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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 & 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, (1998), 1.   Google Scholar

[2]

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

[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.  doi: 10.1002/mma.1670170502.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.   Google Scholar

[8]

E. M. Stein and G. L. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1971).   Google Scholar

[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.  doi: 10.1002/(SICI)1099-1476(19980925)21:14<1297::AID-MMA997>3.0.CO;2-C.  Google Scholar

[10]

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

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition,, Grundlehren der Mathematischen Wissenschaften, 224 (1983).   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).   Google Scholar

[16]

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

[17]

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

[18]

A. Kirsch, Diffraction by periodic structures,, in, 422 (1993), 87.   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.   Google Scholar

[27]

R. Potthast, "Point-Sources and Multipoles in Inverse Scattering Theory,", Chapman & Hall/CRC Research Notes in Mathematics, 427 (2001).   Google Scholar

[28]

A. G. Ramm, "Scattering by Obstacles,", Mathematics and its Applications, 21 (1986).   Google Scholar

[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.   Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

show all references

References:
[1]

C. Athanasiadis, A. G. Ramm and I. G. Stratis, Inverse acoustic scattering by a layered obstacle,, In, (1998), 1.   Google Scholar

[2]

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

[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.  doi: 10.1002/mma.1670170502.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.   Google Scholar

[8]

E. M. Stein and G. L. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1971).   Google Scholar

[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.  doi: 10.1002/(SICI)1099-1476(19980925)21:14<1297::AID-MMA997>3.0.CO;2-C.  Google Scholar

[10]

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

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition,, Grundlehren der Mathematischen Wissenschaften, 224 (1983).   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).   Google Scholar

[16]

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

[17]

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

[18]

A. Kirsch, Diffraction by periodic structures,, in, 422 (1993), 87.   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.   Google Scholar

[27]

R. Potthast, "Point-Sources and Multipoles in Inverse Scattering Theory,", Chapman & Hall/CRC Research Notes in Mathematics, 427 (2001).   Google Scholar

[28]

A. G. Ramm, "Scattering by Obstacles,", Mathematics and its Applications, 21 (1986).   Google Scholar

[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.   Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

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