# American Institute of Mathematical Sciences

October  2021, 15(5): 1269-1286. doi: 10.3934/ipi.2021037

## Inverse obstacle scattering for acoustic waves in the time domain

 School of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Heping Dong

Received  December 2020 Revised  February 2021 Published  October 2021 Early access  April 2021

This paper concerns an inverse acoustic scattering problem which is to determine the location and shape of a rigid obstacle from time domain scattered field data. An efficient convolution quadrature method combined with nonlinear integral equation method is proposed to solve the inverse problem. In particular, replacing the classic Fourier transform with the convolution quadrature method for time discretization, the boundary integral equations for the Helmholtz equation with complex wave numbers can be obtained to guarantee the numerically approximate causality property of the scattered field under some condition. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method.

Citation: Lu Zhao, Heping Dong, Fuming Ma. Inverse obstacle scattering for acoustic waves in the time domain. Inverse Problems & Imaging, 2021, 15 (5) : 1269-1286. doi: 10.3934/ipi.2021037
##### References:
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Chen and X. Wu, Long-time stability and convergence of the uniaxial perfectly matched layer method for time-domain acoustic scattering problems, SIAM J. Numer. Anal., 50 (2012), 2632-2655.  doi: 10.1137/110835268.  Google Scholar [12] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-3-030-30351-8.  Google Scholar [13] Y. Deng, J. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.  Google Scholar [14] Y. Deng, H. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar [15] H. Dong, J. Lai and P. Li, Inverse obstacle scattering for elastic waves with phased orphaseless far-field data, SIAM J. Imaging Sci., 12 (2019), 809-838.  doi: 10.1137/18M1227263.  Google Scholar [16] H. Dong, J. Lai and P. Li, An inverse acoustic-elastic interaction problem with phased or phaseless far-field data, Inverse Problems, 36 (2020), 035014, 36 pp. doi: 10.1088/1361-6420/ab693e.  Google Scholar [17] H. Dong, D. Zhang and Y. Guo, A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data, Inverse Probl. Imaging, 13 (2019), 177-195.  doi: 10.3934/ipi.2019010.  Google Scholar [18] Y. Gao and P. Li, Analysis of time-domain scattering by periodic structures, J. Differential Equations, 261 (2016), 5094-5118.  doi: 10.1016/j.jde.2016.07.020.  Google Scholar [19] Y. Gao and P. Li, Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math. Models Methods Appl. Sci., 27 (2017), 1843-1870.  doi: 10.1142/S0218202517500336.  Google Scholar [20] Y. Gao, P. Li and Y. 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Marmorat, An improved time domain linear sampling method for Robin and Neumann obstacles, Appl. Anal., 93 (2014), 369-390.  doi: 10.1080/00036811.2013.772583.  Google Scholar [26] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: III. Sound-soft obstacle and bistatic data, Inverse Problems, 29 (2013), 085013, 35 pp. doi: 10.1088/0266-5611/26/5/055010.  Google Scholar [27] O. Ivanyshyn and T. Johansson, Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle, J. Integral Equations Appl., 19 (2007), 289-308.  doi: 10.1216/jiea/1190905488.  Google Scholar [28] T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern, IMA J. Appl. Math., 72 (2007), 96-112.  doi: 10.1093/imamat/hxl026.  Google Scholar [29] A. Kirsch and S. Ritter, The Nyström method for solving a class of singular integral equations and applications in 3D late elasticity, Math. Method. Appl. Sci., 22 (1999), 177-197.  doi: 10.1002/(SICI)1099-1476(19990125)22:2<177::AID-MMA36>3.0.CO;2-F.  Google Scholar [30] R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91–S104. doi: 10.1088/0266-5611/19/6/056.  Google Scholar [31] J. Li, H. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746.  doi: 10.1137/130907690.  Google Scholar [32] J. Li, H. Liu, Y. Sun and Q. Wang, Ground detection by a single electromagnetic measurement, J. Comput. Phys., 257, (2014), 554–571. Google Scholar [33] J. Li, H. Liu, H. Sun and J. Zou, Imaging obstacles by hypersingular point sources, Inverse Probl. Imaging, 7 (2013), 545-563.  doi: 10.3934/ipi.2013.7.545.  Google Scholar [34] P. Li and L. Zhang, Analysis of transient acoustic scattering by an elastic obstacle, Commun. Math. Sci., 17 (2019), 1671-1698.  doi: 10.4310/CMS.2019.v17.n6.a8.  Google Scholar [35] Y. Liu, Y. Guo and J. Sun, A deterministic-statistical approach to reconstruct moving sources using sparse partial data, 2021, arXiv: 2101.01290v2. Google Scholar [36] H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 105005, 10 pp. doi: 10.1088/0266-5611/31/10/105005.  Google Scholar [37] D. R. Luke and R. Potthast, The point source method for inverse scattering in the time domain, Math. Methods Appl. Sci., 29 (2006), 1501-1521.  doi: 10.1002/mma.738.  Google Scholar [38] F.-J. Sayas, Retarded Potentials and Time Domain Boundary Integral Equations, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-26645-9.  Google Scholar [39] X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving acoustic emitter, Inverse Problems, 33 (2017), 105009, 19 pp. doi: 10.1088/1361-6420/aa873f.  Google Scholar [40] X. Wang, Y. Guo, J. Li and H. Liu, Two gesture-computing approaches by using electromagnetic waves, Inverse Probl. Imaging, 13 (2019), 879-901.  doi: 10.3934/ipi.2019040.  Google Scholar [41] X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33 (2017), 035001, 18 pp. doi: 10.1088/1361-6420/aa573c.  Google Scholar [42] D. Zhang, Y. Guo, J. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Commun. Comput. Phys., 25 (2019), 1328–1356. arXiv: 1801.05584v1. doi: 10.4208/cicp.oa-2018-0020.  Google Scholar [43] L. Zhao, H. Dong and F. Ma, Time-domain analysis of forward obstacle scattering for elastic wave, Discrete Contin. Dyn. Syst. Ser. B, preprint. doi: 10.3934/dcdsb.2020276.  Google Scholar

