An update on Gabor deconvolution

Gabor deconvolution has been updated and a new ProMAX module is released. The updates are: (1) a new method of spectral smoothing called hyperbolic smoothing; (2) a Gabor transform using compactly supported windows that improves run times by one to two orders of magnitude; and (3) a post-deconvolution time-variant bandpass filter whose maximum frequency tracks along a hyperbola in the time-frequency plane. We discuss the technical details of these improvements and present an overview of the new ProMAX module. INTRODUCTION Time-frequency analysis is a hot topic in applied mathematics with many new textbooks and monographs having appeared in recent years. Seismic data is particularly appropriate for this analysis because of its inherent nonstationarity, characterized by the progressive loss of high frequencies with increasing time. However, it is usually required to go beyond the simple analysis of nonstationarity and to actually process the data to render it more nearly stationary. Usually, this means performing some kind of an operation on the time-frequency decomposition (i.e. filtering) and then synthesizing a new signal by recombining the altered time-frequency components. Of the popular timefrequency analysis techniques, the wavelet transform is probably most popular; however, there is a great deal of interest in windowed Fourier transforms that have become known as Gabor transforms (Gabor, 1946, Feichtinger and Strohmer, 1998, Mertins, 1999). One reason that the Gabor transform remains popular in spite of the many advantages of the wavelet transform is that the latter does not diagonalize a local convolution operator while the former does. Last year, we showed the fundamental concepts behind a new nonstationary deconvolution technique called Gabor deconvolution (Margrave and Lamoureux, 2001). As the name suggests, this method uses the Gabor transform to accomplish a timefrequency decomposition of a seismic trace. This Gabor spectrum is then processed in such a way that the effects of anelastic attenuation and the source signature are approximately removed. In its simplest form, this involves smoothing the magnitude of the Gabor spectrum to estimate the Gabor magnitude spectrum of the propagating wavelet. This is then combined with a minimum phase spectrum, computed in the usual way with the Hilbert transform, to completely specify the propagating wavelet. Then the Gabor spectrum of the seismic signal is pointwise divided by the estimated Gabor spectrum of the propagating wavelet. If the signal were noiseless, then an inverse Gabor transform would complete the process. However, we have found it necessary to precede the inverse Gabor transform with a time-variant bandpass filter to limit the whitened spectrum to the supposed signal band.