  We study an inverse problem for the wave equation where localized wave
sources in random scattering media are to be determined from time resolved
measurements of the waves at an array of receivers. The sources are far from
the array, so the measurements are affected by cumulative scattering in the
medium, but they are not further than a transport mean free path, which is the
length scale characteristic of the onset of wave diffusion that prohibits
coherent imaging. The inversion is based on the Coherent Interferometric (CINT)
imaging method which mitigates the scattering effects by introducing an
appropriate smoothing operation in the image formation. This smoothing
stabilizes statistically the images, at the expense of their resolution. We
complement the CINT method with a convex ($l_1$) optimization in order to
improve the source localization and obtain quantitative estimates of the source
intensities. We analyze the method in a regime where scattering can be modeled
by large random wavefront distortions, and quantify the accuracy of the
inversion in terms of the spatial separation of individual sources or clusters
of sources. The theoretical predictions are demonstrated with numerical
simulations.
