December  2019, 13(6): 1349-1365. doi: 10.3934/ipi.2019059

Scattering by impenetrable scatterer in a stratified ocean waveguide

Shanghai Key Laboratory of Financial Information Technology, Institute of Scientific Computation and Financial Data Analysis, School of Mathematics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China

* Corresponding author: Keji Liu

Received  March 2019 Revised  June 2019 Published  October 2019

In this work, the direct and inverse scattering problems of wave impenetrable scatterers in the three-layered ocean waveguide are under investigation. We have established the well-posedness of forward problem and proposed a novel direct sampling method for the inverse problem. The direct recovery approach only applies the matrix-vector operations to approximate the wave impenetrable obstacle from the received partial data. The method is capable of reconstructing the objects of different shapes and locations, computationally quite cheap and easy to carry out. The theoretical analysis and the novel direct recovery algorithm are expected to have wide applications in the direct and inverse scattering problems of submerged acoustics.

Citation: Keji Liu. Scattering by impenetrable scatterer in a stratified ocean waveguide. Inverse Problems & Imaging, 2019, 13 (6) : 1349-1365. doi: 10.3934/ipi.2019059
References:
[1]

D. S. Ahluwalia and J. B. Keller, Exact and asymptotic representations of the sound field in a stratified ocean, in Wave Propagation and Underwater Acoustics, Lecture Notes in Phys, Springer, Berlin, 70 (1977), 14–85.  Google Scholar

[2]

H. AmmariE. Iakovleva and H. Kang, Reconstruction of a small inclusion in a two-dimensional open waveguide, SIAM J. Appl. Math., 65 (2005), 2107-2127.  doi: 10.1137/040615389.  Google Scholar

[3]

H. AmmariH. KangE. KimK. Louati and M. S. Vogelius, A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements, Numer. Math., 108 (2008), 501-528.  doi: 10.1007/s00211-007-0130-x.  Google Scholar

[4]

H. Ammari, J. Garnier, W. J. Jing, H. Kang, M. Lim, K. Solna and H. Wang, Mathematical and Statistical Methods for Multistatic Imaging, Lecture Notes in Mathematics, 2098. Springer, Cham, 2013. doi: 10.1007/978-3-319-02585-8.  Google Scholar

[5]

L. M. Brekhouvskikh, Waves in Layered Media, Applied Mathematics and Mechanics, Academic Press, New York-London, 1960.  Google Scholar

[6]

J. L. Buchanan, R. P. Gilbert, A. Wirgin and Y. Z. Xu, Marine Acoustics: Direct and Inverse Problems, SIAM, Philadelphia, PA, 2004. doi: 10.1137/1.9780898717983.  Google Scholar

[7]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{rd}$ edition, Applied Mathematical Sciences, 93. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[8]

M. DingJ. L. LiK. J. Liu and J. Q. Yang, Imaging of local rough surfaces by the linear sampling method with near-field data, SIAM J. Imag. Sci., 10 (2017), 1579-1602.  doi: 10.1137/16M1097997.  Google Scholar

[9]

R. P. GilbertM. Wirby and Y. Z. Xu, Determination of a buried object in a two-layered shallow ocean, J. Comput. Acoust., 9 (2001), 1025-1037.  doi: 10.1142/S0218396X0100108X.  Google Scholar

[10]

R. P. Gilbert and Y. Z. Xu, Acoustic imaging in a shallow ocean with a thin ice cap, Inverse Problems, 16 (2000), 1799-1811.  doi: 10.1088/0266-5611/16/6/313.  Google Scholar

[11]

R. P. Gilbert and Y. Z. Xu, The propagation problem and far-field pattern in a stratified finite-depth ocean, Math. Methods Appl. Sci., 12 (1990), 199-208.  doi: 10.1002/mma.1670120303.  Google Scholar

[12]

K. J. Liu, Two effective post-filtering strategies for improving direct sampling methods, Appl. Anal., 96 (2017), 502-515.  doi: 10.1080/00036811.2016.1204441.  Google Scholar

[13]

K. J. Liu, A simple method for detecting scatterers in a stratified ocean waveguide, Comp. Math. Appl., 76 (2018), 1791-1802.  doi: 10.1016/j.camwa.2018.07.030.  Google Scholar

[14]

