# American Institute of Mathematical Sciences

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 and 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. [2] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245. [3] H. Ammari, H. Kang, G. 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. [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. [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. [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. [7] G. Bao, T. 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. [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. [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. [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. [11] G. Bao, P. 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. [12] M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2005), R1–R50. doi: 10.1088/0266-5611/21/2/R01. [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. [14] A. Charalambopoulos, D. 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. [15] T. Cheng, P. 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. [16] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [17] D. Courjon, Near-Field Microscopy and Near-Field Optics, Imperial College Press, London, 2003.  doi: 10.1142/p220. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] P. Li, Y. 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. [28] P. Li, Y. 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. [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. [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. [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. [32] J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces, Institute of Physics Publishing, 1991.

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##### 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. [2] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245. [3] H. Ammari, H. Kang, G. 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. [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. [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. [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. [7] G. Bao, T. 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. [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. [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. [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. [11] G. Bao, P. 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. [12] M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2005), R1–R50. doi: 10.1088/0266-5611/21/2/R01. [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. [14] A. Charalambopoulos, D. 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. [15] T. Cheng, P. 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. [16] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [17] D. Courjon, Near-Field Microscopy and Near-Field Optics, Imperial College Press, London, 2003.  doi: 10.1142/p220. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] P. Li, Y. 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. [28] P. Li, Y. 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. [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. [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. [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. [32] J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces, Institute of Physics Publishing, 1991.
The problem geometry
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 $\rho_1 = 4$
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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