On the Optimum Gain for Estimating the Reverberation Operator from Seismic Data

In seismic data processing various methods of gain control are used in recording the data and to subsequently bring the events in the record up to a nearly uniform level. In carrying out deconvolution, an essential step is estimating the reverberation operator, or, equivalently, its autocorrelation function, from the data. The choice of gain function not only affects the appearance of the data but the reliability of the estimate of the correlation function of the reverberation operator. This paper investigates what choice of gain function yields the most reliable estimate, measured in terms of the signal-to-noise ratio in the estimate. The result, not suprisingly, is that the most reliable estimate is obtained by the gain function which brings the mean-square event strength up to a uniform level. However, this result is true only when certain assumptions are made. The purpose of this paper is to give a rational basis for this intuitively plausible result and to carefully delineate the circumstances under which it is true.