The inverse problem for random sources

Abstract

The problem of deducing the statistical structure of a localized random source ρ (r) of the reduced wave equation from measurements of the field external to the source is addressed for the case when the measurements yield the autocorrelation function of the field at all pairs of points exterior to the source volume and the quantity to be determined is the source’s autocorrelation function R_ρ(r₁,r₂) =〈ρ* (r₁) ρ (r₂) 〉.This problem is shown to be equivalent to that of determining R_ρ from the autocorrelation function of the field’s radiation pattern and is found, in general, not to admit a unique solution due to the possible existence of nonradiating sources within the source volume. Notable exceptions are the class of delta correlated (incoherent) sources whose intensity profiles are shown to be uniquely determined from the data and the class of quasihomogeneous sources whose coherence properties can be determined if their intensity profiles are known and vice versa.