OPTIMUM MULTICHANNEL VELOCITY FILTERS

Abstract

Optimum systems have been developed to correspond to the sub‐optimum moveout discrimination systems presented previously by several authors. The seismic data on the lth trace is assumed to be additive signal S with moveout [Formula: see text], coherent noise N with moveout [Formula: see text], and incoherent noise [Formula: see text], expressed [Formula: see text] where S, N, and [Formula: see text] are independent, second order stationary random processes and [Formula: see text] and [Formula: see text] are random variables with prescribed probability density functions. The signal estimate S⁁ is produced by filtering each trace with its corresponding filter [Formula: see text] and summing the outputs [Formula: see text] We choose the system of filters [Formula: see text] to make the signal estimate optimum in the Wiener sense (minimum mean‐square error of the signal ensemble). For the special cases discussed, the moveouts are linear functions of the trace number l determined by the moveout/trace τ for signal and [Formula: see text] for noise. Thus, the optimum system is determined by the probability densities of τ and [Formula: see text] together with the noise/signal power spectrum ratios [Formula: see text] and [Formula: see text]. In comparison, suboptimum systems are controlled completely by the cut‐off moveout/trace [Formula: see text]. Events whose moveout/trace falls within [Formula: see text] of the expected dip moveout/trace are accepted, and those falling outside this range are suppressed. Suboptimum systems can be derived from optimum systems by choosing probability densities for τ and [Formula: see text] that are uniform within the above ranges and letting [Formula: see text] be very large. Optimum systems have increased flexibility over suboptimum systems due to control over the probability density functions and the power spectrum ratios and allow increased noise suppression in selected regions of f‐k space.