The far‐field approximation in seismic interferometry

Abstract

Green’s function retrieval through seismic interferometry can be derived based on the Rayleigh’s reciprocity theorem of the correlation type. However, the common underlying assumption of theory that the sources are in the far field limits the technology to extracting correctly only the high frequency part of the Green’s function in an open system. This critical approximation can be eliminated using the exact boundary integral equation method. Without the farfield approximation, a crosscorrelation kernel is proposed to recover the exact Green’s function. Symmetric difference kernels are analytically constructed for sources on a plane or on a circle and can be reduced to the known Dirac delta kernel under the far-field approximation. Spurious arrivals can be artificially generated in the construction of Green’s function using wavefield crosscorrelation if a small scattering object is present in the medium. For far-field sources, if the point scattering model is correct, the spurious arrival can be eliminated; however, if the sources are not in the far field, the crosscorrelation kernel must also be used.