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.