General formulation of the source extraction problem

Abstract

A general formulation of the problem of extracting information about random wave sources is presented. The formulation places no restrictions upon the complexity of the propagation and applies to any situation where the propagation may be adequately described by a deterministic Green’s function. It is shown that the general problem may be cast into the form of an integral equation, the kernel of which contains the physics of the propagation and the solution of which describes the location and spectral content of all sources. The special case of a homogeneous medium is examined in detail, and it is shown that conventional processing techniques may be viewed as attempts to solve an approximate version of the integral equation. An exact formal solution for the case of an arbitrary medium is derived. This formal solution, although it may not be implemented per se, is nonetheless expressed in such a form that a useful procedure for generating approximate solutions is suggested. This procedure, which is a generalized version of beamforming, may then be used to investigate limitations on the performance of hypothetical systems imposed by only approximate knowledge of the propagation and by observation of the transmitted wave field over only a limited spatial domain.