Estimation of an optimal mixed-phase inverse filter

Inverse filtering is applied to seismic data to remove the effect of the wa velet and to obtain an estimate of the reflectivity series. In many cases t he wavelet is not known, and only an estimate of its autocorrelation functi on (ACF) can be computed. Solving the Yule-Walker equations gives the inver se 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 alpha 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-lengt h filter. In a previous paper we proposed a mixed-delay inverse filter wher e 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 po lynomial 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 mos t 2(alpha) 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 L-p-norm, with p=5. This is done for incr easing values of alpha 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 appli ed to prestack marine seismic data, resulting in an improved stacked sectio n compared with the one obtained by using a minimum-delay filter.