Title Page
Abstract
Contents
Chapter 1. Introduction 12
1.1. Background 12
1.2. Motivation and research objective 17
1.3. Outline 19
Chapter 2. Wavepath in the Laplace domain 21
2.1. Wave equation in the Laplace domain 22
2.2. Logarithmic objective function for Laplace-domain waveform inversion (WI) 24
2.3. Laplace-domain Green's functions for a homogeneous acoustic unbounded medium 28
2.4. Rytov wavepath in the Laplace domain 31
2.5. Vertical components of wavepath in the Laplace domain considering the geometrical spreading effect 35
2.6. Numerical examples 37
Chapter 3. Truncated Gauss-Newton method for Laplace-domain WI 41
3.1. Gauss-Newton method and ill-conditioned problems 42
3.2. Ill-conditioning of the Laplace-domain WI algorithm 46
3.3. Truncated Gauss-Newton method 49
3.4. Stopping criterion 51
3.5. Numerical examples 53
Chapter 4. Resolution analysis for Laplace-domain WI 61
4.1. Relationship between the number of attenuation constants and model resolution 62
4.2. Relationship between the condition number of wavepath and model resolution 64
4.3. Range of the attenuation constants and condition number of data kernel matrix in the Laplace domain 68
4.4. Numerical examples 70
Chapter 5. An efficient strategy for Laplace constant selection 82
5.1. Continuity and redundancy of attenuation constants 83
5.2. An efficient strategy for Laplace constant selection 86
5.3. A modified Laplace constant selection strategy considering the geometrical spreading 89
5.4. Effectiveness of the Laplace constant selection strategy in a 2D or 3D heterogeneous medium 92
5.5. Numerical examples 95
Chapter 6. Discussions & Conclusions 109
Appendix 7
Appendix A. Rytov wavepath considering the geometrical spreading 112
Appendix B. Truncated Gauss-Newton method 117
References 121
초록 130
Table 1. Description of the set of Laplace constants used in each strategy. 99
Table 2. The relative model misfit of inverted model parameters obtained from... 100
Table 3. The relative model misfit of inverted model parameters obtained from... 101
Table 4. Description of the set of Laplace constants used in each strategy. 106
Figure 1. A schematic diagram describing the relationship between the incident... 27
Figure 2. The Laplace-domain Rytov wavepath where the geometrical spreading... 39
Figure 3. Vertical profiles of the normalized relative amplitudes of wavepaths... 40
Figure 4. Vertical profiles of the normalized relative amplitudes of wavepaths... 40
Figure 5. Contour plots of the objective functions depending on the condition... 44
Figure 6. (a) The true BP P-wave velocity model and (b) the initial model with... 55
Figure 7. The first model updates of the BP benchmark model based on (a) the... 58
Figure 8. Inversion results of the BP benchmark model obtained using (a) the... 59
Figure 9. Depth profiles of the true BP benchmark model (dashed line, Figure... 60
Figure 10. (a) True velocity model with a Gaussian anomaly of 3.50 km/s... 73
Figure 11. (a) True velocity model with a Gaussian anomaly of 3.50 km/s... 74
Figure 12. (a) True velocity model with a Gaussian anomaly of 3.50 km/s... 75
Figure 13. The coverages of scattering attenuation constants in the case of (a)... 79
Figure 14. (a) The velocity model with a Gaussian, high-velocity anomaly, (b)... 81
Figure 15. A diagram of attenuation constants illustrating the importance of... 84
Figure 16. A diagram of attenuation constants illustrating the importance of... 85
Figure 17. Illustration of the Laplace constants selection strategy. 88
Figure 18. (a), (b), and (c) illustrate the Laplace constant selection strategy in... 91
Figure 19. Three-layered velocity model. The relative model misfit of each... 98
Figure 20. (a) The true SEG/EAGE Salt dome velocity model and (b) the initial... 105
Figure 21. Inversion results of the BP benchmark model obtained using the set... 107
Figure 22. The relative model misfit of inverted model parameters obtained... 108
Algorithm 1. Truncated Gauss-Newton method 120