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

List of Figures
2.1 The Inverse and the Forward Problem in waveform inversion. . . . . . . . . . . 14
2.2 Illustration of the Singular Value Decomposition. . . . . . . . . . . . . . . . . 17
2.3 Scheme of the Gauss-Newton method applied to linear problem. . . . . . . . . 22
2.4 Scheme of gradient method applied to linear problem. . . . . . . . . . . . . . . 24
2.5 Construction of the gradient vector from eigenvectors and eigenvalues of the
Hessian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Example of a 2-D, linear inverse problem. . . . . . . . . . . . . . . . . . . . . 30
2.7 Local methods applied to the 2-D, linear inverse problem. . . . . . . . . . . . 31
2.8 Evolution of the misfit function with iterations using local methods. . . . . . . 32
2.9 Scheme of linearised inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.10 Example of a non-linear inverse problem. . . . . . . . . . . . . . . . . . . . . 38
2.11 Results of local methods with an accurate starting model. . . . . . . . . . . . . 40
2.12 Results with a starting model where the quadratic assumption fails. . . . . . . . 41
2.13 Results with a starting model is close to a local minimum. . . . . . . . . . . . . 42
3.1 Wavenumber illumination for a single frequency and source/receiver pair. . . . 57
3.2 Wavenumber illumination for different incident/scattering angle. . . . . . . . . 58
3.3 The 1-D basic scattering experiment. . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Illustration of the frequency discretisation strategy. . . . . . . . . . . . . . . . 64
3.5 Velocity model after Freudenreich and Singh, 2000. . . . . . . . . . . . . . . 66
3.6 Illustration of the computation of the 5 Hz gradient. . . . . . . . . . . . . . . . 67
3.7 A set of amplitude normalized gradient images. . . . . . . . . . . . . . . . . . 69
3.8 Frequency sequence generated by equation (3.18). . . . . . . . . . . . . . . . . 71
3.9 Sequential frequency domain inversion results. . . . . . . . . . . . . . . . . . 72
3.10 Time-like waveform inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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