August  2019, 13(4): 721-744. doi: 10.3934/ipi.2019033

Inverse elastic surface scattering with far-field data

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

2. 

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA

* Corresponding author

Received  April 2018 Revised  December 2018 Published  May 2019

Fund Project: The research of H.-A. Diao was supported in part by the Fundamental Research Funds for the Central Universities under the grant 2412017FZ007

A rigorous mathematical model and an efficient computational method are proposed to solving the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. We demonstrate how an enhanced resolution can be achieved by using more easily measurable far-field data. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangular slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near (FtN) field data conversion and utilizes the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem. Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction. Results show that the proposed method is capable of stably reconstructing surfaces with resolution controlled by the slab's density.

Citation: Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033
References:
[1]

C. Alves and H. Ammari, Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium, SIAM J. Appl. Math., 62 (2001), 94-106. doi: 10.1137/S0036139900369266. Google Scholar

[2]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245. Google Scholar

[3]

H. AmmariH. KangG. Nakamura and K. Tanuma, Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J. Elasticity, 67 (2002), 97-129. doi: 10.1023/A:1023940025757. Google Scholar

[4]

T. Arens, A new integral equation formulation for the scattering of plane elastic waves by diffraction gratings, J. Integral Equations Appl., 11 (1999), 275-297. doi: 10.1216/jiea/1181074278. Google Scholar

[5]

T. Arens, The scattering of plane elastic waves by a one-dimensional periodic surface, Math. Methods Appl. Sci., 22 (1999), 55-72. doi: 10.1002/(SICI)1099-1476(19990110)22:1<55::AID-MMA20>3.0.CO;2-T. Google Scholar

[6]

C. E. Athanasiadis, D. Natroshvili, V. Sevroglou and I. G. Stratis, An application of the reciprocity gap functional to inverse mixed impedance problems in elasticity, Inverse Problems, 26 (2010), 85011, 19pp. doi: 10.1088/0266-5611/26/8/085011. Google Scholar

[7]

G. BaoT. Cui and P. Li, Inverse diffraction grating of maxwell's equations in biperiodic structures, Opt. Express, 22 (2014), 4799-4816. doi: 10.1364/OE.22.004799. Google Scholar

[8]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces, SIAM J. Appl.Math., 73 (2013), 2162-2187. doi: 10.1137/130916266. Google Scholar

[9]

G. Bao and P. Li, Convergence analysis in near-field imaging, Inverse Problems, 30 (2014), 085008, 26pp. doi: 10.1088/0266-5611/30/8/085008. Google Scholar

[10]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867-899. doi: 10.1137/130944485. Google Scholar

[11]

G. BaoP. Li and Y. Wang, Near-field imaging with far-field data, Appl. Math. Lett., 60 (2016), 36-42. doi: 10.1016/j.aml.2016.03.023. Google Scholar

[12]

M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2005), R1–R50. doi: 10.1088/0266-5611/21/2/R01. Google Scholar

[13]

O. P. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries, J. Opt. Soc. Am. A, 10 (1993), 1168-1175. doi: 10.1364/JOSAA.10.001168. Google Scholar

[14]

A. CharalambopoulosD. Gintides and K. Kiriaki, On the uniqueness of the inverse elastic scattering problem for periodic structures, Inverse Problems, 17 (2001), 1923-1935. doi: 10.1088/0266-5611/17/6/323. Google Scholar

[15]

T. ChengP. Li and Y. Wang, Near-field imaging of perfectly conducting grating surfaces, J. Opt. Soc. Am. A, 30 (2013), 2473-2481. doi: 10.1364/JOSAA.30.002473. Google Scholar

[16]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. Google Scholar

[17] D. Courjon, Near-Field Microscopy and Near-Field Optics, Imperial College Press, London, 2003. doi: 10.1142/p220. Google Scholar
[18]

J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings, Math. Methods Appl. Sci., 33 (2010), 1924-1941. doi: 10.1002/mma.1305. Google Scholar

[19]

J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles, Commun. Comput. Phys., 12 (2012), 1434-1460. doi: 10.4208/cicp.220611.130112a. Google Scholar

[20]

J. Elschner and G. Hu, Scattering of plane elastic waves by three-dimensional diffraction gratings, Math. Models Methods Appl. Sci., 22 (2012), 1150019, 34pp. doi: 10.1142/S0218202511500199. Google Scholar

[21]

G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29 (2013), 115005, 25pp. doi: 10.1088/0266-5611/29/11/115005. Google Scholar

[22]

X. Jiang and P. Li, Inverse electromagnetic diffraction by biperiodic dielectric gratings, Inverse Problems, 33 (2017), 085004, 29pp. doi: 10.1088/1361-6420/aa76b9. Google Scholar

