# American Institute of Mathematical Sciences

October  2021, 15(5): 1247-1267. doi: 10.3934/ipi.2021036

## A linear sampling method for inverse acoustic scattering by a locally rough interface

 1 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China 2 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China 3 LSEC, NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 4 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Jiaqing Yang

Received  October 2020 Revised  March 2021 Published  October 2021 Early access  April 2021

Fund Project: The second author is supported by the National Key Research and Development Program of China under Grant 2020AAA0105601 and the NNSF of China under Grant 11771349

This paper is concerned with the inverse problem of time-harmonic acoustic scattering by an unbounded, locally rough interface which is assumed to be a local perturbation of a plane. The purpose of this paper is to recover the local perturbation of the interface from the near-field measurement given on a straight line segment with a finite distance above the interface and generated by point sources. Precisely, we propose a novel version of the linear sampling method to recover the location and shape of the local perturbation of the interface numerically. Our method is based on a modified near-field operator equation associated with a special rough surface, constructed by reformulating the forward scattering problem into an equivalent integral equation formulation in a bounded domain, leading to a fast imaging algorithm. Numerical experiments are presented to illustrate the effectiveness of the imaging method.

Citation: Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems & Imaging, 2021, 15 (5) : 1247-1267. doi: 10.3934/ipi.2021036
##### References:
 [1] G. Bao, J. Gao and P. Li, Analysis of direct and inverse cavity scattering problems, Numer. Math. Theory Methods Appl., 4 (2011), 335-358.  doi: 10.4208/nmtma.2011.m1021.  Google Scholar [2] 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 [3] 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 [4] 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.  Google Scholar [5] G. Bao and J. Lin, Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imaging, 7 (2013), 377-396.  doi: 10.3934/ipi.2013.7.377.  Google Scholar [6] C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems, 26 (2010), 045007, 23 pp. doi: 10.1088/0266-5611/26/4/045007.  Google Scholar [7] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer, Berlin, 2006. Google Scholar [8] F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar [9] S. N. Chandler-Wilde and J. Elschner, Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces, SIAM J. Math. Anal., 42 (2010), 2554-2580.  doi: 10.1137/090776111.  Google Scholar [10] S. N. Chandler-Wilde and R. Potthast, The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 2967-3001.  doi: 10.1098/rspa.2002.0999.  Google Scholar [11] S. N. Chandler-Wilde and B. Zhang, Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers, SIAM J. Math. Anal., 30 (1999), 559-583.  doi: 10.1137/S0036141097328932.  Google Scholar [12] L. Chorfi and P. Gaitan, Reconstruction of the interface between two-layered media using far-field measurements, Inverse Problems, 27 (2011), 075001, 19 pp. doi: 10.1088/0266-5611/27/7/075001.  Google Scholar [13] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar [14] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theorey, 4$^th$ edition, Springer, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar [15] M. Ding, J. Li, K. Liu and J. 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Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley, New York, 1986. doi: 10.1007/978-3-663-10649-4.  Google Scholar [21] P. Li, Coupling of finite element and boundary integral method for electromagnetic scattering in a two-layered medium, J. Comput. Phys., 229 (2010), 481-497.  doi: 10.1016/j.jcp.2009.09.040.  Google Scholar [22] J. Li, G. Sun and R. Zhang, The numerical solution of scattering by infinite rough interfaces based on the integral equation method, Comput. Math. Appl., 71 (2016), 1491-1502.  doi: 10.1016/j.camwa.2016.02.031.  Google Scholar [23] 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.  Google Scholar [24] J. Li, G. Sun and B. Zhang, The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces, Appl. Anal., 96 (2017), 85-107.  doi: 10.1080/00036811.2016.1192141.  Google Scholar [25] 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.  Google Scholar [26] X. Liu, B. Zhang and H. Zhang, A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imaging Sci., 11 (2018), 1629-1650.  doi: 10.1137/18M1166031.  Google Scholar [27] 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.  Google Scholar [28] D. Natroshvili, T. Arens and S. N. Chandler-Wilde, Uniqueness, existence, and integral equation formulations for interface scattering problems, Mem. Differential Equations Math. Phys., 30 (2003), 105-146.   Google Scholar [29] F. Qu, B. Zhang and H. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: The Neumann case, SIAM J. Sci. Comput., 41 (2019), A3673–A3702. doi: 10.1137/19M1240745.  Google Scholar [30] D. G. Roy and S. Mudaliar, Domain derivatives in dielectric rough surface scattering, IEEE Trans. Antennas Propagation, 63 (2015), 4486-4495.  doi: 10.1109/TAP.2015.2463682.  Google Scholar [31] M. Thomas, Analysis of Rough Surface Scattering Problems, Ph.D Thesis, The University of Reading, UK, 2006. Google Scholar [32] X. Xu, B. Zhang and H. Zhang, Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data, SIAM J. Imaging Sci., 12 (2019), 119-152.  doi: 10.1137/18M1210204.  Google Scholar [33] J. Yang, J. Li and B. Zhang, Simultaneous recovery of a locally rough interface and its buried obstacles and homogeneous medium, arXiv: 2102.01855v1. Google Scholar [34] J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 035004, 17 pp. doi: 10.1088/0266-5611/28/3/035004.  Google Scholar [35] J. Yang, B. Zhang and R. Zhang, Reconstruction of penetrable grating profiles, Inverse Problems Imaging, 7 (2013), 1393-1407.  doi: 10.3934/ipi.2013.7.1393.  Google Scholar [36] H. Zhang, Recovering unbounded rough surfaces with a direct imaging method, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 119-133.  doi: 10.1007/s10255-020-0916-5.  Google Scholar [37] B. Zhang and S. N. Chandler-Wilde, Integral equation methods for scattering by infinite rough surfaces, Math. Methods Appl. Sci., 26 (2003), 463-488.  doi: 10.1002/mma.361.  Google Scholar [38] 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.  Google Scholar

