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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

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
[1]

Shixu Meng. A sampling type method in an electromagnetic waveguide. Inverse Problems and Imaging, 2021, 15 (4) : 745-762. doi: 10.3934/ipi.2021012

[2]

Isaac Harris, Dinh-Liem Nguyen, Thi-Phong Nguyen. Direct sampling methods for isotropic and anisotropic scatterers with point source measurements. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022015

[3]

Bo You. Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1579-1604. doi: 10.3934/dcds.2020332

[4]

Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547

[5]

Akram Ben Aissa. Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 983-993. doi: 10.3934/dcdss.2021106

[6]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems and Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[7]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems and Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027

[8]

Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems and Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709

[9]

Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259

[10]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[11]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036

[12]

Elissar Nasreddine. Well-posedness for a model of individual clustering. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2647-2668. doi: 10.3934/dcdsb.2013.18.2647

[13]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241

[14]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1

[15]

David M. Ambrose, Jerry L. Bona, David P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1113-1137. doi: 10.3934/dcdsb.2012.17.1113

[16]

A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469

[17]

Mircea Sofonea, Yi-bin Xiao. Tykhonov well-posedness of a viscoplastic contact problem. Evolution Equations and Control Theory, 2020, 9 (4) : 1167-1185. doi: 10.3934/eect.2020048

[18]

Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527

[19]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations and Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15

[20]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (247)
  • HTML views (142)
  • Cited by (0)

Other articles
by authors

[Back to Top]