Electromagnetic inverse wave scattering in anisotropic media via reduced order modeling.

The inverse wave scattering problem seeks to estimate a heterogeneous, inaccessible medium, modeled by unknown variable coefficients in wave equations, from transient recordings of waves generated by probing signals. It is a widely studied inverse problem with important applications, that is usually formulated as a nonlinear least squares data fit optimization. For typical measurement setups and band-limited probing signals, the least squares objective function has spurious local minima far and near the true solution, so Newton-type optimization methods fail to obtain satisfactory answers. We introduce a different approach, for electromagnetic inverse wave scattering in lossless, anisotropic media. It is an extension of recently developed data driven reduced order modeling methods for the acoustic wave equation in isotropic media. Our reduced order model (ROM) is an algebraic, discrete time dynamical system derived from Maxwell's equations. It has four important properties: (1) It can be computed in a data driven way, without knowledge of the medium. (2) The data to ROM mapping is nonlinear and yet the ROM can be obtained in a non-iterative fashion, using numerical linear algebra methods. (3) The ROM has a special algebraic structure that captures the causal propagation of the wave field in the unknown medium. (4) It is an interpolation ROM i.e., it fits the data on a uniform time grid. We show how to obtain from the ROM an estimate of the wave field at inaccessible points inside the unknown medium. The use of this wave is twofold: First, it defines a computationally inexpensive imaging function designed to estimate the support of reflective structures in the medium, modeled by jump discontinuities of the matrix valued dielectric permittivity. Second, it gives an objective function for quantitative estimation of the dielectric permittivity, that has better behavior than the least squares data fitting objective function. The methodology introduced in this paper applies to Maxwell's equations in three dimensions. To avoid high computational costs, we limit the study to a cylindrical domain filled with an orthotropic medium, so the problem becomes two dimensional. • Quantitative method for electromagnetic inverse wave scattering in lossless anisotropic media outperforming FWI. • Qualitative imaging method for electromagnetic inverse wave scattering in lossless anisotropic media outperforming migration. • Data-driven, interpolating model reduction capable of estimating internal wavefields and facilitating inversion and imaging. [ABSTRACT FROM AUTHOR]