We start with a single, isolated minimum-phase wavelet as in trace (b) of Figure 2.4-6. Assumptions 1 through 5 are satisfied for this wavelet. The ideal result of spiking deconvolution is a zero-lag spike, as indicated by trace (a). In this and the following numerical analyses, we refer to the autocorrelogram and amplitude spectrum (plotted with linear scale) of the output from each deconvolution test to better evaluate the results. In Figure 2.4-6 and the following figures, n, α, and ε refer to operator length of the prediction filter, prediction lag, and percent prewhitening, respectively. The length of the prediction error filter then is n + α.
In Figure 2.4-6, prediction lag is unity and equal to the 2-ms sampling rate, prewhitening is 0%, and operator length varies as indicated in the figure. Short operators yield spikes with small-amplitude and relatively high-frequency tails. The 128-ms-long operator gives an almost perfect spike output. Longer operators whiten the spectrum further, bringing it closer to the spectrum of the impulse response.
The action of spiking deconvolution on the seismogram derived by convolving the minimum-phase wavelet with a sparse-spike series is similar (Figure 2.4-7) to the case of the single isolated wavelet (Figure 2.4-6). Recall that spiking deconvolution basically is inverse filtering where the operator is the least-squares inverse of the seismic wavelet. Therefore, an increasingly better result should be obtained when more and more coefficients are included in the inverse filter.
Now consider the real situation of an unknown source wavelet. Based on assumption 6, autocorrelation of the input seismogram rather than that of the seismic wavelet is used to design the deconvolution operator. The result of using the trace rather than the wavelet autocorrelation is shown in Figure 2.4-8. Deconvolution recovers the gross aspects of the spike series, trace (a). However, note that the deconvolved traces have spurious small-amplitude spikes trailing each of the real spikes. We see that increasing operator length does not indefinitely improve the results; on the contrary, more and more spurious spikes are introduced.
Very short operators produce the same type of noise spikes as in Figures 2.4-7 and 2.4-8. Examine the series of deconvolution tests in Figure 2.4-8 and note that the 94-ms operator does the best job. Compare the autocorrelogram of trace (b) in Figure 2.4-8 with that of trace (b) in Figure 2.4-6. Note that only the first 100-ms portion represents the autocorrelation of the source wavelet. This explains why the 94-ms operator worked best; that is, the autocorrelation lags of trace (b) in Figure 2.4-8 beyond 94 ms do not represent the seismic wavelet.
Consider the seismogram in Figure 2.4-9, where the wavelet is assumed to be unknown. Deconvolution has restored the spikes that correspond to major reflections in the impulse response as in trace (b) with some success. The 64-ms operator is a good choice.
The mixed-phase wavelet in Figure 2.4-10 shows what can happen when assumption 7 is violated. The wavelet in Figure 2.4-6 is the minimum-phase equivalent of the mixed-phase wavelet in Figure 2.4-10. Both wavelets have the same autocorrelograms and amplitude spectra. Hence, the deconvolution operators for both wavelets are identical. Because the minimum-phase assumption was violated, deconvolution does not convert the mixed-phase wavelet to a perfect spike. Instead, the deconvolved output is a complicated high-frequency wavelet. Also note that the dominant peak in the output is negative, while the impulse response has a positive spike. This difference in the sign can happen when a mixed-phase wavelet is deconvolved. Increasing the operator length further whitens the spectrum; however, the 128-ms operator yields a result that cannot be improved further by longer operators.
The seismogram obtained from the mixed-phase wavelet and the sparse-spike series (used in the preceding figures) is shown in Figure 2.4-11. The 94-ms operator gives the best result. This also is the case in Figure 2.4-12, where both assumptions 6 and 7 are violated. The situation with the seismogram in Figure 2.4-13 is not very good. The spikes that correspond to major reflections in the impulse response were restored; however, there are some timing errors and polarity reversals. (Compare these results with those in Figure 2.4-9 for the events between 0.2 and 0.3 s and 0.6 and 0.7 s.) The 64-ms operator gives an output that cannot be improved by longer operators.
What kind of operator length should be used for spiking deconvolution? To select an operator length, ideally we want to use the autocorrelation of the unknown seismic wavelet. Fortunately, the autocorrelation of the input seismogram has the characteristics of the wavelet autocorrelation (assumption 6). Therefore, it seems appropriate that we should use part of the autocorrelation obtained from the input seismogram that most resembles the autocorrelation of the unknown seismic wavelet. That part is the first transient zone in the autocorrelation, as seen by comparing the autocorrelations of trace (b) in Figure 2.4-6 and trace (c) in Figure 2.4-9. The autocorrelations of trace (b) in Figure 2.4-10 and trace (c) in Figure 2.4-13 suggest the same principle.