MORE ROBUST WAVELET DECON

Our inverse wavelet is computed at a coarser sampling interval. After being filled with zeroes to mimic the original sampling interval, we have a higher Nyquist frequency. Frequencies higher than the Nyquist frequency of the inverse wavelet are computed, showing the duplicate spectrum. The order of duplication is the order of the subsampling of the autocorrelation. When the inverse wavelet is convolved with the original trace, the duplicate spectrum amplifies random noise. The cure is to do lowpass filtering of the inverse wavelet to remove the duplicate spectrum. Figure 5 presents the lowpass filtered version of the inverse wavelet of Figure 3, and its spectrum. Convolving this inverse wavelet with the seismogram will not generate the energy near Nyquist that was present in the spectrum shown in Figure 2. The output section of this processing looks roughly the same as the previous deconvolved section; we do not display it here.

Lowpass filtering increases the number of non-zero points in the inverse wavelet filter. Thus it increases the processing cost, while only slightly improving the results. Therefore, we propose not doing the lowpass filtering, leaving the slightly enhanced noise to be suppressed by CDP stacking, which has the effect of lowpass filtering. CDP stacking also attenuates random noise that is buried in the wavelet spectrum band, which could have been enhanced by the wavelet decon.

Gapped predictive error decon (Peacock and Treitel, 1969) does not have good control between wavelet compression and random noise enhancement. For this reason, robust wavelet decon may be a better approach. Interested readers can find classic papers describing in detail the pros and cons of various deconvolution schemes in a monograph ``Deconvolution'' edited by Webster.