Geologic regularization

In Chapter , I introduce a new way to regularize the tomography estimate that takes into account the a priori estimate of geology. I start from a simple missing data problem. I compare the conventional way that geophysicists and geostatisticians characterize covariance and how they apply to a missing data problem. I show how non-stationarity makes both solutions suboptimal. I then introduce another way to approximate model covariance, a steering filter. I show that a steering filter can accurately handle a non-stationary model covariance. I show how to build a steering filter with as little information as early migrated reflector positions. For example, returning to the simply two layer example (Figure 1), if we know the dip of the second layer in the model of Figure 2 we can do a better job constraining the slowness field using steering filter regularization, Figure 9.

 

reg.tomo Figure 9 The same tomography shown in Figure 3 now using steering filters to regularize the velocity estimate.

Using the steering filter I develop a new tomography objective function. I then compare and contrast the results of using a steering filter to regularize the tomography problem versus an isotropic regularizer in both depth and tau. I show that the most reasonable velocity model, and the best migrated image, comes from using steering filters in the tau domain.