Reduced Order Model Approach to Inverse Scattering

TL;DR

This paper introduces a reduced order model (ROM) approach for inverse scattering problems, enabling quantitative imaging of unknown reflectivity by directly deriving the ROM from measured data and leveraging Galerkin projection properties.

Contribution

The paper develops a novel ROM-based inversion method that provides quantitative estimates of reflectivity, extending previous qualitative imaging techniques using Galerkin projection analysis.

Findings

ROM can be computed directly from data

ROM inherits key properties of the wave propagator

New method yields quantitative reflectivity estimates

Abstract

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 wave propagator, because it maps the wave from one time instant to the next, at 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.