Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape

Transcription

1 RADIO SCIENCE, VOL. 38, NO. 3, 1055, doi:10.109/00rs00631, 003 Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape Ibrahim Akduman Istanbul Technical University, Electrical and Electronics Engineering Faculty, Maslak, Istanbul, Turkey Rainer Kress Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, Germany Received 1 March 00; revised 15 August 00; accepted 8 February 003; published 13 June 003. [1] The direct and inverse scattering problems related to objects having inhomogeneous impedance boundaries are addressed by considering cylindrical bodies. In the solution of the direct scattering problem, the scattered field is first expressed in terms of a combined single- and double-layer potential through Green s formula and the boundary condition. By using the jump relations on the boundary of the object, the scattering problem is reduced to a boundary integral equation that can be solved via a Nyström method. The aim of the inverse impedance problem is to reconstruct the inhomogeneous surface impedance of the body from the measured far field data. Here representing the scattered field as a single-layer potential leads to an ill-posed integral equation of the first kind for the density that requires stabilization for its numerical solution; for example, by Tikhonov regularization. With the aid of the jump relations the single-layer potential enables the evaluation of the total field and its derivative on the boundary of the scatterer. Consequently, from the boundary condition the surface impedance can be reconstructed either by direct evaluation or by a minimum norm solution in the least squares sense. The numerical results show that our methods yields good resolution both for the direct and the inverse problem. INDEX TERMS: 0619 Electromagnetics: Electromagnetic theory; 069 Electromagnetics: Inverse scattering; 0669 Electromagnetics: Scattering and diffraction; 0689 Electromagnetics: Wave propagation (475); 698 Radio Science: Tomography and imaging; KEYWORDS: electromagnetic theory, inhomogeneous surface impedance Citation: Akduman, I., and R. Kress, Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape, Radio Sci., 38(3), 1055, doi:10.109/00rs00631, Introduction [] The impedance boundary condition (IBC) which gives a relation between the electric and magnetic field vectors on a given surface in terms of a coefficient called surface impedance is one of the main tools that is used in the solution of electromagnetic scattering problems. The reason for using this type of boundary condition is to simplify the mathematical or numerical difficulties occurring in the solution of scattering problems involving complex structures. The surface impedance is commonly used to model imperfectly conducting scatterers, perfectly conducting objects coated with a penetrable or Copyright 003 by the American Geophysical Union /03/00RS absorbing layer, or scatterers with corrugated or rough surfaces. Its first application to a lossy material surface is generally attributed to Leontovich [1948]. Wait [1990] used this type of boundary condition to simulate the land in studies of ground wave propagation over the earth surface. The simplest form of the IBC is the standard impedance boundary condition (SIBC) which has been used to model coatings and lossy dielectrics. Traditionally, the surface impedance appearing in SIBC is assumed to be independent of the location and associated with a constant coefficient [Senior and Volakis, 1995; Hoppe and Rahmat-Samii, 1995]. This is due to the approximations that are made in the derivation of SIBC. On the other hand, when a more accurate SIBC is considered, the surface impedance may be a function of location, and even may be of tensor form to model anisotropic scatterers [Senior et al., 1997]. For example

