Application of the method of auxiliary sources to the wide-band electromagnetic induction problem

928 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 40, NO. 4, APRIL 2002

Application of the Method of Auxiliary Sources tothe Wide-Band Electromagnetic Induction Problem

Fridon Shubitidze, Associate Member, IEEE, Kevin O’Neill, Member, IEEE, Shah A. Haider, Keli Sun, andKeith D. Paulsen, Member, IEEE

Abstract—The Method of Auxiliary Sources (MAS) is formu-lated and applied to solution of wide-band electromagnetic induc-tion problems involving highly conducting and possibly permeablemetallic objects. Improved remote sensing discrimination of buriedunexploded ordnance (UXO) motivates the study. The method useselementary auxiliary magnetic charges and magnetic current ele-ments to produce the unknown field. Auxiliary sources are locatedon virtual surfaces that usually conform to but do not coincide withthe real surface of the object. Once the source coefficients are de-termined, the secondary field can easily be found. The method in-volves no confrontations with source or Green’s function singular-ities. It is capable of treating penetrable as well as nonpenetrableobjects. Because the solution is composed of fields that automati-cally satisfy the governing equations, by construction, all approxi-mation resides only in the enforcement of boundary conditions atmatching (collocation) points. Accuracy in satisfying the boundaryconditions can be evaluated explicitly using noncollocation pointsover the surface. This in turn allows one to identify problem areason the surface and make intelligent adjustments of the source dis-tributions, to improve solutions at minimal cost. A general 3-D for-mulation is presented, and a version specialized to treat bodies ofrevolution is applied in the specific test cases discussed. Good per-formance of the method is observed, based on a modest number ofdegrees of freedom. Results are given and compare very well withavailable analytical and experimental data. Finally, we illustratethe utility of the method for investigating shape and orientationsensitivities for fundamental geometries as well as such targets asUXO.

Index Terms—Electromagnetic induction, metal, method of aux-iliary sources (MAS), permeable, unexploded ordnance (UXO).

I. INTRODUCTION

T HE investigation of electromagnetic induction (EMI)problems involving highly conducting and permeable

bodies has been a subject of intensive study and research formany years, with new momentum in the present provided bysubsurface sensing applications. Researchers are currentlyinvestigating the application of low-frequency broad-bandinduction ( 30 Hz to 25 kHz) for detection and identificationof buried unexploded ordnance (UXO) (e.g., [1]). This has

Manuscript received November 3, 2000; revised September 19, 2001. Thiswork was supported by the United States Army’s BT25 Program (InstallationRestoration); and by the Strategic Environmental Research and DevelopmentProgram, Project CU1122.

F. Shubitidze, S. A. Haider, K. Sun and K. D. Paulsen are with the ThayerSchool of Engineering, Dartmouth College, Hanover, NH, 03755 USA (e-mail:[email protected]).

K. O’Neill is with the Thayer School of Engineering, Dartmouth College,Hanover, NH, 03755, USA. He is also with the ERDC Cold Regions Researchand Engineering Laboratory, Hanover, NH, 03755 USA.

Publisher Item Identifier S 0196-2892(02)04620-X.

driven the development of new analyses and analytical toolsfor EMI scattering [2], [3]. The broad-band EMI responseof a conducting and permeable sphere is well established[4]. A quasi-magnetostatic solution has been developed onlyrecently for the conducting and permeable spheroid [5], [6].Numerical methods have arisen to address the kind of problemwe consider here, using 3-D Method of Moments (MoM)based on Magnetic Field Integral Equations (MFIE), whichincorporated an impedance boundary condition (IBC) [7],MoM for body of revolution (BOR) [2], and MoM-BORwith hybrid finite element–boundary element formulations(FEM–BEM) [8], [9]. In the first case, the authors combinedthe MFIE with IBC, and solved for surface electric current.This approach has a limited and largely untested range ofvalidity, in terms of (frequency-dependent) skin depth relativeto object dimensions. Further, the method in [7] does not solvefor the internal fields and the method is limited in this respect.We are quite interested in internal fields and currents in orderto understand the underlying mechanisms of response and theirrelation to object shape and composition. This is at the heartof the progress needed for advances in discrimination, and canreadily be investigated by the Method of Auxiliary Sources(MAS).

In more general MoM and FEM-BEM approaches [1], [8],[9], not reliant on the IBC, the Green function applicable to thetarget’s internal material must be resolved on the surface mesh(beyond what is accomplished by any singularity extraction). In[2], the authors use the MoM to solve the full low-frequency EMfield scattering from a BOR. As they have written, reasonableresults are obtained when they use ten subsections per wave-length, in keeping with the requirements of the typefactor in the Green function. This is ironic in that the quantitiesof interest (fields, currents) themselves actually change quitegradually and smoothly over the surface, and as such can beresolved well by a much coarser discretization. That is, the res-olution requirements in the integral formulations in [2], [8], and[9] are dominated by the fact that the observation points for theGreen function are placed on the same surface as the sources.Similarly, for the volume FEM mesh a comparable mesh con-straint applies, such that discretization interval or basis functionsize must be “much smaller than the skin depth” [9]. Becausethe MAS locates sources off the physical surface, the effects ofsource quantities are more smoothly spread over the mesh sur-face. The surface is also devoid of singularities, as source andobservation points never coincide. There is no volume mesh.These facts allow use of many fewer subsections (elements,

0196-2892/02$17.00 © 2002 IEEE

SHUBITIDZE et al.: APPLICATION OF THE MAS TO THE WIDE-BAND EMI PROBLEM 929

patches) on the surface, in keeping with the above-mentionedmild tangential gradients of dependent variables. By and large,existing numerical methods are computation intensive, limitingthe ready application of modeling to problems of practical in-terest, particularly for general geometries. For this reason wehave pursued the MAS, a basic formulation of which we havedeveloped for solution of low frequency EMI problems withhighly conducting and permeable targets [10], [11]. Here wepresent a more general and complete formulation and exploreits realization.

