Seismic interferometry using far-field correlation is a technique to obtain the Green's function between two receivers using passive wave-field recordings, and often it is under the theoretical assumption that sources are in the far field when in fact they may not be. Using a heterogeneous medium enclosed by a closed boundary on which the sources are fired, we offer two views on the meaning of the far-field correlation. In the intrinsic view, the validity of the far-field correlation is based on the far-field approximation for wave scattering, and it is investigated by comparing various physical dimensions in the heterogeneous interior, regardless of the exterior. However, the extrinsic view centers on the medium properties in the exterior only without considering the interior. Previous studies showed that, with the correct scattering model and complete source coverage, no spurious arrivals should be generated if the illuminating sources are in the far field. We investigate the case of near-field cross correlation. This problem is considered in the context of two-dimensional space with a single embedded scatterer that is represented by a cylindrical inclusion with small radius. An analytical solution for the scattered wave is computed. However, if the sources are not in the far field, the cross-correlation kernel must be used to eliminate the spurious arrival, in addition to the correct scattering model and full source aperture. The kernel accounts for the near-field illumination of the region.
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