2 1 - AKDUMAN AND KRESS: DIRECT AND INVERSE SCATTERING Figure 1. Geometry of the problem. when the nonhomogeneous earth surface composed of different parts such as forest, rocky soil, sand, see etc. is modelled by an IBC, the surface impedance becomes a function of location. Higher-order boundary conditions were also proposed for example for thick slabs where the fields vary more rapidly along the surface. The reason for going into higher-order boundary conditions is to develop conditions that take into account strong variations along the surface [Wang, 1987; Senior et al., 1997; Marceaux and Stupfel, 000]. [3] The determination of the IBC for a given scatterer constitutes an important class of problems in the electromagnetic theory and various approximate methods have been established in the literature for special kind of geometries and surfaces [Senior and Volakis, 1995; Hoppe and Rahmat-Samii, 1995; Senior et al., 1997; Marceaux and Stupfel, 000]. In all these methods one first tries to solve the direct scattering problem for a given scattering structure and then express the IBC in terms of the electric and magnetic field on the boundary. The surface impedance of a scattering object can also be obtained by using the scattered data obtained through measurements on a certain domain. In such a case the problem is considered as an inverse scattering problem which aims to get the fields on the boundary of the object in terms of the measured data. A method for the reconstruction of the surface impedances of planar boundaries has been proposed by Akduman and Yapar [001] and Yapar and Akduman [001]. [4] The direct scattering problems involving IBC are of importance and, to our knowledge, most of the available publications are concerned with boundary conditions with constant impedance coefficients. On the other hand, as mentioned above, inhomogeneous impedance boundary conditions are also of interest and importance both from mathematical and physical points of view. Such problems are of practical importance since their results may be used in applications such as antenna design and analysis, detection of buried objects in a known medium, determining the characteristics of the earth surface etc. For example, one may change the radiation characteristics of an existing antenna by introducing an appropriate impedance on its surface. [5] The main objective of this paper is to describe algorithms both for the solution of the direct and inverse scattering problems with scatterers having inhomogeneous impedance boundary conditions. To this aim we consider infinitely long cylindrical bodies of arbitrary cross section and SIBC. For the direct problem we construct the scattered field for the case of plane wave illumination. Our method is based on an integral representation of the scattered field through Green s formula that leads to a boundary integral equation through the jump relations for double- and single-layer potentials. This integral equation is wellposed and can be solved numerically through a Nyström method. [6] The inverse boundary value problem we consider is, for a known shape of the scatterer, to reconstruct the inhomogeneous surface impedance of the cylinder through far field measurements in the case of plane wave illumination. First, the scattered field is represented by a single-layer potential and the density of the single-layer potential is obtained by solving the resulting ill-posed integral equation of the first kind through Tikhonov regularization. The use of the jump relations for singlelayer potentials leads to explicit expressions of the scattered field and its derivative on the impedance surface. Then by using the boundary condition itself one can achieve the reconstruction. Since the use of the boundary condition itself constitutes an ill-posed problem, a regularized solution in a least squares sense is also described. [7] In section the direct scattering problem is formulated and its solution is given. The inverse impedance problem is solved in section 3. The numerical results are presented in section 4. Finally, conclusions and concluding remarks are given in section 5. A time factor exp(iwt) is assumed and omitted throughout the paper.. Solution of the Direct Scattering Problem [8] Consider the electromagnetic scattering problem as described in Figure 1. In this configuration an infinitely long cylindrical body with axis parallel to the x 3 axis and with cross-section D is located in an infinite homogeneous background medium with constitutive parameters e, m, and s. We will assume that D R is a bounded

