The Gabor transform, pseudodifferential operators, and seismic deconvolution

Using the Gabor transform, we describe a technique to correct reflection seismograms for the effects of anelastic attenuation and source signature. Essentially we build a nonstationary deconvolution filter, estimated from the seismic data itself and applied by multiplication in the Gabor domain. In more detail, we estimate the time-frequency magnitude spectrum of the attenuation process and the source signature from the Gabor transform of a seismic signal; the phase then follows under the assumption of minimum phase. The deconvolution filter is the inverse of this estimate and is applied to the Gabor transform of the seismic signal by multiplication. An inverse Gabor transform completes the algorithm and gives a very high resolution estimate for the reflectivity of the earth. As a justification for our algorithm we present a model for a seismic trace that uses a pseudodifferential operator to describe anelastic attenuation. We then argue that the Gabor transform approximately renders this pseudodifferential operator expression into a product of time-frequency dependent factors. Attenuation processes and source signature are removed by multiplication with estimates of their inverses. With both real and synthetic data we illustrate the effectiveness of Gabor deconvolution and demonstrate its superiority over the established Wiener deconvolution.