Introduction

Tomography is inherently non-linear, therefore a standard technique is to linearize the problem by assuming a stationary ray field Stork and Clayton (1991). Unfortunately, we must still deal with the coupled relationship between reflector position and velocity Al-Chalabi (1997); Tieman (1995). As a result, the back projection operator must attempt to handle both repositioning of the reflector and updating the velocity model van Trier (1990) . The resulting back projection operator is sensitive to our current guess at velocity and reflector position. In vertical traveltime space, reflector movement is significantly less. Biondi et al. (1998b) showed that by reformulating the problem in this space, complex velocity functions could be obtained more quickly and are better resolved.

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.