3 AKDUMAN AND KRESS: DIRECT AND INVERSE SCATTERING 1-3 domain with a connected twice continuously differentiable boundary. On the boundary of the cylinder we apply the standard inhomogeneous impedance boundary condition SIBC with a nonconstant continuous impedance coefficient = (x); that is, n ðn EÞ ¼ nh on ; ð1þ where E and H are the total electric and magnetic field vectors and n is the outward unit normal vector of. Note that the coefficient in (1) is the surface impedance of the cylinder that, in general, we assume to be a function of the location on. The cylinder is illuminated by a plane wave whose electric field vector is polarized parallel to the x 3 axis; that is, E i ðþ¼ x 0; 0; u i ðþ x ; u i ðþ¼e x ik xd ; where d = (cos f 0, sin f 0 ) pis the propagation direction with angle f 0 and k ¼ w ffiffiffiffiffiffi e 0 m stands for the wave number of the background medium with the complex dielectric permittivity e 0 = e + is/w. The direct scattering problem consists of finding the scattered field E s = E E i in the exterior of the cylinder. Note that due to the homogeneity of the boundary condition (1) with respect the x 3 axis the total and the scattered electric field vectors also will be polarized parallel to the x 3 axis; that is, E = (0, 0, u) and E s = (0, 0, u s ). Then the problem reduces to a scalar problem in R for the field functions u and u s. The total field u = u i + u s has to satisfy the Helmholtz equation Du þ k u ¼ 0 in IR n D ðþ with the impedance boundary condition u þ h ¼ 0 on ð3þ such that the scattered field u s fulfills the Sommerfeld radiation condition p s lim r ðþiku x s ðþ x ¼ 0; r ¼ jj: x ð4þ Here h is the normalized surface impedance defined by hðþ:¼ x x ðþ ; x ; 0 p where 0 ¼ ffiffiffiffiffiffiffiffi m=e 0 denotes the intrinsic impedance of the background medium. For the scattered field we have the Huygens type Green s formula [see Colton and Kress, 1999] u s ðþ¼ x u s ð ðyþ ðyþgx; ð yþ dsðyþ; x R n D; ð5þ representing u s in terms of the secondary sources on the boundary of the scatterer. Here G(x, y) is the fundamental solution, i.e., the free space Green s function, of the Helmholtz equation in two dimensions given by Gx; ð yþ ¼ i 4 H 0 1 ð kx j yj Þ; ð6þ where H 1 0 denotes the Hankel function of the first kind and of order zero. Similarly, for the incident field u i by Green s integral theorem we have the so-called extinction theorem 0 ¼ u i ð ðyþ ðyþgx; ð yþ dsðyþ; x R n D; ð7þ which together with (3) and (5) yields u yþ ðþ¼ x ðyþ þ ik hðyþ Gx; ð yþ uy ð ÞdsðyÞ; x R n D: ð8þ Here and in the sequel we assume that h(x) 6¼ 0 for all x. The two integrals appearing on the right-hand side of (8) represent double- and single-layer potentials, respectively. By using the jump relations for single- and double-layer potentials the representation (8) can also be extended to the boundary to yield the following boundary integral equation for the total yþ ðyþ þ ik hðyþ Gx; ð yþ uy ð ÞdsðyÞ ¼ u i ðþ; x x : ð9þ ux ðþ For existence and uniqueness of the solution to this integral equation we refer to Colton and Kress [1999, p. 48]. [9] For the numerical treatment of this integral equation we recommend a Nyström method that takes proper care of the logarithmic singularity of the fundamental solution. For its brief description we denote the kernel of the integral equation (9) by K H (x, y) and note that K H ðx; yþ ¼ ik ny ð Þfx yg H ðkx j yjþ jx yj k hðyþ H 0 1 ðkx j yjþ for x, y with x 6¼ y. Analogously we set K J ðx; yþ :¼ ik ny ð Þfx yg J0 0 p jx yj ðkx j yjþ k phðyþ J 0ðkx j yjþ;