[23]

P. Li and J. Shen, Analysis of the scattering by an unbounded rough surface, Math. Methods Appl. Sci., 35 (2012), 2166-2184. doi: 10.1002/mma.2560. Google Scholar

[24]

P. Li and Y. Wang, Near-field imaging of interior cavities, Commun. Comput. Phys., 17 (2015), 542-563. doi: 10.4208/cicp.010414.250914a. Google Scholar

[25]

P. Li and Y. Wang, Near-field imaging of obstacles, Inverse Probl. Imaging, 9 (2015), 189-210. doi: 10.3934/ipi.2015.9.189. Google Scholar

[26]

P. Li, Y. Wang and Y. Zhao, Inverse elastic surface scattering with near-field data, Inverse Problems, 31 (2015), 035009, 27pp. doi: 10.1088/0266-5611/31/3/035009. Google Scholar

[27]

P. LiY. Wang and Y. Zhao, Convergence analysis in near-field imaging for elastic waves, Appl. Anal., 95 (2016), 2339-2360. doi: 10.1080/00036811.2015.1089238. Google Scholar

[28]

P. LiY. Wang and Y. Zhao, Near-field imaging of biperiodic surfaces for elastic waves, J. Comput. Phys., 324 (2016), 1-23. doi: 10.1016/j.jcp.2016.07.030. Google Scholar

[29]

A. Malcolm and D. P. Nicholls, A field expansions method for scattering by periodic multilayered media, J. Acoust. Soc. Am., 129 (2011), 1783-1793. doi: 10.1121/1.3531931. Google Scholar

[30]

D. P. Nicholls and F. Reitich, Shape deformations in rough-surface scattering: Cancellations, conditioning, and convergence, J. Opt. Soc. Am. A, 21 (2004), 590-605. doi: 10.1364/JOSAA.21.000590. Google Scholar

[31]

D. P. Nicholls and F. Reitich, Shape deformations in rough-surface scattering: Improved algorithms, J. Opt. Soc. Am. A, 21 (2004), 606-621. doi: 10.1364/JOSAA.21.000606. Google Scholar

[32]

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces, Institute of Physics Publishing, 1991. Google Scholar

show all references

References:
[1]

C. Alves and H. Ammari, Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium, SIAM J. Appl. Math., 62 (2001), 94-106. doi: 10.1137/S0036139900369266. Google Scholar

[2]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245. Google Scholar

[3]

H. AmmariH. KangG. Nakamura and K. Tanuma, Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J. Elasticity, 67 (2002), 97-129. doi: 10.1023/A:1023940025757. Google Scholar

[4]

T. Arens, A new integral equation formulation for the scattering of plane elastic waves by diffraction gratings, J. Integral Equations Appl., 11 (1999), 275-297. doi: 10.1216/jiea/1181074278. Google Scholar

[5]

T. Arens, The scattering of plane elastic waves by a one-dimensional periodic surface, Math. Methods Appl. Sci., 22 (1999), 55-72. doi: 10.1002/(SICI)1099-1476(19990110)22:1<55::AID-MMA20>3.0.CO;2-T. Google Scholar

[6]

C. E. Athanasiadis, D. Natroshvili, V. Sevroglou and I. G. Stratis, An application of the reciprocity gap functional to inverse mixed impedance problems in elasticity, Inverse Problems, 26 (2010), 85011, 19pp. doi: 10.1088/0266-5611/26/8/085011. Google Scholar

[7]

G. BaoT. Cui and P. Li, Inverse diffraction grating of maxwell's equations in biperiodic structures, Opt. Express, 22 (2014), 4799-4816. doi: 10.1364/OE.22.004799. Google Scholar

[8]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces, SIAM J. Appl.Math., 73 (2013), 2162-2187. doi: 10.1137/130916266. Google Scholar

[9]

G. Bao and P. Li, Convergence analysis in near-field imaging, Inverse Problems, 30 (2014), 085008, 26pp. doi: 10.1088/0266-5611/30/8/085008. Google Scholar

[10]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867-899. doi: 10.1137/130944485. Google Scholar

[11]

G. BaoP. Li and Y. Wang, Near-field imaging with far-field data, Appl. Math. Lett., 60 (2016), 36-42. doi: 10.1016/j.aml.2016.03.023. Google Scholar

[12]

M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2005), R1–R50. doi: 10.1088/0266-5611/21/2/R01. Google Scholar

[13]

O. P. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries, J. Opt. Soc. Am. A, 10 (1993), 1168-1175. doi: 10.1364/JOSAA.10.001168. Google Scholar

[14]

