## 1. Introduction

[2] The two-dimensional scattering problem of electromagnetic waves by buried cylindrical objects has wide application in remote sensing of buried pipes, conduits and cables, in the understanding of mutual interactions between buried objects and surrounding media, in the communication through the earth, in the backscattering from randomly distributed scatterers. For this reason, such problem has been discussed by many authors in the past, both from a theoretical and a numerical point of view, and several different techniques have been proposed to solve it.

[3] *Howard* [1972] solved a two-dimensional Fredholm integral equation for the scattered field from a subterranean cylindrical inhomogeneity excited by a line source employing an eigenfunction expansion. In the pioneering analytical work [*D'Yakonov*, 1959], an interesting solution is obtained as a limiting case of a boundary value problem involving two nonconcentric cylinders, but the results are not suitable for numerical evaluation. D'Yakonov's work is extended by *Ogunade* [1981] in a form appropriate for obtaining numerical data, using conventional eigenfunction expansions; the author considered a conducting circular cylinder inside a dielectric cylinder and took the limit that the radius of the dielectric cylinder goes to infinity; therefore his results cannot be extended to arbitrary configurations and cross sections. *Budko and van den Berg* [1999] modeled a finite-sized object embedded in a lossy half-space in terms of an effective homogeneous circular scatterer, and a Green's function approximate approach is employed. *Bertoncini et al.* [2001] applied the uniform geometrical theory of diffraction to calculate the scattering from a polygonal perfectly conducting object buried in a lossy half-space.

[4] *Ellis and Peden* [1995] computed the scattering from buried inhomogeneous dielectric objects, in the presence of an air-earth interface, employing a two-dimensional method of moment formulation and utilizing cylindrical pulse basis functions and point matching. Surface integral equation methods have been used to treat a buried dielectric cylinder by *Butler and Xu* [1989] and *Diamandi and Sahalos* [1991]. Volume integral equation methods have been used for determining the scattering from a two-dimensional buried lossy dielectric object [*Izadian et al.*, 1984], obtaining both the frequency domain solution and the time domain response. *Xu and Ao* [1997] employed a hybrid finite element and moment method to analyze the scattering by an inhomogeneous lossy dielectric/ferrite cylinder located below a planar interface between two half-spaces of different electromagnetic properties.

[5] The concept of plane wave spectrum of a cylindrical wave [*Cincotti et al.*, 1993] has been used by *Di Vico et al.* [2005] for the case of a finite set of buried conducting cylinders. The reflection properties of such waves in the presence of plane interfaces have been discussed by *Borghi et al.* [1996], by introducing suitable reflected cylindrical functions, while the transmission of the cylindrical waves outside the dielectric half-space has been studied through transmitted cylindrical functions and the relevant spectral integrals by *Di Vico et al.* [2005]. Such integrals have been numerically solved employing suitable adaptive integration techniques of Gaussian type, together with convergence acceleration algorithms [*Borghi et al.*, 1999, 2000].

[6] *Lawrence and Sarabandi* [2002] considered the scattering problem from a dielectric cylinder embedded in a dielectric half-space, in the presence of a slightly rough interface. The solution utilizes the plane wave representation of the fields, taking into account all the multiple interactions between the surface and the cylinder. Asymptotic evaluation of integrals is required for far-field results, and when the cylinder is deeply buried. For the particular case of a flat surface, their method is equivalent to the solution proposed by *Borghi et al.* [1997] for cylinders placed above the interface.

[7] In this paper we extend the work of *Di Vico et al.* [2005] by considering the two-dimensional plane wave scattering problem by a finite set of dielectric circular cylinders, with possibly different radii and arbitrarily placed, buried in a dielectric half-space. We take into account all the cylinder-cylinder interactions and the multiple reflections between the cylinders and the interface.

[8] The present method may be applied for any value of the cylinder size and of the distance between the obstacles and the surface. It deals with both transverse magnetic (TM) and transverse electric (TE) polarization cases and yields results in both the near- and far-field zones. Moreover, it is able to characterize two-dimensional obstacles of arbitrary shape simulated by a suitable array of circular cylinders, as it was proposed for the free-space case by *Elsherbeni and Kishk* [1992].

[9] In section 2 the analytical characteristics of the approach are described. In section 3 details about the numerical solution are given and in section 4 relevant numerical results are reported. Convergence checks are presented. Comparisons are made in terms of scattered field. Examples of scattering by two-dimensional objects of generic shape are shown.