Convex Recovery From Interferometric Measurements

Abstract

This paper discusses some questions that arise when a linear inverse problem involving Ax = b is reformulated in the interferometric framework, where quadratic combinations of b are considered as data in place of b. First, we show a deterministic recovery result for vectors x from measurements of the form (Ax)[subscript i] [bar over (Ax)[subscript j]] for some left-invertible A. Recovery is exact, or stable in the noisy case, when the couples (i, j) are chosen as edges of a well-connected graph. One possible way of obtaining the solution is as a feasible point of a simple semidefinite program. Furthermore, we show how the proportionality constant in the error estimate depends on the spectral gap of a data-weighted graph Laplacian. Second, we present a new application of this formulation to interferometric waveform inversion, where products of the form (Ax)[subscript i] [bar over (Ax)[subscript j]] in frequency encode generalized cross correlations in time. We present numerical evidence that interferometric inversion does not suffer from the loss of resolution generally associated with interferometric imaging, and can provide added robustness with respect to specific kinds of kinematic uncertainties in the forward model A.