4 1-4 AKDUMAN AND KRESS: DIRECT AND INVERSE SCATTERING where J 0 denotes the Bessel function of order zero. Now, assuming a p-periodic parametric representation ¼ fzt ðþ: 0 t pg ð10þ of the boundary curve, we transform the integral equation (9) into the parameterized form p n yðþ t L J ðt; tþln 4 sin t t o þ L H ðt; tþ 0 yt ð Þdt ¼ gt ðþ; 0 t p; ð11þ for the unknown function y(t): = u(z(t)) and the righthand side g(t): = u i (z(t)). The kernels are given by L J ðt; tþ :¼ K J ðzt ðþ; zðtþþjz 0 ðtþj and L H ðt; tþ:¼ K H ðzt ðþ; zðt Þjz 0 ðtþjl J ðt; tþln 4 sin t t : [10] From the fact that v 7! ph 0 1 (v) ln vj 0 (v) isan analytic function, it can be deduced that the kernel functions L J and L H are continuous and p-periodic. If the boundary curve and the impedance function both are analytic, then L J and L H are also analytic. Therefore their integrals can be efficiently approximated by trigonometric interpolatory quadrature formulas. More precisely, we choose N equidistant grid points t n :=pn/n, n =0,..., N 1, and use the quadrature rule p L J ðt m ; t 0 XN1 R ðnþ jmn n¼0 Þln 4 sin t m t j L J ðt m ; t n Þyðt n Þ with the quadrature weights R ð n NÞ :¼ p N and the trapezoidal rule p 0 XN1 m¼1 1 m L H ðt m ; tþyt ð Þdt p N yt ð Þdt cos mnp N N1 X n¼0 1 ð Þn p N L H ðt m ; t n Þyðt n Þ: In the Nyström method for the solution of integral equations of the second kind [see Kress, 1999] this leads to approximating the integral equation (11) by solving the linear system y ð m NÞ XN1 n¼0 y ð n NÞ n R ðnþ jmn j L J ðt m ; t n Þþ p N L Hðt m ; t n ¼ gt ð m Þ; m ¼ 0;...; N 1; o Þ (N) for approximations y n to the values y(t n ) of the solution at the grid points. This Nyström method can be shown to converge for continuous L J and L H, i.e., for twice continuously differentiable boundaries and continuous impedance functions. Moreover, for the case of analytic boundary curves and impedances it enjoys an exponential convergence rate, i.e., doubling the number of grid points doubles the number of correct digits in the approximation. For more details on the Nyström method we refer the readers to Kress [1995, 1999] and Colton and Kress [1999]. [11] Once the boundary integral equation (9) is solved the near and far fields of the scattered wave can be calculated through (8). The scattered far field at large distances from the scatterer is described through the far field pattern u 1 as defined through the asymptotic behavior ðþ¼ eik jj x pffiffiffiffiffi u s x jj x u 1 ðþþo ^x 1 jj x ; jj!1; x ð1þ in the observation direction ^x = x/jxj. From (8) by using the asymptotic expressions of H 0 1 and its derivative it can be deduced that u 1 ^x ðþ¼ eip 4 p ffiffiffiffiffiffiffiffi 8pk ik ðyþ þ ik hðyþ eik ^xy uy ð ÞdsðyÞ ð13þ for the observation direction ^x = (cos q, sin q) with observation angle q [Colton and Kress, 1999]. The scattering cross section per unit length, i.e., the echo width, defined as that is, j sðþ:¼ ^x p lim r us ðr^x Þj r!1 ju i ðr^x Þj ; ð14þ sðþ:¼ ^x pju 1 ðþ ^x j is of particular interest in the scattering problems. 3. Inverse Impedance Problem [1] The inverse impedance problem related to the configuration in Figure 1 consists of reconstructing the impedance function h from the far field pattern u 1 as defined through (1). When the boundary is known a priori, then by Rellich s lemma [see Colton and Kress, 1999] the far field pattern u 1 uniquely determines the scattered field u s and consequently the total field u = u i + u s in the exterior of the scatterer D. Therefore, in principle, in view of (3) the surface impedance can be

