Postcoloring versus prewhitening

The output of a PE filter, as we know, is white (unless it is gapped), but people do not like to look at white signals. Signals are normally sampled at adequate density, which means that they are small anywhere near the Nyquist frequency. There is rarely energy above the half-Nyquist and generally little but marine noises above the quarter-Nyquist. To avoid boosting these noises, the ungapped PE filter is generally altered or accompanied by other filters. Three common approaches follow:

• Use a gapped filter.
• Deconvolve, then apply a filter with the desired spectrum $S(\omega)$.
• Prefilter the input with $S(\omega)^{-1}$,then deconvolve with an ungapped PE filter, and finally postfilter with $S(\omega)$.

The last process is called ``prewhitening'' for some complicated reasons: the idea seems to be that the prefilter removes known color so that the least-squares coefficients are not ``wasted'' on predicting what is already known. Thus the prefilter spectrum $S(\omega)^{-1}$is theoretically the inverse of the prior estimate of the input spectrum. In real life, that is merely an average of estimates from other data. If the desired output spectrum does not happen to be $S(\omega)$,it does not matter, since any final display filter can be used. Although this is a nice idea, I have no example to illustrate it.

There is also the question of what phase the postfilter should have. Here are some cautions against the obvious two choices:

• Zero phase: a symmetrical filter has a noncausal response.
• Causal: if a later step of processing is to make a coherency analysis for velocity versus time, then the effective time will be more like the signal maximum than the first break.

Since the postfilter is broadband, its phase is not so important as that of the deconvolution operator, which tries to undo the phase of a causal and resonant earth.