Estimation of an optimal mixed-phase inverse filter

Abstract

Inverse filtering is applied to seismic data to remove the effect of the wavelet and to obtain an estimate of the reflectivity series. In many cases the wavelet is not known, and only an estimate of its autocorrelation function (ACF) can be computed. Solving the Yule-Walker equations gives the inverse filter which corresponds to a minimum-delay wavelet. When the wavelet is mixed delay, this inverse filter produces a poor result.

By solving the extended Yule-Walker equations with the ACF of lag α on the main diagonal of the filter equations, it is possible to decompose the inverse filter into a finite-length filter convolved with an infinite-length filter. In a previous paper we proposed a mixed-delay inverse filter where the finite-length filter is maximum delay and the infinite-length filter is minimum delay.

Here, we refine this technique by analysing the roots of the Z-transform polynomial of the finite-length filter. By varying the number of roots which are placed inside the unit circle of the mixed-delay inverse filter, at most 2α different filters are obtained. Applying each filter to a small data set (say a CMP gather), we choose the optimal filter to be the one for which the output has the largest Lp-norm, with p=5. This is done for increasing values of α to obtain a final optimal filter. From this optimal filter it is easy to construct the inverse wavelet which may be used as an estimate of the seismic wavelet.

The new procedure has been applied to a synthetic wavelet and to an airgun wavelet to test its performance, and also to verify that the reconstructed wavelet is close to the original wavelet. The algorithm has also been applied to prestack marine seismic data, resulting in an improved stacked section compared with the one obtained by using a minimum-delay filter.