In addition to non-linearity, tomography problems are often under-determined. To create more geologically feasible velocity models and to speed up convergence, Michelena 1991 suggested using varying sized grid cells. Unfortunately, such a parameterization is prone to error when the wrong size blocks are chosen Delprat-Jannaud and Lailly (1992). Other authors have suggested locally clustering grid cells Carrion (1991) or characterizing the velocity model as a series of layers Kosloff et al. (1996). These methods are also susceptible to errors when the wrong parameterization is chosen.

An attractive alternative
approach is to add an additional model regularization term to our
objective function Toldi (1985). In theory, this regularization
term should be the inverse model covariance matrix Tarantola (1987).
The question is how to obtain an estimate of the model covariance matrix.
The obvious answer is through *a priori* information sources such
a geologist's structural model of the area, well log information,
or preliminary stack or migration results. Incorporating
these varied information sources into our objective function has always been
problematic.
For years, geostatisticians have successfully
combined these mixed types of information
to produce variograms Issaks and Srivastava (1989).
Unfortunately, the geostatistical approach does not
easily fit within a standard global tomography problem.

In this paper we follow the course outlined in Clapp and Biondi 1998 to address both the velocity-depth ambiguity and the
problem of adding geologic
constraints.
We formulate our tomography problem in vertical travel-time () coordinates
rather than depth. In this coordinate system, reflectors are significantly
less sensitive to velocity Biondi et al. (1997) and the resulting
back projection operator is less sensitive to the background velocity
model Clapp and Biondi (1999).
We make the assumption that velocity
follows geologic dip or some other known
trend. We then approximate the model covariance matrix by
creating small, plane-wave annihilation filters Claerbout (1992),
or *steering filters* oriented along
geologic dip Clapp et al. (1997, 1999).
To speed up convergence, we reformulate our regularization
problem to a preconditioned problem Claerbout (1998a) using the helix
transform and polynomial division Claerbout (1998c).

We create a synthetic anticline velocity model and compare the inversion result using a symmetric regularization operator in depth, steering filter in depth, and finally steering filter in vertical travel time space. We study the speed and quality of our tomographic estimate using two different synthetic models. We conclude with some preliminary tests on a 2-D marine dataset with gas hydrates. Preliminary migration results are encouraging.

4/20/1999