Robust nonlinear processing of active array data in inverse scattering via truncated reduced order models

TL;DR

This paper presents a data-driven nonlinear processing algorithm for active array data in inverse scattering, transforming complex multiple scattering data into a linearized form suitable for standard imaging techniques.

Contribution

It introduces a novel reduced order model-based algorithm that preprocesses array data to approximate the Born model without prior medium knowledge, applicable to various wave types.

Findings

Transforms array data to Born approximation for improved imaging

Applicable to acoustic, electromagnetic, and elastic waves

Balances stability and accuracy through regularization

Abstract

We introduce a novel algorithm for nonlinear processing of data gathered by an active array of sensors which probes a medium with pulses and measures the resulting waves. The algorithm is motivated by the application of array imaging. We describe it for a generic hyperbolic system that applies to acoustic, electromagnetic or elastic waves in a scattering medium modeled by an unknown coefficient called the reflectivity. The goal of imaging is to invert the nonlinear mapping from the reflectivity to the array data. Many existing imaging methodologies ignore the nonlinearity i.e., operate under the assumption that the Born (single scattering) approximation is accurate. This leads to image artifacts when multiple scattering is significant. Our algorithm seeks to transform the array data to those corresponding to the Born approximation, so it can be used as a pre-processing step for any linear inversion method. The nonlinear data transformation algorithm is based on a reduced order model defined by a proxy wave propagator operator that has four important properties. First, it is data driven, meaning that it is constructed from the data alone, with no knowledge of the medium. Second, it can be factorized in two operators that have an approximately affine dependence on the unknown reflectivity. This allows the computation of the Fr\'{e}chet derivative of the reflectivity to the data mapping which gives the Born approximation. Third, the algorithm involves regularization which balances numerical stability and data fitting with accuracy of the order of the standard deviation of additive data noise. Fourth, the algebraic nature of the algorithm makes it applicable to scalar (acoustic) and vectorial (elastic, electromagnetic) wave data without any specific modifications.