The electromagnetic characterization of one or more conductive obstacles buried in a conductive subsoil from low-frequency data (induction regime) is expected to benefit from a fast yet fairly accurate solution of the direct problem associated to it. Here, an approximate model is proposed to that effect. It is based on the localized nonlinear approximation, coupling between obstacles (whenever two or more) being approached via the Lax-Foldy theory of multiple diffraction. The obstacles are modeled as generic ellipsoids, with spheres as the limit case. The exact primary field due to an electrical loop (or a magnetic dipole) is calculated, and its low-frequency expansion is introduced. A semianalytical solution of the direct problem is developed from the exact field formulation, analytical solutions being proposed in the most simple cases of either sources or configurations. The model is validated by comparison with numerical experimentations carried out by tools available from the CIVA platform and by the FEKO code. Limits of the model are explored also, notably about the retrieval of the pertinent geometric and electric parameters of an ellipsoid buried in a half-space from sparse data collected above it.
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 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  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.
 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.
 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. , 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. . Chen et al.  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.
 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.
2. Modeling of the Electromagnetic Interaction
2.1. Single-Obstacle Case
 Referring to Figure 1 from now on, let us consider a single nonmagnetic (permeability μ0) 3-D bounded obstacle which is buried in a homogeneous, nonmagnetic subsoil-like half-space (or, in the limit case, a homogeneous infinite space) with conductivity σe. Its volume is denoted as V1, its dimensions being small versus the skin depth in the embedding medium at circular frequency ω, and its conductivity is σ1. In order to keep the possibility to extend the frequency domain of the model, one considers σn as the complex conductivity defined as:
where σ′1 is the static conductivity and ε1 is the dielectric permittivity of the obstacle. So, the wave number kn at a given point can be written as
with henceforth implied time dependence e−jωt, obstacle and host skin depth respectively being δe ≈ and δ1 ≈ .
 A circular transmitter loop T with constant electric current I and a circular receiver loop R (it could be T itself in a single-loop experiment, or another loop) are operated together. Application of reciprocity yields a well known formulation of the variation of the transfer impedance ΔZRT associated to the induced power as
where is the field that would be due to the receiver loop R operated in a fictitious transmitter mode with same current I, and where (T) are the electric source currents induced within the obstacle when it is illuminated by the transmitter loop T. The latter are such that
where the difference of conductivity is Δσ1 = σ1 − σe and (T) is the total electric field (i.e., the sum of the primary field (T)inc due to the transmitter loop in the absence of obstacle, plus the field (T)scat that is scattered by the obstacle):
 To calculate the above current sources, upon the assumption that the contrast of conductivity between the obstacle and the embedding medium is low enough (say, ratios of less than 100 could be handled) and in addition that the primary field does not vary too much in the domain of the obstacle (which means in particular that any exterior source should be not too close to it, that distance being in particular appreciated in terms of the skin depth of the host medium and the dimensions of the obstacle), one comes back to these two issues from the numerical results, the localized nonlinear approximation [Habashy et al., 1993] can be used.
 One has
where 1 is the depolarization tensor (see next) of the obstacle. So, (T) become
Since the dimensions of the obstacle have been hypothesized small enough versus the skin depth of the host medium, carrying out the volume integral is equivalent to taking the value at the center 1 of the obstacle times its volume V1:
Within the above framework, the obstacle simply behaves as an electric dipole which would be set at the center of the (then absent) obstacle. To recapitulate,
The magnetic field scat that is scattered off the obstacle is equivalent to the magnetic field radiated by the dipole 1, so it writes as
where me is the Green magnetic-electric tensor characteristic of the host space, which yields the magnetic field observed at point due to an electric current source taken successively along the three axes of Cartesian coordinates at point . It is the same as well for the electric field scat scattered off the obstacle, which writes as
where is the Green electric-electric tensor characteristic of the host space, which yields the electric field observed at point due to an electric current source taken successively along the three axes of Cartesian coordinates at point .
