Identification of obstacles using only the scattered P-waves or the scattered S-waves

Previous Article
Positive definiteness of Diffusion Kurtosis Imaging
IPI Home
This Issue
Next Article
On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

February 2012, 6(1): 39-55. doi: 10.3934/ipi.2012.6.39

Identification of obstacles using only the scattered P-waves or the scattered S-waves

Drossos Gintides ^1, and Mourad Sini ^2,

1.	Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens
2.	RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria

Received October 2010 Revised October 2011 Published February 2012

Abstract
Related Papers

In this work, we are concerned with the inverse scattering by obstacles for the linearized, homogeneous and isotropic elastic model. We study the uniqueness issue of detecting smooth obstacles from the knowledge of elastic far field patterns. We prove that the 'pressure' parts of the far field patterns over all directions of measurements corresponding to all 'pressure' (or all 'shear') incident plane waves are enough to guarantee uniqueness. We also establish that the shear parts of the far field patterns corresponding to all the 'shear' (or all 'pressure') incident waves are also enough. This shows that any of the two different types of waves is enough to detect obstacles at a fixed frequency. The proof is reconstructive and it can be used to set up an algorithm to detect the obstacle from the mentioned data.

Keywords: linear elasticity., Inverse scattering problem.

Mathematics Subject Classification: Primary: 35R30, 35J40, 74B0.

Citation: Drossos Gintides, Mourad Sini. Identification of obstacles using only the scattered P-waves or the scattered S-waves. Inverse Problems and Imaging, 2012, 6 (1) : 39-55. doi: 10.3934/ipi.2012.6.39

References:

[1]	C. J. Alves and R. Kress, On the far-field operator in elastic obstacle scattering, IMA Journal of Applied Mathematics, 67 (2002), 1-21. doi: 10.1093/imamat/67.1.1.
[2]	H. Ammari and H. Kang, "Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory," Applied Mathematical Sciences, 162, Springer, New York, 2007.
[3]	T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464. doi: 10.1088/0266-5611/17/5/314.
[4]	K. Baganas, B. B. Guzina, A. Charalambopoulos and D. George, A linear sampling method for the inverse transmission problem in near-field elastodynamics, Inverse Problems, 22 (2006), 1835-1853. doi: 10.1088/0266-5611/22/5/018.
[5]	M. Bonnet, "Boundary Integral Methods for Solids and Fluids," Wiley & Sons, 1995.
[6]	A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity, Inverse Problems, 18 (2002), 547-558. doi: 10.1088/0266-5611/18/3/303.
[7]	A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for non-absorbing penetrable elastic bodies, Inverse Problems, 19 (2003), 549-561. doi: 10.1088/0266-5611/19/3/305.
[8]	D. Gintides and K. Kiriaki, The far-field equations in linear elasticity-An inversion scheme, ZAMM Z. Angew. Math. Mech., 81 (2001), 305-316. doi: 10.1002/1521-4001(200105)81:5<305::AID-ZAMM305>3.0.CO;2-T.
[9]	B. B. Guzina and A. I. Madyarov, A linear sampling approach to inverse elastic scattering in piecewise-homogeneous domains, Inverse Problems, 23 (2007), 1467-1493. doi: 10.1088/0266-5611/23/4/007.
[10]	P. Hahner and G. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534. doi: 10.1088/0266-5611/9/5/002.
[11]	N. Honda, R. Potthast, G. Nakamura and M. Sini, The no-response approach and its relation to non-iterative methods for the inverse scattering, Ann. Mat. Pura Appl. (4), 187 (2008), 7-37.
[12]	M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations, 23 (1998), 1459-1474.
[13]	V. Isakov, On uniqueness in the inverse transmission scattering problem, Commun. Part. Diff. Equ., 15 (1990), 1565-1587.
[14]	A. Kirsch and R. Kress, Uniqueness in the inverse obstacle scattering problem, Inverse Problems, 9 (1993), 285-299. doi: 10.1088/0266-5611/9/2/009.
[15]	V. D. Kupradze, "Potential Methods in the Theory of Elasticity," Israel Program for Scientific Translations, Jerusalem, Daniel Davey & Co., Inc., New York, 1965.
[16]	V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili and T. V. Burchuladze, "Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity," North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland Publishing Co., Amsterdam-New York, 1979.
[17]	J. J. Liu, G. Nakamura and M. Sini, Reconstruction of the shape and surface impedance from acoustic scattering data for an arbitrary cylinder, SIAM J. Appl. Math., 67 (2007), 1124-1146. doi: 10.1137/060654220.
[18]	G. Nakamura and M. Sini, Obstacle and boundary determination from scattering data, SIAM J. Math. Anal., 39 (2007), 819-837. doi: 10.1137/060658667.
[19]	G. Nakamura, R. Potthast and M. Sini, Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test, Comm. Partial Differential Equations, 31 (2006), 1505-1528.
[20]	F. Nintcheu and B. B. Guzina, Elastic scatterer reconstruction via the adjoint sampling method, SIAM J. Appl. Math., 67 (2007), 1330-1352. doi: 10.1137/060653123.
[21]	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.
[22]	F. Simonetti, Localization of pointlike scatterers in solids with subwavelength resolution, Applied Physics Letter, 89 (2006), 094105. doi: 10.1063/1.2338888.
[23]	V. A. Solonnikov, On Green's matrices for elliptic boundary value problems. I, Trudy Mat. Inst. Steklov., 110 (1970), 107-145.
[24]	V. A. Solonnikov, The Green's matrices for elliptic boundary value problems. II, Boundary Value Problems of Mathematical Physics, 7, Trudy. Math. Inst. Steklov., 116 (1971), 181-216.

show all references

References:

[1]	C. J. Alves and R. Kress, On the far-field operator in elastic obstacle scattering, IMA Journal of Applied Mathematics, 67 (2002), 1-21. doi: 10.1093/imamat/67.1.1.
[2]	H. Ammari and H. Kang, "Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory," Applied Mathematical Sciences, 162, Springer, New York, 2007.
[3]	T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464. doi: 10.1088/0266-5611/17/5/314.
[4]	K. Baganas, B. B. Guzina, A. Charalambopoulos and D. George, A linear sampling method for the inverse transmission problem in near-field elastodynamics, Inverse Problems, 22 (2006), 1835-1853. doi: 10.1088/0266-5611/22/5/018.
[5]	M. Bonnet, "Boundary Integral Methods for Solids and Fluids," Wiley & Sons, 1995.
[6]	A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity, Inverse Problems, 18 (2002), 547-558. doi: 10.1088/0266-5611/18/3/303.
[7]	A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for non-absorbing penetrable elastic bodies, Inverse Problems, 19 (2003), 549-561. doi: 10.1088/0266-5611/19/3/305.
[8]	D. Gintides and K. Kiriaki, The far-field equations in linear elasticity-An inversion scheme, ZAMM Z. Angew. Math. Mech., 81 (2001), 305-316. doi: 10.1002/1521-4001(200105)81:5<305::AID-ZAMM305>3.0.CO;2-T.
[9]	B. B. Guzina and A. I. Madyarov, A linear sampling approach to inverse elastic scattering in piecewise-homogeneous domains, Inverse Problems, 23 (2007), 1467-1493. doi: 10.1088/0266-5611/23/4/007.
[10]	P. Hahner and G. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534. doi: 10.1088/0266-5611/9/5/002.
[11]	N. Honda, R. Potthast, G. Nakamura and M. Sini, The no-response approach and its relation to non-iterative methods for the inverse scattering, Ann. Mat. Pura Appl. (4), 187 (2008), 7-37.
[12]	M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations, 23 (1998), 1459-1474.
[13]	V. Isakov, On uniqueness in the inverse transmission scattering problem, Commun. Part. Diff. Equ., 15 (1990), 1565-1587.
[14]	A. Kirsch and R. Kress, Uniqueness in the inverse obstacle scattering problem, Inverse Problems, 9 (1993), 285-299. doi: 10.1088/0266-5611/9/2/009.
[15]	V. D. Kupradze, "Potential Methods in the Theory of Elasticity," Israel Program for Scientific Translations, Jerusalem, Daniel Davey & Co., Inc., New York, 1965.
[16]	V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili and T. V. Burchuladze, "Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity," North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland Publishing Co., Amsterdam-New York, 1979.
[17]	J. J. Liu, G. Nakamura and M. Sini, Reconstruction of the shape and surface impedance from acoustic scattering data for an arbitrary cylinder, SIAM J. Appl. Math., 67 (2007), 1124-1146. doi: 10.1137/060654220.
[18]	G. Nakamura and M. Sini, Obstacle and boundary determination from scattering data, SIAM J. Math. Anal., 39 (2007), 819-837. doi: 10.1137/060658667.
[19]	G. Nakamura, R. Potthast and M. Sini, Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test, Comm. Partial Differential Equations, 31 (2006), 1505-1528.
[20]	F. Nintcheu and B. B. Guzina, Elastic scatterer reconstruction via the adjoint sampling method, SIAM J. Appl. Math., 67 (2007), 1330-1352. doi: 10.1137/060653123.
[21]	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.
[22]	F. Simonetti, Localization of pointlike scatterers in solids with subwavelength resolution, Applied Physics Letter, 89 (2006), 094105. doi: 10.1063/1.2338888.
[23]	V. A. Solonnikov, On Green's matrices for elliptic boundary value problems. I, Trudy Mat. Inst. Steklov., 110 (1970), 107-145.
[24]	V. A. Solonnikov, The Green's matrices for elliptic boundary value problems. II, Boundary Value Problems of Mathematical Physics, 7, Trudy. Math. Inst. Steklov., 116 (1971), 181-216.

[1]	Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems and Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009
[2]	Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems and Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[3]	Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[4]	Mikko Orispää, Markku Lehtinen. Fortran linear inverse problem solver. Inverse Problems and Imaging, 2010, 4 (3) : 485-503. doi: 10.3934/ipi.2010.4.485
[5]	Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems and Imaging, 2021, 15 (5) : 1015-1033. doi: 10.3934/ipi.2021026
[6]	Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems and Imaging, 2021, 15 (5) : 1247-1267. doi: 10.3934/ipi.2021036
[7]	Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems and Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
[8]	Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems and Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951
[9]	Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems and Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681
[10]	Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems and Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035
[11]	Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems and Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013
[12]	John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[13]	Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062
[14]	Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems and Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004
[15]	Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029
[16]	Yang Yang, Jian Zhai. Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity. Inverse Problems and Imaging, 2019, 13 (6) : 1309-1325. doi: 10.3934/ipi.2019057
[17]	Daniele Boffi, Franco Brezzi, Michel Fortin. Reduced symmetry elements in linear elasticity. Communications on Pure and Applied Analysis, 2009, 8 (1) : 95-121. doi: 10.3934/cpaa.2009.8.95
[18]	Armin Lechleiter, Marcel Rennoch. Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media. Inverse Problems and Imaging, 2017, 11 (1) : 151-176. doi: 10.3934/ipi.2017008
[19]	Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
[20]	Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343

2021 Impact Factor: 1.483

Tools

Metrics

Other articles
by authors

on AIMS
Drossos Gintides Mourad Sini
on Google Scholar
Drossos Gintides Mourad Sini

[Back to Top]