Local spectral analysis of short-pulse excited scattering from weakly inhomogeneous media-part II: Inverse scattering

This paper is concerned with the reconstruction of a weakly inhomogeneous scattering profile from data generated by a short-pulse incident plane wave, which is postprocessed so as to localize the interrogated region to a space-time resolved scattering cell. The phase-space localization due to postprocessing is brought about by applying local (i.e., windowed) slant-stack transforms to the time-dependent scattered fields. In the domain of the scatterer, this processing corresponds to applying windowed Radon transforms to the induced field distribution, which, in turn, generates pulsed-beam (PB) wave packets traveling toward the observer. The forward analysis parameterizing this new form of time-domain (TD) diffraction tomography has been performed in a companion paper and furnishes the framework for the investigation here. Via the forward parameterization, the three-dimensional (3-D) global scattering phenomenology has been reduced to scattering from an equivalent one-dimensional (1-D) scattering cell oriented along the bisector between the direction of the incident plane pulse and the direction of the scattered pulsed beam (PB) to the observer. For the inverse problem, this process is reversed by windowing the scattered field and backpropagating the resulting PB's so as to form local images of any selected region in the scattering domain. The phase-space signature of the scattering cell is related to the Radon transform of the medium in the cell so that the local profile function can be recovered by Radon inversion. An illustrative numerical example is included. Also discussed is the ultimate localization achieved by incident PB excitation and PB postprocessing of the scattered field.