2.2. Two-Obstacle Case
 To account for the interaction between two (or more) obstacles, one starts from the Lax-Foldy theory as is described, e.g., in the work of Braunisch ; the primary field impinging upon a given obstacle among N ones is equal to the primary field generated by the loop T, plus the sum of the secondary fields radiated by all other N − 1 obstacles, those being themselves equivalent to electric dipoles in our setting as seen in the above. Upon introduction of mscat as the secondary field that is due to the presence of the mth obstacle, one has for the nth obstacle the primary field
 Here, let us focus onto the case of two obstacles only, respectively centered at , , and with conductivity contrasts Δσ1, Δσ2. After combining equations (6) and (12) in the particular case N = 2, it suffices to solve
In the above, ee(, ′) is the Green dyad that yields the electric field at observation point for a point-like electric current source set at source point ′. Thus, the field at the center of the first obstacle (the one at the center of the second obstacle has a similar expression) reads as
where is the identity matrix; 1 accounts for only one part of the interaction, the other part being hidden in 1inc. Equation (14) looks the same as the LNL one (6) in which 1−11 is the new depolarization tensor of the first obstacle, an equivalent expression being found for the second obstacle by substitution of index 2 to index 1.
 As for the impedance variation, it satisfies
If one neglects possible reflections from the interface back onto the obstacle, the expression of the magnetic field scattered by both obstacles is as
 At this stage no hypothesis has yet been enforced upon the Green dyads. For simplicity, one henceforth assumes that they are reduced to the ones of a homogeneous infinite space, i.e.,
where g(, ′) is the scalar Green function in an infinite homogeneous space of wave number ke. This assumption means that interaction at the half-space interface is accounted for in the primary field (be it generated by the transmitter, or fictitiously, by the receiver when one is modeling the transfer impedance), but is neglected in the secondary field (it is assumed that the obstacles are buried far enough from the interface to neglect the multiple reflection on the soil/air interface), as it has been validated from numerical experiments (not shown here) in typical configurations of study.
2.3. Ellipsoidal Model and Corresponding Low-Frequency Expansions
 The above is valid for any smooth enough obstacle shape. A main hurdle however remains, the calculation of the depolarization tensor n:
with relative contrast χn = . The tensor is known analytically for a sphere (radius a):
For ellipsoids with semiaxis lengths (a1, a2, a3), only the first three nonzero terms of a low-frequency expansion [Kleinman and Dassios, 2000] of the tensor as a function of integer powers of jke are effectively available in closed form (in the eigensystem of the ellipsoid) [Perrusson et al., 2000]:
letting R be the matrix of rotation from the fixed Cartesian system to the one set along the three semiaxes of the ellipsoid. In the ellipsoid's system each term of the expansion of the full tensor n is diagonal. The term of order zero reads as
where I1i(a1) (i = 1, 2, 3) is a Carlson elliptic integral of the second kind in a1. The term of order one is null:
The term of order two reads as
involving coefficients dii and cii that can be found in the work of Perrusson et al. ; ρ is one of the three ellipsoidal coordinates, as solution of
h2 and h3 are semifocal distances defined as:
The last known term, of order three, only depends upon the volume Vn of the ellipsoid:
From this point onward, two different calculations can be carried out: one denoted as “partial expansion,” the tensor of depolarization n being the only quantity expanded at low frequency; one denoted as “total expansion,” for which all terms involved are expanded and the result truncated at third order.
