In ray-based migration velocity analysis, Kirchhoff migration is normally used to construct CRP gathers and residual moveout. Unfortunately, conventional implementation of Kirchhoff methods have trouble handling complex wave behavior. Wave equation methods are an attractive alternative to Kirchhoff. Clayton and Stolt (1981) and later Prucha et al. (1999) showed that wave equation methods can form CRP gathers in terms of reflection angle.
Tomography problems are also often under-determined. By adding an additional model regularization term to our objective function Toldi (1985) we can stabilize the inversion. In theory, this regularization term should be the inverse model covariance matrix Tarantola (1987). Clapp et al. (1998a) constructed a series of small plane wave annihilators, called steering filters, using a priori information to produce changes to our velocity model that were more geologically reasonable.
In this paper we show how to apply steering filters to smooth the slowness, rather than the change in slowness. We use wave equation CRP gathers as the basis for ray based tomography. We apply the technique on a 2-D line taken from a 3-D marine dataset from the North Sea. We show that we can obtain a velocity model that improves the focusing of the data while being geologically reasonable.