show all references

##### References:
 [1] L. Banjai and S. Sauter, Rapid solution of the wave equation in unbounded domains, SIAM J. Numer. Anal., 47 (2008), 227-249.  doi: 10.1137/070690754.  Google Scholar [2] G. Bao, Y. Gao and P. Li, Time domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 229 (2018), 835-884.  doi: 10.1007/s00205-018-1228-2.  Google Scholar [3] G. Bao, B. Hu, P. Li and J. Wang, Analysis of time-domain Maxwell's equations in biperiodic structures, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 259-286.  doi: 10.3934/dcdsb.2019181.  Google Scholar [4] C. Burkard and R. Potthast, A time-domain probe method for three-dimensional rough surface reconstructions, Inverse Probl. Imaging, 3 (2009), 259-274.  doi: 10.3934/ipi.2009.3.259.  Google Scholar [5] F. Cakoni, H. Haddar and A. Lechleiter, On the factorization method for a far field inverse scattering problem in the time domain, SIAM J. Math. Anal., 51 (2019), 854-872.  doi: 10.1137/18M1214809.  Google Scholar [6] Z. Chen, Convergence of the time-domain perfectly matched layer method for acoustic scattering problems, Int. J. Numer. Anal. Model., 6 (2009), 124-146.   Google Scholar [7] Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001, 17 pp. doi: 10.1088/0266-5611/26/8/085001.  Google Scholar [8] B. Chen, Y. Guo, F. Ma and Y. Sun, Numerical schemes to reconstruct three dimensional time-dependent point sources of acoustic waves, Inverse Problems, 36 (2020), 075009, 21 pp. doi: 10.1088/1361-6420/ab8f85.  Google Scholar [9] B. Chen, F. Ma and Y. Guo, Time domain scattering and inverse scattering problems in a locally perturbed half-plane, Appl. Anal., 96 (2017), 1303-1325.  doi: 10.1080/00036811.2016.1188288.  Google Scholar [10] Z. Chen and J.-C. N$\acute{e}$d$\acute{e}$lec, On Maxwell equations with the transparent boundary condition, J. Comput. Math., 26 (2008), 284-296.   Google Scholar [11] Z. Chen and X. Wu, Long-time stability and convergence of the uniaxial perfectly matched layer method for time-domain acoustic scattering problems, SIAM J. Numer. Anal., 50 (2012), 2632-2655.  doi: 10.1137/110835268.  Google Scholar [12] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-3-030-30351-8.  Google Scholar [13] Y. Deng, J. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.  Google Scholar [14] Y. Deng, H. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar [15] H. Dong, J. Lai and P. Li, Inverse obstacle scattering for elastic waves with phased orphaseless far-field data, SIAM J. Imaging Sci., 12 (2019), 809-838.  doi: 10.1137/18M1227263.  Google Scholar [16] H. Dong, J. Lai and P. Li, An inverse acoustic-elastic interaction problem with phased or phaseless far-field data, Inverse Problems, 36 (2020), 035014, 36 pp. doi: 10.1088/1361-6420/ab693e.  Google Scholar [17] H. Dong, D. Zhang and Y. Guo, A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data, Inverse Probl. Imaging, 13 (2019), 177-195.  doi: 10.3934/ipi.2019010.  Google Scholar [18] Y. Gao and P. Li, Analysis of time-domain scattering by periodic structures, J. Differential Equations, 261 (2016), 5094-5118.  doi: 10.1016/j.jde.2016.07.020.  Google Scholar [19] Y. Gao and P. Li, Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math. Models Methods Appl. Sci., 27 (2017), 1843-1870.  doi: 10.1142/S0218202517500336.  Google Scholar [20] Y. Gao, P. Li and Y. Li, Analysis of time-domain elastic scattering by an unbounded structure, Math. Methods Appl. Sci., 41 (2018), 7032-7054.  doi: 10.1002/mma.5214.  Google Scholar [21] Y. Gao, P. Li and B. Zhang, Analysis of transient acoustic-elastic interaction in an unbounded structure, SIAM J. Math. Anal., 49 (2017), 3951-3972.  