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November  2011, 5(4): 793-813. doi: 10.3934/ipi.2011.5.793

Uniqueness in inverse transmission scattering problems for multilayered obstacles

1. 

Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany, Germany

Received  November 2010 Revised  March 2011 Published  November 2011

Assume a time-harmonic electromagnetic wave is scattered by an infinitely long cylindrical conductor surrounded by an unknown piecewise homogenous medium remaining invariant along the cylinder axis. We prove that, in TM mode, the far field patterns for all incident and observation directions at a fixed frequency uniquely determine the unknown surrounding medium as well as the shape of the cylindrical conductor. A similar uniqueness result is obtained for the scattering by multilayered penetrable periodic structures in a piecewise homogeneous medium. The periodic interfaces and refractive indices can be uniquely identified from the near field data measured only above (or below) the structure for all quasi-periodic incident waves with a fixed phase-shift. The proofs are based on the singularity of the Green function to a two dimensional elliptic equation with piecewise constant leading coefficients.
Citation: Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems and Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
References:
[1]

C. Athanasiadis, A. G. Ramm and I. G. Stratis, Inverse acoustic scattering by a layered obstacle, In "Inverse Problem, Tomography and Image Processing" (Newark, DE, 1997), Plenum, New York, (1998), 1-8.

[2]

G. Bruckner and J. Elschner, The numerical solution of an inverse periodic transmission problem, Math. Methods Appl. Sci., 28 (2005), 757-778. doi: 10.1002/mma.588.

[3]

A.-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem, Math. Meth. Appl. Sci., 17 (1994), 305-338. doi: 10.1002/mma.1670170502.

[4]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.

[5]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-R66. doi: 10.1088/0266-5611/22/3/R01.

[6]

D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inverse Problems, 21 (2005), 383-398. doi: 10.1088/0266-5611/21/1/023.

[7]

V. L. Druskin, The unique solution of the inverse problem in electrical surveying and electric well-logging for piecewise-continuous conductivity, Izvestiya Earthy Physics, 18 (1982), 51-53.

[8]

E. M. Stein and G. L. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.

[9]

J. Elschner and G. Schmidt, Diffraction in periodic structures and optimal design of binary gratings. I. Direct problems and gradient formulas, Math. Meth. Appl. Sci., 21 (1998), 1297-1342. doi: 10.1002/(SICI)1099-1476(19980925)21:14<1297::AID-MMA997>3.0.CO;2-C.

[10]

J. Elschner and M. Yamamoto, Uniqueness results for an inverse periodic transmission problem, Inverse Problems, 20 (2004), 1841-1852. doi: 10.1088/0266-5611/20/6/009.

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.

[12]

P. Hähner, A uniqueness theorem for an inverse scattering problem in an exterior domain, SIAM J. Math. Anal., 29 (1998), 1118-1128. doi: 10.1137/S0036141097318614.

[13]

F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures, Inverse Problems, 13 (1997), 351-361. doi: 10.1088/0266-5611/13/2/010.

[14]

V. Isakov, On uniqueness in the inverse transmission scattering problem, Comm. Part. Diff. Equat., 15 (1990), 1565-1587.

[15]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[16]

V. Isakov, On uniqueness in the general inverse transmission problem, Comm. Math. Phys., 280 (2008), 843-858. doi: 10.1007/s00220-008-0485-6.

[17]

D. Kammler, "A First Course in Fourier Analysis," Second edition, Cambridge University Press, Cambridge, 2007.

[18]

A. Kirsch, Diffraction by periodic structures, in "Inverse Problems in Mathematical Physics" (Saariselkä, 1992), Lecture Notes in Phys., 422, Springer, Berlin, (1993), 87-102.

[19]

A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci., 21 (1998), 619-651. doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.

[20]

A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), 285-299. doi: 10.1088/0266-5611/9/2/009.

