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### Keywords:

• inverse scattering;
• integral equation method;
• conductive boundary condition

### Abstract

[1] In this paper, we derive a numerical solution of an inverse obstacle scattering problem with conductive boundary condition. The aim of the direct problem is the computation of the scattered field for a given arbitrarily shaped cylinder with conductive boundary condition on its surface.The inverse problem considered here is the reconstruction of the conductivity function of the scatterer from meausurements of the far field. A potential approach is used to obtain boundary layer integral equations both for the solution of the direct and the inverse problem. The numerical solutions of the integral equations which contain logarithmically singular kernels are evaluated by a Nyström method and Tikhonov regularization is used to solve the first kind of integral equations occuring in the solution of the inverse problem. Finally, numerical simulations are carried out to test the applicability and the effectiveness of the method.

### 1. Introduction

[2] Magnetotellurics (MT) is an electromagnetic geopyhsical prospecting method of imaging stuctures below the earth surface. In the MT studies, geophysicists explore different types of media by using their electrical properties. In both oil and mineral prospecting one can expect large resistivity contrasts in earth materials as mentioned in the work of Cagniard [1953]. Along this line, the electrical properties of the scatterers play an important role for their recognition process. In these applications a conductive boundary condition (CBC) is commonly used. For the physical explanation of CBC, see Schmucker [1971] and Vasseur and Weidelt [1977].

[3] From the electromagnetic point of view, electrical fields cannot penetrate into an ideal conductor with positive thickness. However, the CBC occurs when the obstacle is covered by an infinitely thin non-ideal type conductor. This physical phenomenon is interesting both for geophysicists and mathematicians. Recovering the specific function that defines the electrical properties (conductivity function) of the structures by measurements of the scattered field can be used to distinguish the types of rocks or sediments in an inaccessible medium. Mathematically, this type of problems can be considered as a generalization of classical transmission boundary value problems by using an impedance boundary condition. In this context, the derivation of the CBC for time-harmonic electromagnetic waves in MT applications is described by Angell and Kirsch [1992]. In the paper of Hettlich [1994], Kirsch-Kress and Colton-Monk methods are applied to reconstruct the shape of the scatterers, where CBC is satisfied on their surface. Uniqueness theorems for the inverse obstacle scattering with CBC are proven for time harmonic electromagnetic and acoustic waves by Hettlich [1996] and by Gerlach and Kress [1996], respectively and also for the latter case, the properties of the far field operator is investigated by Torun and Ates [2006]. Additionally, the problem of determining some information on the surface conductivity without assuming that the boundary is known based on the linear sampling method using multistatic data (many incident waves) is considered by Cakoni et al. [2005] and by Colton and Monk [2006].

[4] In this paper we present a new method for reconstructing the conductivity function. Its main objective is to describe a numerical algorithm based on the so-called potential approach by using a Nyström method both for the solution of the direct and the inverse problems. In order to use the standard equivalence of the acoustic and electromagnetic problems in two dimensions, an infinitely long and arbitrarily shaped cylinder which CBC satisfied on its boundary is chosen for the case of plane wave illumination.

[5] The direct problem considered is to find the scattered field at large distances from the obstacle, provided that the interior and the exterior total fields both satisfy the Helmholtz equation in the corresponding media and the CBC on the boundary of the scatterer under the Sommerfeld radiation condition. In the direct scattering problem, the fields are represented with combined single- and double-layer potentials. Afterwards, an integral equation system is obtained by using the jump relations and regularity properties of the potentials [see Colton and Kress, 1998]. This equation system is well-posed and the existence and the uniqueness of the solution is guaranteed by Gerlach and Kress [1996].

[6] The inverse boundary value problem is to determine the conductivity function, from the far field pattern for a given shape of the scatterer. According to this aim we followed the main idea of the method suggested by Akduman and Kress [2003] for impedance cylinders and propose an algorithm to the cylinders with conductive boundary condition on their surface. Furthermore, in order to avoid an inverse crime, the scattered and the interior total field are represented by using only single-layer potentials. One of the densities is obtained by solving the far field equation through Tikhonov regularization since it is an ill-posed integral equation of the first kind. Once the density is known one can compute the total field outside of the scatterer. Then from the continuity of the field across the boundary the interior total field can be calculated via reconstruction of the second density. In the last step the conductivity function is obtained from the CBC with a least squares regularization.