In the MAS, boundary value problems are solved numeri-cally by representing the electromagnetic fields in each domainof the structure under investigation by a finite linear combina-tion of analytical solutions of the relevant field equations, cor-responding to sources situated at some distance away from theboundaries of each physical domain. The “auxiliary sources”producing these analytical solutions are chosen to be elementarydipoles, charges, or current configurations located on fictitiousauxiliary surface(s), usually conforming to the actual surface(s)of the structure. In practice, at least in the simplest variant of themethod applied to most of the problems here, we only requirepoints on the auxiliary and actual surfaces, without resortingto the detailed mesh structures as required by other methods.In general, auxiliary surfaces are set up inside and outside thescattering object (“target”). Specifically, the fields outside of thestructure are considered to originate from distribution of auxil-iary magnetic charges placed inside the object, and the fieldsinside the object arise from a set of auxiliary magnetic currentelements placed outside the object. These fields, generated frominside and outside of the object, are required to obey funda-mental electromagnetic interface conditions: continuity of tan-gential magnetic field components and the jump condition forthe normal magnetic field components, enforced at an array ofsubregions or selected points on the physical surface(s) of thestructure. The result is a matrix equation from which the am-plitudes of auxiliary sources are to be determined. Once theamplitudes of auxiliary sources are found the solution is com-plete. The magnetic or electromagnetic field and related param-eters can easily be computed throughout the interior and exte-rior domains. Thus the MAS formulation we present here of-fers a number of advantages: field singularities at source loca-tions need not be confronted directly since the auxiliary surfacescontaining the sources are separated from the physical surfacewhere conditions are applied. In the simplest MAS formulation,no discretization into elements or coordinate lines of either thesurfaces or volumes of interest is required. Only locations of ob-servation (testing) points on the real surface and source locationson the auxiliary surfaces are needed, together with associatedtangents and normals. In more complex MAS formulations, wemay trade the greater programming overhead involved in moreelaborate mesh and source specification for higher resolutionand efficiency, as illustrated below.

The examples pursued in this paper show the accuracy ofthe method and its considerable potential for inferring links be-tween characteristics of the target and features in the scatteredfield. In the most recent related work (e.g., [2], [8], [9]), the mainintent is to provide insight into the existence and nature of res-

onant frequencies (preferential signal decay modes) of metallicobjects under EMI excitation. As such these groundbreaking pa-pers admirably explicate the concepts of characteristic modes inEMI responses. At the same time, only very limited geometriesare considered and other aspects of the EMI response besidesresonance are largely unexplored. The intention in our paper isto investigate systematic and fairly comprehensive variation oftarget properties that are key determinants of EMI response. Wetreat sensitivities to both geometry and material properties andidentify the kind of distinct trends in scattering phenomenologythat are needed for development of intelligent inversion strate-gies. Our results are viewed in terms of signal behavior over theentire EMI frequency band , thereby containing the effects of allparticipating modes of response.

Our exploration of (target) cause and (signal) effect are in-tended to have value particularly as an extension of the largelyempirical work that has appeared, which constitutes the prin-cipal window into applicable EMI scattering phenomenologyto date, e.g., [12]–[14]. Results in these papers are cast interms of relations between the EMI backscattering strengthsof various objects along their principle axes. Relationships aresought which might be useful for distinguishing one class ofobjects from another, e.g., UXO relative to metallic clutter orUXO-fragments. These first attempts at UXO discriminationwith EMI provide valuable insight and a good orientation inthe kind of parameter space that will inevitably be relevant forfuture progress. However, each discrimination class that wasconsidered contains rather diverse objects in terms of detailedshape and composition. Further, signals are typically averagedor integrated over intervals of time or bands of frequency.Therefore, it is not surprising that key signal parameters mayvary over an order of magnitude for a given class and overlapwith results from a different class. Elementary models were ap-plied to assist in reduction of the data, largely by idealizing thetargets (and sensors) as elementary tri-axial magnetic dipoleswith strengths to be determined empirically. This does not re-veal the general target characteristics responsible for the partic-ular scattering behaviors observed. Systematic analyses of theeffects of target properties are needed to provide the kind ofdetailed understanding that will allow us to distinguish moreprecisely between objects of concern and innocuous scrap.

In this paper, we see the effects of aspect ratio, orientation,and magnetic permeability and investigate wide-band EMIresponse from highly conducting and permeable objects, suchas UXO. In related work, we have shown the effects of sharpedges and points, material heterogeneity, and the degree ofelectrical contact between dissimilar parts (e.g., [11]). Thiswork describes the computational basis of our approachexplicitly enough so that it can be applied readily by othersand generalizes the approach we have presented elsewhere[10], [11]. The paper is organized as follows. In Section II, theformulation of the problem is developed. Section III describesthe numerical implementation. Finally, in Section IV, severalexample EMI scattering problems are solved, demonstratingthe accuracy of the method, its usefulness for characterizingscattering trends from variation of fundamental shapes, and itsapplicability to specific practical targets, such as UXO.


Fig. 1. Geometry of the problem.

II. FORMULATION OF THE PROBLEM

A. General Formulation

Let us consider that a permeable and conducting, metallic 3-Dobject is placed in a primary magnetic field with time depen-dence (Fig. 1). The primary magnetic field penetrates theobject to some degree, inducing eddy currents within it and asecondary or scattered field outside. For future convenience, werefer to the region external to the object as region 1, to the ob-ject region as region 2. Ampere’s Law is applicable within bothregions in the form

(1)

where is the magnetic field (A/m), is the conduction cur-rent density (A/m), is electric displacement (C/m), andis time. As is well established in the EMI frequency regime,we may assume that displacement currents are negli-gible in both regions. Thus, we operate in the quasi-magneto-static regime. Further, both electric fieldand soil conductivity( S/m) are small and combine to produce negligiblecurrent within the soil. Thus, outside the object thefield is irrotational and can be represented in terms of a scalarpotential

(2)

In general, outside the object we distinguish between the scat-tered secondary field, , and the transmitted primary or exci-tation field , which is specified. For

(3)

where represents magnetic source intensity (A/m),i.e., is proportional to magnetic “charge” concentration, and weassume that the exterior space in nonmagnetic:is the mag-netic permeability of free space ( H/m). In general,materials are characterized here by permeability ,where is the dimensionless relative permeability and

is the intrinsic permeability of free space (H/m). The scalarpotential thus satisfies Poisson’s equation