5 AKDUMAN AND KRESS: DIRECT AND INVERSE SCATTERING 1-5 obtained from the values of the total field u and its normal on via hðþ¼ik x ux ; x : ð15þ ðþ x Possible zeros in the denominator on the right hand side of (15) will be taken care off by a least squares regularization. In the sequel we will describe a method for reconstructing the required field values on the boundary from the far field data. To this aim we first represent the scattered field as a single-layer potential of the form u s ðþ¼ x Gx; ð yþjðyþdsðyþ; x R n D; ð16þ with an unknown density function j. For the sake of simplicity we assume that k is not at interior resonance; that is, k is not a Dirichlet eigenvalue for the negative Laplacian in D. In this case, any solution to the Helmholtz equation in the exterior of D that satisfies the radiation condition indeed can be represented as a single-layer potential [Colton and Kress, 1983]. In order to avoid the above restriction we would need to replace the single-layer potential by a combined single- and double-layer potential u s ðþ¼ x Gx; ð y ð yþ ðyþ x R n D; jðyþdsðyþ; with an unknown function j, since any radiating solution to the Helmholtz equation in the exterior of D can be represented in this form [Colton and Kress, 1983]. [13] Analogously to (13) we have that the far field pattern of (6) is given by u 1 ðþ¼ ^x p eip=4 ffiffiffiffiffiffiffiffi e ik ^xy jðyþdsðyþ ð17þ 8kp for the observation direction ^x = (cos q, sin q) with observation angle q. Hence given a far field pattern u 1, we need to solve the integral equation of the first kind Aj ¼ u 1 ð18þ for the density j, where the integral operator A is given by ðajþðþ:¼ ^x p eip=4 ffiffiffiffiffiffiffiffi e ik ^xy jðyþdsðyþ: ð19þ 8kp The operator A has an analytic kernel and therefore the equation (18) is severely ill-posed. For that reason some kind of stabilization such as Tikhonov regularization has to be applied. For a regularized solution in the sense of Tikhonov we solve the equation aj þ A*Aj ¼ A*u 1 ð0þ with a regularization parameter a > 0 and the adjoint A* of A as given by ða*gþðyþ ¼ p eip=4 ffiffiffiffiffiffiffiffi e ik ^xy gðþds ^x ðþ; ^x y : 8kp V For a given set of data points ^x m = (cos q m, sin q m ), m = 1,..., M, we first discretize the integral operator A as defined in (19) by using the parametric representation (10) and the trapezoidal rule with N grid points to obtain an M N-matrix approximation for A. Approximating A* by the adjoint of this matrix then reduces (0) to a N N-matrix system that can be solved by any of the well-known techniques. [14] Once the single-layer density j is known, the values u of the total field on the boundary can be recovered through the jump relations for the single-layer potential [Kress, 1999; Colton and Kress, 1999]; that is, by ux ðþ¼u i ðþþ x ðþ¼@ui x ðþþ x Gx; ð yþ ðþ x x : x ; ð1þ jðyþdsðyþ 1 jðþ; x ðþ For the numerical evaluation of these singular integrals we make use of the same quadrature formulas as in the previous section in connection with the Nyström method. The surface impedance can now be reconstructed from (15) in terms of the values of u and its normal derivative for each point x. [15] It is obvious that this solution will be sensitive to errors in the normal derivative of u in the vicinity of zeros. To obtain a more stable solution, we express the unknown impedance function in terms of some basis functions f n, n =1,..., N, as a linear combination h ¼ XN n¼1 a n f n on : ð3þ A possible choice of basis functions consists of splines or trigonometric polynomials. Then we satisfy (15) in the least squares sense; that is, we determine the coefficients

6 1-6 AKDUMAN AND KRESS: DIRECT AND INVERSE SCATTERING a 1,..., a N in (3) such that for a set of grid points x 1,..., x M on the least squares sum X M ux ð m Þþ XN a n f n ðx m ð x mþ ð4þ m¼1 n¼1 is minimized. The number of basis functions N in (3) can be considered as some kind of regularization parameter. 