Abstract

This work is motivated by an inverse problem for the acoustic wave equation, where an array of sensors probes an unknown medium with pulses and measures the scattered waves. The goal of the inversion is to determine from these measurements the structure of the scattering medium, modeled by a spatially varying acoustic impedance function. Many inversion algorithms assume that the mapping from the unknown impedance to the scattered waves is approximately linear. This linearization, known as the Born approximation, is not accurate in strongly scattering media, where the waves undergo multiple reflections before they reach the sensors in the array. This results in the artifacts in the reconstructions of the impedance obtained via linearized approaches (e.g., various migration algorithms). Our main result is a novel, linear-algebraic algorithm that uses a reduced order model (ROM) to map the multiply scattered data to those corresponding to the single scattering (Born) model, the so-called Data-to-Born transform. The ROM is a proxy for the wave propagator operator, that propagates the wave in the unknown medium over the duration of the time sampling interval. Its construction is based only on the measurements at the sensors in the array.  The output of the algorithm can be passed to any off-the-shelf inversion software that incorporates state of the art linear inversion algorithms to reconstruct the unknown acoustic impedance.