 Let us consider this in more detail, as follows. The low-frequency expansion of the tensor of depolarization n reads as
First, the tensor n is defined as the inverse of the tensor of depolarization n, which needs the expansion of the contrast of conductivity and thus of the tensor n;
Here one assumes that Δ = 0 (the contribution of the imaginary part of the conductivity (equation 1) is at least 100 times lower than the one of the real part for the considered induction frequencies ∈ [100 Hz; 300 kHz]), so the expression simplifies to
The first three orders of the low-frequency expansion of n then follow as
The expansion of Γn proceeds from relationship nn = . By identification one easily arrives at
2.4. Low-Frequency Expansion of the Green Dyads
 Let us denote = − , R = ∥ − ′∥, and g(, ′) = . The electric-electric dyad is given by equation (19):
its analytical expression follows as
Using the low-frequency expansion of jωμ0 as
one can calculate the first three nonzero terms (the term of order one is null) of the expansion of the electric-electric Green dyad [ee(, ′)](p)(jke)p:
Since the magnetic-electric Green dyad can be expressed as in equation (20), its low-frequency expansion up to order three reads as
2.5. Calculation of the Primary Electric Field
 The primary electric field due to an electric loop (possibly reduced to a magnetic dipole) above a homogeneous half-space or within a homogeneous infinite space is essential to the above, low-frequency expansions being an asset in that matter, whether available. For reasons of symmetry, one considers a cylindrical coordinate system (r, ϕ, z). The only component of the primary electric field which is nonzero thus is Eϕ(T)inc (the primary magnetic field has two nonzero components, Hr and Hz, but one does not work upon it). Its expansion starts from the second order (the static electric field due to a circular loop is obviously zero), and one will develop it up to the fifth order in practice (to have the same number of orders than for the depolarization tensor):
 A horizontal circular loop with radius a placed at z = h above the (x, y) plane, with fixed current I, radiates within a homogeneous infinite space
J0 and J1 being first-kind Bessel functions order 0 and 1. This reduces to
for a vertical magnetic dipole (following ) equivalent to a small enough horizontal loop (a → 0, moment M = Iπa2).
 Expanding the above fields into integer power series of jke in accord with Wait's approach [Wait, 1982], terms of odd order, and the zeroth order as well, are found to cancell out: E() = E() = E() = E() = 0. The remaining terms of the second and fourth order of the expansion of Eϕ(T)inc in the general case of the loop read as
For a vertical magnetic dipole, the analysis simplifies. One starts from the well known relationship
yielding the terms of second, fourth and fifth order of the expansion as
The fifth order term is null according to the Wait-type calculation, so the two low-frequency expansions are equivalent up to the fourth order.
 In the case of a homogeneous half-space the primary electric field transmitted into it, i.e., under the interface air/subsoil, satisfies
which involves the transmission coefficient in the spectral domain
Only the even orders of its low-frequency expansion are nonzero. However, there does not exist any simple expression for them, so one has to resort to numerical Hankel transforms even if one wishes to perform an approximate calculation, letting
In contrast, for a vertical magnetic dipole, there exists an analytical solution up to the fourth order, the two nonzero terms being
2.6. Calculation of the Interaction Tensor
 At this stage it remains to expand the tensor of interaction n, which reads up to order three as
The inverse of i (i = 1, 2), is thus defined by
By identification one has
Yet, i(1) = , consequently i = . After proper simplification the terms of second and third order are given as a function of the one of zeroth order:
For the numerical computations the low-frequency expansion of the tensor of interaction requires the inversion of i(0). The different terms of this expansion can be expressed as follows:
 For two spheres 1 simplifies. One has
In addition, one can apply the reciprocity theorem which tells that ee(2, 1) = ee(1, 2), letting eet(1, 2) be the transpose of ee(1, 2). So, one arrives at the expression of the static term of the tensor of interaction in the two-sphere case:
One could similarly simplify the terms of higher order which (in concatenated form) read as
where l = 2, 3 and n, m, p, q = 0, …, l, enforcing n + m + p + q = l.
2.7. Total Low-Frequency Expansion
 In the case of the homogeneous infinite space, the total low-frequency expansion goes from the second to the fifth order, since the expansion of the primary field is started from the second order and the depolarization tensor is only known from the static term to the term of third order. The scattered electric field scat at the center of the different obstacles reads as
It can be written as a sum of three terms:
where l = 4, 5 and a, b, c, d, e, f = 0, …, l, enforcing a + b + c + 2 = l and a + b + d + e + f + 2 = l.