(4)

and the solution outside the object can then be written as

(5)

where

(6)

and the integration surface contains the source distribution, as explained below. satisfies the equation

(7)

Ultimately, combining (2) and (5) produces the final expressionfor

(8)

This satisfies the governing equations (3) and (4).Inside the target, we cannot apply the same simplification of

the Maxwell’s equations as we did outside. The governing equa-tion for is

(9)

assuming the existence of a magnetic current density(A/m ), where is the wave-number in the object’s material(m ). Within the EMI frequency range

(10)

where is the magnetic permeability of scatterer’s metal. Wemay conveniently express the solution for in terms of byusing the Hertzian magnetic potential. This potential satisfiesthe equation

(11)

the solution for which is

(12)

where the function

(13)

satisfies the equation

(14)

This means that the internal solution can ultimately be writtenas

(15)

which satisfies (9). The integration surface contains thesource distribution , as explained below. Thus overall theproblem is cast in terms of Poisson’s equations (3), (4) outside


the object with solution (8), coupled to the solution of (9)expressed by (15) and (12) inside the target.

The boundary conditions on the scatterer’s surfacecor-respond to enforcement of continuous tangential components of

and of the normal component of

(16)

(17)

where is a unit normal vector on the real surface andis therelative magnetic permeability of the target. For succinctness inwhat follows, (16) and (17) may be written as

(18)

where the vectors consist of two independent tangentialand one normal vector at each point on the surface, and the sub-script indicates where the vector is on the outside or in-side of the surface. In the case of the tangential vectors,

while for the normal vectors .The principle idea of the MAS is based on the concept of dis-

tributing a set of auxiliary sources with unknown coefficients onvirtual surfaces that usually conform to the real surfaces of thescattering structures under examination [15]. By construction,based on (5) and (12) above, the fields from these sources sat-isfy the governing equations. Thus, we need only concentrateon enforcement of the boundary conditions (18) to calculate thesolution of the problem and also to evaluate the accuracy of thesolution. The auxiliary sources are distributed on auxiliary sur-faces , , lying inside and outside the object(Fig. 1). For perfectly conducting objects, , since only one(internal) surface is required to describe the external field [16],[17] while for penetrable objects, , since two surfaces arerequired to describe both the external and internal fields [18],[19]. For composite or multiple objects, we may construct agreater number of auxiliary surfaces to produce fields associ-ated with all regions [11]. Regardless of the number of regions,note that the sourcesinsidea region produce the fieldsoutsidethe region, and vice versa.

For the problems here, we will restrict the formulation to twoauxiliary surfaces and , over which sources and

are distributed, respectively. These sources produce theinternal field and external field at observationpoints on the object surface . The source distributions aredecomposed into sets of simple distributions, each determinedby discrete coefficients and , , andeach associated with a basis or interpolation function

(19)

(20)

As an elucidation of the fundamental properties of these so-lutions produced by sets of independent sources removed fromthe object surface , it is shown in [15] that the functionsand as derived above possess the following properties.

1) For each function , which satisfies Poisson’s equation(4), a new nonsingular function on the physicalsurface always exists. Similarly, for each function ,which satisfies Helmholtz equation (14), a new function

on the physical surface always exists.2) The functions and are complete and linearly inde-

pendent on the physical surface for the functional space.

3) The weights (coefficients) and of the aux-iliary sources, which are the scalar and the vector coeffi-cients, can be calculated by enforcing boundary condition(18) for the best (by measure) decomposition ofand

, using a linear combination of source functions toproduce those functions. Thus the secondary field on thephysical surface can be represented by linear combina-tions of solutions to (4) and (11).

The properties 1–3 of the fundamental solutions guaranteethe existence of the scalar and vector coefficients,which ensure the validity of (4) and (14). For an appropriatedistribution of basis functions, (19) and (20) tend to the exactsolutions as . Using (19) and (20), the solutions (8) and(15) can be written as

(21)

(22)

where is the identity tensor. Note that the vectorand tensor depend for

their specification on which sets of basis functions arechosen.

Our general strategy for obtaining the unknown coefficientsand will be to enforce the boundary condition (18)

via a weighted residual integral statement

(23)

where is a weighting function, to be defined. Substituting(21) and (22) into (23) yields the ultimate general formulationof the numerical system

(24)


where we have taken advantage of the fact that issymmetrical.

Different variants of the formulation are produced by dif-ferent selections of and . In the simplest, both and

are delta functions:

(25)

so that

(26)

where it is understood that the operations inand are eval-uated first, then the result evaluated at the discrete points.Numerically, this formulation is tantamount to point matchingat points on the scatterer surface ; the fields are generatedby point sources consisting of magnetic charges atonand magnetic dipoles at on . This formulation requiresthe least in terms of programming complexity and mesh detailin particular. While surface vectorsare required at points on

and , otherwise only mesh points are required as no nu-merical integrations are performed.

In some instances, it has been advantageous to use sourcesdistributed in piecewise continuous sheets as opposed topoint sources. Numerical investigation has shown that usingdistributed sources can achieve the same accuracy with 2–3times fewer auxiliary sources than using point sources. Thishas a great practical interest for EMI scattering from com-posites 3-D objects and sharp curvature, in view of CPU timeand memory requirements. In this variant of the formulation

elsewhere(27)

where indicates the th subregion on the surfacewhere the basis function is applied. For the sake of speci-ficity and illustration of computational practice, details aregiven in the following section for the approach obtainedfrom (26). In particular, the system is reduced for applicationto a body of revolution (BOR), an important case for simu-lation of UXO.

Before proceeding, we note that, in general, solution of the ul-timate algebraic system requires much more CPU time than gen-eration and insertion of matrix elements in that system. With thisin mind, one can achieve considerable efficiency by reducingthe number of equations via a Galerkin-type formulation. In this

Fig. 2. Three-dimensional geometry of the real and the auxiliary surfaces.

case, , defined above, is used for both and ,and (24) becomes

(28)

where the formulation now requires subdomain integrationswith respect to both and . However, this extra computationaleffort appears to be more than overbalanced by the increasedefficiency in applications to 3-D (non-BOR) geometries. Inthe results below, we show the extraordinary efficiency of aformulation based on (28) for the benchmark case of an EMIscattering from a sphere. Detailed presentation of the resultsfrom simulations of irregular 3-D shapes are presented in asubsequent publication.

