Approximate inverses

The approximation of equation (7) at first gives a visually appealing result. But if we take a closer look, we see that it isn't as close to the true inverse as we might hope. To see how, let's look at a simpler problem. The circles in Figure  show a series of irregular data points sub-sampled from the solid line curve. The dotted lines show the solution when applying equation (7). The dashed line represents the solution using fitting goals (3). Note how the approximate solution approach has the correct low frequency shape but varies significantly from the full inverse solution.

 

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Figure 1 The solid line represents the input signal. The circles represent our data points. The dashed line shows the solution with fitting goals (3). The dotted lines show the solution using the approximate method in equation (7).

With better choices of $\bf p$, 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.

 

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Figure 2 A portion of the regularized data, (t,cmp_x,cmp_y) cube, estimated using the approximate solution in equation (7). Note the dimming and brightening due to the irregular sampling of the input data.

 

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Figure 3 A portion of the regularized data, (t,cmp_x,cmp_y) cube, estimated by five conjugate gradient iterations using (3). Note how the improved amplitude continuity.

 

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Figure 4 Migration of one line of dataset regularized with the approximation solution (7). Note the difference in amplitude at `A' and `B' caused by the approximate solution.

Solving the full inverse introduces its own problems. First, we have now significantly increased the cost. The approximate solution (7) required calling both $\bf L$ and $\bf B$ 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.