[7] The organization of the paper is as follows: In the general formulation section, the problem is mathematically defined and the fundamental formulation is given. Section 3 and section 5 are devoted to the formulation of the direct and inverse problems, respectively, and in section 4 the parametrization of the integral operators that are used both for the direct and inverse problems is introduced. To illustrate the applicability and the effectiveness of the proposed method some numerical simulations are presented in section 6. In the final section, conclusions and concluding remarks are given. Note, that a time factor eiωt is assumed and omitted throughout the paper.

### 2. General Formulation

[8] Let D2 be a bounded domain with connected twice continuously differentiable boundary Γ, and v the unit normal of the boundary directed into the exterior of D. The complex wave number for the interior/exterior medium is defined as

Here, ω is the radial frequency, j,μj, σj, are the dielectric permittivity, magnetic permeability and the conductivity of the background medium, respectively. It is assumed that k0 and k1 are given positive complex numbers, where Re kj > 0 and Im kj ≥ 0. Furthermore, the complex valued continuous conductivity function λ is given by,

The parameter τ, appearing in the equation 2 is the so-called integrated conductivity given by

where σδ is the space dependent conductivity parameter given in the study of Angell and Kirsch [1992]. For the rest of the paper it is also assumed that Im λ ≤ 0 on Γ [see Gerlach and Kress, 1996].

[9] In this configuration, an arbitrarily shaped cylinder is illuminated by a plane wave whose electric field vector is polarized parallel to the x3-axis, that is,

d = (cosϕ0, sinϕ0) is the propagation direction with angle ϕ0. Owing to the symmetry and homogeneity along the x3-axis the total electric field vector will be polarized both inside and outside of the cylinder parallel to the x3-axis, that is, E0 = (0, 0, u0) and E1 = (0, 0, u1), respectively. Then the problem is reduced to a scalar one in terms of total fields that have to satisfy Helmholtz equations

with the conductive boundary condition

At infinity the Sommerfeld radiation condition is imposed that the scattered wave us is outgoing such that

where us = u1ui in \. It is also well-known that every radiating solution us has an asymptotic behaviour of an outgoing wave

uniformly in all directions = x/∣x∣, where the function u defined on the unit circle Ω is called the far field pattern of us.

[10] A potential approach is chosen to formulate the direct and the inverse problems throughout this paper. Physically speaking, the single- and double-layer potential correspond to a layer of monopoles and dipoles, respectively. In this context, let Γ0 be a closed bounded curve and given an integrable function f, the integrals

and

are called, respectively, single- and double-layer potentials with density f for the corresponding background medium m, for m = 0,1. Here, Φm(x, y) is the fundamental solution to the two dimensional Helmholtz equation, given in terms of the Hankel function H0(1) of the first kind and zero order by

The single- and double-layer potentials are solutions to the Helmholtz equation in the interior/exterior domain of the scatterer that satisfy the Sommerfeld radiation condition. Any solution to the Helmholtz equation can be represented as a combination of single- and double layer potentials in terms of the boundary values and the normal derivatives on the boundary. The behaviour of the surface potentials at the boundary is described by regularity and jump relations.

[11] At this point, for the sake of brevity throughout the paper, the single- and double-layer operators Sm and Km and the normal derivative operators Km and Tm are defined as

[12] For all integral operators above, x ∈ Γ0 and (Sm, Km, Km, Tm): C0) C0), m = 0,1. However, we note that the hypersingular operator Tm is only defined on subspaces LC0) of sufficiently smooth functions. Detailed explanations of the boundary integral operators are given by Colton and Kress [1983, 1998].

### 3. Direct Problem

[13] We follow the existence and the uniqueness theorems in Gerlach and Kress [1996] omitting the technical details and use the same integral representation of the fields under the assumptions of the referred paper. We represent the fields as a combination of single- and double-layer potentials

Then we can write the integral equation system in an abbreviated form as

Here, I is the unity matrix, the operators E and A are defined by

and Λ and H are given as

Then, using the jump relations for single- and double-layer potentials [see Colton and Kress, 1998], it can be seen that the fields (16) and (17) satisfy the conductive boundary value problem provided the densities ψ and ϕ satisfy the integral equation (18). This integral equation system has to be solved for the densities and then the interior total field u0, the exterior total field u1, and the far-field u given by

can be computed.

