Linear one-dimensional inverse profiling: the role of a perfect reflecting plane in the background

One-dimensional (1D) inverse profiling aims to reconstruct the dielectric profile of 1D scatterers from the scattered field data. While the problem can be significantly simplified within a linearized framework (albeit at the cost of obvious performance limitations), the solution remains challenging due to the inherent ill-posedness of the problem, which is related to the "filtering" introduced by the linear scattering operator. To improve reconstruction accuracy, a host medium with known inhomogeneities can be exploited. Such a medium enriches the scattering environment by introducing additional reflections and scattering, which alters the properties of the scattering operator, resulting, in general, in less severe filtering. In this study, we investigate the effect of a perfect reflecting plane on the 1D linear inverse profiling problem using multi-frequency data. While this scenario has previously been explored through somewhat qualitative studies, we focus on the mathematical features of the involved scattering operator via its singular value decomposition (SVD). In particular, we derive an analytical estimation of the singular system, which in turn allows us to identify the influence of the configuration parameters on the achievable performance. In more detail, the point-spread function (PSF) can be analytically determined and linked to the configuration parameters. Results obtained by using a truncated-SVD regularized inversion scheme show that the reflecting plane doubles the band, with respect to the homogeneous host medium case, of the band-pass filtering that the unknown undergoes during the reconstruction process. Moreover, the presence of the reflecting plane facilitates the reconstruction of the continuous component (mean value) of the unknown dielectric profile as well. Overall, the inclusion of a reflecting plane enables the stable reconstruction of a broader class of dielectric profiles, improving both the resolution and the accuracy of the inverse profiling process.