Special 2006 | Volume 31 Issue Special | View Issue
This article is about our nonstationary seismic deconvolution algorithm, which we have named Gabor deconvolution and described in a previous RECORDER article (Margrave et al. 2003). We begin by presenting conceptual descriptions of stationary Wiener deconvolution, as it is accomplished by Fourier transform, and nonstationary deconvolution as it is accomplished by Gabor transform. Then we will examine some of the physical models of nonstationary behaviour and discuss how they fit, or in some cases conflict, with our Gabor process. Next we will discuss in some detail the mathematical machinery which we find useful in the analysis and processing of nonstationary signals. Finally we will summarize the present state of our experience with Gabor deconvolution on both synthetic and real seismic data.
Stationary spiking (Wiener) deconvolution
Though we usually speak of stationary spiking deconvolution as Wiener deconvolution, it was actually brought to exploration seismology by Enders Robinson and Sven Treitel (Robinson and Treitel, 1967, and Peacock and Treitel, 1969). Although it was developed in the time domain as a least squares algorithm, we describe here a complete Fourier domain analog that has more obvious linkages to our Gabor process. The stationary seismic trace model is
EQUATION 01
where w is the source signature or wavelet, r is the reflectivity (a time series containing the reflection coefficients of reflectors at their proper two-way traveltimes), and the bullet (•) denotes convolution. If we Fourier transform equation (1) we have
EQUATION 02
Author Biography
Gary Margrave is Associate Professor of Geophysics and Mathematics at the University of Calgary. He is also Associate Director of CREWES and the U of C Site Director for PIMS. Before joining the University of Calgary in 1995, he spent fifteen years with Chevron Corporation where he held a variety of geophysical positions. He obtained his Ph.D. in geophysics from the University of Alberta in 1981. Dr. Margrave’s expertise is in the research, teaching and application of exploration seismology, including seismic data processing, migration and mathematical signal theory. His research interests include signal band estimation, seismic attenuation, deconvolution, imaging and seismic wave propagation.
Michael Lamoureux received a B.Sc. (1982) in Honours Mathematics from the University of Alberta, a M.Sc. (1983) in Mathematics from Stanford University and a Ph.D. (1988) in Mathematics from the University of California, Berkeley. He has been a Professor of Mathematics at the University of Calgary since 1992. His research interests include functional and harmonic analysis, operator algebras, and their applications to mathematical modeling and signal processing, in particular seismic imaging. He is a member of AMS, CMS, and SIAM.