 The scattered magnetic field can be handled from its general expression
which is more compactly written upon introduction of the two electric dipoles 1, 2:
their low-frequency expansion coming from the one of the scattered electric field
where i(l) = ΔσiViidiff(l)(i), i = 1, 2, l = 2, 4, 5. Then, letting
 As for the variation of the transfer impedance induced by the two obstacles, its expansion starts from the fourth order and is truncated at the seventh order, because of the double occurrence of the primary field, which starts from the second order:
Rather involved expressions follow as
where l = 6, 7 and g, a, b, c, d, e, f = 0, …, l, such as g + a + b + c + 4 = l and g + a + b + d + e + f + 4 = l.
2.8. Generalization to N Obstacles
 To model the variation of the transfer impedance due to an arbitrary number of buried obstacles, one starts from the Lax-Foldy theory and equation (5), which enables to write down a coupled system, its nth equation reading as
Then, one can combine the equations including the LNL approximation (see (6)) and Lax-Foldy multiple diffraction (see (12)):
Replacing scat by its expression, like for the coupled system (13) describing the problem at N = 2, one obtains a system whose nth equation is
 To solve this system, one applies an iterative method. New notations are introduced for handiness:
where index (0) means that the indexed term is describing the decoupled case. Then, one starts from the last equation which depends only upon the first N−1 fields:
Replacing N() by its value given in equation (85) into the first N − 1 equations, a system of N−1 coupled equations with N − 1 unknowns follows. After factorizing by n, a new system is obtained, where the nth equation is
with n = 1, 2, …, N−1.
 One iterates the procedure to get the field at the center of obstacle numbered 1:
The field at the center of each one of the N obstacles follows as
This solution specialized to the case of two obstacles can be shown to reproduce the solution as described in equations (14) and (16).
3. Numerical Simulations
 The proposed LNL model has been benchmarked by using the FEKO code (http://www.feko.info/) for spheres buried in a homogeneous infinite space, and by using the eddy-current method of moments tools of the CIVA platform (http://www-civa.cea.fr) for both spheres and ellipsoids buried in the homogeneous half-space in several configurations. Other cases have been validated from data courtesy of CCE Nanjing simulated with a CG-FFT type code initialized by the extended Born solution [Cui et al., 2003a; Wang et al., 2004; Habashy et al., 1993].
 In addition to the full CIVA tool (denoted as CIVA exact) one considers the particular version of the CIVA tool involving a localized nonlinear approximation in which the tensor of depolarization is estimated not for the obstacle as a whole but for each one of the cubical voxels which it is described with.
 Let us notice here that with the two CIVA tools one can simulate cubes or parallelepipeds with cubical meshes, so one makes the following approximation on the dimensions of the obstacles in order to get an equivalent volume between spheres and cubes, and ellipsoids and parallelepipeds:
where ai are the lengths of the semiaxes of the ellipsoid (the radius of the sphere), ci are the different side lengths of the parallelepiped (the side of the cube), and i corresponds to the xi axis in the Cartesian coordinate system of the ellipsoid.
 As for what concerns the FEKO code the sphere is meshed by cubic voxels (with maximum side of a/10), so its total volume is smaller than expected, the field produced by the code then being multiplied by the ratio of the true volume of the sphere to the volume of the cubically meshed one Vexact/Vfeko sphere ≈ 1.117.
3.1. Single-Sphere Case
 As mentioned in section 2.1, the difference between the impedance variation simulated with CIVA exact and the one provided by the LNL model herein depends upon the contrast of conductivity q and the dimensions of the obstacle (always smaller than the skin depth). The variation of the relative discrepancy between models can be characterized, to an extent at least, by the gap indicator
where one refers the results of our model (denoted as ΔZMODEL) to those obtained by means of the full CIVA method of moments tool (denoted as ΔZCIVA exact).
