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 powerful 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.

翻译：在无源成像中，人们试图从观测到的波动方程随机激发解的相关性中，重构该波动方程中的某些系数。迄今为止，针对这类逆问题提出的许多方法仅具有定性性质，例如仅尝试识别扰动的支撑区域。主要挑战在于：在预处理步骤中从原始数据计算相关性时维度的增加，以及通常极低的逐点信噪比。本文提出一种同时应对这两大挑战的方法：该方法仅基于原始数据运行，同时隐式利用相关性数据中包含的全部信息，并通过迭代实现定量估计与收敛。我们的研究受日震全息术启发——这是一种用于绘制太阳内部非均匀性与流场的强大成像方法。研究表明，经典日震全息术中使用的反向传播可解释为将太阳内部性质映射至表面相关性数据的算子的弗雷歇导数的伴随算子。本文发展的无源成像问题理论与数值框架将日震全息术扩展至非线性问题，并实现定量重构。我们在均匀介质中给出了概念验证。