B. Numerical Implementation

In this section we present details of the formulation corre-sponding to (26), that is, point matching with point sources.In all that follows we consider that the surface of the body issmooth . The real (physical) surface (Fig. 1) is divided intosubsurfaces , . For each th subsur-face, a location point , the surface normal there , and sur-face tangential vectors and are determined, with

as in Fig. 2. Similarly, is divided into a finitenumber of subsurfaces each surrounding , with the

th one shown in Fig. 2. The secondary field in the region 1 dueto the target will be generated from magnetic point charges,

, placed on the surface at thepoints . We emphasize that this auxiliary surface is enclosedby the physical surface and these charges radiate asif in an unbounded space, with region 1 characteristics, givingrise to the secondary field, .


The total field in region 1, external to the target, is simply thesum of the primary and secondary fields where thelatter is represented at point as

(29)

where and . Thus, the operator ,as given above, calculates the magnetic field produced atby a single magnetic charge at point . For simplicity hereand in what follows for the treatment of point sources and pointmatching, we drop the superscript “” in and .

The induced field inside the target is assumed to be gen-erated by a finite number of vector magnetic dipole pointsources, , , distributed over the sur-face , which encloses the target. These dipole sources areassumed to radiate in unbounded homogeneous space filled withthe target’s material properties. The induced field at point

can be given as

(30)

where it is understood again that the differentiations areperformed first, then the expressions evaluated at and

. Thus, the operator calculates magnetic fields atcaused by a point magnetic dipole placed at point .is arbitrarily oriented relative to the global Cartesiancoordinate system (Fig. 2).

To define , a set of orthogonal local coordinate basis vec-tors is defined, , , and , normal and tangential to theauxiliary subsurface . According to the demands of theapplication, we may reduce the number of unknowns in theproblem by specifying with only two independent compo-nents along and , i.e., and . This is similar tothe way in which fields are expressed using Huygens’ Principleintegral statements involving only tangential surface currents.Here we simply displace the source currents from the physicalsurface. The ultimate expression for the magnetic field in region2 at a point then becomes

(31)

where the expressions in (31) are given explicitly by

(32)

and the functions and are given by

(33)

with .is expressed analogously, with in

place of . Altogether, expression of the magnetic field in therespective regions has been reduced to specification of threesets of unknown scalar coefficients, , , and ,

. Continuity of tangential componentsof magnetic field and normal components of magnetic fluxdensity vector is enforced at points over thephysical boundary by substituting the above expressions inthe boundary condition (26), with and . Thisprovides a sufficient number of independent linear equationsto determine the unknown sets of coefficients, assuming that

. For the inequality we must apply some sort of errorminimization scheme to identify the desired coefficients. Oncethe unknown coefficients of the auxiliary sources are obtained,the approximate field can be evaluated easily in either regionby simple summation of the source contributions, i.e., (29),with an arbitrary in place of .

Formulation for BOR: Many UXO and objects handily rep-resentative of UXO can be modeled as BOR. Moreover, while aBOR is ultimately 3-D and produces 3-D scattered fields, usingrotational symmetry reduces the computational problem to 2-D.Details of typical BOR formulations for other techniques, suchas MoM, appear in [20]–[23], to which the reader is referred.Here we only indicate the particular expressions needed in termsof auxiliary sources.

Fig. 3 shows a conducting and permeable BOR, whichis formed by rotating a generating line about the axisof a Cartesian coordinate system. The object resides in atime-harmonic primary magnetic field, which need not itselfbe rotationally symmetric (time dependence ). The az-imuthal dependence of all fields and sources is expressed viaa Fourier series with terms of the form . Since theazimuthal field variation is thereby accounted for analytically,no boundary condition matching points are distributed in theazimuthal direction. The resulting sets of simultaneousequations from (18) may be represented in matrix form as

for (34)

where is the impedance matrix for theth Fourier mode,is a column vector containing the unknown amplitude of the

auxiliary sources for theth Fourier mode, and is a drivingvector for the th Fourier mode.

The generating line for scatterer geometry is divided into seg-ments by points. With reference to the conventional cylin-drical coordinate system , a unit vector is defined,tangentially along the curve such that and .For each segment on the real surface, an az-imuthal belt (ring) is defined, and corresponding azimuthal beltsof source locations are determined on the two auxiliary surfaces.The secondary and total fields in regions 1


Fig. 3. Geometry of the generating line for the BOR case.

and 2, respectively, can be generated by sources on eachth ring of auxiliary sources, with

(35)

(36)

where in (35)

(37)

is the scattered magnetic field in region 1 at pointproduced by magnetic charge , which is

distributed on the inside auxiliary th belt (ring) at, where

is inner auxiliary th belt’s (ring’s) radius. Similarly in (36)

(38)

is the induced magnetic field in region 2 produced by themagnetic dipole, which is distributed on theth outer

auxiliary belt, ,is outer auxiliary th belt’s (ring’s) radius.

For the BOR problem at hand, we include thedependenceof in (34), in (37) and in (38) by expandingthem in Fourier series in

(39)

(40)

(41)

where is the mode number, and , , , ,, and are the Fourier coefficients. These coefficients

are dependent only onand and not .Finally, the formulation is achieved by substituting (40) and

(41) into (37) and (38), and then into (35) and (36) , so that thescattered and induced fields from all Fouriermodes are represented as

(42)

(43)

– – – (44)

When the primary field is uniformly parallel to the-(sym-metry-) axis, we have axial excitation and need only deal withthe zeroth component; and . Thus, onlyand enter into the problem, and we choose matching points

at on the real surface. This mode permits consid-eration of loop sources as well as spatially uniform excitationfields. The algebraic system is provided by applying only (42)and (43) in (18) for . For spatially uniform transverse ex-citation , we only treat the component, all of

, , , and enter the problem, and we choose matchingpoints at and on the real surface. By combiningaxial and transverse excitation cases, we can treat any orienta-tion of the object relative to a uniform primary field. In bothcases, we must sum the effects of sources distributed both lon-gitudinally (in the direction) and in the direction, though thevariation of the sources in the latter direction is simply dictatedby the value of considered.

