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

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

30 CHAPTER 2. INVERSE THEORY a) b) 3 7 6 2 5 4 3 1 2 ÉËÊ$Ì]Í#Î ÒÓ ÜÝ 1 0 0 1 2 3 0 1 2 3 4 5 6 7 ÔÖÕ ×ÙØ Figure 2.6: Example of a 2-D, linear inverse problem where a) the solution is the intersection point m = (1, 1), between two straight lines, b) the corresponding misfit function with the two eigenvectors v 1 , v 2 are shown centred on the minimum. ÏÑÐ ÚÑÛ Since there are no zero singular values, we have p = n m = n d . There is neither a model nor a data null space, the solution of the inverse problem yields the true solution: m † = m true (see Table 2.2). The misfit function is shown Figure 2.6b), with the model eigenvectors centred on the minimum of the misfit function. Figure 2.7 shows the minimisation of the misfit function using (a) Gauss-Newton , (b) gradient , (c) preconditioned gradient and (d) conjugated gradient methods. As expected, the Gauss- Newton method locates the minimum in one iteration whereas the gradient methods converges to the minimum in several iterations. The convergence rate is improved by preconditioning the gradient direction with the diagonal term of the inverse Hessian and the conjugated gradient convergences in 2 iterations. The evolution of the misfit function with iterations is shown Figure 2.8 for each method .
2.2. INVERSION OF LINEAR PROBLEMS 31 a) b) 7 7 6 6 5 5 4 3 −H −1 ∇E 4 3 &(' )+*-,/.¡0£1¥23,/.¡0 "# 2 2 1 1 $% 7 0 1 2 3 4 5 6 7 c) d) 7 0 1 2 3 4 5 6 7 ! 6 6 5 5 öç÷ 4 4 åçæ úüû@ýÿþ¡ £¢¥¤ ýþ¡ 3 3 áâ 2 2 èêéìëîí]ï9ðñ@ëîí]ï ôõ 1 1 ¦¨§©¡© 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ã&ä ø&ù Figure 2.7: Local methods applied to a 2-D, linear inverse problem for a) Gauss-Newton , b) Gradient, c) Preconditioned gradient and d) Conjugated gradient methods. The gradient in c) was preconditioned with the diagonal term of the inverse Hessian. Þàß òàó
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 12 and 13: xii CONTENTS 3 A strategy for selec
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
Page 26 and 27: 6 CHAPTER 1. INTRODUCTION quantify
Page 28 and 29: 8 CHAPTER 1. INTRODUCTION technique
Page 30 and 31: 10 CHAPTER 1. INTRODUCTION 1.8 Summ
Page 32 and 33: 12 CHAPTER 1. INTRODUCTION
Page 34 and 35: 14 CHAPTER 2. INVERSE THEORY Data M
Page 36 and 37: 16 CHAPTER 2. INVERSE THEORY Unknow
Page 38 and 39: 18 CHAPTER 2. INVERSE THEORY where
Page 40 and 41: 20 CHAPTER 2. INVERSE THEORY Where
Page 42 and 43: 22 CHAPTER 2. INVERSE THEORY The mi
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Page 46 and 47: œS œ *ž œ7Ÿ• ›&œ žu¡[¢
Page 48 and 49: 28 CHAPTER 2. INVERSE THEORY Adding
Page 52 and 53: 32 CHAPTER 2. INVERSE THEORY Misfit
Page 54 and 55: 34 CHAPTER 2. INVERSE THEORY Gauss-
Page 56 and 57: O P 36 CHAPTER 2. INVERSE THEORY St
Page 58 and 59: 38 CHAPTER 2. INVERSE THEORY a) YZ
Page 60 and 61: 40 CHAPTER 2. INVERSE THEORY a) b)
Page 62 and 63: € 42 CHAPTER 2. INVERSE THEORY 10
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Page 74 and 75: 54 CHAPTER 3. STRATEGY FOR SELECTIN
Page 78 and 79: ë ¸ · 0 0.1 0.2 0.3 0.4 0.5 0.6
Page 84 and 85: „ 64 CHAPTER 3. STRATEGY FOR SELE
Page 92 and 93: a) Depth (Km) Depth (Km) 0 0.5 1.0
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80 CHAPTER 3. STRATEGY FOR SELECTIN
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92 CHAPTER 4. STARTING FROM REALIST
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a) b) Depth (km) 0 0.5 1.0 1.5 2.0
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100 CHAPTER 4. STARTING FROM REALIS
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a) b) Depth (km) 0 0.5 1.0 1.5 2.0
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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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Sirgue, Laurent, 2003. Inversion de la forme d'onde dans le ...

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