doi: 10.1137/16M1090326.  Google Scholar [22] Y. Guo, D. Hömberg, G. Hu, J. Li and H. Liu, A time domain sampling method for inverse acoustic scattering problems, J. Comput. Phys., 314 (2016), 647-660.  doi: 10.1016/j.jcp.2016.03.046.  Google Scholar [23] Y. Guo, P. Monk and D. Colton, Toward a time domain approach to the linear sampling method, Inverse Problems, 29 (2013), 095016, 17 pp. doi: 10.1088/0266-5611/29/9/095016.  Google Scholar [24] Y. Guo, P. Monk and D. Colton, The linear sampling method for sparse small aperture data, Appl. Anal., 95 (2016), 1599-1615.  doi: 10.1080/00036811.2015.1065317.  Google Scholar [25] H. Haddar, A. Lechleiter and S. Marmorat, An improved time domain linear sampling method for Robin and Neumann obstacles, Appl. Anal., 93 (2014), 369-390.  doi: 10.1080/00036811.2013.772583.  Google Scholar [26] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: III. Sound-soft obstacle and bistatic data, Inverse Problems, 29 (2013), 085013, 35 pp. doi: 10.1088/0266-5611/26/5/055010.  Google Scholar [27] O. Ivanyshyn and T. Johansson, Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle, J. Integral Equations Appl., 19 (2007), 289-308.  doi: 10.1216/jiea/1190905488.  Google Scholar [28] T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern, IMA J. Appl. Math., 72 (2007), 96-112.  doi: 10.1093/imamat/hxl026.  Google Scholar [29] A. Kirsch and S. Ritter, The Nyström method for solving a class of singular integral equations and applications in 3D late elasticity, Math. Method. Appl. Sci., 22 (1999), 177-197.  doi: 10.1002/(SICI)1099-1476(19990125)22:2<177::AID-MMA36>3.0.CO;2-F.  Google Scholar [30] R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91–S104. doi: 10.1088/0266-5611/19/6/056.  Google Scholar [31] J. Li, H. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746.  doi: 10.1137/130907690.  Google Scholar [32] J. Li, H. Liu, Y. Sun and Q. Wang, Ground detection by a single electromagnetic measurement, J. Comput. Phys., 257, (2014), 554–571. Google Scholar [33] J. Li, H. Liu, H. Sun and J. Zou, Imaging obstacles by hypersingular point sources, Inverse Probl. Imaging, 7 (2013), 545-563.  doi: 10.3934/ipi.2013.7.545.  Google Scholar [34] P. Li and L. Zhang, Analysis of transient acoustic scattering by an elastic obstacle, Commun. Math. Sci., 17 (2019), 1671-1698.  doi: 10.4310/CMS.2019.v17.n6.a8.  Google Scholar [35] Y. Liu, Y. Guo and J. Sun, A deterministic-statistical approach to reconstruct moving sources using sparse partial data, 2021, arXiv: 2101.01290v2. Google Scholar [36] H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 105005, 10 pp. doi: 10.1088/0266-5611/31/10/105005.  Google Scholar [37] D. R. Luke and R. Potthast, The point source method for inverse scattering in the time domain, Math. Methods Appl. Sci., 29 (2006), 1501-1521.  doi: 10.1002/mma.738.  Google Scholar [38] F.-J. Sayas, Retarded Potentials and Time Domain Boundary Integral Equations, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-26645-9.  Google Scholar [39] X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving acoustic emitter, Inverse Problems, 33 (2017), 105009, 19 pp. doi: 10.1088/1361-6420/aa873f.  Google Scholar [40] X. Wang, Y. Guo, J. Li and H. Liu, Two gesture-computing approaches by using electromagnetic waves, Inverse Probl. Imaging, 13 (2019), 879-901.  doi: 10.3934/ipi.2019040.  Google Scholar [41] X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33 (2017), 035001, 18 pp. doi: 10.1088/1361-6420/aa573c.  Google Scholar [42] D. Zhang, Y. Guo, J. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Commun. Comput. Phys., 25 (2019), 1328–1356. arXiv: 1801.05584v1. doi: 10.4208/cicp.oa-2018-0020.  Google Scholar [43] L. Zhao, H. Dong and F. Ma, Time-domain analysis of forward obstacle scattering for elastic wave, Discrete Contin. Dyn. Syst. Ser. B, preprint. doi: 10.3934/dcdsb.2020276.  Google Scholar
