Fluctuations in diffuse field-field correlations and the emergence of the Green's function in open systems

Recent intense interest in diffuse field correlation functions, with applications to passive imaging in underwater acoustics and seismology, has raised questions about the degree with which a retrieved waveform can be expected to conform to the Green's function, and in particular the degree with which a ray arrival may be discerned. On considering a simple scalar wave model consisting of fields with distributed random sources, the difffuse field-field correlation function R is defined as a sum of correlation integrals, one for each of the many distinct distributed sources. It is then shown that this ensemble of fields has a correlation function with expectation (R) equal to the Green's function. This model also lends itself to calculations of the variance of R, and thus to estimates of the degree to which an R calculated using finite amounts of data will conform to the Green's function. The model predicts that such conformation is strongest at low frequencies. Ray arrivals are detectable if sufficient data have been collected, but the amount of data needed scales in three dimensions with the square of the source-receiver separation, and the square of the frequency. Applications to seismology are discussed.