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

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

16 CHAPTER 2. INVERSE THEORY Unknowns/Equations G −1 exists if G has Solution n m < n d linearly independent columns At most one n m = n d linearly independent columns Only one n m > n d n d linearly independent columns More than one Table 2.1: Condition of existence of G −1 in equation 2.5 (p. 48 of Scales et al. (2001)). 2.2.1 Singular value decomposition The Singular Value Decomposition (SVD) method was introduced by Lanczos (1961). I briefly review in this section the principle results of the SVD. Further details on the method may be found in Aki and Richards (1980); Menke (1989); Scales et al. (2001). The singular value decomposition of any matrix G of dimension (n d × n m ) is expressed as G = UΛV t (2.7) where the columns of U (dimension (n d × n d )) are the data space eigenvectors of GG t , and the column of V (dimension (n m × n m )) are the model space eigenvectors of G t G. The matrix Λ of dimension (n d × n m ) is diagonal where the diagonal elements are called singular values. There are min(n d , n m ) number of singular values which are defined as the square roots of the first min(n d , n m ) eigenvalues of G t Gv i = λ 2 i v i i = 1, n m GG t u i = λ 2 i u i i = 1, n d (2.8) Each data eigenvector u i and model eigenvector v i are columns of U and V respectively such as U and V form a basis of orthonormal vectors. Any vector of the data or model space can thus be expressed as a linear combination of eigenvectors and we have U t U = UU t = I nd (2.9) V t V = VV t = I nm (2.10) Note that the eigenvalues λ 2 i defined in equation (2.8) may be positive or zero real numbers so that the diagonal terms of Λ may contain zeros. If n m ≠ n d , the number of eigenvectors of the data and model space are different. If p ≤ min(n d , n m ) is the number of non-zero singular values, the data and model space have p eigenvectors with associated non-zero eigenvalues.
£ 2.2. INVERSION OF LINEAR PROBLEMS 17 ¤¦¥ §©¨ ¡ ¢ Figure 2.2: Illustration of the Singular Value Decomposition of the forward operator G. The SVD identifies the model and data null spaces V o and U o . G can be reduced to a projection operator from V p to U p . Therefore, there will be (n m − p) space eigenvectors and (n d − p) data eigenvectors with the eigenvalue zero. The projection of eigenvectors from the model to the data space is ⎧ ⎨ and from the data to the model space is ⎩ ⎧ ⎨ ⎩ Gv i = λ i u i for i = 1, p Gv i = 0 for i = p, n, m (2.11) G t u i = λ i v i for i = 1, p G t u i = 0 for i = p, n d . (2.12) Equation (2.11) and (2.12) implies that, for the p non-zero eigenvalues, an eigenvector of the model space is coupled with an eigenvector of the data space which have the same eigenvalue. The projection of an eigenvector with a zero eigenvalue has no component in the corresponding space. Understanding the forward problem The matrix U and V can be decomposed into two sub-spaces represented by the eigenvectors with non-zero and zero eigenvalues (as shown Figure 2.2) U = (U p , U 0 ) V = (V p , U 0 ) (2.13)
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 38 and 39: 18 CHAPTER 2. INVERSE THEORY where
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66 CHAPTER 3. STRATEGY FOR SELECTIN
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a) Depth (Km) Depth (Km) 0 0.5 1.0
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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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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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