THE CORRECTIVE GRADIENT PROJECTION METHOD AND RELATED ALGORITHMS APPLIED TO SEISMIC ARRAY PROCESSING

Abstract

A general iterative method is described for the problem of optimizing a nonlinear function of multiple parameters subject to a system of linear constraint equations. This scheme, derived by Frost (1972), is called the “corrective gradient projection (CGP)” method. It is obtained by modifying Rosen’s (1960) “gradient projection (GP)” method to correct for any deviation of the parameters from the constraint equations. The CGP method can be applied to the problem of multichannel tapped‐delay‐line (TDL) sensor array processing as used in the fields of seismic, radar and acoustic communications. A variety of algorithms associated with signal processing problems—such as those developed by Frost (1972), Rosen (1960), Booker and Ong (1971), Widrow et al. (1967), and Griffiths (1969)—can be derived from or related to the CGP method. They are obtained by making some basic assumptions and statistical approximations. Comparative simulation experiments have been made using a digital computer to examine the performance of some of the algorithms for real seismic data. The algorithms examined are the constrained (unbiased) adaptive algorithm of Frost (1972), the unconstrained adaptive algorithm of Widrow et al. (1967) and the constrained (unbiased) optimum processor (Claerbout, 1968; Frost, 1972). The adaptive algorithms are capable of discriminating against noise signals in real time. The Frost algorithm strictly maintains a specified constrained frequency response for the array, while the Widrow algorithm alters the equivalent array response during the adaptations. For the Frost algorithm, the adaptive processor approaches the optimum one as the number of adaptations increases because it progressively learns the noise statistics.