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A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise
A non-iterative sampling method for inverse elastic wave scattering by rough surfaces
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China |
Consider the two-dimensional inverse elastic wave scattering by an infinite rough surface with a Dirichlet boundary condition. A non-iterative sampling method is proposed for detecting the rough surface by taking elastic field measurements on a bounded line segment above the surface, based on reconstructing a modified near-field equation associated with a special surface, which generalized our previous work for the Helmholtz equation (SIAM J. Imag. Sci. 10(3) (2017), 1579-1602) to the Navier equation. Several numerical examples are carried out to illustrate the effectiveness of the inversion algorithm.
References:
[1] |
T. Arens,
Uniqueness for elastic wave scattering by rough surfaces, SIAM J. Math. Anal., 33 (2001), 461-476.
doi: 10.1137/S0036141099359470. |
[2] |
T. Arens,
Existence of solution in elastic wave scattering by unbounded rough surfaces, Math. Methods Appl. Sci., 25 (2002), 507-528.
doi: 10.1002/mma.304. |
[3] |
G. Bao, J. Gao and P. Li,
Analysis of direct and inverse cavity scattering problems, Numer. Math. Theor. Meth., 4 (2011), 335-358.
doi: 10.4208/nmtma.2011.m1021. |
[4] |
G. Bao and P. Li,
Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187.
doi: 10.1137/130916266. |
[5] |
G. Bao and P. Li,
Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imag. Sci., 7 (2014), 867-899.
doi: 10.1137/130944485. |
[6] |
G. Bao and J. Lin,
Imaging of local surface displacement on an infinite ground plane: The multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.
doi: 10.1137/110824644. |
[7] |
G. Bao and J. Lin,
Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imag., 7 (2013), 377-396.
doi: 10.3934/ipi.2013.7.377. |
[8] |
G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16 pp.
doi: 10.1088/0266-5611/32/8/085002. |
[9] |
C. Burkard and R. Potthast,
A time-domain probe method for three-dimensional rough surface reconstructions, Inverse Probl. Imag., 3 (2009), 259-274.
doi: 10.3934/ipi.2009.3.259. |
[10] |
F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-31230-7. |
[11] |
F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, 2011.
doi: 10.1137/1.9780898719406. |
[12] |
S. N. Chandler-Wilde and P. Monk,
Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), 598-618.
doi: 10.1137/040615523. |
[13] |
M. Ding, J. Li, K. Liu and J. Yang,
Imaging of local rough surfaces by the linear sampling method with near-field data, SIAM J. Imag. Sci., 10 (2017), 1579-1602.
doi: 10.1137/16M1097997. |
[14] |
J. Elschner and G. Hu,
Elastic scattering by unbounded rough surfaces, SIAM J. Math. Anal., 44 (2012), 4101-4127.
doi: 10.1137/12086203X. |
[15] |
J. Elschner and G. Hu,
Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces, Appl. Anal., 94 (2015), 252-279.
doi: 10.1080/00036811.2014.887695. |
[16] |
G. Hu, X. Liu, B. Zhang and H. Zhang, A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaces, Inverse Problems, 35 (2019), 025007, 20 pp.
doi: 10.1088/1361-6420/aaf3d6. |
[17] |
V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israeli Program for Scientific Translations, Jerusalem, 1965. |
[18] |
A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, Ph.D thesis, Universitätsverlag Karlsruhe, 2008. |
[19] |
J. Li, X. Liu, B. Zhang, and H. Zhang, The Nyström method for elastic wave scattering by unbounded rough surfaces, preprint, arXiv: 2108.02600. |
[20] |
J. Li and G. Sun,
A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces, Inverse Probl. Sci. Eng., 23 (2015), 557-577.
doi: 10.1080/17415977.2014.922077. |
[21] |
J. Li, G. Sun and B. Zhang,
The Kirsch-Kress method for inverse scattering by infinite locally, Appl. Anal., 96 (2017), 85-107.
doi: 10.1080/00036811.2016.1192141. |
[22] |
C. D. Lines and S. N. Chandler-Wilde,
A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180.
doi: 10.1007/s00607-004-0109-8. |
[23] |
X. Liu, B. Zhang and H. Zhang,
A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imag. Sci., 11 (2018), 1629-1650.
doi: 10.1137/18M1166031. |
[24] |
X. Liu, B. Zhang and H. Zhang,
Near-field imaging of an unbounded elastic rough surface with a direct imaging method, SIAM J. Appl. Math., 79 (2019), 153-176.
doi: 10.1137/18M1181407. |
[25] |
P. A. Martin,
On the scattering of elastic waves by an elastic inclusion in two dimensions, Q. J. Mech. Appl. Math., 43 (1990), 275-291.
doi: 10.1093/qjmam/43.3.275. |
[26] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. |
[27] |
A. Meier, T. Arens, S. N. Chandler-Wilde and A. Kirsch,
A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces, J. Integral Equ. Appl., 12 (2000), 281-321.
