SPACE-VARIABLE DECONVOLUTION

Because of the massive increase in the number of filter coefficients, allowing these many filters takes us from overdetermined to very undetermined. We can estimate all these filter coefficients by the usual deconvolution fitting goal ()

(15)

but we need to supplement it with some damping goals, say   where is a roughening operator to be chosen.

Experience with missing data in Chapter shows that when the roughening operator is a differential operator, the number of iterations can be large. We can speed the calculation immensely by ``preconditioning''. Define a new variable $\bold m$ by and insert it into (16) to get the equivalent preconditioned system of goals.

The fitting (17) uses the operator .For we can use subroutine nhconest() ; for the smoothing operator we can use nonstationary polynomial division with operator npolydiv():

Figure 15 shows a synthetic data example using these programs. As we hope for deconvolution, events are compressed. The compression is fairly good, even though each event has a different spectrum. What is especially pleasing is that satisfactory results are obtained in truly small numbers of iterations (about three). The example is for two free filter coefficients (1,a₁,a₂) per output point. The roughening operator was taken to be (1,-2,1) which was factored into causal and anticausal finite difference.

 

tvdecon90
Figure 15 Time variable deconvolution with two free filter coefficients and a gap of 6. Four iterations is better than many.

I hope also to find a test case with field data, but experience in seismology is that spectral changes are slow, which implies unexciting results. Many interesting examples should exist in two- and three-dimensional filtering, however, because reflector dip is always changing and that changes the spatial spectrum.

In multidimensional space, the smoothing filter can be chosen with interesting directional properties. Sergey, Bob, Sean and I have joked about this code being the ``double helix'' program because there are two multidimensional helixes in it, one the smoothing filter, the other the deconvolution filter. Unlike the biological helixes, however, these two helixes do not seem to form a symmetrical pair.

EXERCISES:

1. Is nhconest the inverse operator to npolydiv ? Do they commute?
2. Sketch the matrix corresponding to operator nhconest . HINTS: Do not try to write all the matrix elements. Instead draw short lines to indicate rows or columns. As a ``warm up'' consider a simpler case where one filter is used on the first half of the data and another filter for the other half. Then upgrade that solution from two to about ten filters.