Abstract

Passive imaging problems, arising in seismology, ocean acoustics, ultrasonics, and local helioseismology, among others, consist in finding coefficients of wave equations from correlation data of measured randomly excited waves. Most of the methods used in this field so far only yield qualitative results, but no quantitative information. The present study has mainly been motivated by helioseismic holography, a powerful method for feature detection in the interior of at far-side of the Sun, which is successfully used in space weather prediction, among others. In helioseismic holography, as for many other methods, improvements are mainly achieved by clever choices of the backward propagators. In this work we show that commonly used back-propagators can be interpreted as the adjoints of Fréchet derivative of forward operators mapping unknown interior coefficients to covariance operators of wave fields on the surface. Based on this interpretation, we can achieve improvements and convergence for nonlinear inverse problems by iteration rather than improved back-propagation, and in particular turn helioseismic holography into a quantitative imaging method. Our approach uses the full information contained in cross-correlation data implicitly without the need of pre-computing cross-correlations explicitly, which is typically infeasible due to the high dimensionality. We show results of the proposed method for the estimation of meridional flows and differential rotation of the Sun.Finally, we show uniqueness results for some passive imaging problems in the Sun.