Reduced Order Model Approach to Inverse Scattering

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. * One can also consider truncation of the whole space, as long as the medium is known and non-scattering on one side of the array of sensors. P(q) = cos τ L(q)L(q) T .( 1.9) The ROM is defined by a pair of matrices P ROM (q) ∈ R nm×nm and b ROM ∈ R nm×m , which are proxies of P(q) and the initial wave b(x). These matrices are calculated from the data (1.8) (i.e., the ROM is data-driven) and they capture physical aspects of the wave propagation that are needed for inversion. The ROM was introduced in [12,6] and was used in [13] for imaging, and in [5] for transforming the data (1.8) to that corresponding to the single scattering (Born) approximation. The new results in this paper are:1. We show that the ROM propagator P ROM (q) is a Galerkin projection of the operator (1.9), and use the Galerkin framework to prove properties of the ROM that facilitate the solution of the inverse scattering problem. † The kinematic model (smooth wave speed) appears in the coefficients of the operators L(q) and L(q) T (see [6] and sections 3-4). We suppress the dependence on the known kinematic model in our notation.3

Reduced order model approach for imaging with waves

“…To our knowledge, the first sensor array data driven ROM for wave propagation was introduced in [13] for the one-dimensional wave equation. The extension to higher dimensions was obtained in [7] and was analyzed in [8]. The latter study showed that wave propagation can be viewed as a discrete time dynamical system governed by a "propagator operator", where the time step τ is the data sampling interval.…”

“…The latter study showed that wave propagation can be viewed as a discrete time dynamical system governed by a "propagator operator", where the time step τ is the data sampling interval. This propagator operator maps the wave from the states at instants (j − 1)τ and jτ to the future state at time (j + 1)τ , for any j ∈ N. The ROM in [7,8] is an algebraic analogue of the dynamical system. Its evolution is controlled by an nm × nm propagator matrix, given by the Galerkin projection of the propagator operator on the function space spanned by the first n snapshots of the wave, assuming that the array records for the duration (2n − 1)τ .…”

“…The ROM introduced in [7,8] has been used so far to: (1) Approximate the Fréchet derivative of the reflectivity to sensor array data map [6]. This gives the single scattering (Born) forward map used in conventional imaging in radar [12,11], seismic inversion [4] and elsewhere.…”

“…This gives the single scattering (Born) forward map used in conventional imaging in radar [12,11], seismic inversion [4] and elsewhere. (2) Obtain a fast converging, iterative inverse scattering method for the acoustic impedance in a medium with known and smooth wave speed [8]. (3) Develop a non-iterative "backprojection" imaging method that is free of multiple scattering artifacts [14].…”

“…The paper is organized as follows: We begin in section 2 with the mathematical formulation of the imaging problem and review briefly from [8] the relevant facts about the ROM, needed in the next sections. The estimation of the internal wave is described in section 3.…”