Fortunately, we often have other sources of information, such as well logs, stacked sections, or geologist's interpretation, that we can use to construct anisotropic operators (steering filters) Clapp et al. (1997, 1998) that fill the null space with more geologically reasonable velocities. Convergence speed can be improved by changing from a regularized to a preconditioned problem Claerbout and Nichols (1993). By forming the regularization operator in a helical coordinate system we can efficiently obtain an inverse operator by polynomial division Claerbout (1998). This new operator can be used as a preconditioner, creating an equivalent optimization problem Fomel et al. (1997) that converges significantly faster.
Another major difficulty in depth tomography is the strong connection between depth and velocity. We can avoid some of the problems caused by this connection, by transforming the whole problem into vertical-traveltime coordinates (,). In the time domain reflector position is less sensitive to velocity changes. This modified coordinate system still allows for complex velocity structures, but significantly reduces the map migration term in tomography Biondi et al. (1997).
We construct a synthetic anticline velocity model and apply a standard ray based tomography technique to estimate velocity. We show that the inversion result is improved by the use of steering filters to precondition our tomography operator over a more standard isotropic regularization technique. We then apply the same basic tomography method in (,) space again significantly improving our velocity estimate.