K. J. Liu, Direct imaging of inhomogeneous obstacles in a three-layered ocean waveguide, Commun. Comput. Phys., 26 (2019), 700-718.  doi: 10.4208/cicp.OA-2018-0127.  Google Scholar

[15]

K. J. Liu, Near-field imaging of inhomogeneities in a stratified ocean waveguide, J. Comput. Phys., 398 (2019), 108901.  doi: 10.1016/j.jcp.2019.108901.  Google Scholar

[16]

K. J. Liu and J. Zou, A multilevel sampling algorithm for locating inhomogeneous media, Inverse Problems, 29 (2013), 095003, 19 pp. doi: 10.1088/0266-5611/29/9/095003.  Google Scholar

[17]

K. J. LiuY. Z. Xu and J. Zou, Imaging wave-penetrable objects in a finite depth ocean, Appl. Math. Comput., 235 (2014), 364-376.  doi: 10.1016/j.amc.2014.02.048.  Google Scholar

[18]

K. J. LiuY. Z. Xu and J. Zou, A multilevel sampling method for detecting sources in a stratified ocean waveguide, J. Comput. Appl. Math., 309 (2017), 95-110.  doi: 10.1016/j.cam.2016.06.039.  Google Scholar

[19]

K. J. LiuY. Z. Xu and J. Zou, Direct recovery of wave-penetrable scatterers in a stratified ocean waveguide, J. Comput. Appl. Math., 338 (2018), 239-257.  doi: 10.1016/j.cam.2018.01.031.  Google Scholar

[20]

X. Lutron, An Introduction to Underwater Acoustics, 1$^{st}$ edition, Springer Science & Business Media, 2002. Google Scholar

[21]

Y. Z. XuC. Mawata and W. Lin, Generalized dual space indicator method for underwater imaging, Inverse Problems, 16 (2000), 1761-1776.  doi: 10.1088/0266-5611/16/6/311.  Google Scholar

[22]

Y. Z. Xu, The propagation solutions and far-field patterns for acoustic harmonic waves in a finite depth ocean, Appl. Anal., 35 (1990), 129-151.  doi: 10.1080/00036819008839908.  Google Scholar

[23]

J. Q. Yang and K. J. Liu, Detecting buried wave-penetrable scatterers in a two-layered medium, J. Comput. Appl. Math., 330 (2018), 318-329.  doi: 10.1016/j.cam.2017.08.021.  Google Scholar

show all references

References:
[1]

D. S. Ahluwalia and J. B. Keller, Exact and asymptotic representations of the sound field in a stratified ocean, in Wave Propagation and Underwater Acoustics, Lecture Notes in Phys, Springer, Berlin, 70 (1977), 14–85.  Google Scholar

[2]

H. AmmariE. Iakovleva and H. Kang, Reconstruction of a small inclusion in a two-dimensional open waveguide, SIAM J. Appl. Math., 65 (2005), 2107-2127.  doi: 10.1137/040615389.  Google Scholar

[3]

H. AmmariH. KangE. KimK. Louati and M. S. Vogelius, A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements, Numer. Math., 108 (2008), 501-528.  doi: 10.1007/s00211-007-0130-x.  Google Scholar

[4]

H. Ammari, J. Garnier, W. J. Jing, H. Kang, M. Lim, K. Solna and H. Wang, Mathematical and Statistical Methods for Multistatic Imaging, Lecture Notes in Mathematics, 2098. Springer, Cham, 2013. doi: 10.1007/978-3-319-02585-8.  Google Scholar

[5]

L. M. Brekhouvskikh, Waves in Layered Media, Applied Mathematics and Mechanics, Academic Press, New York-London, 1960.  Google Scholar

[6]

J. L. Buchanan, R. P. Gilbert, A. Wirgin and Y. Z. Xu, Marine Acoustics: Direct and Inverse Problems, SIAM, Philadelphia, PA, 2004. doi: 10.1137/1.9780898717983.  Google Scholar

[7]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{rd}$ edition, Applied Mathematical Sciences, 93. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[8]

M. DingJ. L. LiK. J. Liu and J. Q. Yang, Imaging of local rough surfaces by the linear sampling method with near-field data, SIAM J. Imag. Sci., 10 (2017), 1579-1602.  doi: 10.1137/16M1097997.  Google Scholar

[9]

