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October  2021, 15(5): 975-997. doi: 10.3934/ipi.2021024

Near-field imaging for an obstacle above rough surfaces with limited aperture data

School of Mathematical Sciences, Heilongjiang University, Heilongjiang Provincial Key Laboratory of Complex Systems Theory and Computation, Harbin, 150080, China

* Corresponding author: Lei Zhang

Received  August 2020 Revised  February 2021 Published  October 2021 Early access  March 2021

This paper is concerned with the scattering and inverse scattering problems for a point source incident wave by an obstacle embedded in a two-layered background medium. It is a nontrivial extension of the previous theoretical work on the inverse obstacle scattering in an unbounded structure [Commun. Comput. Phys., 26 (2019), 1274-1306]. By the potential theory of boundary integral equations, we derive a novel integral equation formula for the scattering problem, then the well-posedness of the system is proved. Based on the singularity analysis of integral kernels, we presented a numerical method for the integral equations. Furthermore, we developed a reverse time migration method for the corresponding composite inverse scattering problem with the limited aperture data. Numerical experiments show that the proposed method is effective to recover the support of an unknown obstacle and the shape, location of the surfaces.

Citation: Lei Zhang, Luming Jia. Near-field imaging for an obstacle above rough surfaces with limited aperture data. Inverse Problems and Imaging, 2021, 15 (5) : 975-997. doi: 10.3934/ipi.2021024
References:
[1]

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

[2]

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.

[3]

G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002. doi: 10.1088/0266-5611/32/8/085002.

[4]

G. BaoH. LiuP. Li and L. Zhang, Inverse obstacle scattering in an unbounded structure, Commun. Comput. Phys., 26 (2019), 1274-1306.  doi: 10.4208/cicp.2019.js60.01.

[5]

C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems, 26 (2010), 045007. doi: 10.1088/0266-5611/26/4/045007.

[6]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, 188. Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.

[7]

S. N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math., 58 (1998), 1774-1790.  doi: 10.1137/S0036139996309722.

[8]

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.

[9]

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.

[10]

S. N. Chandler-WildeE. Heinemeyer and R. Potthast, Acoustic scattering by mildly rough unbounded surfaces in three dimensions, SIAM J. Appl. Math., 66 (2006), 1002-1026.  doi: 10.1137/050635262.

[11]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Acoustic waves, Inverse Problems, 29 (2013), 085005. doi: 10.1088/0266-5611/29/8/085005.

[12]

Z. Chen and G. Huang, Reverse time migration for reconstructing extended obstacles in the half space, Inverse Problems, 31 (2015), 055007. doi: 10.1088/0266-5611/31/5/055007.

[13]

F. J. Claerbout, Toward a unified theory of reflector mapping, Geophysics, 36 (1971), 467-481.  doi: 10.1190/1.1440185.

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, Springer-Verlag, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[15]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, SIAM, Philadelphia, 2013. doi: 10.1137/1.9781611973167.ch1.

[16]

Y. HeP. Li and J. Shen, A new spectral method for numerical solution of the unbounded rough surface scattering problem, J. Comput. Phys., 275 (2014), 608-625.  doi: 10.1016/j.jcp.2014.07.026.

[17]

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. doi: 10.1088/1361-6420/aaf3d6.

[18]

K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003. doi: 10.1088/0266-5611/28/2/025003.

[19]

A. Kirsch, Surface gradients and continuity properties for some integral operators in classical scattering theory, Math. Method Appl. Sci., 11 (1989), 789-804.  doi: 10.1002/mma.1670110605.

[20]

R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comput. Appl. Math., 61 (1995), 345-360.  doi: 10.1016/0377-0427(94)00073-7.

[21]

R. Kress and T. Tran, Inverse scattering for a locally perturbed half-plane, Inverse Problems, 16 (2000), 1541-1559.  doi: 10.1088/0266-5611/16/5/323.

[22]

J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Probl. Imaging, 7 (2013), 757-775.  doi: 10.3934/ipi.2013.7.757.

[23]

J. LiG. 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.

[24]

P. LiH. Wu and W. Zheng, Electromagnetic scattering by unbounded rough surfaces, SIAM J. Math. Anal., 43 (2011), 1205-1231.  doi: 10.1137/100806217.

[25]

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.

[26]

P. Li, J. Wang and L. Zhang, Inverse obstacle scattering for Maxwell's equations in an unbounded structure, Inverse Problems, 35 (2019), 095002. doi: 10.1088/1361-6420/ab1f1b.

[27]

X. LiuB. 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.

[28]

X. LiuB. 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.

[29]

Y. Lu and B. Zhang, Direct and inverse scattering problem by an unbounded rough interface with buried obstacles, preprint, arXiv: 1610.03515v1.

[30]

A. MeierT. ArensS. N. Chandler-Wilde and A. Kirsch, A Nystr$\ddot{o}$m method for a class of integral equationson the real line with applications to scattering by diffraction gratings and rough surfaces, J. Integral Equations Appl., 12 (2000), 281-321.  doi: 10.1216/jiea/1020282209.

