We consider an inverse problem for the wave equation, where we wish to detect and localize (image) a compact object in a strongly scattering medium, using measurements of the scattering matrix. Entries in this matrix are echoes measured in some time window, at a remote array of sensors. A variety of methods that have emerged recently use the spectral decomposition of the scattering matrix to image the scatterer. These methods are usually applied in homogeneus media or at most in inhomogeneous, weakly scattering media. Our goal in this talk is to show how one can develop a detection and imaging approach in strongly scattering media, that uses the spectral analysis of the scattering matrix in combination with adaptive time windowing. Mathematically, we model the medium as a random field. The spectral analysis of the scattering matrix becomes a problem in random matrix theory. We present some numerical simulations that illustrate our detection and imaging approach in a variety of random media. We also present a detailed analysis in randomly layered media.
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