4. Numerical Results [16] In this section we present the results of some illustrative examples in order to show the effectiveness and the accuracy of the methods presented in the previous sections. To this aim we considered cylinders both with an elliptical and a drop-shaped cross sections given by the parameterizations and e ¼ fð0:6cost; sin tþ : t ½0; pšg n d ¼ sin t ; sin t : t ½0; p o Š ; respectively. The results for the direct scattering problem are presented in terms of the echo width as defined in (14). Figure shows the variation of the echo width for the elliptical cylinder e for two different impedances, i.e., h 1 ¼ 0:5 exp ðt pþ ; h ¼ 0:5 þ 0:3 sin t þ i0:6 cos t: Figure. Echo widths of the elliptical cylinder for two different surface impedances. Figure 3. Echo widths of the drop-shaped cylinder for two different surface impedances. We note that the impedance h 1 is continuous whereas h is analytic. In all cases the operating frequency is f = 33 MHz and the incidence angle of the plane wave is f 0 = p/. The echo widths corresponding to the dropshaped cylinder for the same parameters and impedance functions are illustrated in Figure 3. In these examples the boundary integral equation (9) is solved through the Nyström method with a discretization parameter N = 100. The integral appearing in the far field expression (13) is evaluated numerically by using the trapezoidal rule. In the case of circular cylindrical scatterers the solution of the direct scattering problem can be obtained by using series representation for the scattered field. In such a case one expands also the impedance function into a Fourier series and reduces the problem to the solution of a system of linear equations. We compared the results obtained through the method presented here with those obtained by the Fourier series method mentioned above and observed that both results match very accurately. For that reason we do not present them here. [17] The reconstructions of the surface impedance have been obtained by using both the direct method (15) and the least squares solution (3), (4). In the application of the least squares solution the basis functions are chosen as f n (x(t)) = e int, n = 0, ±1,.. ±N. The grid points x 1... x M appearing in (4) are chosen as equidistant with respect to the parameterization parameter t with M = 100. Figures 4 and 5 show variations of the real and imaginary parts of the exact and reconstructed values of the surface impedance versus the boundary parameter t for the elliptical cylinder e. The reconstructions are obtained by using far field data which are collected at 100 equally spaced directions all around the object,

7 AKDUMAN AND KRESS: DIRECT AND INVERSE SCATTERING 1-7 Figure 4. Real parts of the exact and reconstructed values of the normalized surface impedance for the elliptical cylinder. namely q m = mp/50, m =0,..., 99. The wavelength and incident angle are l = 1 m and f 0 = p/, respectively. It is obvious that both the direct and the least squares method yield very accurate results. The distortions in the direct method correspond to points in the shadow region of the cylinder. The second regularization in terms of least squares for N = 5 suppresses these errors. [18] The results in Figures 4 and 5 are obtained by using the data collected all around the scatterer for one single illumination. In other words, it has been achieved with full aperture measurements. The method yields also Figure 6. The exact and reconstructed values of a real surface impedance in the case of limited angle measurements. satisfactory results in the case of limited aperture data. Figure 6 illustrates the exact and reconstructed values of a purely real surface impedance for the elliptical cylinder. The body is illuminated from the direction f 0 = 0 and the far field data are collected at 50 equally spaced points of the semi circle f p ; 3p. The number of basis functions in the least square solution is N = 5. Obviously the least squares solution still yields satisfactory results. [19] Finally, Figures 7 and 8 illustrate the exact and reconstructed values of the real and imaginary parts of the surface impedance in the case of a drop-shaped cylinder for the aperture measurements of far field Figure 5. Imaginary parts of the exact and reconstructed values of the normalized surface impedance for the elliptical cylinder. Figure 7. Real parts of the exact and reconstructed values of the normalized surface impedance for the dropshaped cylinder.