 Similarly, one might wish to appreciate the difference between the scattered magnetic field Hscat computed with FEKO and the one obtained with the LNL model as a function of the contrast q = and the radius of the sphere. The relative discrepancy between models can thus be characterized by the following gap indicator, computed for each component x, y, z:
where one refers again the results of a given model (denoted as HiMODEL) to the ones obtained with FEKO (denoted as HiFEKOscat).
 In both configurations (infinite space and half-space), the evolution of these gaps between models has been considered for different values of contrast q (1/50, 5, 50, 500), sizes of the equivalent cube (0.5 m, 4 m, 8 m, 12 m, 16 m), see Table 1, or radii of the sphere for the infinite space case (0.5 m, 4 m, 8 m, 12 m), see Table 2. Furthermore, one has investigated the influence of frequency (100 Hz, 10 kHz, 100 kHz), of the conductivity of the host medium σe (2.10−4 S/m to 0.1 S/m), and the depth of burial (30 m, 50 m, 100 m).
Results are similar with the imaginary part. Maximum of the gap computed with the LNL version of the CIVA tool and with our LNL model, as a function of the size of the equivalent cube and its ratio of conductivity q with respect to the host medium.
q = 1/50
q = 5
q = 50
q = 500
Table 2. Complex-Valued Components of the Scattered Magnetic Fielda
Maximum of the gap computed with the FEKO code and with our LNL model, as a function of the size of the equivalent cube and its ratio of conductivity q with respect to the host medium.
q = 1/50
q = 5
q = 50
q = 500
 In the half-space case, one has considered the impedance variation of only one electrical loop, at 21 points (each 5 m) on a 100 m long line, 1 m above the soil interface, shifted by 50 m along y from the center of the obstacle. The measurement configuration is described in Figures 23–4 for the half-space; for the infinite space one computes the three complex components of the scattered magnetic field in the same configuration but without the air/soil interface.
 Illustrative results for the half-space case are provided in Table 1 in the configuration under study (a buried sphere assimilated as needed to a cube of same volume). They consist of the above gaps as a function of the side (from 4 to 16 m) of the equivalent cube, computed via our LNL model (denoted as LNL in Table 1) and via the CIVA LNL.
 As said before, a similar study has been carried out on the complex-valued components of the scattered magnetic field collected in the same configuration as in Figures 2 and 3 but in the absence of the interface, for a single buried sphere. Results are summarized in Table 2. One compares the field simulated by the FEKO code with the one provided by the LNL model (see equation 94). The gap overall appears smaller than the one in Table 1, similar increases versus ratio of conductivity and size being observed.
 To summarize, the proposed LNL model appears suitable for a conductivity contrast q between a spherical obstacle and its host medium lower than about 100 and an obstacle radius of 10 m (the maximum of the gap is 18.8% for q = 50 and a = 10 m). So, the model is valid as long as the sphere is penetrable in full, i.e., its radius is of the order of or smaller than the skin depth within the sphere material which corresponds to the condition of validity of the LNL approach. The choice of frequency, depth and host conductivity ranges (say, 100 Hz to 100 kHz, 30 m to 100 m, 2.10−4 S/m to 10−1 S/m) does not influence the discrepancy between models.
3.2. Single-Ellipsoid Case
 The former analysis can be generalized to the case of a conductive ellipsoid for different orientations, ratios of conductivity, and frequencies. The comparison as summarized in Table 3, letting aside its evolution as a function of the ratio of conductivity (as the one in the sphere case), is between the impedance variation computed with the CIVA code for an equivalent parallelepiped and the one computed with our LNL model for the ellipsoid. Each row in Table 3 is referring to a certain set of side lengths (given in meters) of the equivalent parallelepipeds: 4 × 1 × 1 (prolate spheroid); 4 × 4 × 1 (oblate spheroid); 4 × 2 × 1 (triaxial ellipsoid); and 10 × 2 × 1 (strongly elongated ellipsoid). Each column is referring to a certain set of orientations according to the rule: x1 following Ox, x2 following Oy and x3 following Oz; x1 following Oy, x2 following Oz and x3 following Ox; x1 following Oz, x2 following Ox and x3 following Oy.
