Windowing and estimation variance in deconvolution

Abstract

Spiking deconvolution operators are computed using the Levinson and orthonormal lattice filter algorithms. By comparing the amplitude and phase spectra of the operators, the differences between the two methods are shown to be strictly tied to the differences in how the algorithms handle the windowing problem. Also, the two methods are equivalent if the windowing problem can be overcome through the use of multipass deconvolution prior to the application of an interpretive wavelet. The most significant effect of windowing is to introduce errors in the estimate of the phase spectrum. When a filter is computed from a single trace, estimation variance in the filter is high. Spatial averaging in the filter design process can overcome a large part of this problem and produce sections with better lateral continuity. The parameters averaged spatially are the autocorrelations and the inner products for the Levinson and orthonormal lattice filter algorithms, respectively. This suggests that short window, time, and space varying deconvolution is undesirable and that the geology is better defined by using operators which result from averaging over several independent estimates. Analysis of the phase spectra of the deconvolution operators computed for synthetic and real data indicates that selection of the interpretive wavelet based on the phase spectrum of the deconvolution operator is the best approach to reject the frequencies which will have a large phase error in the deconvolved section.