## 1. Introduction

[2] The work summarized in the present paper aims at developing a fast approximate model of the electromagnetic behavior of obstacles buried close to one another within a given infinite or half-space host medium, in the particular case of low frequencies of operation (induction regime) and homogeneous conductive obstacles and host media. The configuration of study is sketched in Figure 1. Our main concern is the modeling of the variation of the so-called transfer impedance observed between two horizontal electrical loops, one set as transmitter, one as receiver, moved together at some fixed distance above the surface of a subsoil-like half-space wherein one or a small number of obstacles are buried. Thus, one will focus on this surface-to-surface probing situation, studied for the detection of UXO by *Huang and Won* [2003] or for geophysical applications as in the work of *Cui et al.* [2003a]. Yet another situation of interest (for which the same approach applies for the most part) is the modeling of the response of a small receiver loop displaced along a borehole with obstacles nearby when a horizontal electrical loop is set above the surface [*Bourgeois et al.*, 2000], and examples will be given about it as well.

[3] To begin with, one considers the case of a single obstacle (taken homogeneous, its conductivity differing from the one of its host medium). The model of the electromagnetic interaction (time harmonic, at a single low frequency of exploration for which diffusive phenomena are predominant) is based on the localized nonlinear (LNL) approximation [*Habashy et al.*, 1993] (which is known to work well for moderate contrasts of conductivity between the host medium and the obstacle), in addition to the hypothesis of small obstacle dimensions with respect to the skin depth in the host medium. Then, one extends the modeling to the case of two homogeneous obstacles close to one another, the geometric and electric properties of which differ, via a proper account of their interaction by means of the theory of Lax-Foldy on multiple diffraction [*Tsang et al.*, 1985; *Braunisch*, 2001]. Finally, one generalizes the approach to an arbitrary finite number *N* of obstacles via an iterative method based on the same tools.

[4] In the induction regime, the modelization by canonical obstacles as spheres [*Wang and Chew*, 2004; *Kaufman and Keller*, 1985] or spheroids [*Chen et al.*, 2007, 2004; *Nag and Sinha*, 1995; *Sinha and MacPhie*, 1977] is simple but quite enough for large wavelength measurements. Solving this problem can be achieved using exact methods, as method of moment or the CGFFT [*Cui et al.*, 2003b] or approximate methods, as extended Born [*Habashy et al.*, 1993] or modified extended Born [*Tseng et al.*, 2003; *Zhdanov et al.*, 2007; *D'Urso et al.*, 2007] that use meshed obstacles. The LNL approximation of *Habashy et al.* [1993], that is the base of extended Born methods, is here used differently with low-frequency approximation [*Kleinman and Dassios*, 2000], in order to obtain simple analytical solutions that are low cost and allows high-speed numerical computations. Focus of the investigation is on the case of ellipsoids and their low-frequency expansions of their polarization tensors and associated electromagnetic fields are here proposed in harmony with earlier work of *Perrusson et al.* [2000]. *Chen et al.* [2007] has modeled UXO as spheroid and in order to find analytical solutions of the scattered field, but it was approximated by the static field that provides the main behavior of the fields at low frequency. Here low-frequency expansions allow us to provide more than the static behavior while adding the three first terms of the low-frequency expansion.

[5] A variety of numerical results helps us to evaluate pros and cons of the approach, with specific attention given to coupling effects, and the determination of possible bounds on the proposed model of interaction. The ill-posedness of the characterization of buried obstacles in the surface-to-surface configuration is also considered briefly, different ellipsoids generating similar impedance variations.