Prewhitening - book

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 11
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

In geophysical signal processing, most signals are truncated, which means that the tails of the signals are cut off and lost. If a valid autocorrelation is truncated, the resulting power spectrum might have negative values. In such a case, the truncated autocorrelation function is not an autocorrelation function in its own right. Such a truncated autocorrelation would lead to significant errors in processing. Thus, the truncated autocorrelation must be modified to make it legitimate. One way to modify the truncated autocorrelation is to taper it. Another way, called prewhitening, is to boost the value of its zero-lag coefficient. The goal is to make the power spectrum positive for all frequencies. Treitel and Lines (1982)[1] show the relationship of prewhitening to inverse theory.

When the true autocorrelation, shown in Figure 10a, is truncated to the curve shown in Figure 10b, the spectrum is no longer positive for all values of the frequency. Such a situation is untenable. To correct this problem, the truncated autocorrelation can be tapered, as shown in Figure 10c, so that it damps out more rapidly. This increased damping is enough to make the corresponding spectrum positive for all frequencies.

As an alternative to tapering, the truncated autocorrelation shown in Figure 10b can have its zero-lag value boosted. Such boosting is called prewhitening. Figure 10d shows that 50% prewhitening (that is, increasing the zero-lag value by 50%) makes the spectrum positive for nearly all frequencies. The present case represents an extreme. In actual seismic processing, the use of 0.1% to 5% prewhitening is usually enough.

Prewhitening can be understood in several ways. Figure 11a shows a minimum-phase wavelet and its inverse, and Figures 11b and 11c show, respectively, the autocorrelations and the amplitude spectra of these wavelets. Next, the autocorrelation of the minimum-phase wavelet has its zero-lag autocorrelation coefficient boosted (the prewhitening step), which produces the wavelets, autocorrelations, and amplitude spectra shown in Figure 12a through 12c, respectively. We see that the amplitude spectrum of the inverse of the prewhitened wavelet, which is shown in Figure 12c, has better characteristics than the characteristics of its counterpart depicted in Figure 11c.

Figure 10.  (a) A true autocorrelation has a positive spectrum. (b) The truncated autocorrelation, however, has a spectrum with negative values. (c) If the truncated autocorrelation is tapered, the resulting spectrum is positive. (d) If the zero-lag value of the truncated autocorrelation is given a 50% boost, the resulting power spectrum is almost positive.
Figure 11.  (a) A minimum-phase wavelet and its inverse wavelet. (b) The autocorrelations of the wavelets shown in (a). (c) The amplitude spectra of the wavelets shown in (a).
Figure 12.  (a) A prewhitened version of the minimum-phase wavelet shown in Figure 11a and the inverse of this prewhitened version. (b) The autocorrelations of the prewhitened wavelets shown in (a). Note that the prewhitening was done by increasing the zero-lag value of the wavelet autocorrelation shown in Figure 11b to obtain the autocorrelation of this wavelet shown here. (c) The amplitude spectra of the wavelets shown in (a). Note that the amplitude spectrum of the inverse shown here has better characteristics than does its counterpart in Figure 11c.


References

  1. Treitel, S., and L. Lines, 1982, Linear inverse theory and deconvolution: Geophysics, 47, 1153-1159.

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