauss-Newton and gradient techniques. A useful link is then established between the SVD and local methods. The resolution of non-linear inverse problem using the local methods is then detailed. Numerical examples of resolution of a linear and non-linear problem using local methods are shown. This review of inverse theory provides the elements for the introduction of the frequency domain waveform inversion. In Chapter 3, I aim to further understand the relation between temporal frequencies, offset and wavenumber reconstruction of the model. A strategy for selecting frequencies is derived that assures a continuous reconstruction of the wavenumber spectrum of the model. This strategy makes use of the redundancy of information present when a range of offset is available and that allows to discretise temporal frequencies with a much larger sampling interval than that dictated by the sampling theorem. It is shown that the larger the maximum effective offset is, the fewer frequencies are required. The accuracy of the strategy, exact in a 1-D earth, is validated on a 1-D numerical experiment. The strategy is also useful in more general complex media as it is successfully applied on the 2-D Marmousi model using a starting model that was obtained from traveltime tomography. In this chapter, all waveform inversion however start from unrealistically low frequencies. A minimum frequency of 5 Hz is used in the 2-D Marmousi experiment. Such a frequency is considered too low to be present in real seismic data. In Chapter 4, I therefore investigate waveform inversion starting from a more realistic frequency: 7Hz. It is shown that at this frequency, waveform inversion can easily converge into a local minimum. The aim of this chapter is therefore to further understand the mechanism governing the inversion process in order to design a strategy that improves the linearity of the inverse problem. A SVD is applied on simple 1-D models and indicates that the gradient vector is dominated by the high wavenumbers. A linearity study is also carried out showing that the low wavenumbers are the more non-linear components of the model. As a result, the gradient vector must be smoothed in order to assure the proper reconstruction of the low wavenumber components. In order to mitigate the risk of convergence into a local minimum during the recovery of the low wavenumbers, adequate preconditioning of the data residuals may be applied that focus on the inversion of the early arrivals.
1.8. SUMMARY OF THE THESIS 11 In Chapter 5, I explore the feasibility of an application of waveform inversion to the subbasalt imaging problem. A non-linearity study is carried out that aims to define the requirements of the starting model that ensure the success of waveform inversion. This study is performed on a 1-D velocity model and emphasises the importance of the determination of the starting model in the waveform inversion process. It is shown that the requirements on the starting model depends on the minimum frequency available in the data: the higher this frequency is, the more accurate the starting model needs to be.
Page 1 and 2: Thèse de Doctorat de l’Universit
Page 3 and 4: Abstract The standard imaging appro
Page 5 and 6: Résumé de thèse L’approche sta
Page 7 and 8: vii l’inversion de la forme d’o
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Page 14 and 15: xiv CONTENTS 6 Conclusions 155 6.1
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Page 20 and 21: xx LIST OF TABLES
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136 CHAPTER 5. WAVEFORM INVERSION A
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156 CHAPTER 6. CONCLUSIONS The wave
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158 CHAPTER 6. CONCLUSIONS 6.3 Wave
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162 APPENDIX A. IMAGE STRETCH AND N
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164 BIBLIOGRAPHY Bleistein, N., 198
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166 BIBLIOGRAPHY Forgues, E., Scala
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168 BIBLIOGRAPHY Lafond, C., Kaculi
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170 BIBLIOGRAPHY Pratt, R. G., and
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172 BIBLIOGRAPHY Tarantola, A., 198
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Sirgue, Laurent, 2003. Inversion de la forme d'onde dans le ...

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