Retrieval of the green's function from cross correlation:: The canonical elastic problem

In realistic materials, multiple scattering takes place and average field intensities or energy densities follow diffusive processes. Multiple P to S energy conversions by the random inhomogeneities result in equipartition of elastic waves, which means that in the phase space the available elastic energy is distributed among all the possible states of P and S waves, with equal amounts in average. In such diffusive regimes, the P to S energy ratio equilibrates in a universal way independent of the particular details of the scattering. We study the canonical problem of isotropic plane waves in an elastic medium and show that the Fourier transform of azimuthal average of the cross correlation of motion between two points within an elastic medium is proportional to the imaginary part of the exact Green's tensor function between these points, provided the energy ratio E-S/E-P is the one predicted by equipartition in two and three dimensions, respectively. These results clearly show that equipartition is a necessary condition to retrieve the exact Green's function from correlations of the elastic field. However, even if there is not an equipartitioned regime and correlations do not allow to retrieve precisely the exact Green's function, the correlations may provide valuable results of physical significance by reconstructing specific arrivals.