A. CharalambopoulosD. Gintides and K. Kiriaki, On the uniqueness of the inverse elastic scattering problem for periodic structures, Inverse Problems, 17 (2001), 1923-1935. doi: 10.1088/0266-5611/17/6/323. Google Scholar

[15]

T. ChengP. Li and Y. Wang, Near-field imaging of perfectly conducting grating surfaces, J. Opt. Soc. Am. A, 30 (2013), 2473-2481. doi: 10.1364/JOSAA.30.002473. Google Scholar

[16]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. Google Scholar

[17] D. Courjon, Near-Field Microscopy and Near-Field Optics, Imperial College Press, London, 2003. doi: 10.1142/p220. Google Scholar
[18]

J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings, Math. Methods Appl. Sci., 33 (2010), 1924-1941. doi: 10.1002/mma.1305. Google Scholar

[19]

J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles, Commun. Comput. Phys., 12 (2012), 1434-1460. doi: 10.4208/cicp.220611.130112a. Google Scholar

[20]

J. Elschner and G. Hu, Scattering of plane elastic waves by three-dimensional diffraction gratings, Math. Models Methods Appl. Sci., 22 (2012), 1150019, 34pp. doi: 10.1142/S0218202511500199. Google Scholar

[21]

G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29 (2013), 115005, 25pp. doi: 10.1088/0266-5611/29/11/115005. Google Scholar

[22]

X. Jiang and P. Li, Inverse electromagnetic diffraction by biperiodic dielectric gratings, Inverse Problems, 33 (2017), 085004, 29pp. doi: 10.1088/1361-6420/aa76b9. Google Scholar

[23]

P. Li and J. Shen, Analysis of the scattering by an unbounded rough surface, Math. Methods Appl. Sci., 35 (2012), 2166-2184. doi: 10.1002/mma.2560. Google Scholar

[24]

P. Li and Y. Wang, Near-field imaging of interior cavities, Commun. Comput. Phys., 17 (2015), 542-563. doi: 10.4208/cicp.010414.250914a. Google Scholar

[25]

P. Li and Y. Wang, Near-field imaging of obstacles, Inverse Probl. Imaging, 9 (2015), 189-210. doi: 10.3934/ipi.2015.9.189. Google Scholar

[26]

P. Li, Y. Wang and Y. Zhao, Inverse elastic surface scattering with near-field data, Inverse Problems, 31 (2015), 035009, 27pp. doi: 10.1088/0266-5611/31/3/035009. Google Scholar

[27]

P. LiY. Wang and Y. Zhao, Convergence analysis in near-field imaging for elastic waves, Appl. Anal., 95 (2016), 2339-2360. doi: 10.1080/00036811.2015.1089238. Google Scholar

[28]

P. LiY. Wang and Y. Zhao, Near-field imaging of biperiodic surfaces for elastic waves, J. Comput. Phys., 324 (2016), 1-23. doi: 10.1016/j.jcp.2016.07.030. Google Scholar

[29]

A. Malcolm and D. P. Nicholls, A field expansions method for scattering by periodic multilayered media, J. Acoust. Soc. Am., 129 (2011), 1783-1793. doi: 10.1121/1.3531931. Google Scholar

[30]

D. P. Nicholls and F. Reitich, Shape deformations in rough-surface scattering: Cancellations, conditioning, and convergence, J. Opt. Soc. Am. A, 21 (2004), 590-605. doi: 10.1364/JOSAA.21.000590. Google Scholar

[31]

D. P. Nicholls and F. Reitich, Shape deformations in rough-surface scattering: Improved algorithms, J. Opt. Soc. Am. A, 21 (2004), 606-621. doi: 10.1364/JOSAA.21.000606. Google Scholar

[32]

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces, Institute of Physics Publishing, 1991. Google Scholar

Figure 1.  The problem geometry
Figure 2.  Example 1: the reconstructed surface (dashed line) is plotted against the exact surface (solid line). (a) $ \rho_1 = 1 $; (b) $ \rho_1 = 2 $; (c) $ \rho_1 = 4 $; (d) 1 step of nonlinear correction when $ \rho_1 = 4 $; (e) 2 steps of nonlinear correction when $ \rho_1 = 4 $; (f) 3 steps of nonlinear correction when $ \rho_1 = 4 $
Figure 3.  Example 2: the reconstructed surface (dashed line) is plotted against the exact surface (solid line). (a) $ \rho_1 = 1 $; (b) $ \rho_1 = 2 $; (c) $ \rho_1 = 4 $; (d) 1 step of nonlinear correction when $ \rho_1 = 4 $; (e) 2 steps of nonlinear correction when $ \rho_1 = 4 $; (f) 3 steps of nonlinear correction when $ \rho_1 = 4 $
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