[21]

A. Lechleiter, Imaging of periodic dielectrics, BIT, 50 (2010), 59-83. doi: 10.1007/s10543-010-0255-7.

[22]

X. Liu, B. Zhang and G. Hu, Uniqueness in the inverse scattering problem in a piecewise homogeneous medium, Inverse Problems, 26 (2010), 015002, 14 pp.

[23]

X. Liu and B. Zhang, Direct and inverse obstacle scattering problems in a piecewise homogeneous medium, SIAM J. Appl. Math., 70 (2010), 3105-3120. doi: 10.1137/090777578.

[24]

X. Liu, B. Zhang and J. Yang, The inverse electromagnetic scattering problem in a piecewise homogeneous medium, Inverse Problems, 26 (2010), 125001, 19 pp.

[25]

A. Nachman, L. Päivärinta and A. Teirilä, On imaging obstacles inside inhomogeneous media, J. Funct. Anal., 252 (2007), 490-516. doi: 10.1016/j.jfa.2007.06.020.

[26]

J.-C. Nédélec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equation, SIAM J. Math. Anal., 22 (1991), 1679-1701.

[27]

R. Potthast, "Point-Sources and Multipoles in Inverse Scattering Theory," Chapman & Hall/CRC Research Notes in Mathematics, 427, Chapman & Hall/CRC, Boca Raton, FL, 2001.

[28]

A. G. Ramm, "Scattering by Obstacles," Mathematics and its Applications, 21, D. Reidel Publishing Co., Dordrecht, 1986.

[29]

A. G. Ramm, Fundamental solutions to some elliptic equations with discontinuous senior coefficients and an inequality for these solutions, Math. Inequalities and Applic., 1 (1998), 99-104.

[30]

B. Strycharz, An acoustic scattering problem for periodic, inhomogeneous media, Math. Methods Appl. Sci., 21 (1998), 969-983. doi: 10.1002/(SICI)1099-1476(19980710)21:10<969::AID-MMA982>3.0.CO;2-Y.

[31]

B. Strycharz, Uniqueness in the inverse transmission scattering problem for periodic media, Math. Methods Appl. Sci., 22 (1999), 753-772. doi: 10.1002/(SICI)1099-1476(199906)22:9<753::AID-MMA50>3.0.CO;2-U.

[32]

F. Yaman, Location and shape reconstruction of sound-soft obstacles buried in penetrable cylinders, Inverse Problems, 25 (2009), 065005, 17 pp.

[33]

G. Yan, Inverse scattering by a multilayered obstacle, Computers and Mathematics with Applications, 48 (2004), 1801-1810. doi: 10.1016/j.camwa.2004.09.003.

show all references

References:
[1]

C. Athanasiadis, A. G. Ramm and I. G. Stratis, Inverse acoustic scattering by a layered obstacle, In "Inverse Problem, Tomography and Image Processing" (Newark, DE, 1997), Plenum, New York, (1998), 1-8.

[2]

G. Bruckner and J. Elschner, The numerical solution of an inverse periodic transmission problem, Math. Methods Appl. Sci., 28 (2005), 757-778. doi: 10.1002/mma.588.

[3]

A.-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem, Math. Meth. Appl. Sci., 17 (1994), 305-338. doi: 10.1002/mma.1670170502.

[4]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.

[5]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-R66. doi: 10.1088/0266-5611/22/3/R01.

[6]

D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inverse Problems, 21 (2005), 383-398. doi: 10.1088/0266-5611/21/1/023.

[7]

V. L. Druskin, The unique solution of the inverse problem in electrical surveying and electric well-logging for piecewise-continuous conductivity, Izvestiya Earthy Physics, 18 (1982), 51-53.

[8]

E. M. Stein and G. L. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.

[9]

J. Elschner and G. Schmidt, Diffraction in periodic structures and optimal design of binary gratings. I. Direct problems and gradient formulas, Math. Meth. Appl. Sci., 21 (1998), 1297-1342. doi: 10.1002/(SICI)1099-1476(19980925)21:14<1297::AID-MMA997>3.0.CO;2-C.