Results are similar with the imaginary part. Maximum of the gap computed with our LNL model as a function of the equivalent parallelepiped size (c1 × c2 × c3) and orientation (i × j × k) with i, j, k = 1, 2, 3, for a ratio of conductivity q = 50 with respect to the host medium at 10 kHz frequency (results are of same order for 100 Hz and 100 kHz).
4 × 1 × 1
4 × 4 × 1
4 × 2 × 1
10 × 2 × 1
 Let us observe that the orientation a1; a2; a3 is associated to the lowest gap, and this is also in that orientation (assuming that the configuration of the measurements is the one described in Figures 2 and 3) that the coupling between the source and the obstacle is the most important; the electric loop is horizontal so the coupling is built only on the x and y components.
 As a further result of interest, one compares the variation of impedance due to a buried ellipsoid (or a sphere) calculated with the LNL partial expansion up to the zeroth order (denoted as LNL 0) and with the LNL partial expansion up to the third order (denoted as LNL 3). This result is illustrated in Figure 5. The zeroth order suffices at induction frequencies, this being true whatever be the dimensions of the obstacle and the ratio of conductivity (in the previously introduced ranges).
3.3. Contribution of the Coupling Effect: Case of Two Obstacles
 In this section one attempts to estimate the error introduced by the LNL model for various obstacles (spheres, ellipsoids). Two identical obstacles are buried in a rather resistive soil of conductivity σe = 10−3 S/m. The ratio of conductivity q is equal to 1/50, 5 or 50, the frequency of the source still chosen as 100 Hz, 10 kHz, or 100 kHz. Since the contribution of coupling between two obstacles depends upon their distance of separation, one introduces d as the distance side to side. One writes d as a function of the side length (c for a cube, ci, i = 1, 2, 3 for a parallelepiped) of the obstacle parallel to , letting d = 3c, c, c/2, and c/4. The effect of coupling is appreciated via the quantity
where (I2ΔZ)coup is the variation of power when accounting for coupling and (I2ΔZ)decoup without accounting for it (the two obstacles behaving independently).
 In Tables 4 and 5 the maximum of the above quantity is displayed as a function of q and of d for two identical cubes of 4 m size centered at 30 m depth and observed at a 10 kHz frequency (results are the same for 100 Hz and 100 kHz) when the cubes are aligned along x (Table 4) and y (Table 5).
Results are similar with the imaginary part. Maximum of the relative contribution of the coupling effect between two identical cubes of c = 4 m side, computed with the full CIVA tool (CIVA exact) and with our LNL model, as a function of the ratio of conductivity q = σi/σe with respect to the host medium σe = 10−3 S/m and their distance dx of separation. The configuration of the cubes is the same as the one in Figures 6–8; they are centered at z1 = z2 = 30 m depth, and frequency is 10 kHz.
 The coupling effect is negligible for a distance larger than about three times the size of the cubes, and becomes quite significant when this distance is smaller than the size of the cubes. The results do not vary depending up on the size as long as one keeps the same ratio between it and the distance d. As an example for d = c/2 and q = 50 the coupling effect is 9.5% for (in meters) c = 0.5, 9.4% for c = 4 and c = 8 with the LNL model, whilst one has the same kind of results with the full CIVA tool, 8.5% for c = 0.5, 8.4% for c = 4, and 8.2% for c = 8.
 The coupling as expected depends upon the orientation of the vector between centers. It is the most important when = O1O2 in the configuration described in Figures 67–8, and is the least one when = O1O2. When the obstacles are aligned along the coupling effect is only impacting onto the tensor (see equation (17)) since the component z of the primary electric field is null for a horizontal loop.
