In this paper, we solve for as a time domain convolution operator by minimizing the residual, in a least squares (L2) sense. The degree of spectral matching, essentially the number of degrees of freedom, is then controlled by the length of the time domain operator. By working with a short operator of a similar length to the two wavelets being matched, the operator can provide the ``right amount'' of spectral shaping: a close enough spectral and phase match to compensate for differences in wavelets and residual statics between the two surveys, while avoiding an over-match that can zero out differences in the data sets caused by petrophysical changes during reservoir production.
Figure 3 shows the amplitude spectra for the two datasets before and after matched-filtering. The frequency content of the 1991 spectrum is much wider than the 1979 data, suggesting that the 1991 survey has the higher useful bandwidth. As an initial pre-processing step, we low-pass filtered the 1979 survey to 60 Hz and resampled from 4 ms to 6 ms to coincide with the 1991 survey.
Figure 3 Amplitude spectra before (left) and after (right) matched-filtering. The solid line corresponds to the 1979 survey and the dashed line to the 1991 survey.
The matched-filtering operators, were designed to map the higher quality 1991 survey to the 1979 survey. Figure 4 shows example operators. Separate filters were then designed for each trace, by considering a design window from 0.5 s to 2.0 s depth (above the reservoir zones) and three traces wide in the in-line direction. The filters show a consistent shape in the left-hand side of the Figure, but become noisy in the salt, where there are no reflectors. Just before the salt you can see the static shift associated with the filters change due to the different imaging of the dipping reflectors.
Figure 4 Matched-filters from an in-line slice. The salt begins at approximately 2/3 of the way across the panel, coinciding with the noisy filters.
Close inspection of the matched-filters reveals an average phase-shift between 45 and 90, and an average residual static correction of about one time sample (6 ms). The matched-filters equalize the amplitude spectra of the two datasets, as shown in the right of Figure 3.
As well as matching wavelets and small residual static shifts, a matched-filter also has an associated amplitude correction. However this amplitude correction may be significantly biased by the presence of noise in Rickett et al. (1997), so an additional amplitude balancing step is required.