8 1-8 AKDUMAN AND KRESS: DIRECT AND INVERSE SCATTERING Figure 8. Imaginary parts of the exact and reconstructed values of the normalized surface impedance for the drop-shaped cylinder. pattern. These results are obtained with the same parameters in Figures 4 and 5 with the exception that the incidence direction is f 0 = p. 5. Conclusions [0] The scattering problems related to IBC are of special interest in the electromagnetic theory since the use of IBC reduces some of the mathematical and numerical difficulties occurring in the solution. By the use of IBC a three-dimensional scatterer can be represented by a two-dimensional surface. According to the geometrical and physical properties of the scattering bodies the surface impedance may be a function of location. The problems involving inhomogeneous IBC are of importance from both mathematical and physical points of view. These kind of problems may have practical applications such as antenna design for specific purposes. By choosing an appropriate surface impedance, one can obtain a certain radiation pattern. In this case, the geometry of the scatterer is known as a priori and one tries to get an appropriate impedance function which can be obtained by locating an appropriate coating on its surface. In this paper an electromagnetic scattering problem for an inhomogeneous cylinder of arbitrary shape is considered and solved by reducing the problem to a boundary integral equation. To this aim the scattered field is represented in terms of combined single- and double-layer potentials. The numerical solution of the boundary integral equation which contains a singular kernel is obtained by a Nyström method. The method developed here can be used in the solution of scattering problems related to any cylindrical geometry of arbitrary shape. The method can also be extended to three-dimensional scattering problems. [1] The inverse impedance problem whose aim is to recover the impedance function of the boundary of the scatterer through far field measurements of the scattered field is also considered in this paper. Since the surface impedance contains all the physical and geometrical properties of the scattering object, its reconstruction leads to recover information about its inner structure. The reconstruction of the surface impedance can be achieved by using the impedance condition itself which requires to know the total field and its normal derivative on the surface. These required field values are obtained from the far field pattern by expressing the scattered field in terms of a single-layer potential. This leads to a Fredholm integral equation of the first kind that is solved by Tikhonov regularization. To reconstruct the surface impedance a second regularization scheme in the sense of a least squares fit is applied. One of the important properties of the method is that it yields also good resolution with limited aperture data. The method given here can be used to obtain the IBC for any scattering object in two dimensions. In other words, it is a new method for the determination of IBC. It can be extended to the three-dimensional cases. [] Acknowledgments. This research was carried out while Ibrahim Akduman was visiting the University of Göttingen as a fellow of the Alexander von Humboldt Foundation. The hospitality of the University of Göttingen and the support of the Alexander von Humboldt Foundation are gratefully acknowledged. References Akduman, I., and A. Yapar, Surface impedance determination of a planar boundary by the use of scattering data, IEEE Trans. Antennas Propagat., 49(), , 001. Colton, D., and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley, New York, Colton, D., and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, nd ed., Springer-Verlag, New York, Hoppe, D. J., and Y. Rahmat-Samii, Impedance Boundary Conditions in Electromagnetics, Taylor and Francis, London, Kress, R., On the numerical solution of a hypersingular integral equation in scattering theory, J. Comput. Appl. Math., 61, , Kress, R., Linear Integral Equations, nd ed. Springer-Verlag, New York, Leontovich, M. A., Investigations of Radio Wave Propagation, Part, Sov. Acad. of Sci., Moscow, Marceaux, O., and B. Stupfel, Higher order impedance boundary conditions for multilayer coated 3-D objects, IEEE Trans. Antennas Propagat., 46(3), , 000.

9 AKDUMAN AND KRESS: DIRECT AND INVERSE SCATTERING 1-9 Senior,T.B.A.,andJ.L.Volakis,Approximate Boundary Conditions In Electromagnetics, Inst. of Electr. Eng., London, Senior, T. B. A., J. L. Volakis, and S. R. Legualt, Higher order impedance and absorbing boundary conditions, IEEE Trans. Antennas Propagat., 45(1), , Wait, J. R., The scope of impedance boundary conditions in radio propagation, IEEE Trans. Geosci. Remote Sens., 8, 71 73, Wang, D. S., Limits and validity of the impedance boundary condition on penetrable surfaces, IEEE Trans. Antennas Propagat., 37(4), , Yapar, A., and I. Akduman, Reconstruction of the surface impedance of an inhomogeneous impedance boundary beyond layered media, Radio Sci., 36(4), , 001. I. Akduman, Istanbul Technical University, Electrical and Electronics Engineering Faculty, Maslak Istanbul, Turkey. R. Kress, Institut für Numerische und Angewandte Mathematik, Universität Göttingen, D Göttingen, Germany.