The idea of using a **gap** in a prediction filter
is to relax the goal of converting
realistic signals into perfect impulses.
Figure 7 shows synthetic data,
sparse noise into a leaky integrator,
and deconvolutions with prediction-error filters.

Figure 7

Theoretically, the filters should turn out to be .Varying degrees of success are achieved by the filters obtained on the different traces, but overall, the results are good.

To see what happens when an unrealistic **deconvolution** goal
is set for prediction error,
we can try to compress a wavelet
that is resistant to compression--for example,
the impulse response of a Butterworth bandpass filter.
The perfect filter to compress any wavelet is its inverse.
But a wide region of the spectrum
of a **Butterworth filter** is nearly zero,
so any presumed inverse
must require nearly dividing by that range of zeros.
Compressing a Butterworth filter is so difficult
that I omitted the random numbers used in Figure 7
and applied prediction error to the Butterworth response itself,
in Figure 8.

dbutter
Butterworth deconvolution by prediction error.
Figure 8 |

Thus, we have seen that gapped PE filters sometimes are able to compress a wavelet, and sometimes are not. In real life, resonances arise in the earth's shallow layers; and as we will see, the resonant filters can be shortened by PE filters.

10/21/1998