The nonlinear (quadratic) distorted approximation of the inverse scattering of dielectric cylinders is investigated, with the aim of pointing out the influence of the background medium. We refer to a canonical geometry consisting of a radially symmetric dielectric cylinder illuminated at a single frequency. We discuss how the spatial variations of those unknown dielectric profile functions that can be reconstructed by a stable inversion procedure are related to the permittivity of the background cylinder. First, results for the linear distorted approximation, obtained by means of the singular-value decomposition, are recalled and compared with the Born approximation. It turns out that the distorted model provides a smoother behavior of the singular values, and thus the inversion is more sensitive to the presence of uncertainties in the data. Furthermore, a stable inversion procedure can reconstruct only a very limited class of unknowns in correspondence with fast spatial variations related to the background permittivity and the excitation frequency. On the other hand, the quadratic model improves the approximation in the distorted case. This can be traced not only to the higher allowable level of permittivity but mainly to the fact that the model makes it possible to reconstruct different spatial features as the solution space changes. Numerical results show that the quadratic inversion performs better than the linear one for the same amount of uncertainty in the data.
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