Velocity analysis in the presence of uncertainty
Velocity analysis resolves relatively long scales of earth structure, on the order of 1 km. Migration produces images with length scales (wavelengths) on the order of 10's of m. In between these two scale regimes lies another, corresponding roughly to structures between 60 to 300m in extent, in which the resolution of velocity analysis is uncertain and the energy of images is small to non-existent. This thesis aims at assessing the impact on velocity analysis of uncertainty at these intermediate length scales, using ideas on time reversal and imaging in randomly inhomogeneous media developed by Borcea and colleagues, in combination with velocity estimation methods of differential semblance type. The main result of this thesis is that the noise in seismic reflection data associated with the middle scales in velocity heterogeneity does not strongly affect the estimates of the long-scale component of velocity, if these estimates are obtained using a statistically stable formulation of differential semblance optimization. Hence the nonlinear influence of uncertainty in the middle scales does not propagate down the length scale. This is in contrast with the results of Borcea and colleagues, who have shown that prestack images are strongly affected, implying that the uncertainty in the middle scales does certainly propagate up the length scale. Random perturbations associated with the middle scales of velocity heterogeneity yield measurable phase shifts in the reflection data. However, it is known that cross-correlations of neighboring seismic traces are stable against these perturbations, under some circumstances. The main theoretical achievement of this thesis, presented in Chapter 3, is to extend this stability result to laterally homogeneous background velocity models, and to cross-correlations containing slowly-varying weights. Chapter 4 shows that differential semblance functionals, specialized to layered modeling, can be written entirely in terms of weighted cross-correlations, and therefore argues that the velocity analysis algorithm and the associated velocity estimates inherit the statistical stability property. A quantitative verification of the stability claims is provided in Chapter 5.
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