R. P. GilbertM. Wirby and Y. Z. Xu, Determination of a buried object in a two-layered shallow ocean, J. Comput. Acoust., 9 (2001), 1025-1037.  doi: 10.1142/S0218396X0100108X.  Google Scholar

[10]

R. P. Gilbert and Y. Z. Xu, Acoustic imaging in a shallow ocean with a thin ice cap, Inverse Problems, 16 (2000), 1799-1811.  doi: 10.1088/0266-5611/16/6/313.  Google Scholar

[11]

R. P. Gilbert and Y. Z. Xu, The propagation problem and far-field pattern in a stratified finite-depth ocean, Math. Methods Appl. Sci., 12 (1990), 199-208.  doi: 10.1002/mma.1670120303.  Google Scholar

[12]

K. J. Liu, Two effective post-filtering strategies for improving direct sampling methods, Appl. Anal., 96 (2017), 502-515.  doi: 10.1080/00036811.2016.1204441.  Google Scholar

[13]

K. J. Liu, A simple method for detecting scatterers in a stratified ocean waveguide, Comp. Math. Appl., 76 (2018), 1791-1802.  doi: 10.1016/j.camwa.2018.07.030.  Google Scholar

[14]

K. J. Liu, Direct imaging of inhomogeneous obstacles in a three-layered ocean waveguide, Commun. Comput. Phys., 26 (2019), 700-718.  doi: 10.4208/cicp.OA-2018-0127.  Google Scholar

[15]

K. J. Liu, Near-field imaging of inhomogeneities in a stratified ocean waveguide, J. Comput. Phys., 398 (2019), 108901.  doi: 10.1016/j.jcp.2019.108901.  Google Scholar

[16]

K. J. Liu and J. Zou, A multilevel sampling algorithm for locating inhomogeneous media, Inverse Problems, 29 (2013), 095003, 19 pp. doi: 10.1088/0266-5611/29/9/095003.  Google Scholar

[17]

K. J. LiuY. Z. Xu and J. Zou, Imaging wave-penetrable objects in a finite depth ocean, Appl. Math. Comput., 235 (2014), 364-376.  doi: 10.1016/j.amc.2014.02.048.  Google Scholar

[18]

K. J. LiuY. Z. Xu and J. Zou, A multilevel sampling method for detecting sources in a stratified ocean waveguide, J. Comput. Appl. Math., 309 (2017), 95-110.  doi: 10.1016/j.cam.2016.06.039.  Google Scholar

[19]

K. J. LiuY. Z. Xu and J. Zou, Direct recovery of wave-penetrable scatterers in a stratified ocean waveguide, J. Comput. Appl. Math., 338 (2018), 239-257.  doi: 10.1016/j.cam.2018.01.031.  Google Scholar

[20]

X. Lutron, An Introduction to Underwater Acoustics, 1$^{st}$ edition, Springer Science & Business Media, 2002. Google Scholar

[21]

Y. Z. XuC. Mawata and W. Lin, Generalized dual space indicator method for underwater imaging, Inverse Problems, 16 (2000), 1761-1776.  doi: 10.1088/0266-5611/16/6/311.  Google Scholar

[22]

Y. Z. Xu, The propagation solutions and far-field patterns for acoustic harmonic waves in a finite depth ocean, Appl. Anal., 35 (1990), 129-151.  doi: 10.1080/00036819008839908.  Google Scholar

[23]

J. Q. Yang and K. J. Liu, Detecting buried wave-penetrable scatterers in a two-layered medium, J. Comput. Appl. Math., 330 (2018), 318-329.  doi: 10.1016/j.cam.2017.08.021.  Google Scholar

Figure 1.  The demonstrations of (a) one-layered waveguide and (b) two-layered waveguide
Figure 2.  The geometrical demonstration of stratified ocean waveguide
Figure 3.  The geometrical demonstration of the impenetrable obstacle $ D $ buried in the cylindrical region $ \Omega_R $
Figure 4.  The demonstration of sources and receivers for direct sampling method
Figure 5.  The geometrical demonstrations of (a) sources and (b) receivers; the numerical reconstructions by the NDSM under (c) 10% random noise and (d) 15% random noise in the Example 1
Figure 6.  The numerical reconstructions by the NDSM under (a) 10% random noise and (b) 15% random noise in the Example 2
Figure 7.  The numerical reconstructions by the NDSM under (a) 10% random noise and (b) 15% random noise in the Example 3
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