Quantitative passive imaging by iterative holography: The example of helioseismic holography

TL;DR

This paper introduces an iterative, quantitative passive imaging method based on helioseismic holography, addressing challenges of high dimensionality and low signal-to-noise ratio, enabling nonlinear and quantitative reconstructions from primary data.

Contribution

It develops a new iterative approach that uses only primary data to achieve quantitative, nonlinear imaging, extending helioseismic holography beyond qualitative support identification.

Findings

Method successfully reconstructs interior properties in uniform media.

Framework extends helioseismic holography to nonlinear regimes.

Provides convergence guarantees for the iterative process.

Abstract

In passive imaging, one attempts to reconstruct some coefficients in a wave equation from correlations of observed randomly excited solutions to this wave equation. Many methods proposed for this class of inverse problem so far are only qualitative, e.g., trying to identify the support of a perturbation. Major challenges are the increase in dimensionality when computing correlations from primary data in a preprocessing step, and often very poor pointwise signal-to-noise ratios. In this paper, we propose an approach that addresses both of these challenges: It works only on the primary data while implicitly using the full information contained in the correlation data, and it provides quantitative estimates and convergence by iteration. Our work is motivated by helioseismic holography, a well-established imaging method to map heterogenities and flows in the solar interior. We show that the back-propagation used in classical helioseismic holography can be interpreted as the adjoint of the Fr\'echet derivative of the operator which maps the properties of the solar interior to the correlation data on the solar surface. The theoretical and numerical framework for passive imaging problems developed in this paper extends helioseismic holography to nonlinear problems and allows for quantitative reconstructions. We present a proof of concept in uniform media.