Nonstationarity was discussed in detail in the entries for gain applications and the convolutional model in the time domain. The time-variant character of the seismic wavelet (Figure 2.1-2) often requires a multiwindow deconvolution. Figure 2.6-4 is a field record that was deconvolved by using three time gates. The autocorrelograms from gates 1, 2, and 3 are shown in Figure 2.6-5. Note the difference in character of the reverberatory energy from one gate to another. The shallow gate (1) has more high-frequency signal than the middle gate (2); while the middle gate has more high-frequency signal than the deeper gate (3). For best results, we must design different deconvolution operators from different parts of the record and apply them to the corresponding time gates. Up to three windows usually are sufficient to handle the nonstationary character of the seismic signal.
Figure 2.6-4 Three-window deconvolution. The solid bars indicate window boundaries. With data from each window, a deconvolution operator is designed and applied to the data in that window. The operators are blended across the window boundaries. (a) Input gather. Deconvolution using operator length of 160 ms and prediction lags of (b) 4 ms, (c) 12 ms, (d) 32 ms.
Figure 2.6-7 Three-window deconvolution on the same shot record as in Figure 2.6-6. In this case, there is no significant difference between the characters of the autocorrelograms estimated from three windows. (a) Input gather; (b) autocorrelograms before and after spiking deconvolution; (c) three-window spiking deconvolution on (a); (d) band-pass filtering on (c).
Another example of single- and multiwindow deconvolution is shown in Figures 2.6-6 and 2.6-7. Here, autocorrelograms from different gates do not show significant variations. Therefore, it probably does not make any difference whether a single or multigate deconvolution is used. In Figures 2.6-6 and 2.6-7, the record is shown after deconvolution followed by a wide bandpass filter application. Since the amplitude spectrum of the input data is flattened as a result of spiking deconvolution, both the high-frequency ambient noise as much as the high-frequency components of the signal are boosted. Therefore, the output of spiking deconvolution often is filtered with a wide band-pass operator.
A practical problem with time-variant deconvolution is limiting design gates to small time windows. Consider, for instance, a three-window deconvolution of a 5s data. This means that at best an average gate kength of 1.5 s at near offset and less than 1 s at far offset can be used to design a deconvolution operator. To attain good statistics in an autocorrelation estimate, an operator length no more than one-eighth to one-tenth of the design gate, say 150 ms, should be considered. Hence, if you need to use a longer operator, time-variant deconvolution may have limited effectiveness in attenuating reverberations and short-period multiples. A way to account for nonstationarity while avoiding the short-operator effect of multiwindow deconvolution follows.
- Time-variant spectral whitening
- Frequency-domain deconvolution
- Inverse Q filtering
- Deconvolution strategies
- The problem of nonstationarity