Joseph geology and seismic deconvolution

Abstract

Common practice in seismic deconvolution is to assume that the reflection sequence is uncorrelated, that is, that the sequence has a white power spectrum and a delta function autocorrelation. A white spectrum implies that the acoustic impedance function has a power spectrum proportional to [Formula: see text], which is characteristic of a nonstationary Brownian process (f is frequency). However, the maximum power spectrum permissible for the acoustic impedance function is 1/f; we call a spectrum of this kind a Joseph spectrum. A Joseph spectrum corresponds to a reflection sequence with a power spectrum proportional to f and a negative autocorrelation at small lags. Joseph spectrum behavior for reflection sequences has been seen before and we show it again in a well off Newfoundland and in two wells from Quebec. If the power spectrum is proportional to f, then the first term of the discretized autocorrelation function is −0.405 of the zero‐lag term and higher terms are negligible. We construct a Joseph filter analogous to the prediction error filter (PEF) using this extra term. The method requires one additional term in the normal equations, equations which are solved iteratively. When used to deconvolve artificial seismograms from the wells, the Joseph filter recovered the reflection sequences with as little as one‐tenth the error of the PEF.