In areas of complex geology, prestack depth migration is the only way to produce an accurate image of the subsurface. Prestack depth migration requires an accurate interval velocity model. With few exceptions, the subsurface velocities are not known beforehand and must be estimated. When the velocity structure is complex, with significant lateral variations, reflection tomography methods must be applied. Unfortunately, reflection tomography often converges slowly, to a model that geologically unreasonable, or not at all.; One reason for this slow or non-convergence is that reflection tomography attempts to simultaneously estimate reflector position (mapping velocity) and image the data (focusing velocity). In this dissertation, I present a new approach to finding an acceptable interval velocity model for prestack migration. By performing tomography in vertical travel-time space, I avoid estimating mapping velocity, instead concentrating on focusing velocity.; The large null space of reflection tomography problems forces a sparse parameterization of the model and/or regularization criteria to be added to the estimation. Standard tomography schemes tend to create isotropic features in velocity that are inconsistent with geology. These isotropic features are due in large part to using symmetric regularization operators or by choosing a poor model parameterization. By replacing these symmetric operators with operators that tend to spread information along structural dips, I can generate velocity models that are more geologically reasonable. In addition, by forming these operators in helical 1-D space and performing polynomial division, I can find the inverse of these space-varying anisotropic operators. These inverse operators can be used as a preconditioner to a standard tomography problem, significantly improving convergence speed compared to the typical, regularized inversion problem. Results from synthetic, 2-D field, and 3-D field data are shown. In each case the velocity obtained improves the focusing of the migrated image.
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