Simultaneous Spherical Divergence Correction and Optima Deconvolution

The effects of spherical divergence on reflection seismograms destroys the stationarity assumption implicit in Weiner smoothing by distorting the amplitudes of the reflection coefficient sequence. The traditional correction for spherical divergence consists of an immediate scaling of the observed signal by an expected attenuation factor. It is this correction that violates the stationarity assumption, for it distorts the amplitudes of the observation noise process. Attempts to adapt the Weiner smoother to time-varying signals by time gating are cumbersome and suboptimal. In this paper, we show how to implement a Kalman deconvolution smoother which compensates for the effects of spherical divergence. The implementation is simple and straightforward, and estimates of the reflection coefficient sequence are optimal and corrected in amplitude. A computational example which compares our approach to an ad hoc Weiner smoother is provided for a simulated seismic signal. By means of this example, we demonstrate better performance for our approach over the Weiner smoother approach.