doi: 10.1216/jiea/1020282209. |
[28] |
H. Zhang and B. Zhang,
A novel integral equation for scattering by locally rough surfaces and application to the inverse problem, SIAM J. Appl. Math., 73 (2013), 1811-1829.
doi: 10.1137/130908324. |
show all references
References:
[1] |
T. Arens,
Uniqueness for elastic wave scattering by rough surfaces, SIAM J. Math. Anal., 33 (2001), 461-476.
doi: 10.1137/S0036141099359470. |
[2] |
T. Arens,
Existence of solution in elastic wave scattering by unbounded rough surfaces, Math. Methods Appl. Sci., 25 (2002), 507-528.
doi: 10.1002/mma.304. |
[3] |
G. Bao, J. Gao and P. Li,
Analysis of direct and inverse cavity scattering problems, Numer. Math. Theor. Meth., 4 (2011), 335-358.
doi: 10.4208/nmtma.2011.m1021. |
[4] |
G. Bao and P. Li,
Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187.
doi: 10.1137/130916266. |
[5] |
G. Bao and P. Li,
Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imag. Sci., 7 (2014), 867-899.
doi: 10.1137/130944485. |
[6] |
G. Bao and J. Lin,
Imaging of local surface displacement on an infinite ground plane: The multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.
doi: 10.1137/110824644. |
[7] |
G. Bao and J. Lin,
Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imag., 7 (2013), 377-396.
doi: 10.3934/ipi.2013.7.377. |
[8] |
G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16 pp.
doi: 10.1088/0266-5611/32/8/085002. |
[9] |
C. Burkard and R. Potthast,
A time-domain probe method for three-dimensional rough surface reconstructions, Inverse Probl. Imag., 3 (2009), 259-274.
doi: 10.3934/ipi.2009.3.259. |
[10] |
F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-31230-7. |
[11] |
F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, 2011.
doi: 10.1137/1.9780898719406. |
[12] |
S. N. Chandler-Wilde and P. Monk,
Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), 598-618.
doi: 10.1137/040615523. |
[13] |
M. Ding, J. Li, K. Liu and J. Yang,
Imaging of local rough surfaces by the linear sampling method with near-field data, SIAM J. Imag. Sci., 10 (2017), 1579-1602.
doi: 10.1137/16M1097997. |
[14] |
J. Elschner and G. Hu,
Elastic scattering by unbounded rough surfaces, SIAM J. Math. Anal., 44 (2012), 4101-4127.
doi: 10.1137/12086203X. |
[15] |
J. Elschner and G. Hu,
Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces, Appl. Anal., 94 (2015), 252-279.
doi: 10.1080/00036811.2014.887695. |
[16] |
G. Hu, X. Liu, B. Zhang and H. Zhang, A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaces, Inverse Problems, 35 (2019), 025007, 20 pp.
doi: 10.1088/1361-6420/aaf3d6. |
[17] |
V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israeli Program for Scientific Translations, Jerusalem, 1965. |
[18] |
A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, Ph.D thesis, Universitätsverlag Karlsruhe, 2008. |
[19] |
J. Li, X. Liu, B. Zhang, and H. Zhang, The Nyström method for elastic wave scattering by unbounded rough surfaces, preprint, arXiv: 2108.02600. |
[20] |
J. Li and G. Sun,
A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces, Inverse Probl. Sci. Eng., 23 (2015), 557-577.
doi: 10.1080/17415977.2014.922077. |
[21] |
J. Li, G. Sun and B. Zhang,
The Kirsch-Kress method for inverse scattering by infinite locally, Appl. Anal., 96 (2017), 85-107.
doi: 10.1080/00036811.2016.1192141. |
[22] |
C. D. Lines and S. N. Chandler-Wilde,
A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180.
doi: 10.1007/s00607-004-0109-8. |
[23] |
X. Liu, B. Zhang and H. Zhang,
A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imag. Sci., 11 (2018), 1629-1650.
doi: 10.1137/18M1166031. |
[24] |
X. Liu, B. Zhang and H. Zhang,
Near-field imaging of an unbounded elastic rough surface with a direct imaging method, SIAM J. Appl. Math., 79 (2019), 153-176.
doi: 10.1137/18M1181407. |
[25] |
P. A. Martin,
On the scattering of elastic waves by an elastic inclusion in two dimensions, Q. J. Mech. Appl. Math., 43 (1990), 275-291.
doi: 10.1093/qjmam/43.3.275. |
[26] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. |
[27] |
A. Meier, T. Arens, S. N. Chandler-Wilde and A. Kirsch,
A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces, J. Integral Equ. Appl., 12 (2000), 281-321.
doi: 10.1216/jiea/1020282209. |
[28] |
H. Zhang and B. Zhang,
A novel integral equation for scattering by locally rough surfaces and application to the inverse problem, SIAM J. Appl. Math., 73 (2013), 1811-1829.
doi: 10.1137/130908324. |










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