In this work, we propose a new approach to recover sparse obstacles q from measurements b of the scattered wave field in a slowly varying medium. In the Born approximation the problem is linear (Aq = b) but the unknown wave speed variation induces a phase error on the data b. Classical approaches fail at recovering the scatters in this case. Hence we propose an alternative inversion that consists in considering an interferometric quadratic problem

The interest of such system comes from the better robustness property of some cross-correlations b k b with respect to the medium uncertainties.