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

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

xii CONTENTS 3 A strategy for selecting temporal frequencies 53 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Gradient vector and wavenumber illumination . . . . . . . . . . . . . . . . . . 55 3.2.1 Analysis of the gradient within the Born approximation . . . . . . . . . 56 3.3 The 1-D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.1 Wavenumber illumination for the 1-D case . . . . . . . . . . . . . . . 60 3.4 Strategy for choosing frequencies for 1-D imaging . . . . . . . . . . . . . . . 62 3.5 Numerical test I: 1-D models . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5.1 The synthetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.5.2 Validation of the plane wave predictions . . . . . . . . . . . . . . . . . 65 3.5.3 A remark on “image stretch” . . . . . . . . . . . . . . . . . . . . . . . 68 3.5.4 1-D waveform inversion . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.5.5 Sensitivity to noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6 Numerical test II: 2-D models . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.6.1 Marmousi velocity model . . . . . . . . . . . . . . . . . . . . . . . . 76 3.6.2 Determination of the starting model by traveltime inversion . . . . . . 77 3.6.3 Waveform inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.7.1 Efficiency of the strategy for 2-D heterogeneous media . . . . . . . . . 85 3.7.2 Practical implementation of the frequency selection strategy . . . . . . 87 3.7.3 The equivalence between gradient images and migration . . . . . . . . 87 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4 Waveform inversion starting from realistic frequencies 91 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 SVD of the Fréchet derivative matrix . . . . . . . . . . . . . . . . . . . . . . . 93 4.2.1 1-D medium in homogeneous background . . . . . . . . . . . . . . . 93 4.2.2 1-D medium with a constant vertical velocity gradient . . . . . . . . . 100 4.3 Non-linearity of the waveform inverse problem . . . . . . . . . . . . . . . . . 108 4.3.1 The effect of cycle skipping . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.2 Analytic linearity study of the 3D homogeneous Green’s function . . . 108 4.3.3 Non-linearity and wavenumber domain . . . . . . . . . . . . . . . . . 110 4.3.4 Non-linearity and offset . . . . . . . . . . . . . . . . . . . . . . . . . 112
CONTENTS xiii 4.4 Tools for the mitigation of non-linearities . . . . . . . . . . . . . . . . . . . . 114 4.4.1 Gradient preconditioning by wavenumber filtering . . . . . . . . . . . 114 4.4.2 Time damping of the data residuals . . . . . . . . . . . . . . . . . . . 116 4.4.3 f-k filtering of the data residuals . . . . . . . . . . . . . . . . . . . . . 122 4.4.4 Offset windowing of the data residuals . . . . . . . . . . . . . . . . . . 124 4.4.5 Model parameterisation . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.5 Definition of a strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6.1 1-D velocity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6.2 The 2-D extended Marmousi model . . . . . . . . . . . . . . . . . . . 128 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5 Waveform inversion and starting models: the sub-basalt imaging problem 135 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.2 The sub-basalt imaging problem . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.1 Transmission loss at the top basalt interface and multiples . . . . . . . 136 5.2.2 Interface scattering at the top of the basalt . . . . . . . . . . . . . . . 138 5.2.3 Body wave scattering within the basalt . . . . . . . . . . . . . . . . . . 138 5.2.4 Sub-basalt imaging and waveform inversion . . . . . . . . . . . . . . . 138 5.3 The 1-D sub-basalt numerical experiment . . . . . . . . . . . . . . . . . . . . 139 5.4 Inversion starting from a “correct” macro model . . . . . . . . . . . . . . . . . 141 5.4.1 Misfit function and model smoothness . . . . . . . . . . . . . . . . . . 141 5.4.2 The top basalt and bottom basalt interfaces . . . . . . . . . . . . . . . 141 5.5 Wavefield separation by layer stripping . . . . . . . . . . . . . . . . . . . . . . 143 5.5.1 The sediment layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.5.2 The basalt layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.5.3 The sub-basalt layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.6 Waveform inversion and starting model requirements . . . . . . . . . . . . . . 146 5.6.1 Frequency dependence of the starting model requirements . . . . . . . 146 5.6.2 Waveform inversion at 7 Hz . . . . . . . . . . . . . . . . . . . . . . . 148 5.7 Discussion: starting model and standard methods . . . . . . . . . . . . . . . . 151 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
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
Page 9: Ackowledgments First and foremost,
Page 14 and 15: xiv CONTENTS 6 Conclusions 155 6.1
Page 16 and 17: xvi LIST OF FIGURES 3.11 Imapact of
Page 18 and 19: xviii LIST OF FIGURES
Page 20 and 21: xx LIST OF TABLES
Page 22 and 23: 2 CHAPTER 1. INTRODUCTION is made t
Page 24 and 25: 4 CHAPTER 1. INTRODUCTION the data
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Page 28 and 29: 8 CHAPTER 1. INTRODUCTION technique
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Page 34 and 35: 14 CHAPTER 2. INVERSE THEORY Data M
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52 CHAPTER 2. INVERSE THEORY
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54 CHAPTER 3. STRATEGY FOR SELECTIN
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ë ¸ · 0 0.1 0.2 0.3 0.4 0.5 0.6
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„ 64 CHAPTER 3. STRATEGY FOR SELE
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a) Depth (Km) Depth (Km) 0 0.5 1.0
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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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160 CHAPTER 6. CONCLUSIONS
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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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