III. N UMERICAL RESULTS

In this section, numerical results are presented for inducedsecondary magnetic fields from various objects subjected to ex-citation by an oscillating, spatially uniform primary magnetic


Fig. 4. Geometry of the spheroid.

field. To demonstrate the potential efficiency of the method, weconsider as a first example low frequency EM field scatteringby a sphere using the 3-D Galerkin-type MAS code. All sub-sequent calculations were performed with the point matching,point source BOR code. Examination of the 3-D sphere case isfollowed by treatment of the permeable and highly conductingspheroid; computed results are compared with analytical values[5], [6]. Next we consider scattering from permeable and con-ducting cylinders, with comparison of numerical results to ex-perimental data. In this connection, we explore how trends inthe secondary magnetic field depend on the combination of ob-ject elongation, electromagnetic parameters (permeability, fre-quency), and orientation. Lastly we apply the method to someexample small UXO. Simulations agree very well with data, andthe same orientation trends seen in numerical experiments onthe cylinders appear here. In the UXO case we again investigatehow the secondary magnetic field depends on the target’s ori-entation relative to the primary field, and verify a characteristicorientation dependent frequency shift.

A. Permeable and Conducting Sphere 3-D Case

The sphere case corresponds to that in Fig. 4, with ,cm. We examine the special choice of equal number of

the auxiliary sources and matching points, that is , con-ductivity [S/m], and permeability . For thisparticular numerical experiment, we employ piecewise constantbasis and testing functions (Galerkin technique). Inner and outerauxiliary surfaces consist of spheres with diameters equal to0.65 and 1.5 times that of the target, respectively. All integra-tions relevant to the subdomain basis functions were carried outby means of a four-point Gaussian quadrature formula. Fig. 5shows scattered field versus frequency from 1 to 100 kHz, at theobservation point , m. The circles indicatesthe result obtained using MAS, and the solid line shows the an-alytical solution [4]. Clearly the results match virtually exactly.The same problem was considered in [7] where the authors statethat for the sphere they required 120 surface patches. Becausethe formulation in [7] is based on an impedance boundary condi-tion formulation, it is only reliably applicable to the higher fre-quency EMI range. Using MAS with piecewise constant basis

Fig. 5. Real and imaginary parts of the secondary magnetic field from thesphere, diameter 10 cm,� = 1, � = 10 S/m. Circles: 3-D-MAS (piecewiseconstant basis and testing function; 49 subsurfaces).

Fig. 6. Normalized secondary magnetic field from the spheroid (Fig. 4) withb=a = 2, at the observation pointz = 10b. Primary magnetic field is parallelto thez axis,� = 10 S/m,� = 10.

and weighting functions, we obtain virtually exact agreementwith analytical solution, over the entire EMI frequency range,using just 49 subsurfaces. Based on the resolution requirementsoutlined above in the Introduction, the other methods cited thatare capable of solving over a broader frequency range than [7]would require orders of magnitude more unknowns over at leastsome portion of that range. The evident broad range and high ef-ficiency of the MAS represent a significant advantage for widerapplication, i.e., to more complicated 3-D geometries. In this re-gard we emphasize the utility of MAS for error assessment. Inmost applications, one does not possess an exact solution suchas that displayed in Fig. 5, against which to evaluate the solution.However, with the MAS we can still obtain a rigorous numer-ical error measure very easily by determining the residual error,either locally or globally, in matching the boundary condition


Fig. 7. Normalized secondary magnetic field from the spheroid (Fig. 4)b=a =2 at the pointx = 10b. Primary magnetic field is perpendicular toz-axis,� =

10 [S/m], � = 10.

(18). Beyond geometrical approximation of the scatterer, that isthe only relation that our method approximates.

B. Permeable and Conducting Prolate Spheroid

For this case, example parameters are ,S/m, and the ratio of semi-major to semi-minor axes is ,

cm (Fig. 4). Fig. 6 shows normalized secondary magneticfield versus induction number , , where

is frequency. The spheroid is excited by an axial (parallel tothe axis symmetry line) primary magnetic field .The number of auxiliary sources along the generating arc was101 in all calculations using point matching BOR code about anorder of magnitude smaller than is required for the approachesin [2], [8], and [9].

The sources are situated on auxiliary spheroidal surfacesconcentrically inside and outside the real surface. Specifically,the semi-axes of the inside auxiliary spheroid wereand ,

, and those of the outside auxiliary spheroidwere and , . The figure shows thatresults obtained with MAS have excellent agreement withthe analytical results. We also observe that numerical methodworks for high induction numbers , where evaluation ofthe analytical expression breaks down. For validity of MASresults for high induction numbers, comparisons are made withcalculations using the Thin Skin Approximation (TSA) in [24].At least in this test, comparisons show that results obtainedfrom both methods converge in the high-frequency/high-induc-tion number regime.

Fig. 7 shows results for the same object under transverse ex-citation, that is, with , with the same arrangement ofauxiliary surfaces and sources. Clearly here, as well, results areexcellent relative to the analytical value. Further comparisonsto TSA results for the transverse case are also very good in thehigher induction number range [24].

Fig. 8. Geometry of the cylinder.

C. Permeable and Conducting Cylinder

We consider a permeable and conducting cylinder under axialprimary magnetic field. The geometry of the cylinder is givenin Fig. 8. Fig. 9 shows corresponding results, versus frequency,under axial [Fig. 9(a)] and transverse [Fig. 9(b)] primary mag-netic fields, for a brass cylinder , assumed to be equalto 1.2 10 S/m, and a length to diameter ratio ,

in. Results are obtained using point-matching BORcode. The number of auxiliary sources along the generating arcwas 100. Fig. 9 also contains some experimental data for com-parison, over a limited frequency range where data were avail-able. Clearly numerical and experimental data agree very well.Fig. 10 shows corresponding results under axial [Fig. 10(a)]and transverse [Fig. 10(b)] primary magnetic field, for a steelcylinder with , conductivity 4 10 S/m, and

, in. The number of auxiliary sourcesalong the generating arc was 130. Fig. 10 also shows experi-mental data for comparison; again agreement is very good. Be-cause the electromagnetic properties of the steel were unknown,reasonable estimates were made and results checked againstthe measurements for this axial case. The transverse case forthis same target [Fig. 10(b)] produces substantially different re-sponses across the frequency spectrum, in both real and imagi-nary parts. The fact that the same parameters produce very goodagreement in both orientations argues for the position that thequality of the results is not an artifact of parameter selection.

