  We consider imaging in a scattering medium where the illumination goes
through this medium but there is also an auxiliary, passive receiver array that
is near the object to be imaged. Instead of imaging with the source-receiver
array on the far side of the object we image with the data of the passive array
on the near side of the object. The imaging is done with travel time migration
using the cross correlations of the passive array data. We showed in [J.
Garnier and G. Papanicolaou, Inverse Problems {28} (2012), 075002] that if (i)
the source array is infinite, (ii) the scattering medium is modeled by either
an isotropic random medium in the paraxial regime or a randomly layered medium,
and (iii) the medium between the auxiliary array and the object to be imaged is
homogeneous, then imaging with cross correlations completely eliminates the
effects of the random medium. It is as if we imaged with an active array,
instead of a passive one, near the object. The purpose of this paper is to
analyze the resolution of the image when both the source array and the passive
receiver array are finite. We show with a detailed analysis that for isotropic
random media in the paraxial regime, imaging not only is not affected by the
inhomogeneities but the resolution can in fact be enhanced. This is because the
random medium can increase the diversity of the illumination. We also show
analytically that this will not happen in a randomly layered medium, and there
may be some loss of resolution in this case.
