General representation theorem for perturbed media and application to Green's function retrieval for scattering problems

Green's function reconstruction relies on representation theorems. For acoustic waves, it has been shown theoretically and observationally that a representation theorem of the correlation-type leads to the retrieval of the Green's function by cross-correlating fluctuations recorded at two locations and excited by uncorrelated sources. We extend the theory to any system that satisfies a linear partial differential equation and define an 'interferometric operation' that is more general than cross-correlation for the reconstruction. We analyse Green's function reconstruction for perturbed media and establish a representation theorem specifically for field perturbations. That representation is then applied to the general treatment of scattering problems, enabling interpretation of the contributions to Green's function reconstruction in terms of direct and scattered waves. Perhaps surprising, Green's functions that account for scattered waves cannot be reconstructed from scattered waves alone. For acoustic waves, retrieval of scattered waves also requires cross-correlating direct and scattered waves at receiver locations. The addition of cross-correlated scattered waves with themselves is necessary to cancel the spurious events that contaminate the retrieval of scattered waves from the cross-correlation of direct with scattered waves. We illustrate these concepts with numerical examples for the case of an open scattering medium. The same reasoning holds for the retrieval of any type of perturbations and can be applied to perturbation problems such as electromagnetic waves in conductive media and elastic waves in heterogeneous media.