Preface |
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viii | |
1 Basics of seismic inversion |
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1 | (10) |
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1.1 The linear inverse problem |
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1 | (2) |
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1.2 Data, model and mapping |
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3 | (2) |
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5 | (1) |
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6 | (5) |
2 Linear systems for inversion |
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11 | (13) |
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2.1 The governing equation and its solution |
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11 | (4) |
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15 | (2) |
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17 | (1) |
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2.4 Seismic downward continuation |
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18 | (2) |
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2.5 Seismic data processing |
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20 | (4) |
3 Least-squares solutions |
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24 | (18) |
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24 | (4) |
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3.2 The inverse of a square matrix |
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28 | (1) |
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3.3 LU decomposition and Cholesky factorisation |
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29 | (6) |
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3.4 Least-squares solutions of linear systems |
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35 | (3) |
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3.5 Least-squares solution for a nonlinear system |
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38 | (1) |
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3.6 Least-squares solution by QR decomposition |
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39 | (3) |
4 Singular value analysis |
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42 | (11) |
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4.1 Eigenvalues and eigenvectors |
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42 | (3) |
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4.2 Singular value concept |
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45 | (2) |
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4.3 Generalised inverse solution by SVD |
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47 | (1) |
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48 | (5) |
5 Gradient-based methods |
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53 | (15) |
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54 | (2) |
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5.2 The steepest descent method |
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56 | (3) |
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5.3 Conjugate gradient method |
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59 | (3) |
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5.4 Biconjugate gradient method |
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62 | (3) |
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5.5 The subspace gradient method |
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65 | (3) |
6 Regularisation |
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68 | (16) |
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6.1 Regularisation versus conditional probability |
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68 | (3) |
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6.2 The 4, norm constraint |
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71 | (3) |
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6.3 The maximum entropy constraint |
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74 | (3) |
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6.4 The Cauchy constraint |
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77 | (3) |
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6.5 Comparison of various regularisations |
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80 | (4) |
7 Localised average solutions |
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84 | (10) |
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85 | (1) |
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86 | (1) |
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86 | (3) |
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7.4 The Backus-Gilbert stable solution |
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89 | (5) |
8 Seismic wavelet estimation |
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94 | (18) |
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8.1 Wavelet extraction from seismic-to-well correlation |
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95 | (4) |
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8.2 Generalised wavelet constructed from power spectrum |
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99 | (4) |
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8.3 Kurtosis matching for a constant-phase wavelet |
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103 | (4) |
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8.4 Cumulant matching for a mixed-phase wavelet |
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107 | (5) |
9 Seismic reflectivity inversion |
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112 | (15) |
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9.1 The least-squares problem with a Gaussian constraint |
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112 | (2) |
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9.2 Reflectivity inversion with an Li, norm constraint |
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114 | (2) |
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9.3 Reflectivity inversion with the Cauchy constraint |
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116 | (3) |
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9.4 Multichannel reflectivity inversion |
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119 | (3) |
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9.5 Multichannel conjugate gradient method |
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122 | (5) |
10 Seismic ray-impedance inversion |
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127 | (12) |
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10.1 Acoustic and elastic impedances |
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127 | (3) |
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130 | (3) |
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10.3 Workflow of ray-impedance inversion |
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133 | (1) |
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10.4 Reflectivity inversion in the ray-parameter domain |
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134 | (2) |
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10.5 Ray-impedance inversion with a model constraint |
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136 | (3) |
11 Seismic tomography based on ray theory |
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139 | (17) |
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139 | (1) |
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11.2 Velocity-depth ambiguity in reflection tomography |
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140 | (3) |
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11.3 Ray tracing by a path bending method |
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143 | (4) |
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11.4 Geometrical spreading of curved interfaces |
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147 | (2) |
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11.5 Joint inversion of traveltime and amplitude data |
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149 | (7) |
12 Waveform tomography for the velocity model |
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156 | (17) |
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12.1 Inversion theory for waveform tomography |
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157 | (3) |
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12.2 The optimal step length |
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160 | (2) |
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12.3 Strategy for reflection seismic tomography |
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162 | (4) |
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12.4 Multiple attenuation and partial compensation |
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166 | (2) |
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12.5 Reflection waveform tomography |
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168 | (5) |
13 Waveform tomography with irregular topography |
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173 | (14) |
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13.1 Body-fitted grids for finite-difference modelling |
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173 | (3) |
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13.2 Modification of boundary points |
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176 | (1) |
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13.3 Pseudo-orthogonality and smoothness |
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177 | (3) |
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13.4 Wave equation and absorbing boundary condition |
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180 | (4) |
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13.5 Waveform tomography with irregular topography |
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184 | (3) |
14 Waveform tomography for seismic impedance |
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187 | (15) |
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14.1 Wave equation and model parameterisation |
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189 | (2) |
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14.2 The impedance inversion method |
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191 | (2) |
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14.3 Inversion strategies and the inversion flow |
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193 | (5) |
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14.4 Application to field seismic data |
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198 | (2) |
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200 | (2) |
Appendices |
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202 | (13) |
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A Householder transform for QR decomposition |
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202 | (3) |
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B Singular value decomposition algorithm |
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205 | (5) |
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C Biconjugate gradient method for complex systems |
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210 | (1) |
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D Gradient calculation in waveform tomography |
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211 | (4) |
Exercises and solutions |
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215 | (20) |
References |
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235 | (7) |
Author index |
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242 | (2) |
Subject index |
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244 | |