[31]

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.

[32]

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.

[33]

L. ZhangF. Ma and J. Wang, Regularized conjugate gradient method with fast multipole acceleration for wave scattering from 1D fractal rough surface, Wave Motion, 50 (2013), 41-56.  doi: 10.1016/j.wavemoti.2012.06.005.

[34]

Y. Zhang and G. Zhang, One-step extrapolation method for reverse time migration, Geophysics, 74 (2009), A29–A33. doi: 10.1190/1.3123476.

show all references

References:
[1]

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

[2]

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.

[3]

G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002. doi: 10.1088/0266-5611/32/8/085002.

[4]

G. BaoH. LiuP. Li and L. Zhang, Inverse obstacle scattering in an unbounded structure, Commun. Comput. Phys., 26 (2019), 1274-1306.  doi: 10.4208/cicp.2019.js60.01.

[5]

C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems, 26 (2010), 045007. doi: 10.1088/0266-5611/26/4/045007.

[6]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, 188. Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.

[7]

S. N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math., 58 (1998), 1774-1790.  doi: 10.1137/S0036139996309722.

[8]

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.

[9]

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.

[10]

S. N. Chandler-WildeE. Heinemeyer and R. Potthast, Acoustic scattering by mildly rough unbounded surfaces in three dimensions, SIAM J. Appl. Math., 66 (2006), 1002-1026.  doi: 10.1137/050635262.

[11]

J. Chen, Z. Chen and G. Huang, Reverse time migration for extended obstacles: Acoustic waves, Inverse Problems, 29 (2013), 085005. doi: 10.1088/0266-5611/29/8/085005.

[12]

Z. Chen and G. Huang, Reverse time migration for reconstructing extended obstacles in the half space, Inverse Problems, 31 (2015), 055007. doi: 10.1088/0266-5611/31/5/055007.

[13]

F. J. Claerbout, Toward a unified theory of reflector mapping, Geophysics, 36 (1971), 467-481.  doi: 10.1190/1.1440185.

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, Springer-Verlag, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[15]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, SIAM, Philadelphia, 2013. doi: 10.1137/1.9781611973167.ch1.

[16]

Y. HeP. Li and J. Shen, A new spectral method for numerical solution of the unbounded rough surface scattering problem, J. Comput. Phys., 275 (2014), 608-625.  doi: 10.1016/j.jcp.2014.07.026.

[17]

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. doi: 10.1088/1361-6420/aaf3d6.

[18]

K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003. doi: 10.1088/0266-5611/28/2/025003.

[19]

A. Kirsch, Surface gradients and continuity properties for some integral operators in classical scattering theory, Math. Method Appl. Sci., 11 (1989), 789-804.  doi: 10.1002/mma.1670110605.

[20]

R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comput. Appl. Math., 61 (1995), 345-360.  doi: 10.1016/0377-0427(94)00073-7.

[21]

R. Kress and T. Tran, Inverse scattering for a locally perturbed half-plane, Inverse Problems, 16 (2000), 1541-1559.  doi: 10.1088/0266-5611/16/5/323.

[22]

J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Probl. Imaging, 7 (2013), 757-775.  doi: 10.3934/ipi.2013.7.757.

[23]

J. LiG. 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.

[24]

P. LiH. Wu and W. Zheng, Electromagnetic scattering by unbounded rough surfaces, SIAM J. Math. Anal., 43 (2011), 1205-1231.  doi: 10.1137/100806217.

[25]

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.

[26]

P. Li, J. Wang and L. Zhang, Inverse obstacle scattering for Maxwell's equations in an unbounded structure, Inverse Problems, 35 (2019), 095002. doi: 10.1088/1361-6420/ab1f1b.

[27]

X. LiuB. 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.

[28]

X. LiuB. 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.

[29]

Y. Lu and B. Zhang, Direct and inverse scattering problem by an unbounded rough interface with buried obstacles, preprint, arXiv: 1610.03515v1.

[30]

A. MeierT. ArensS. N. Chandler-Wilde and A. Kirsch, A Nystr$\ddot{o}$m method for a class of integral equationson the real line with applications to scattering by diffraction gratings and rough surfaces, J. Integral Equations Appl., 12 (2000), 281-321.  doi: 10.1216/jiea/1020282209.

[31]

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.

[32]

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.

[33]

L. ZhangF. Ma and J. Wang, Regularized conjugate gradient method with fast multipole acceleration for wave scattering from 1D fractal rough surface, Wave Motion, 50 (2013), 41-56.  doi: 10.1016/j.wavemoti.2012.06.005.

[34]

Y. Zhang and G. Zhang, One-step extrapolation method for reverse time migration, Geophysics, 74 (2009), A29–A33. doi: 10.1190/1.3123476.

Figure 1.  Reconstruction of an obstacle and the rough surface
Figure 2.  Reconstruction of an obstacle and the rough surface
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