[10]

J. Elschner and M. Yamamoto, Uniqueness results for an inverse periodic transmission problem, Inverse Problems, 20 (2004), 1841-1852. doi: 10.1088/0266-5611/20/6/009.

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.

[12]

P. Hähner, A uniqueness theorem for an inverse scattering problem in an exterior domain, SIAM J. Math. Anal., 29 (1998), 1118-1128. doi: 10.1137/S0036141097318614.

[13]

F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures, Inverse Problems, 13 (1997), 351-361. doi: 10.1088/0266-5611/13/2/010.

[14]

V. Isakov, On uniqueness in the inverse transmission scattering problem, Comm. Part. Diff. Equat., 15 (1990), 1565-1587.

[15]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[16]

V. Isakov, On uniqueness in the general inverse transmission problem, Comm. Math. Phys., 280 (2008), 843-858. doi: 10.1007/s00220-008-0485-6.

[17]

D. Kammler, "A First Course in Fourier Analysis," Second edition, Cambridge University Press, Cambridge, 2007.

[18]

A. Kirsch, Diffraction by periodic structures, in "Inverse Problems in Mathematical Physics" (Saariselkä, 1992), Lecture Notes in Phys., 422, Springer, Berlin, (1993), 87-102.

[19]

A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci., 21 (1998), 619-651. doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.

[20]

A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), 285-299. doi: 10.1088/0266-5611/9/2/009.

[21]

A. Lechleiter, Imaging of periodic dielectrics, BIT, 50 (2010), 59-83. doi: 10.1007/s10543-010-0255-7.

[22]

X. Liu, B. Zhang and G. Hu, Uniqueness in the inverse scattering problem in a piecewise homogeneous medium, Inverse Problems, 26 (2010), 015002, 14 pp.

[23]

X. Liu and B. Zhang, Direct and inverse obstacle scattering problems in a piecewise homogeneous medium, SIAM J. Appl. Math., 70 (2010), 3105-3120. doi: 10.1137/090777578.

[24]

X. Liu, B. Zhang and J. Yang, The inverse electromagnetic scattering problem in a piecewise homogeneous medium, Inverse Problems, 26 (2010), 125001, 19 pp.

[25]

A. Nachman, L. Päivärinta and A. Teirilä, On imaging obstacles inside inhomogeneous media, J. Funct. Anal., 252 (2007), 490-516. doi: 10.1016/j.jfa.2007.06.020.

[26]

J.-C. Nédélec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equation, SIAM J. Math. Anal., 22 (1991), 1679-1701.

[27]

R. Potthast, "Point-Sources and Multipoles in Inverse Scattering Theory," Chapman & Hall/CRC Research Notes in Mathematics, 427, Chapman & Hall/CRC, Boca Raton, FL, 2001.

[28]

A. G. Ramm, "Scattering by Obstacles," Mathematics and its Applications, 21, D. Reidel Publishing Co., Dordrecht, 1986.

[29]

A. G. Ramm, Fundamental solutions to some elliptic equations with discontinuous senior coefficients and an inequality for these solutions, Math. Inequalities and Applic., 1 (1998), 99-104.

[30]

B. Strycharz, An acoustic scattering problem for periodic, inhomogeneous media, Math. Methods Appl. Sci., 21 (1998), 969-983. doi: 10.1002/(SICI)1099-1476(19980710)21:10<969::AID-MMA982>3.0.CO;2-Y.

[31]

B. Strycharz, Uniqueness in the inverse transmission scattering problem for periodic media, Math. Methods Appl. Sci., 22 (1999), 753-772. doi: 10.1002/(SICI)1099-1476(199906)22:9<753::AID-MMA50>3.0.CO;2-U.

[32]

F. Yaman, Location and shape reconstruction of sound-soft obstacles buried in penetrable cylinders, Inverse Problems, 25 (2009), 065005, 17 pp.

[33]

G. Yan, Inverse scattering by a multilayered obstacle, Computers and Mathematics with Applications, 48 (2004), 1801-1810. doi: 10.1016/j.camwa.2004.09.003.

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