It is revealing that, while the experimental cylinders had flat-tened ends with sharp edges, we first used smoothed ends inthe simulations, as shown in Fig. 10, out of numerical cau-tion. However, variations in the cylinder end geometry, from flatto slightly rounded to very rounded, produced insignificant ef-fects on the results. This also applied to some transverse casesbelow, where we might expect more of an effect. The implica-tions touch both physical and modeling realms. It appears thatsuch sharp edges do not disconcert the numerical method, if han-dled properly. Also, the details of the edges do not seem to haveany substantial physical significance, in terms of the magnetic


Fig. 9. Normalized secondary magnetic field from the brass cylinder (Fig. 8)with L=2a = 2:4, � = 1.2� 10 S/m,� = 1, 2a = 3:175 cm. Circles:MAS, solid line: experimental under (a) axial excitation, (b) transverseexcitation.

response. We explore this latter point in greater detail in a sub-sequent publication. Another striking feature in the results forthe cylinders in the very different responses as a function of ori-entation. While the patterns for the steel cylinder shift dramat-ically in frequency, the aluminum cylinder shows little contrastin response between the two orientations. Considering the algo-rithm validated at both high and low frequencies, from the com-parisons to the above combinations of results from analytical,experimental, and other numerical data [24], we now performwide-band sensitivity tests on the cylinder to explore orienta-tion related parameter sensitivities.

Beginning with axial excitation. Fig. 11 shows the imaginarypart of the secondary magnetic field versus permeability andfrequency, for a cylinder with cm and

5.5 10 S/m. We see that, for this elongation, peaks

Fig. 10. Normalized secondary magnetic field from the steel cylinder (Fig. 8)L=2a = 2:4,2a = 3:175 cm,� = 4� 10 S/m,� = 50. (a) Axial excitationand (b) transverse excitation.

in the imaginary part shift higher in frequency, while the peakbecomes flatter and wider, as permeability increases. Fig. 12shows results under similar conditions, but for , wherethe dimension is the same as for the previous length. With thisgreater elongation, the peaks in the imaginary part move in theopposite direction relative to the previous case, shifting lowerin frequency and again becoming broader asincreases. Ba-sically, the short cylinder behaves more like a sphere, for whichthe quadrature peak shifts upwards in frequency as permeabilityis increased.

Fig. 13 illustrates the effects of elongation, holding the di-mension cm fixed, for fixed values of the other param-eters: For a cylinder with and S/m, againunder axial excitation, we see the peak in the quadrature com-ponent shift lower in frequency, as elongation increases, only


Fig. 11. Imaginary part of the secondary magnetic field [A/m] fromthe cylinder (Fig. 8), withL=2a = 1:2, as a function of frequency andpermeability. Axial excitation,� = 5.5� 10 S/m.

Fig. 12. Imaginary part of the secondary magnetic field [A/m] from thecylinder (Fig. 8), withL=2a = 7, as a function of frequency and permeability.Axial excitation,� = 5.5� 10 S/m.

converging on the value for the infinite case at . Forthese same electrical properties, Fig. 14 shows the relationshipbetween the real (inphase) and imaginary (quadrature) parts ofthe secondary magnetic field, as a function of frequency forelongations , 4, 6. These curves were normalized rel-ative to the high-frequency response. Interestingly, as elonga-tion increases for the axial orientation, the crossing of real andimaginary curves (“resonance point”) remains essentially fixed.However, while the picture remains unchanged in the higher fre-quency region, there is much activity in the lower frequency re-gion: the asymptotic static limit of the real component dropsmore deeply with greater , as one would expect, while thepeak of the imaginary part rises and moves lower in frequency,as in the previous figure. Among other things, this suggests that

Fig. 13. Normalized imaginary part of the secondary magnetic field from thecylinder (Fig. 8) as a function of frequency and elongation. Arrows show theshift in location of its maximum. Axial excitation,� = 5.5� 10 S/m,� =

40.

Fig. 14. Normalized secondary magnetic field components versus frequency,from the cylinder (Fig. 8), normalized with respect to real component at the highfrequency limit. Axial excitation,� = 5.5� 10 S/m,� = 40.

the resonance point might help in isolating permeability and di-ameter, under axial excitation: For the given combination ofand , aspect ratio has no effect.

The next figures illustrate wide-band scattering from thecylinder under transverse excitation. Fig. 15 shows the imagi-nary part of the secondary magnetic field versus permeabilityand frequency, for a cylinder with and5.5 10 S/m at observation point m.We see that, for this small degree of elongation, peaks in theimaginary part shift higher in frequency, similar to the trendseen in the comparable axial case (Fig. 11). However, greaterelongation produces the greatest contrast between trends for thedifferent orientations, when the material is magnetic ( ).


Fig. 15. Imaginary part of the secondary magnetic field [A/m] from thecylinder (Fig. 8), withL=2a = 1:2, as a function of frequency and relativepermeability. Transverse excitation,� = 5.5� 10 S/m.

Fig. 16. Imaginary part of the secondary magnetic field [A/m] from thecylinder (Fig. 8), withL=2a = 7, as a function of frequency and relativepermeability. Transverse excitation,� = 5.5� 10 S/m.

In Fig. 16, for a much longer cylinder undertransverse excitation, the peaks in the imaginary part shift inthe opposite direction relative to the axial case Fig. 12. Underaxial excitation, we see quite different (opposite) behaviors inpeak shift for the short and elongated cylinders as a function of

. Under transverse excitation, this is not the case. Aside fromthe trends in terms of shift in these peak locations, we note adramatic effect of change in orientation in terms of the peaklocationsper se, for greater elongations. Compare the axialcase in Fig. 12 with the transverse case in Fig. 16, for examplefor . The peak location changes from about 10 Hz inthe axial case to about 1000 Hz for the transverse.