### 4. Parametrization of the Integral Operators

[14] The boundary curve Γ is assumed to parametrized in the form

where z is 2π–periodic and twice continuosly differentiable function satisfying z′(t) ≠ 0, t ∈ [0, 2π]. For any vector a = (a1, a2), the orthogonal vector a, is obtained as a = (a2, −a1) by a clockwise rotation of 90 degrees. Then via = ∣z′∣f ˆ z for m = 0, 1 we introduce parametrized operators. The parametrized single layer operator m: C[0, 2π] C[0, 2π] is given by

and the corresponding parametrized normal derivative operator m: C[0, 2π] C[0, 2π] by

The parametrized double-layer operator m, and the parametrized normal derivative double-layer operator m, are given via = fz,

where

and

Here we made use of H0(1)′ = −H1(1) with the Hankel function H1(1) of order zero and of the first kind. The parametrized form of the integral equation system (18) is now given via = ψz and = ∣z′∣z,

[15] For these integral equations with singular kernels, a combined collocation and quadrature methods based on trigonometric interpolation as described in section 3.5 of Colton and Kress [1998] are applied. For a related error analysis we refer to Kress [1998]. Moreover, analytic boundary curves such as ellipses Γ(e), kite shaped Γ(k), drop shaped Γ(d) and rounded triangles Γ(r) are considered to obtain exponential convergence. The parametrization of these curves are given by

for t ∈ [0, 2π]. Here e0 and e1 are constants. The parametric representation of the far-field is obtained as

where γ = e/4/ and ∈ Ω.

[16] To illustrate the exponential convergence of the direct problem clearly some numerical results are given in Table 1 by choosing the shape of the scatterer as kite, and a continuous conductivity function as

In this table some approximate values are given for the far field pattern u (d) and u (−d) in the forward direction d and the backward direction −d. The direction d of the incident wave is d = (1, 0). The number of quadrature points in Nyström method is 2n.

Table 1. Numerical Example for Direct Scattering Problem
Wave NumbersnRe u (d)Im u (d)Re u (−d)Im u (−d)
8−0.849504320.94133306−1.407396950.42737369
k0 = 1 + i16−0.849389700.94141239−1.407996110.42745603
32−0.849390760.94141434−1.407973520.42745599
k1 = 0.5 + i0.564−0.849390750.94141434−1.407973540.42745598
128−0.849390750.94141434−1.407973540.42745598
81.159745942.59430512−1.846004900.58468379
k0 = 3 + i2161.152586102.58983201−1.920747840.64224335
321.152586292.58983202−1.920750710.64226952
k1 = 1 + i641.152586292.58983202−1.920750720.64226952
1281.152586292.58983202−1.920750720.64226952
81.68845437−1.94677475−2.143392240.61626281
k0 = 5 + i161.47319387−1.70565385−2.152611130.72987802
321.47316833−1.70571814−2.155609810.73398480
k1 = 2 + i641.47316833−1.70571814−2.155609830.73398478
1281.47316833−1.70571814−2.155609830.73398478

[17] The conductive boundary value problem is reduced to the well-known transmission problem by choosing λ = 0 and to the scattering problem for perfect electric conductors (PEC) via λ ≫ 0. In order to test the reduced problem we solve the transmission problem with two different methods. The first method is based on series representation of the fields for circular cylindrical geometry as given in chapter 7.1 of Eom [2004]. The second method is the solution of the transmission problem via a potential approach where the definition and the investigation of the problem is described in chapter 3.8 of Colton and Kress [1983]. For the solution of the PEC scattering problem, chapter 3.5 of Colton and Kress [1983] is followed. We observed that in each comparison case both numerical results match accurately.

### 5. Inverse Problem

[18] The inverse scattering problem considered here is the reconstruction of the conductivity function from the far field pattern for one incident wave, assuming that the shape of scatterer is known.

[19] Assumption 1 is as follows: The wave number k02 is not a Dirichlet eigenvalue for the negative Laplacian in the interior of Γ. The boundary curve Γ is analytic.

