Figure 1

With better choices of , or additional diagonals Guitton (2003), a better solution might be possible, but the potential is limited. The full matrix can not be adequately described by the limited description we are allowing.

We can see the same effect in our regularization problem. Figure shows the regularization result of a small portion of a 3-D land dataset. Figure shows the result of doing five steps of conjugate gradient solving the fitting goals in (3). Note how we have a smoother, more believable, amplitude behavior as a function of midpoint. Unfortunately, the effect of the approximate solution translates directly to an effect on the amplitudes in our migration. Figure is the result of migrating a single line from the 3-D dataset regularized by the method described in Biondi and Vlad (2001). Note the stripes of high and low amplitude indicated by `A' and `B'. Wavefront healing helps minimize the effect at depth but there is still a noticable effect on the amplitude.

Figure 2

Figure 3

Figure 4

Solving the full inverse introduces its own problems. First, we have now significantly increased the cost. The approximate solution (7) required calling both and three times. Even using a minimal number of iteration (3-5) increases the cost by a factor of two or three. In addition, we have significantly increased our disk space requirement. If we set up the inverse problem shown in fitting goals (3), we now must store six copies of our model space. We are quickly approaching the point of impracticality even for a small dataset.

10/14/2003