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Abstract
The nonlinear inverse scattering problem of estimating the locations and scattering strengths or reflectivities of a number of small, point-like inhomogeneities (targets) to a known background medium from single-snapshot active wave sensor array data is investigated in connection with time-reversal multiple signal classification and an alternative signal subspace method which is based on search in high-dimensional parameter space and which is found to outperform the time-reversal approach in number of localizable targets and in estimation variance. A noniterative formula for the calculation of the target reflectivities is derived which completes the solution of the nonlinear inverse scattering problem for the general case when there is significant multiple scattering between the targets. The paper includes computer simulations illustrating the theory and methods discussed in the paper.
Highlights
This research is concerned with signal subspace frameworks for inverse scattering with active wave sensor arrays of small, point-like inhomogeneities or perturbations to a background medium whose constitutive properties relevant to the particular remote sensing modality (e.g., permittivity, permeability, conductivity, sound speed, diffusion coefficient in radiative transfer-based sensing, etc.) are known
The problem under consideration comprises both localization of the inhomogeneities as well as determination of the perturbation strengths or target reflectivities from single-snapshot entries of a noisy scattering or multistatic response (MSR) matrix gathered by a generally noncoincident array of Nt point transmitters and Nr point receivers
It is shown that if one implements a different approach based on multiple signal classification (MUSIC)-like steering not of a single target (as in (26)–(28)) but of all the M targets simultaneously it is possible to locate up to NrNt − n(n − 1)/2 − 1 targets, where n is the number of coincident elements, as long as the targets are approximately describable by the Born approximation
Summary
This research is concerned with signal subspace frameworks for inverse scattering with active wave sensor arrays of small, point-like inhomogeneities or perturbations to a background medium whose constitutive properties relevant to the particular remote sensing modality (e.g., (electromagnetic) permittivity, permeability, conductivity, (acoustic) sound speed, diffusion coefficient in (e.g., optic) radiative transfer-based sensing, etc.) are known. We emphasize the particular scalar Helmholtz operator context, but the general developments apply in forms which differ only on the specifics of the Green function and the scattering potential operator [10, Chapter 9] to a variety of partial differential equations governing the source-field systems of interest This includes the diffusion equation [11, Chapter 9] which is relevant to certain random media and has been used in time-reversal studies [12]. For the general multiple scattering regime the associated inversion is less straightforward due to the resulting nonlinearity of the reflectivities-to-MSR matrix mapping which traditionally would be handled via nonlinear optimization Despite this nonlinearity, the latter problem is solved in this paper analytically, noniteratively (unlike in [13] which adopts the more conventional numerical iterations route). Appendix C presents the Fisher information matrix/CRB calculations relevant to the estimation of target positions and reflectivities under general multiple scattering conditions
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