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.

翻译：在被动成像中，研究者试图从观测到的随机激发波动解的相关性中，重建波动方程中的某些系数。目前针对这类反问题提出的许多方法仅能提供定性结果，例如试图识别扰动的支撑集。主要挑战在于：在预处理步骤中从原始数据计算相关性时会增加维度，且通常存在极差的逐点信噪比。本文提出的方法同时解决了这两项挑战：它仅对原始数据进行处理，同时隐式利用相关性数据中包含的全部信息，并通过迭代提供定量估计与收敛性保证。我们的研究受日震全息术启发——这是一种成熟的成像方法，用于绘制太阳内部的不均匀性和流动。研究表明，经典日震全息术中的反传播可解释为将太阳内部特性映射至太阳表面相关性数据的算子的Fréchet导数的伴随算子。本文针对被动成像问题建立的理论与数值框架，将日震全息术扩展至非线性问题，并实现了定量重建。我们在均匀介质中给出了概念验证。