 By comparison between the coupling effect evaluated with the full CIVA tool and the LNL model in the different configurations studied, one can conclude that the model of the coupling between two obstacles holds for both cubes and spheres.
 The same kind of evaluation has been performed for ellipsoids. One considers those introduced already, having the three orientations as described in the single ellipsoid case. Now, unlike the case of cubes and spheres the model of coupling reaches a limit; indeed, as is shown in Table 6, the model does not work when dx is lower than c1 (this is true as well when the obstacles are aligned along y or z).
Results are similar with the imaginary part. Maximum of the relative contribution of the coupling effect between two identical parallelepipeds with sizes 4 × 2 × 1 m, and ratio of conductivity q = 50 with respect to the host medium σe = 10−3 S/m, computed with the full CIVA tool, in comparison with the contribution computed for two identical ellipsoids by the LNL model, as a function of distance dx and the orientation of the obstacles. The latter are centered at z1 = z2 = 30 m, and frequency is 10 kHz.
4 × 2 × 1
1 × 4 × 2
2 × 1 × 4
 From these results, one deduces that the model holds only if the distance O1O2 is larger than the length of the longest semiaxis, a1. If this condition is not satisfied the coupling effect is overestimated. One can explain this by analyzing the tensor , which exhibits a discontinuity when = V1V2 Δσ1Δσ21ee(1, 2) 2ee(2, 1) (from equation (16)). This equality is never satisfied for cubes or spheres, which is the reason why the model then works regardless of the configuration.
 Like with the single obstacle situation, the partial expansion encompassing order 0 only is still sufficient as seen in Figure 9, which displays the variation of impedance at 10 kHz due to two identical 4 m sided cubes aligned along y, with ratio of conductivity q = 50, centered at same depth of z1 = z2 = 30 m.
 As an additional contribution to the evaluation, in the same configuration as described in Figures 6–8 (without interface air/soil), one compares the complex components of the magnetic fields scattered by two identical coupled spheres with those calculated as the sum of the scattered fields due each sphere separately, i.e., without coupling.
 Simulations are carried out with the FEKO code and the LNL model. The fields are displayed in Figures 1011–12. When the two spheres are separated by a distance dx = c/2, the coupling contribution is about 6% on Hxscat, 12% on Hyscat and Hzscat. Both models fit to less than 0.5% on each component of the scattered magnetic field (using the aforementioned gap indicator). In brief, the model of coupling works for spheres buried in a homogeneous infinite space (as for the half-space) for all three field components.
3.4. A Few Examples for Three Obstacles
 In order to evaluate the generalized model for N obstacles, one simulates several configurations for three cubes with the CIVA tools and one compares the variations of impedances with those computed with the LNL model. Like with two obstacles, one considers aligned obstacles: parallel to the measurement lines (see Figure 13), along x; orthogonal to them, along y or along z.
 In each case, the coupling contribution is larger than the one observed for two obstacles (the maximum of the coupling effect increases to 18% instead of 13% when the obstacles are aligned along x), and the discrepancy between the models represents about 1% of the variation of impedance. Typical impedance variations are shown in Figure 14. Results look alike with or without the coupling effect. The offset between the two models remains also the same (about 15% for q = 50).
 If the three obstacles are not aligned as is exemplified in Figure 15, the coupling effect might be weaker than the one between two obstacles. Indeed, in this case its maximum decreases to 7% instead of 13% previously obtained for two obstacles.
 Thus, the model of the coupling between obstacles is still suitable for more than two obstacles and the gap between models remains the same as long as the value of the ratio q does not change. As for the decrease of the coupling effect when the obstacles are shifted, the sign of the coupling is positive when the obstacles are aligned along the measurement line and negative when they are perpendicular to it. This means constructive coupling when all obstacles are aligned, destructive otherwise.