[20] In our approach we first construct the total fields u1 and u0 from the given far field pattern. Then we determine the unknown function λ from the conductive boundary condition,

Indeed, by Rellich's lemma the scattered wave us, and consequently also the total field u1, are uniquely determined by the far field pattern. Owing to the Assumption 1 and the boundary condition u0 = u1 on Γ, the field u0 is also uniquely determined in D. Therefore we can use (32) to determine λ. Since Γ is analytic u1 cannot vanish on open subsets of Γ otherwise it would vanish on all of Γ. But then from the boundary condition u1 = u0 on Γ and our assumption on k02 we obtain u0 = 0 in all of D. This implies that ∂u0/∂v = 0 on Γ, and therefore the CBC yields ∂u1/∂v = 0 on Γ. By Holmgren's theorem u1 = ∂u1/∂v = 0 on Γ implies that u1 = 0 in all \. But this means that ui + us = 0 vanishes. This is a contradiction since the plane wave ui does not satisfy the Sommerfeld radiation condition.

[21] In order to proceed along these lines we seek the fields in the form of single-layer potentials,

By the first CBC, namely, u0 = u1 on Γ the densities ϕ and χ are coupled by the integral equation

The density ϕ is determined from the far-field pattern of the scattered field which is obtained by using the asymptotic behaviour of the Hankel function as

for the observation direction , where = {(cos θ, sin θ ) : θ ∈ [0, 2π]}. Hence, given the far field pattern u, the integral equation of the first kind has to be solved,

Here the integral operator S is given by

[22] Theorem 1 is as follows: The far field integral operator S, defined by (38), is injective and has dense range provided k12 is not a Dirichlet eigenvalue for the negative Laplacian in the interior of Γ. For the proof, see Theorem 5.17 in the work of Colton and Kress [1998].

[23] The operator S has an analytic kernel, therefore the equation (37) is severely ill-posed. For this reason some kind of stabilization such as Tikhonov regularization has to be applied. For a regularized solution in the sense of Tikhonov the equation is solved as

with a regularization parameter α > 0 and the adjoint S* of S is given by

Once the density ϕ is known from the equation (39), one can compute the density χ by using the integral equation (35).

[24] Finally, to reconstruct the conductivity function λ, with the equation (32), the total field expressions over the boundary Γ are given under the jump conditions as

The reconstruction of the λ will be sensitive to errors u1 in the vicinity of zeros. To obtain a more stable solution, we express the unknown function λ in terms of exponential basis functions κn(z(t)) = eint, n = 0, ± 1, ± 2,…, ± N, as a linear combination

Then equation (32) has to be satisfied in the least squares sense, that is, the coefficients τN,…,τN in (44) are determined such that for a set of grid points x1, x2, …, xM on Γ the least squares sum

is minimized. The number of basis functions N in (44) can be considered as some kind of additional regularization parameter.

### 6. Numerical Examples

[25] This final section is devoted to describe the details of the numerical implementation of the method and to show the applicability of the reconstruction algorithm both for noisy and exact data case by presenting some illustrative examples. In these examples, the integral equation systems are solved through the Nyström method with a discretization number n = 64. The integral appearing in the far field expression (30) is evaluated numerically by using the trapezoidal rule. In all examples the regularization parameters α and N were chosen by trial and error.

[26] In order to obtain a synthetic noisy far field , random errors were added pointwise to the exact far field data u, that is,

Here, β denotes a random variable with {Re β, Im β} ∈ (0, 1). The parameter η is called the noise ratio and in this study for the following three experiments it was chosen as η = 0.03 to have a noise level of 3%. However, in the final simulation 5% noise level was used. The reconstruction with distorted type far field data is indicated in the legends of Figures 1–4by noisy data. It is also noted that the reconstructions presented in this study with noisy field data were selected from the reconstructions having average quality.

[27] In the first numerical application, an elliptic scatterer Γ = Γ(e) with semiaxis e0 = 2 and e1 = 1.5 is considered. The conductivity function over the boundary is given by

Both for the exact and noisy data cases, the wave numbers and the illumination angle are given by k0 = 1 + i0.25, k1 = 0.5, ϕ0 = 180°, respectively. To reconstruct λ with exact data, the regularization parameters are chosen as α = 10−9 and N = 4 whereas for noisy data α = 0.2 and N = 3 gives acceptable results. Here, the conductivity function was chosen as a rational function and representing the conductivity function in terms of exponential basis functions yields poor approximation. Therefore, we could not observe the advantage of solving the equation (32) in least squares sense. However, for this example especially the reconstruction with the exact data can be considered rather impressive.

