In order to compute stable solutions to ill-conditioned systems it is often necessary to apply regularization methods. Claerbout 1997 writes: `` In geophysical fitting we generally have two goals, the first being data ``fitting'' and the second being the ``damping'' or regularization goal''. One way to achieve such objective is to augment the problem with a second regression that adds assumptions about the model (e.g. roughness, smoothness, curvature, energy in one dip, etc...).
For the data-space inverse, one now solves the problem:
The solution represents an equalized data vector that is unevenly sampled. I chose the penalty operator, , to be the identity matrix. This is the standard Tikhonov regularization where the solution solves the problem:
Similar to the data-space solution, to regularize the model-space inverse one seeks a solution to the system of regressions:
Since the model is regularly sampled, I chose to be the Laplacian operator, which represents differentiation in the midpoint-space. The parameter controls the smoothness of the solution and is again an estimate of the smallest resolved singular value of .At the time of writing the dissertation, I didn't investigate a robust strategy for estimating . However, by processing a single frequency (e.g the dominant frequency of the survey), I was able to iteratively guess a good estimate for . Ideally, one might assume a different value for should be used for each frequency inversion. Results showed that a single good estimate of produced a reasonably smooth solution while still preserving the high frequency components of the reflectivity function.