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

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 Fréchet 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.

26 pages, 6 figures

References in corpus (5)

• Detection and imaging in a random medium: A matrix method to overcome multiple scattering and aberration
• Direct, nonlinear inversion algorithm for hyperbolic problems via projection-based model reduction
• Untangling the nonlinearity in inverse scattering with data-driven reduced order models
• A nonlinear method for imaging with acoustic waves via reduced order model backprojection
• Time and direction of arrival detection and filtering for imaging in strongly scattering random media

Cited by in corpus (6)

• Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems
• Reduced order models for spectral domain inversion: Embedding into the continuous problem and generation of internal data
• Reduced order model approach for imaging with waves
• On extension of the data driven ROM inverse scattering framework to partially nonreciprocal arrays
• Practical shift choice in the shift-and-invert Krylov subspace evaluations of the matrix exponential
• Geometry-based approximation of waves in complex domains