[28] The second example is devoted to test the method with limited aperture far field data using complex valued wave numbers both for the interior and the exterior mediums. To this aim, a rounded triangular shaped cylinder with the plane wave incidence of ϕ0 = 180° is chosen and the far field data are collected in 64 equidistant points on the semicircle ϕ ∈ [0, π]. The wave numbers of the background media are given as k0 = 2 + 0.5i and k1 = 0.5 + 1.5i and the conductivity function is

The Tikhonov regularization parameters and the degree of the polynomials are chosen for exact data as α = 10−12, N = 2 and for noisy data as α = 10−2, N = 1.

[29] It is observed that the treatment of the reconstructions do not change significantly using the half far field pattern. Moreover, choosing exponential basis functions in the least square regularization provides a major advantage on the quality of the reconstructions when the conductivity function has a nature of trigonometric polynomials.

[30] In the third application, we wanted to observe the quality of the reconstructions using real valued wave numbers that satisfy the condition k0 < k1. This condition physically states the obstacle is buried in a more dense medium than itself. For this a drop shaped scatterer having wave number k0 = 2 and conductivity function

is buried in an infinite background medium k1 = 3, and illuminated by a plane wave with an incident angle ϕ0 = 0°. In the exact data case the regularization parameters α = 10−15 and N = 3 are chosen and for the noisy case α = 10−5 and N = 2. Here, the main difference of the treatment of the inversion algortihm as compare to previous two experiments is in order to get acceptable reconstructions one needs smaller Tikhonov regularization parameters. Moreover, especially the imaginary parts of the reconstructions do not match exactly to the imaginary part of the original function even by using exact full far field data. Hence we conclude that having at least one complex valued wave number for any mediums (interior/exterior) increases the accuracy of the reconstructions. To observe the affect of the shape of the obstacle and the type of the conductivity function to our last conclusion some other numerical experiments also done by changing shape and conductivity functions. However, we observed that our proposed result is still valid for different type of problem configurations.

[31] In the final example, reconstructions are obtained by using full near field data. Therefore we compute the scattered field using the equation (16) on a measurement circle having radius rm = 3. For the solution of the inverse problem instead of using far field equation given in (36) the integral representation of the scattered field in (33) is substituted to the equation (37). Then the same procedure is followed. The integral appearing in (33) is computed by trapezoidal rule with a discretization number, n = 64. The geometry of the scatterer is chosen as a kite. The incident angle is ϕ0 = 180° and the wave numbers for corresponding mediums are k0 = 4 and k1 = 3 + i. The conductivity function is

The parameters chosen are α = 10−15, N = 4 for exact data and α = 10−7, N = 2 for noisy data.

[32] As a main result, using near field data allows us to obtain acceptable reconstructions with a noise level of 5%. In this case we choose smaller Tikhonov regularization parameters as compare to the other experiments done with far field data. As to be expected, the quality of the reconstructions are affected by the selection of the radius of the measurement circle. Furthermore, if we increase the radius of the measurement circle then smaller regularization parameters are needed for good reconstructions.

### 7. Conclusion

[33] In this paper, an algorithm is investigated to reconstruct the conductivity function of a scatterer which is located in a medium either with a complex or real valued wave number from the far or near full/half field data for one incident wave. To obtain a satisfactory reconstruction, the noise ratio, the conductivity function, the regularization parameters, the wave numbers, the shape of the scatterer, and the illumination angle have to be chosen appropriately.

[34] It is observed that using far field data when the noise level exceeds 3%−4%, reconstructions start to deteriorate. Smooth and round scatterers with an optimal illumination angle and at least one real valued wave number increase the accuracy of inversion. Higher wave numbers, rapid varying conductivity functions and obstacles having rough parts cause rapid oscillations in the scattered field and this effect also deteriorates the reconstrunctions.

[35] Additionally, variation of the real parts of the wave numbers do not affect neither the real nor the imaginary part of the reconstructed conductivity function dramatically. However, the variation of the imaginary parts of the wave numbers affect the imaginary part of the reconstructed function significantly. We also conclude that we need smaller Tikhonov regularization parameters and a higher degree of trigonometric polynomials for the exact data case, whereas we need stronger regularization parameters and a smaller degree of polynomials for noisy data. The latter result is the same as given by Kress et al. [2008]. Finally, we expect that the method can be extended for the obstacles buried in layered cylinders.

### Acknowledgments

[36] The author would like to thank Rainer Kress for his interest and helpful discussions on the topic of this work. This research was carried out while the author was visiting the University of Göttingen as a fellow of Deutsche Akademische Auslandsdienst (DAAD). The hospitality of the University of Göttingen and the support of the DAAD are gratefully acknowledged.

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