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##### References:
 [1] G. Bao, J. Gao and P. Li, Analysis of direct and inverse cavity scattering problems, Numer. Math. Theory Methods Appl., 4 (2011), 335-358.  doi: 10.4208/nmtma.2011.m1021.  Google Scholar [2] 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 [3] 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 [4] 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.  Google Scholar [5] G. Bao and J. Lin, Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imaging, 7 (2013), 377-396.  doi: 10.3934/ipi.2013.7.377.  Google Scholar [6] C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems, 26 (2010), 045007, 23 pp. doi: 10.1088/0266-5611/26/4/045007.  Google Scholar [7] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer, Berlin, 2006. Google Scholar [8] F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar [9] S. N. Chandler-Wilde and J. Elschner, Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces, SIAM J. Math. Anal., 42 (2010), 2554-2580.  doi: 10.1137/090776111.  Google Scholar [10] S. N. Chandler-Wilde and R. Potthast, The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 2967-3001.  doi: 10.1098/rspa.2002.0999.  Google Scholar [11] S. N. Chandler-Wilde and B. Zhang, Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers, SIAM J. Math. Anal., 30 (1999), 559-583.  doi: 10.1137/S0036141097328932.  Google Scholar [12] L. Chorfi and P. Gaitan, Reconstruction of the interface between two-layered media using far-field measurements, Inverse Problems, 27 (2011), 075001, 19 pp. doi: 10.1088/0266-5611/27/7/075001.  Google Scholar [13] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar [14] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theorey, 4$^th$ edition, Springer, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar [15] M. Ding, J. Li, K. Liu and J. Yang, Imaging of locally rough surfaces by the linear sampling method with the near-field data, SIAM J. Imaging Sci., 10 (2017), 1579-1602.  doi: 10.1137/16M1097997.  Google Scholar [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.  Google Scholar [17] A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, Ph.D thesis. KIT, Germany, 2008. Google Scholar [18] A. Lechleiter and R. Zhang, A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces, SIAM J. Sci. Comput., 39 (2017), B819–B839. doi: 10.1137/16M1104111.  Google Scholar [19] A. Lechleiter and R. Zhang, Reconstruction of local perturbations in periodic surfaces, Inverse Problems, 34 (2018), 035006, 17 pp. doi: 10.1088/1361-6420/aaa7b1.  Google Scholar [20] R. Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley, New York, 1986. doi: 10.1007/978-3-663-10649-4.  Google Scholar [21] P. Li, Coupling of finite element and boundary integral method for electromagnetic scattering in a two-layered medium, J. Comput. Phys., 229 (2010), 481-497.  doi: 10.1016/j.jcp.2009.09.040.  Google Scholar [22] J. Li, G. Sun and R. Zhang, The numerical solution of scattering by infinite rough interfaces based on the integral equation method, Comput. Math. Appl., 71 (2016), 1491-1502.  doi: 10.1016/j.camwa.2016.02.031.  Google Scholar [23] 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.  Google Scholar [24] J. Li, G. Sun and B. Zhang, The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces, Appl. Anal., 96 (2017), 85-107.  doi: 10.1080/00036811.2016.1192141.  Google Scholar [25] 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.  Google Scholar [26] X. Liu, B. Zhang and H. Zhang, A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imaging Sci., 11 (2018), 1629-1650.  doi: 10.1137/18M1166031.  Google Scholar [27] 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.  