IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 48, NO. 2, FEBRUARY 2000 293 Two-Step Inverse Scattering Method for One-Dimensional Permittivity Profiles Valeri A. Mikhnev and Pertti Vainikainen, Member, IEEE Abstract—A numerical method to invert the dielectric permit- tivity profile from the Riccati equation using the Newton–Kan- torovich iterative scheme is described. Instead of handling the equations in terms of usual geometrical depth, we determine the profile as a function of the electromagnetic path length since the convergence and the stability of the solution are found to be significantly better in this case. The initial profile used as a starting point for the inversion is obtained by another method employing successive reconstruction of dielectric interfaces and homogeneous layers in a step-like form. This method, though not always accurate, is fast and well suited for the approximate re- construction of the profile, thus creating ideal starting conditions for the previous approach. As a result, the computation time is considerably reduced without using any a priori information. The approach is applicable to both continuous and discontinuous pro- files of high contrast and exhibits a good stability of the solution with respect to noisy input data. A lossy medium profile can also be inverted provided the overall thickness of the inhomogeneous slab and the background permittivity are known. Index Terms—Inverse scattering, nonhomogeneous media. I. INTRODUCTION V ARIOUS methods have been used to reconstruct one-di- mensional permittivity profiles from electromagnetic re- flection coefficient data. The first-order Born and Rytov approx- imations assuming that the medium acts as a small perturbation on the incident wave [1]–[3], can be applied to accurate imaging of the quasi-homogeneous objects and produce qualitative im- ages in other cases. Nevertheless, the methods of this kind are mostly used in practice because of their simplicity and stability. A large number of investigations have been carried out using the Gel’fand–Levitan–Marchenko theory [4]–[7]. Unfortu- nately, this in principle exact approach is actually very difficult to implement due to considerable mathematical complexity. This leads sometimes to failures especially when faced with the discontinuous profiles of high contrast [6]. A nonlinear approximation of the Riccati equation allowing its solution in closed form with subsequent inversion yields in some cases very accurate reconstructions [8]. Iterative numerical methods based on the exact equations are also widely used in microwave imaging [9]–[12]. They do not principally have contrast limitations. Unfortunately, because of Manuscript received December 27, 1996; revised March 12, 1999. V. A. Mikhnev was with the Helsinki University of Technology, IRC/Radio Laboratory, Espoo, FIN 02015 HUT Finland. He is now with the Institute of Applied Physics, Minsk, 220072 Belarus. P. Vainikainen is with Helsinki University of Technology, IRC/Radio Labo- ratory, Espoo, FIN 02015 HUT Finland. Publisher Item Identifier S 0018-926X(00)01279-5. ill-posedness of the inverse problem, the convergence, and the stability of solution essentially depend on the actual contrast values, deteriorating for highly contrasted discontinuous pro- files. To improve the reliability of the solution and the conver- gence rate, as much as possible of a priori knowledge of the object under test should be included in the inversion procedure. However, this is not always convenient in practice. The purpose of this paper is to improve the convergence and the stability of the iterative optimization scheme for compli- cated highly contrasted dielectric profiles. No a priori infor- mation of the reconstructed profile is used in a lossless case, whereas minimal additional information is needed when con- ductive losses are taken into account. Besides, angular depen- dency of the reflected signal is excluded from consideration. Although multiangle measurements [9], [13], [14] yield addi- tional input data allowing to get more reliable reconstruction and to retrieve, e.g., material dispersion, their implementation results in large and expensive antenna arrangements. Further- more, the measurement routine becomes time consuming. This is not always appropriate for many practical applications such as ground penetrating radar, nondestructive testing in civil engi- neering, etc. Consequently, the consideration is restricted here to the case of normal incidence only. The reconstruction of dielectric half-space is performed using a new two-step approach. The Newton–Kantorovich iterative method applied to the Riccati equation is used as the basic re- construction algorithm. It is shown that a linear integral equa- tion to obtain the next iterate to the profile is more accurate in case if the derivation is accomplished in terms of electromag- netic path length rather than in usual spatial coordinate. This allows improving the convergence and the stability of the so- lution. The initial profile needed for the iterative procedure is obtained by another method employing discrete reconstruction. At this step, the permittivities and the thicknesses of the layers can be determined one after another by minimizing the max- imum of the reflection coefficient in the frequency band of op- eration. The method yields the exact reconstruction for simple one- and two-layered profiles, if the reflection data are given in a wide enough frequency band. Otherwise, the inversion is approximate. Nevertheless, the difference between the recon- structed profile and the exact one is quite small. Hence, good starting conditions are created for the previous approach. Only a few iterations are needed now to complete the reconstruction, saving the computation time. The numerical simulations demonstrate good accuracy, fast convergence, and robustness of the algorithm for complicated continuous and discontinuous profiles of high contrast. For a lossy medium, simultaneous determination of the permittivity 0018–926X/00$10.00 © 2000 IEEE