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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Bonnet, "Boundary Integral Methods for Solids and Fluids," Wiley & Sons, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations, 23 (1998), 1459-1474.  Google Scholar

[13]

V. Isakov, On uniqueness in the inverse transmission scattering problem, Commun. Part. Diff. Equ., 15 (1990), 1565-1587.  Google Scholar

[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.  Google Scholar

[15]

V. D. Kupradze, "Potential Methods in the Theory of Elasticity," Israel Program for Scientific Translations, Jerusalem, Daniel Davey & Co., Inc., New York, 1965.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

F. Simonetti, Localization of pointlike scatterers in solids with subwavelength resolution, Applied Physics Letter, 89 (2006), 094105. doi: 10.1063/1.2338888.  Google Scholar

[23]

V. A. Solonnikov, On Green's matrices for elliptic boundary value problems. I, Trudy Mat. Inst. Steklov., 110 (1970), 107-145.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Bonnet, "Boundary Integral Methods for Solids and Fluids," Wiley & Sons, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations, 23 (1998), 1459-1474.  Google Scholar

[13]

V. Isakov, On uniqueness in the inverse transmission scattering problem, Commun. Part. Diff. Equ., 15 (1990), 1565-1587.  Google Scholar

[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.  Google Scholar

[15]

V. D. Kupradze, "Potential Methods in the Theory of Elasticity," Israel Program for Scientific Translations, Jerusalem, Daniel Davey & Co., Inc., New York, 1965.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

F. Simonetti, Localization of pointlike scatterers in solids with subwavelength resolution, Applied Physics Letter, 89 (2006), 094105. doi: 10.1063/1.2338888.  Google Scholar

[23]

V. A. Solonnikov, On Green's matrices for elliptic boundary value problems. I, Trudy Mat. Inst. Steklov., 110 (1970), 107-145.  Google Scholar

[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.  Google Scholar

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