## 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 *e*^{−iωt} is assumed and omitted throughout the paper.