350 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 10, 2011 Reconstruction of Unknown Surface Profiles in Multilayered Media by Complex Images Green’s Functions Technique Behzad Yektakhah, Student Member, IEEE, and Reza Faraji-Dana, Senior Member, IEEE Abstract—Reconstruction of unknown surfaces profiles in mul- tilayered media with a computationally efficient scheme is studied in this letter. The proposed inverse scattering scheme combines a fast and accurate forward scattering solver with an efficient opti- mization algorithm. The application of method of moments (MoM) to the integral equation formulation of the forward scattering problem employing complex images Green’s functions has re- sulted in a fast and accurate method for determining the scattered field from unknown surface profiles in multilayered media. The unknown surface profiles are parameterized by B-spline functions with a finite number of unknown coefficients. The simulated annealing (SA) algorithm is then employed to solve the inverse scattering problem. Numerical results show the accuracy of the proposed scheme in the reconstruction of the unknown surface profiles even in noisy environments. Index Terms—Complex images Green’s functions, integral equa- tion technique, inverse scattering problems, simulated annealing, surface profiles. I. INTRODUCTION R ECONSTRUCTION of unknown surface profiles is an important problem in inverse scattering theory that arises in a wide variety of applications such as nondestructive testing, ground-penetrating radar (GPR), underground imaging, geosciences, etc. Nonhorizontal surfaces cause disorders in the scattered field and make it difficult to determine buried objects in subsurface-sensing applications, hence a fast and accurate inversion algorithm to determine the unknown profile of a surface in order to compensate for the distortion on the scattered field is necessary [1]. Several techniques have been proposed to reconstruct unknown surface profiles in 2-D or 3-D problems with or without buried objects. Computational-based algorithms usually combine a fast forward solver and an effi- cient optimization algorithm [1]–[4]. In this letter, the method of moments (MoM) in combination with the complex images technique is employed as a fast and accurate forward solver of the volume integral equation (VIE) formulation of the problem. Since the problem under investiga- tion is usually large with respect to the wavelength, the conven- tional complex images technique described in [5] and [6] cannot Manuscript received March 28, 2011; accepted April 10, 2011. Date of pub- lication April 21, 2011; date of current version May 02, 2011. B. Yektakhah is with the School of Electrical & Computer Engi- neering, College of Engineering, University of Tehran, Tehran, Iran (e-mail: b.yektakhah@ece.ut.ac.ir). R. Faraji-Dana is with the Center of Excellence on Applied Electromagnetic Systems, School of Electrical & Computer Engineering, College of Engi- neering, University of Tehran, Tehran, Iran (e-mail: reza@ut.ac.ir). Digital Object Identifier 10.1109/LAWP.2011.2144560 be used without the surface-wave extraction process prescribed there, otherwise large errors will be imposed on the complex im- ages Green’s function when the field point moves away from the source point in the horizontal direction. The reason is that the path used in approximating the spectral domain Green’s func- tion by a finite sum of exponentials does not provide enough in- formation about the surface wave poles. Since in our application the surface wave poles extraction process prescribed in [5] and [6] is not suitable due to its analytical complexity, a new com- plex images representation of the Green’s function valid for both the near- and far field-point-to-source-point distances is used in our forward scattering formulation. Recently, a new path has been introduced for approximating the spectral domain Green’s function in terms of a finite number of exponentials without extracting the surface wave poles [7]. This will result in an accurate complex images representation of the Green’s function for the field points in the near field up to far field (distances ). However, this method is only applicable when images are placed at the upper medium of the layered media. In our proposed method, the complex images are assumed in the medium where the field point exists. As it is shown in Section III, this will bring us a considerable reduc- tion in the number of spectral Green’s functions to be approx- imated. On the other hand, when the field point is not placed at the upper medium, it requires a new path for approximating the spectral Green’s function [8], [9]. In Section III, this new path is also introduced, and the accuracy of the complex im- ages Green’s function, without the need for pole extraction, is demonstrated by comparing its results to the results of direct nu- merical integration. The success of the reconstruction process of the unknown sur- face profile depends very much on the parameterization of the profile by appropriate basis functions with unknown parame- ters that can represent the curvatures of the surface. The number of unknown parameters to be estimated in the inverse problem should be as low as possible because of the limited number of data collected in the receivers [1]. In our proposed method, the B-spline basis functions are employed to represent the surface profile [1], [4]. In this way, one has to find the unknown co- efficients of these B-spline basis functions to solve the inverse problem, i.e., to determine the unknown surface profile. This is done by introducing an appropriate cost function and making use of a robust optimization algorithm. Since the cost function to be minimized may have several local minima, a global optimiza- tion algorithm should be used to converge to the global min- imum of the problem. To this end, the simulated annealing (SA) algorithm has been employed due to its versatility to deal with the large number of parameters to be optimized. 1536-1225/$26.00 © 2011 IEEE