A systematic study of theoretical relations between spatial correlation and Green's function in one-, two- and three-dimensional random scalar wavefields

It has been demonstrated theoretically and experimentally that Green's function between two receivers can be retrieved from the spatial correlation of an ambient noise field. While studies on the spatial correlation of ambient noises have been mainly conducted in the frequency domain, the retrieval of Green's function has been done in the time domain. Recently, it was pointed out that both studies could be connected via the inverse Fourier transform. In this paper, a systematic study of theoretical relations is conducted between the spatial correlation and the Green's function of wave equation in 1-, 2- and 3-D random scalar fields under an assumption that mutually uncorrelated plane waves are incident on two receivers from various directions. Different relation is obtained for a different dimensional case: Integration, Hilbert transform and differentiation have to be operated on the normalized spatial correlation to retrieve Green's function for 1-, 2- and 3-D, respectively. It is reconfirmed that the Green's function can be retrieved from the spatial correlation between two receivers only for isotropic random fields. The effect of the anisotropy of incident waves on the retrieval of Green's function is evaluated analytically. Even for the anisotropic incident waves, it is possible to retrieve Green's function from the spatial correlation by taking an azimuthal average in 2-D and a spherical average in 3-D.