Elongation itself produces quite different behavior in thequadrature peak location for transverse relative to the axial

Fig. 17. Imaginary part of the secondary magnetic field, from the cylinder(Fig. 8), normalized with respect to the maximum, as a function of frequencyand elongation. Transverse excitation,� = 5.5� 10 S/m,� = 40.

Fig. 18. Secondary magnetic field from the cylinder (Fig. 8) versusfrequency, normalized with respect to real component at the high frequencylimit; transverse excitation;� = 5.5� 10 S/m,� = 40.

cases, other things being equal. Fig. 17 shows quadraturecomponent peaks for fixed permeability and con-ductivity S/m. For this transverse case, thepeak locations are essentially fixed as a function of elongation;compare the axial case in Fig. 14, where elongation produceda very strong effect. In terms of relations between the different(real and imaginary) components, we also see a contrast in thetrends that appear for the two orientations, as a function ofelongation. Fig. 18 shows both components of the secondarymagnetic field, as a function of frequency, for , 4, 6.These curves were normalized relative to the high frequencyresponse, as in Fig. 14. Interestingly, as elongation increasesfor the transverse orientation, the quadrature peak location does


Fig. 19. Secondary magnetic field from the UXO, diameter 2.5 cm, length11.25 cm,� = 10 S/m,� = 90, axial case.

not change. Comparing Figs. 14 and 18 shows that, under thisnormalization, the lower frequency portion of the response inthe transverse case declines with increased elongation, and alsoconvergesto a limiting curve. In contrast to the axial case, theresonance point shifts lower with elongation for the transversecase, though this effect is slight.

One should note that, in all these cases involving changesin elongation, we have proceeded by changing only the lengthof the uniform mid-section of the cylinder (Fig. 8), leaving thegeometry of the ends unaltered. In elongating more complexgeometries—e.g., increasing for the prolate spheroid whilekeeping fixed, or somehow stretching a UXO shape—onealters more than one basic property of the shape, therebymaking it more difficult to infer cause and effect. Lengtheningthe prolate spheroid with fixed sharpens its tip. Especiallyfor the transverse case, in which induced electric currents must(presumably) pass over this smaller radius of curvature, itmay be difficult to separate tip effects and overall aspect ratioeffects. Results here suggest a strong role of gross aspect ratio.We investigate this kind of issue more deeply in a subsequentpublication.

D. UXO

In this section we consider application specifically to UXO.As in the case of the steel cylinder, it is necessary to infer theelectromagnetic parameters. These were then tested for consis-tency under change of orientation. Results are shown for twodeactivated shells, one 2.5 cm at its widest diameter with

cm, and the other is 2.0 cm with cm. Simula-tions agree well with the measured data (Figs. 19–22) and allcases reinforce the same insights into the orientation effect out-lined above. The frequency of the peak in the quadrature com-ponent rises dramatically, by almost two orders of magnitude,when the orientation switches from axial to transverse: in thefirst case from about 200 to 9000 Hz; in the second from about300 to 23 000 Hz. Overall, the details of this frequency shift may

Fig. 20. Secondary magnetic field from the UXO, diameter 2.5 cm, length11.25 cm,� = 10 S/m,� = 90, transverse case.

Fig. 21. The secondary magnetic field [A/m] from the UXO, diameter 2.0 cm,length 7.68 cm,� = 4� 10 S/m,� = 50, axial case.

provide clues to both target permeability and aspect ratio. Notethe long taper in the high-frequency portion of the quadraturepeak for the 20-mm shell, axial case. More detailed examinationof these peaks and curves, e.g., in terms of width and symmetryof the quadrature peak, may also provide information on detailsof target structure. We pursue this in a subsequent publication.

IV. SUMMARY AND CONCLUSION

In this paper, the MAS is formulated to develop a simple andefficient numerical procedure for treating electromagnetic in-duction problems, with permeable and highly conducting tar-gets of arbitrary shape. The procedure of synthesizing the un-known fields using a distribution of sources separated from theactual target surfaces is easily realized, providing very efficient


Fig. 22. The secondary magnetic field [A/m] from the UXO, diameter 2.0 cm,length 7.68 cm,� = 4� 10 S/m,� = 50, transverse case.

and accurate solutions over a wide frequency range for 3-Dbodies. Specialized for application to bodies of revolution, themethod was verified against analytical solutions and experi-mental data for cases of spheroidal, cylindrical, and UXO tar-gets, under axial and transverse excitation. Supported by thevery good agreement in those tests, we performed some sensi-tivity analyzes on elongated targets with various metallic elec-tromagnetic properties.

Overall, the characteristics of the scattered field dependstrongly on the geometry of the object, its orientation, andelectromagnetic properties. Still, it is possible to identifysome distinct trends which could be useful in classification ofunknown targets.

• For elongated magnetic targets, we see a dramatic upwardshift in the frequency of the peak in the quadrature (imag-inary) component, when the object is rotated from axial totransverse orientation, relative to the primary field. Thisappears both in data and simulation, for both cylindersand UXO. Nonmagnetic targets show approximately un-changed peak locations

• In the axial case, increases in permeability tend to push thequadrature peak into higher frequencies for cylinders withaspect ratio near unity, while for longer cylinders the peakmoves in the opposite direction, into lower frequencies.For the transverse case, increasing permeability pushes thequadrature peak upwards in frequency, for both long andshort cylinders.

• For an example, magnetic metal increases in elongationdepress the quadrature peak frequency in the axial case,while we see little effect of elongation in the transversecase.

• In terms of behavior of both components relative to oneanother over the entire spectrum, elongation of examplemagnetic cylinders with fixed diameter shows oppositetrends in transverse and axial orientations. Under normal-ization by high-frequency values, in the axial case the

resonance point (real and imaginary curve crossing) re-mains fixed at a relatively high frequency, while withinthe lower frequencies distinctive changes appear in bothcomponents. For the transverse case, the resonance pointshifts downwards only slightly, while the curves respondmuch less over the lower frequencies and diminish wherethey increased in the axial case.

In future work, we explore these effects more comprehen-sively, organizing them into a suitable basis for inversion ortarget classification. More complex 3-D geometrical featuresare treated, including thin elements and material heterogeneity.We also investigate the linkages between particular target fea-tures, such as sharpness of peaks and edges and angularity, andsignal features such as peak location, width, symmetry, asymp-totic values, and relations between the components.

