Interferometric inversion: a robust approach to linear inverse problems

This note describes an interferometric reformulation of linear wave-based inverse problems, such as inverse source and inverse Born scattering. Instead of fitting data directly, we propose to match cross-correlations and other quadratic data combinations that generalize cross-correlations. This modification exhibits surprising robustness to modeling uncertainties of a kinematic nature. Three optimization formulations are proposed: (1) a simple nonconvex problem, (2) a convex relaxation via lifting and a semidefinite constraint, and (3) a fast, relaxed scheme that empirically bypasses the lack of convexity. We describe sufficient conditions for recovery from the lifted scheme (2). We illustrate the robustness of interferometric inversion on two numerical examples, with uncertainty either in the model velocity or the sensor locations.