 As an example of validation with FEKO of the LNL model for N spheres buried in the homogeneous infinite space, three identical spheres aligned along the measurement line shifted by 50 m, centered at a depth of 31 m, with radius 4 m and ratio q = 50 at 10 kHz have been considered. The larger gap between the two models represents 4% on the imaginary part of Hzscat. The maximum of the coupling effect over all components is 17.8% of Hyscat and 17.0% of Hzscat (9% of Hxscat) and the gap on the coupling effect between FEKO and the LNL model is smaller than 0.5%.
3.5. Limit of the Total Low-Frequency Expansion
 The complete low-frequency expansion yields an approximate analytical solution when the primary source is a magnetic dipole (homogeneous infinite space and half-space), an analytical solution being also obtained for the horizontal electric loop in infinite space. The computation time is consequently divided by a factor of about 20. Yet this expansion holds only within a limited frequency range. Indeed, the main limitation is about the low-frequency expansion of the source term. In the infinite space, its accuracy is good down to 100 m if the frequency is lower than 10 kHz, the frequency range being increased if the space range is reduced. This is reduced to 1 kHz in the case of the half-space (the primary source term shows up twice in the calculation of the impedance variation, which appears to explain it). Again, one is able to increase the frequency by reducing the space range.
3.6. On an Ill-Posed Retrieval of Ellipsoids in Half-Space
 Let us consider a single ellipsoid of semiaxes a1, a2, a3 orientated according to x1, x2, x3, buried in a half-space and observed from five parallel lines separated by a distance equal to a quarter of the skin depth in the subsoil. As an example, let two ellipsoids of dimensions (in meters) 10 × 2 × 1, orientated as a1; a2; a3 and a2; a3; a1, with respective ratios of conductivity of 5 and 50. Figures 16 and 17 show that the variation of impedance is the same along all five lines. One obtains a similar result with two ellipsoids of dimensions 4 × 2 × 1 orientated as a1; a2; a3 and a2; a3; a1, with respective ratios of 7 and 50.
 So, to retrieve an ellipsoidal obstacle and overcome ill-posedness as illustrated here, it appears that multifrequency data should be used (the risk being, say, above 300 kHz, that the contribution of polarization currents is not anymore negligible), or (at possibly a higher degree of complexity) one or two additional electric loops perpendicular to the receiver one should be added to be sensitive to other magnetic field components (Hx and/or Hy).
 A model based on a low-frequency expansion and the localized nonlinear approximation yields the scattered magnetic fields or the impedance variations induced by the one or several ellipsoidal or spherical obstacles. The coupling effect between obstacles is taken into account via the Lax-Foldy theory of multiple diffraction.
 The simulations show that one reaches a good accuracy even for three close by obstacles. In particular, comparison with results acquired by the FEKO code illustrates that the coupling effect is suitably modeled for all components of the scattered magnetic field. In addition, one exhibits a limit of the validity of the model for this coupling effect. Furthermore, one exemplifies that the inverse problem is possibly ill posed for an ellipsoid in a half-space even with multiline measurements, i.e., two different ellipsoids might induce the same variation of impedance (associated to a single component of the scattered magnetic field, Hzscat in the considered configuration, at a single frequency); multifrequency data and variations of impedance associated to additional field components would be required to overcome that ill-posedness.
 One has also developed both a partial low-frequency expansion and a total one. The partial one holds throughout the frequency range of study (100 Hz to 100 kHz) even if one only uses the static depolarization tensor; the (much faster) total one is less applicable, since it is accurate only below 10 kHz for an infinite host space and below 1 kHz for a half-space.
 To conclude, the model allows fast field and impedance calculations, so it appears appropriate to tackle the characterization of buried obstacles with a hybrid differential evolution as studied in the work of Bréard et al. .
 The authors are grateful to CEA LIST Saclay for giving them full access to the eddy-current modeling tools of the CIVA platform. Complementary checks have been enabled by data provided by the Center for Computational Electromagnetics, Southeast University, Nanjing.