Google Scholar [28] D. Natroshvili, T. Arens and S. N. Chandler-Wilde, Uniqueness, existence, and integral equation formulations for interface scattering problems, Mem. Differential Equations Math. Phys., 30 (2003), 105-146.   Google Scholar [29] F. Qu, B. Zhang and H. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: The Neumann case, SIAM J. Sci. Comput., 41 (2019), A3673–A3702. doi: 10.1137/19M1240745.  Google Scholar [30] D. G. Roy and S. Mudaliar, Domain derivatives in dielectric rough surface scattering, IEEE Trans. Antennas Propagation, 63 (2015), 4486-4495.  doi: 10.1109/TAP.2015.2463682.  Google Scholar [31] M. Thomas, Analysis of Rough Surface Scattering Problems, Ph.D Thesis, The University of Reading, UK, 2006. Google Scholar [32] X. Xu, B. Zhang and H. Zhang, Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data, SIAM J. Imaging Sci., 12 (2019), 119-152.  doi: 10.1137/18M1210204.  Google Scholar [33] J. Yang, J. Li and B. Zhang, Simultaneous recovery of a locally rough interface and its buried obstacles and homogeneous medium, arXiv: 2102.01855v1. Google Scholar [34] J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 035004, 17 pp. doi: 10.1088/0266-5611/28/3/035004.  Google Scholar [35] J. Yang, B. Zhang and R. Zhang, Reconstruction of penetrable grating profiles, Inverse Problems Imaging, 7 (2013), 1393-1407.  doi: 10.3934/ipi.2013.7.1393.  Google Scholar [36] H. Zhang, Recovering unbounded rough surfaces with a direct imaging method, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 119-133.  doi: 10.1007/s10255-020-0916-5.  Google Scholar [37] B. Zhang and S. N. Chandler-Wilde, Integral equation methods for scattering by infinite rough surfaces, Math. Methods Appl. Sci., 26 (2003), 463-488.  doi: 10.1002/mma.361.  Google Scholar [38] 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.  Google Scholar
The physical configuration of the scattering problem
Reconstructions of the locally rough interface given in Example 1 from data with no noise (a), 2% noise (b) and 5% noise (c)
Reconstructions of the locally rough interface given in Example 2 from data with no noise (a), 2% noise (b) and 5% noise (c)
Reconstructions of the locally rough interface given in Example 3 from data with no noise (a), 2% noise (b) and 5% noise (c)
Reconstructions of the locally rough interface given in Example 4 from data with $N = 201$ (a), $N = 401$ (b) and $N = 601$ (c) at the same level of 2% noise
Reconstructions of the locally rough interface given in Example 5 from data with $b = 0.25$ (a), $b = 0.65$ (b) and $b = 1.05$ (c) at the same level of 2% noise
Reconstructions of the locally rough interface given in Example 5 from data with $a = 2$ (a), $a = 8$ (b) and $a = 14$ (c) at the same level of 2% noise
Numerical solutions of $G_r^{\rm s}(x,z)$ as the radius $r\to\infty$
 $r$ $G_r^{\rm s}(x_1,z)$ $G_r^s(x_2,z)$ $G^s(x_3,z)$ $10^2$ 1.0e-02$\cdot$(-2.23-0.45i) 1.0e-02$\cdot$(-2.41-0.51i) 1.0e-2$\cdot$(-2.23-0.45i) $10^4$ 1.0e-03$\cdot$(0.80+2.89i) 1.0e-03$\cdot$(0.80+2.89i) 1.0e-03$\cdot$(0.80+2.89i) $10^6$ 1.0e-03$\cdot$(-0.03+0.13i) 1.0e-03$\cdot$(-0.03+0.13i) 1.0e-03$\cdot$(-0.03+0.13i) $10^8$ 1.0e-05$\cdot$(-1.69-0.80i) 1.0e-05$\cdot$(-1.69-0.80i) 1.0e-05$\cdot$(-1.69-0.80i) $10^{10}$ 1.0e-06$\cdot$(0.82+1.79i) 1.0e-06$\cdot$(0.82+1.79i) 1.0e-06$\cdot$(0.82+1.79i)
 $r$ $G_r^{\rm s}(x_1,z)$ $G_r^s(x_2,z)$ $G^s(x_3,z)$ $10^2$ 1.0e-02$\cdot$(-2.23-0.45i) 1.0e-02$\cdot$(-2.41-0.51i) 1.0e-2$\cdot$(-2.23-0.45i) $10^4$ 1.0e-03$\cdot$(0.80+2.89i) 1.0e-03$\cdot$(0.80+2.89i) 1.0e-03$\cdot$(0.80+2.89i) $10^6$ 1.0e-03$\cdot$(-0.03+0.13i) 1.0e-03$\cdot$(-0.03+0.13i) 1.0e-03$\cdot$(-0.03+0.13i) $10^8$ 1.0e-05$\cdot$(-1.69-0.80i) 1.0e-05$\cdot$(-1.69-0.80i) 1.0e-05$\cdot$(-1.69-0.80i) $10^{10}$ 1.0e-06$\cdot$(0.82+1.79i) 1.0e-06$\cdot$(0.82+1.79i) 1.0e-06$\cdot$(0.82+1.79i)
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