Reconstructions of an apple-shaped obstacle at different levels of noise, the radius of the initial guess is $r_0 = 0.3$, the position of the initial guess is $(c_1^{(0)},c_2^{(0)}) = (-0.7,0.3)$ and the launch position of the incident wave is $(1.5\cos\theta,1.5\sin\theta )$, $\theta = 7\pi/10$
Reconstructions of a peanut-shaped obstacle at different levels of noise, the radius of the initial guess is $r_0 = 0.3$, the position of the initial guess is $(c_1^{(0)},c_2^{(0)}) = (0.75,-0.55)$ and the launch position of the incident wave is $(1.5\cos\theta,1.5\sin\theta )$, $\theta = 7\pi/20$
Reconstructions of an apple-shaped obstacle at different positions of the initial guess $(c_1^{(0)},c_2^{(0)}) = (-0.75.-0.4)$ and $(c_1^{(0)},c_2^{(0)}) = (1,0.2)$, the radius of the initial guess is $r_0 = 0.3$, the noise level is 1$\%$ and the launch position of the incident wave is $(1.5\cos\theta,1.5\sin\theta )$, $\theta = \pi/2$
Reconstructions of a peanut-shaped obstacle at different positions of the initial guess $(c_1^{(0)},c_2^{(0)}) = (-0.5,-0.65)$ and $(c_1^{(0)},c_2^{(0)}) = (0.46,0.7)$, the radius of the initial guess is $r_0 = 0.3$, the noise level is 1$\%$ and the launch position of the incident wave is $(1.5\cos\theta,1.5\sin\theta )$, $\theta = 13\pi/20$
Reconstructions of an apple-shaped obstacle at different launch positions of the incident wave $(1.5\cos\theta,1.5\sin\theta)$, the values of $\theta$ are given in figure. The radius of the initial guess is $r_0 = 0.3$, the position of the initial guess is $(c_1^{(0)},c_2^{(0)}) = (-0.9,-0.2)$ and the noise level is 1$\%$
Reconstructions of a peanut-shaped obstacle at different launch positions of the incident wave $(1.5\cos\theta,1.5\sin\theta)$, the values of $\theta$ are given in figure. The radius of the initial guess is $r_0 = 0.3$, the position of the initial guess is $(c_1^{(0)},c_2^{(0)}) = (0.7,0.3)$ and the noise level is 1$\%$
Reconstructions of an apple-shaped obstacle at different numbers of launch position or different levels of noise, the radius of the initial guess is $r_0 = 0.3$, the position of the initial guess is $(c_1^{(0)},c_2^{(0)}) = (0.9,-0.2)$ and the launch position of the incident wave is $(1.5\cos\theta,1.5\sin\theta )$, $\theta_1 = 17\pi/20$ and $\theta_2 = 37\pi/20$
Reconstructions of a peanut-shaped obstacle at different numbers of launch position or different levles of noise, the radius of the initial guess is $r_0 = 0.3$, the position of the initial guess is $(c_1^{(0)},c_2^{(0)}) = (-0.7,-0.4)$ and the launch position of the incident wave is $(1.5\cos\theta,1.5\sin\theta )$, $\theta_1 = 11\pi/20$ and $\theta_2 = 31\pi/20$
Parametrization of the exact boundary curves
 Type Parametrization apple-shaped $p_D(\theta)= \frac{0.55(1+0.9\cos{\theta}+0.1\sin{2\theta})}{1+0.75\cos{\theta}}(\cos{\theta},\sin{\theta}), \quad \theta\in [0,2\pi]$ peanut-shaped $p_D(t)=0.5\sqrt{0.25\cos^2{\theta}+\sin^2{\theta}}(\cos{\theta},\sin{\theta}), \quad \theta\in[0,2\pi]$
 Type Parametrization apple-shaped $p_D(\theta)= \frac{0.55(1+0.9\cos{\theta}+0.1\sin{2\theta})}{1+0.75\cos{\theta}}(\cos{\theta},\sin{\theta}), \quad \theta\in [0,2\pi]$ peanut-shaped $p_D(t)=0.5\sqrt{0.25\cos^2{\theta}+\sin^2{\theta}}(\cos{\theta},\sin{\theta}), \quad \theta\in[0,2\pi]$
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