Low-frequency dipolar excitation of a perfect ellipsoidal conductor

Authors:
Gaële Perrusson, Panayiotis Vafeas and Dominique Lesselier

Journal:
Quart. Appl. Math. **68** (2010), 513-536

MSC (2000):
Primary 78A45; Secondary 78A25

Published electronically:
May 27, 2010

MathSciNet review:
2676974

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the scattering by a perfectly conductive ellipsoid under magnetic dipolar excitation at low frequency. The source and the ellipsoid are embedded in an infinite homogeneous conducting ground. The main idea is to obtain an analytical solution of this scattering problem in order to have a fast numerical estimation of the scattered field that can be useful for real data inversion. Maxwell equations and boundary conditions, describing the problem, are firstly expanded using low-frequency expansion of the fields up to order three. It will be shown that fields have to be found incrementally. The static one (term of order zero) satisfies the Laplace equation. The next non-zero term (term of order two) is more complicated and satisfies the Poisson equation. The order-three term is independent of the previous ones and is described by the Laplace equation. They constitute three different scattering problems that are solved using the separated variables method in the ellipsoidal coordinate system. Solutions are written as expansions on the few analytically known scalar ellipsoidal harmonics. Details are given to explain how those solutions are achieved with an example of numerical results.

**1.**J. Björkberg, G. Kristensson,*Three-dimensional subterranean target identification by use of optimization techniques*, PIER, vol. 15, 1997, pp. 141-164.**2.**B. Bourgeois, D. Legendre, M. Lambert, G. Hendrickson,*Three Dimensional Electromagnetics*, SEE, 1999, pp. 625-657.**3.**T. Yu, L. Carin,*Analysis of the electromagnetic inductive response of a void in a conducting-soil background*, IEEE Transactions on Geoscience and Remote Sensing, vol. 38, no. 3, 2000, pp. 1320-1327.**4.**H. Huang, I. J. Won,*Detecting metal objects in magnetic environments using a broadband electromagnetic method*, Geophysics, vol. 68, no. 6, 2003, pp. 1877-1887.**5.**X. Chen, K. O'Neill, B. E. Barrowes, T. M. Grzegorczyk, J. A. Kong,*Application of a spheroidal mode approach and differential evolution in inversion of magneto-quasistatic data for UXO discrimination*, Inverse Problems, vol. 20, no. 6, 2004, pp. 527-540.**6.**T. J. Cui, W. C. Chew, D. L. Wright, D. V. Smith,*Three dimensional imaging of buried objects in very lossy earth by inversion of VETEM data*, IEEE Transactions on Geoscience and Remote Sensing, vol. 41, no. 10, 2003, pp. 2197-2210.**7.**H. Tortel,*Electromagnetic imaging of a three-dimensional perfectly conducting object using a boundary integral formulation*, Inverse Problems**20**(2004), no. 2, 385–398. MR**2065429**, 10.1088/0266-5611/20/2/005**8.**G. Perrusson, M. Lambert, D. Lesselier, A. Charalambopoulos, G. Dassios,*Electromagnetic scattering by a triaxial homogeneous penetrable ellipsoid : low-frequency derivation and testing of the localized non-linear approximation*, Radio Science, vol. 35, no. 2, 2000, pp. 463-481.**9.**G. L. Wang, W. C. Chew, T. J. Cui, D. L. Wright, D. V. Smith,*3D near-to-surface conductivity reconstruction by inversion of VETEM data using the distorted Born iterative method*, Inverse Problems, vol. 20, 2004, pp. 195-216.**10.**C. O. Ao, H. Braunisch, K. O'Neill, J. A. Kong,*Quasi-magnetostatic solution for a conducting and permeable spheroid with arbitrary excitation*, IEEE Transactions on Geoscience and Remote Sensing, vol. 40, no. 4, 2002, pp. 887-897.**11.**G. Perrusson, P. Vafeas, D. Lesselier,*Low-frequency modeling of the interaction of magnetic dipoles and ellipsoidal bodies in a conductive medium*, 2004 URSI International Symposium on Electromagnetic Theory, Pisa, Proceedings, pp. 1017-1019 (+ CD-ROM), May, 2004.**12.**T. Habashy, R. Groom, B. Spies,*Beyond the Born and the Rytov approximations: a non-linear approach to electromagnetic scattering*, Journal of Geophysical Research, vol. 98, 1993, pp. 1759-1775.**13.**E. W. Hobson,*The Theory of Spherical and Ellipsoidal Harmonics*, Chelsea, New York, 1965.**14.**George Dassios and Kiriakie Kiriaki,*The rigid ellipsoid in the presence of a low frequency elastic wave*, Quart. Appl. Math.**43**(1986), no. 4, 435–456. MR**846156****15.**Panayiotis Vafeas and George Dassios,*Stokes flow in ellipsoidal geometry*, J. Math. Phys.**47**(2006), no. 9, 093102, 38. MR**2263654**, 10.1063/1.2345474**16.**P. Vafeas, G. Perrusson, D. Lesselier,*Low-frequency solution for a perfectly conducting sphere in a conductive medium with dipolar excitation*, PIER, vol. 49, 2004, pp. 87-111.**17.**P. Moon and D. E. Spencer,*Field theory handbook*, 2nd ed., Springer-Verlag, Berlin, 1988. Including coordinate systems, differential equations and their solutions. MR**947546**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
78A45,
78A25

Retrieve articles in all journals with MSC (2000): 78A45, 78A25

Additional Information

**Gaële Perrusson**

Affiliation:
Département de Recherche en Electromagnétisme - Laboratoire des Signaux et Systèmes, (Univ. Paris-Sud, CNRS, SUPELEC) UMR8506, 3 rue Joliot-Curie, Gif-sur-Yvette, F-91192, France

Email:
perrusson@lss.supelec.fr

**Panayiotis Vafeas**

Affiliation:
Division of Applied Mathematics and Mechanics - Department of Engineering Sciences, School of Engineering - University of Patras, Patras 265 04, Greece

Email:
vafeas@des.upatras.gr

**Dominique Lesselier**

Affiliation:
Département de Recherche en Electromagnétisme - Laboratoire des Signaux et Systèmes, (CNRS, Univ. Paris-Sud, SUPELEC) UMR8506, 3 rue Joliot-Curie, Gif-sur-Yvette, F-91192, France

Email:
lesselier@lss.supelec.fr

DOI:
http://dx.doi.org/10.1090/S0033-569X-2010-01171-5

Keywords:
Low-frequency expansion,
ellipsoidal harmonics,
dipole excitation

Received by editor(s):
November 20, 2008

Published electronically:
May 27, 2010

Article copyright:
© Copyright 2010
Brown University

The copyright for this article reverts to public domain 28 years after publication.