ACKNOWLEDGMENT

The authors would like to thank Geophex, Ltd. for assistingin the acquisition of the data.

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[6] H. Braunisch, C. O. Ao, K. O’Neill, and J. A. Kong, “Magnetoquasistaticresponse of conducting and permeable prolate spheroid under axial ex-citation,” IEEE Trans. Geosci. Remote Sensing, vol. 39, pp. 2689–2701,Dec. 2001.

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Fridon Shubitidze (A’98) received the degree ofDiploma radio physicist (M.S.) from the Sukhumibranch of Tbilisi State University, Republic ofGeorgia, in 1994 and Candidate of Sciences Ph.D.degree in radio physics (applied electromagnetics)from its physics department, Tbilisi State University,Republic of Georgia, in 1997.

Beginning in 1994, he was on the Research Staff ofthe Laboratory of Applied Electrodynamics, TbilisiState University, Department of Physics, Republic ofGeorgia. At the same time he joined the Department

of Physics and Mathematics, Sukhumi branch of Tbilisi State University as aSenior Teacher and became Associate Professor there in 1998. From 1998 to1999 he held a post-doctoral fellowship with the National Technical Univer-sity of Athens, Greece, performing research in connection with computer sim-ulation of electrostatic discharge, electrodynamics aspects of EMC, numericalmodeling of conformal antennas, electromagnetic wave scattering, field visual-ization and identification of objects by scattered field analysis, investigation ofwave propagation through anisotropy, plasma and chiral media, and innovativenumerical methods. He is currently working as Senior Research Associate at theThayer School of Engineering, Dartmouth College, Hanover, NH. His currentwork focuses on numerical modeling of electromagnetic scattering by subsur-face metallic objects.

Kevin O’Neill (M’98) received the B.A. degree(magna cum laude) from Cornell University, Ithaca,NY, and the M.A., M.S.E., and Ph.D. degrees fromthe Department of Civil Engineering, PrincetonUniversity, Princeton, NJ.

After an NSF post-doctoral fellowship at theThayer School of Engineering at Dartmouth College,Hanover, NH, and the U.S. Army Cold RegionsResearch and Engineering Laboratory (CRREL),he joined CRREL as a Research Civil Engineer.His research has focused on numerical modeling

of porous media transport phenomena and of geotechnically relevant elec-tromagnetic problems. He has been a Visiting Fellow in the Department ofAgronomy at Cornell University, continues since 1989 as a Visiting Scientistat the Center for Electromagnetic Theory and Applications at MIT, and since1984 has been on the adjunct faculty of the Thayer School. Current workcenters on electromagnetic remote sensing of surfaces, layers, and especiallyburied objects such as unexploded ordnance.

Shah A. Haider received the B.Sc.Eng. degreein electrical engineering and electronics from theBangladesh University of Engineering and Tech-nology, Dhaka, Bangladesh, and the M.S. degree andthe Ph.D. degree in electrical engineering from theUniversity of Arizona, Tucson, in 1988 and 1996,respectively.

He was an Assistant Engineer, Dhaka ElectricSupply in Bangladesh Power Development Board incharge of operation and maintenance of the electricalpower supply system Dhaka. He worked as an Elec-

trical Engineer at the Ministry of Electricity, and at the National PetrochemicalCompany, Marsa Al-Brega, both in Libya. Subsequently, as a graduate studentat the University of Arizona, he performed research on hyperthermia for cancertherapy. At the Thayer School of Engineering, Dartmouth College, Hanover,NH, he worked as a Senior Research Associate, concentrating on finite elementand analytical modeling of electromagnetic phenomena.

Dr. Haider was awarded the First Prize in the Nineteenth All Pakistan ScienceFair, 1970, the National Science and Technological Fellowship (1977–1978) toconduct research in Industrial Physics Division at Bangladesh Council of Indus-trial and Scientific Laboratories, Dhaka, and the Curtis Carl Johnson MemorialAward of Bio-Electromagnetics Society (BEMS) for the Best Paper presentedin its 11th Annual Meeting (1989). He is a member of Eta Kappa Nu.

Keli Sun received the B.S., M.S., and Ph.D. degreesin computational and biofluid mechanics fromthe Department of Mechanics and EngineeringSciences, Peking University, Beijing, China, in1991, 1994, and 1997, respectively, and the M.S.degree in computational electromagnetics from theThayer School of Engineering, Dartmouth College,Hanover, NH, in 2001.

As an exchange student, he also worked inthe school of Pure and Applied Sciences, TokyoUniversity, Tokyo, Japan, from December 1995 to

December 1996, studying the mobility and mechanical properties of membraneproteins in living cells. After receiving the Ph.D. degree in 1997, he worked onthe faculty of Tsinghua University, Beijing, performing research and teachingbiomechanics. He is currently a Research Associate in the Numerical MethodsLaboratory in the Thayer School of Engineering, furthering his research incomputational electromagnetics and its applications in remote sensing.

Keith D. Paulsen (M’86) received the B.S. degreefrom Duke University, Durham, NC, and the M.S.and Ph.D. degrees from the Thayer School of En-gineering, Dartmouth College, Hanover, NH, all inbiomedical engineering.

He was an Assistant Professor of Electricaland Computer Engineering at the University ofArizona, Tucson, and jointly an Assistant Professorin Radiation Oncology at the University of ArizonaHealth Sciences Center. He is now a Professor ofEngineering at the Thayer School. A recipient of

numerous academic and research awards and fellowships, he has carried outsponsored research for the National Science Foundation, the National CancerInstitute, the Whitaker Foundation, and the National Institute of Health. He hasserved on more than ten advisory committees for the National Cancer Institute,has chaired or organized five symposia on hyperthermic cancer treatment,and has contributed chapters on electromagnetic power deposition patternsto five books. At Thayer School, he is co-founder and co-manager of theNumerical Methods Laboratory. He performs research and teaches courses incomputational methods for engineering and scientific problems, with particularapplications in electromagnetics, subsurface object sensing, and biomedicalengineering.

Documents

Application of the method of auxiliary sources to the wide-band electromagnetic induction problem