# Percent prewhitening

The reasons for prewhitening were discussed in optimum Wiener filters. Consider the single, isolated minimum-phase wavelet in Figure 2.4-24. Keep the operator length and prediction lag constant and vary the percent prewhitening. Note that the effect of varying prewhitening is similar to that of varying the prediction lag; that is, the spectrum increasingly becomes less broadband as the percent prewhitening is increased. Compare Figure 2.4-14 with Figure 2.4-24. Note that prewhitening narrows the spectrum without changing much of the flatness character, while larger prediction lag narrows the spectrum and alters its shape, making it look more like the spectrum of the input seismic wavelet. These characteristics also can be inferred from the shapes of the output wavelets. Prewhitening preserves the spiky character of the output, although it adds a low-amplitude, high-frequency tail (Figure 2.4-24). On the other hand, increasing prediction lag produces a wavelet with a duration equal to the prediction lag (Figure 2.4-14).

The effect of prewhitening on the sparse-spike train seismogram with a known and unknown minimum-phase wavelet is shown in Figures 2.4-25 and 2.4-26, respectively. The effect of prewhitening on deconvolution of the synthetic seismogram obtained from the sonic log (Figure 2.1-1a) is shown in Figures 2.4-27 and 2.4-28 for known and unknown minimum-phase wavelets. Prewhitening tests using the mixed-phase wavelet are shown in Figure 2.4-29. Finally, the combined effects of a prediction lag that is greater than unity and prewhitening for the single, isolated wavelet are shown in Figure 2.4-30. These figures demonstrate that prewhitening narrows the output spectrum, making it band-limited. In particular, the tests in Figures 2.4-24 and 2.4-29 using the single, isolated minimum- and mixed-phase wavelets suggest that spiking deconvolution with some prewhitening is somewhat equivalent to spiking deconvolution without prewhitening followed by post-deconvolution broad band-pass filtering. However, this is not exactly true, for prewhitening still leaves some relatively suppressed energy at the high-frequency end of the spectrum. From Figure 2.4-30, we infer that predictive deconvolution with a prediction lag greater than unity and with some prewhitening yields a result somewhat equivalent to a spiking deconvolution followed by band-pass filtering.

In conclusion, we can say that prewhitening yields a band-limited output. However, the effect is less controllable when compared to varying the prediction lag. By varying prediction lag, we have some idea of the output bandwidth, since it is related to prediction lag. The smaller the prediction lag, the broader the output bandwidth. Prewhitening is used only to ensure that numerical instability in solving for the deconvolution operator (equation 32) is avoided. In practice, typically 0.1 to 1% prewhitening is standard.

 ${\displaystyle {\begin{pmatrix}\beta r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&\beta r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&\beta r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &\beta r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}1\\0\\0\\\vdots \\0\end{pmatrix}},}$ (32)