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Identification of partially coated anisotropic buried objects using electromagnetic Cauchy data. (English) Zbl 1134.78008
Summary: We consider the three dimensional electromagnetic inverse scattering problem of determining information about a target buried in a known inhomogeneous medium from a knowledge of the electric and magnetic fields corresponding to time harmonic electric dipoles as incident fields. The scattering object is assumed to be an anisotropic dielectric that is (possibly) partially coated by a thin layer of highly conducting material. The data is measured at a given surface containing the object in its interior. Our concern is to determine the shape of this scattering object and some information on the surface conductivity of the coating without any knowledge of the index of refraction of the inhomogeneity. No a priori assumption is made on the extent of the coating, i.e., the object can be fully coated, partially coated or not coated at all. Our method, introduced in [{\it F. Cakoni, M. Fares} and {\it H. Haddar}, Inverse Probl. 22, No. 3, 845--867 (2006; Zbl 1099.35167); {\it D. Colton} and {\it H. Haddar}, Inverse Probl. 21, No. 1, 383--398 (2005; Zbl 1086.35129)], is based on the linear sampling method and reciprocity gap functional for reconstructing the shape of the scattering object. The algorithm consists in solving a set of linear integral equations of the first kind for several sampling points and three linearly independent polarizations. The solution of these integral equations is also used to determine the surface conductivity.

MSC:
78A46Inverse scattering problems
78A45Diffraction, scattering (optics)
WorldCat.org
Full Text: DOI
References:
[1] S. Andrieux and A. Ben Abda, Identification of planar cracks by complete over-determined data : Inversion formulae , Inverse Problems 12 (1996), 553-563. · Zbl 0858.35131 · doi:10.1088/0266-5611/12/5/002
[2] T.S. Angell and A. Kirsch, The conductive boundary condition for Maxwell’s equations , SIAM J. Appl. Math. 52 (1992), 1597-1610. JSTOR: · Zbl 0786.35127 · doi:10.1137/0152092 · http://links.jstor.org/sici?sici=0036-1399%28199212%2952%3A6%3C1597%3ATCBCFM%3E2.0.CO%3B2-Q&origin=euclid
[3] T. Arens, Why linear sampling method works , Inverse Problems 20 (2004), 163-173. · Zbl 1055.35131 · doi:10.1088/0266-5611/20/1/010
[4] C. Baum, Detection and identification of visually obscured targets , Taylor and Francis, London, 199-9.
[5] A. Buffa and P. Jr. Ciarlet, On traces for functional spaces related to Maxwell’s equations. Part I: An integration by parts formula in Lipschitz polyhedra , Math. Meth. Appl. Sci. 24 (2001), 9-30. · Zbl 0998.46012 · doi:10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
[6] A. Buffa, M. Costabel and C. Schwab, Boundary element methods for Maxwell’s equations on non-smooth domains , Numer. Math. 92 (2002) 679-710. · Zbl 1019.65094 · doi:10.1007/s002110100372
[7] F. Cakoni and D. Colton, Combined far field operators in electromagnetic inverse scattering theory , Math. Methods Appl. Sci. 26 (2003), 413-429. · Zbl 1069.35091 · doi:10.1002/mma.360
[8] --------, Target identification of buried coated objects , J. Appl. Comput. Math., · Zbl 1182.35229 · doi:10.1590/S0101-82052006000200009 · http://www.scielo.br/scielo.php?script=sci_abstract&pid=S1807-03022006000200009&lng=en&nrm=iso&tlng=en
[9] --------, The determination of the surface impedance of a partially coated obstacle from far field data , SIAM J. Appl. Math. 64 (2004), 709-723. · Zbl 1059.35163 · doi:10.1137/S0036139903424254
[10] F. Cakoni and D. Colton, A uniqueness theorem for an inverse electomagnetic scattering problem in inhomogeneous anisotropic media , Proc. Edinburgh Math. Soc. 46 (2003), 293-314. · Zbl 1051.78013 · doi:10.1017/S0013091502000664 · http://journals.cambridge.org/bin/bladerunner?REQUNIQ=1087480707&REQSESS=3723236&118000REQEVENT=&REQINT1=163437&REQAUTH=0
[11] --------, Qualitative methods in inverse scattering theory , Springer, Berlin, 200-6.
[12] F. Cakoni and H. Haddar, A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media , · Zbl 1149.35078 · doi:10.3934/ipi.2007.1.443
[13] F. Cakoni, D. Colton and P. Monk, The determination of the surface conductivity of a partially coated dielectric , SIAM J. Appl. Math. 65 (2005), 767-789. · Zbl 1083.78007 · doi:10.1137/040604224
[14] F. Cakoni, F.MB. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imagining of burried objects , Inverse Problems 22 (2006), 845-867. · Zbl 1099.35167 · doi:10.1088/0266-5611/22/3/007
[15] F. Collino, F.MB. Fares and H. Haddar, Numerical and analytical studies of the linear sampling method in electromagnetic scattering problems , Inverse Problems 19 (2003), 1279-1299. · Zbl 02072195 · doi:10.1088/0266-5611/19/6/004
[16] D. Colton, Partial differential equations, An introduction , Dover Publications, Mineola, New York, 200-4.
[17] D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory , Inverse Problems 21 (2005), 383-398. · Zbl 1086.35129 · doi:10.1088/0266-5611/21/1/023
[18] D. Colton, H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse scattering problem , SIAM J. Sci. Comp. 24 (2002), 719-731. · Zbl 1037.78008 · doi:10.1137/S1064827501390467
[19] D. Colton and M. Piana, Inequalities for inverse scattering problems in absorbing media , Inverse Problems 17 (2001), 597-605. · Zbl 0991.35106 · doi:10.1088/0266-5611/17/4/302
[20] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory , 2 -1998. · Zbl 0760.35053
[21] D. Colton and P. Monk, Target identification of coated objects , IEEE Trans. Antennas Prop. 54 (2006), 1232-1242.
[22] P. Cutzach and C. Hazard, Existence, uniqueness and analyticity properties for electromagnetic scattering in a two layered medium , Math. Meth. Appl. Sci. 21 (1998), 433-461. · Zbl 0916.35119 · doi:10.1002/(SICI)1099-1476(19980325)21:5<433::AID-MMA960>3.0.CO;2-8
[23] H. Haddar, The interior transmission problem for anisotropic Maxwell’s equations and its applications to the inverse problem , Math. Meth. Appl. Sci. 27 (2004), 2111-2129. · Zbl 1062.35168 · doi:10.1002/mma.465
[24] H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem , Inverse Problems 18 (2002), 891-906. · Zbl 1006.35102 · doi:10.1088/0266-5611/18/3/323
[25] P. Hähner, A uniqueness theorem for the Maxwell equations with $L^2$ Dirichlet boundary conditions , Methoden Verfahren Math. Phys. 37 (1991), 85-96. · Zbl 0745.35043
[26] A. Kirsch, An integral equation for Maxwell’s equations in a layered medium with an application to the factorization method , J. Integral Equations and Appl., · Zbl 1136.78310 · doi:10.1216/jiea/1190905490
[27] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem , Inverse Problems 21 (2005), 1207-1223. · Zbl 1086.35139 · doi:10.1088/0266-5611/21/4/002
[28] P. Monk, Finite element methods for Maxwell’s equations , Oxford University Press, Oxford, 200-3.
[29] J.C. Nédélec, Acoustic and electromagnetic equations. Integral representations for harmonic problems , Springer Verlag, New York, 200-1.