SIAM Journal on Imaging Sciences, Volume 13, Issue 2, Page 685-723, January 2020.
We study an inverse scattering problem for a generic hyperbolic system of equations with an unknown coefficient called the reflectivity. The solution of the system models waves (sound, electromagnetic, or elastic), and the reflectivity models unknown scatterers embedded in a smooth and known medium. The inverse problem is to determine the reflectivity from the time resolved scattering matrix (the data) measured by an array of sensors. We introduce a novel inversion method, based on a reduced order model (ROM) of an operator called a wave propagator, because it maps the wave from one time instant to the next, at an interval corresponding to the discrete time sampling of the data. The wave propagator is unknown in the inverse problem, but the ROM can be computed directly from the data. By construction, the ROM inherits key properties of the wave propagator, which facilitate the estimation of the reflectivity. The ROM was introduced previously and was used for two purposes: (1) to map the scattering matrix to that corresponding to the single scattering (Born) approximation and (2) to image, i.e., obtain a qualitative estimate of the support of the reflectivity. Here we study further the ROM and show that it corresponds to a Galerkin projection of the wave propagator. The Galerkin framework is useful for proving properties of the ROM that are used in the new inversion method which seeks a quantitative estimate of the reflectivity.

中文翻译：

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逆散射的降阶模型方法

SIAM影像科学杂志，第13卷，第2期，第685-723页，2020年1月。
我们研究具有未知系数（称为反射率）的一般双曲方程组的逆散射问题。该系统的解决方案对波（声音，电磁波或弹性波）进行建模，而反射率对嵌入在已知平滑介质中的未知散射体进行建模。反问题是从由传感器阵列测量的时间分辨散射矩阵（数据）确定反射率。我们介绍一种新颖的反演方法，该方法基于称为波传播器的运算符的降阶模型（ROM），因为它以与数据的离散时间采样相对应的时间间隔将波从一个时刻映射到下一个时刻。波传播器在逆问题中是未知的，但是可以直接从数据中计算出ROM。通过施工，ROM继承了波传播器的关键特性，这有助于估计反射率。ROM是先前介绍的，用于两个目的：（1）将散射矩阵映射到与单次散射（Born）近似相对应的矩阵，以及（2）进行成像，即获得对反射率支持的定性估计。在这里，我们进一步研究ROM，并显示它对应于波传播器的Galerkin投影。Galerkin框架可用于证明新反演方法中使用的ROM属性，该方法可对反射率进行定量估计。（1）将散射矩阵映射到与单次散射（Born）近似相对应的矩阵，并将（2）映射到图像，即获得反射率支持的定性估计。在这里，我们进一步研究ROM，并显示它对应于波传播器的Galerkin投影。Galerkin框架可用于证明新反演方法中使用的ROM属性，该方法可对反射率进行定量估计。（1）将散射矩阵映射到与单次散射（Born）近似相对应的矩阵，并将（2）映射到图像，即获得反射率支持的定性估计。在这里，我们进一步研究ROM，并显示它对应于波传播器的Galerkin投影。Galerkin框架可用于证明新反演方法中使用的ROM属性，该方法可对反射率进行定量估计。