Direct imaging of inhomogeneities in a 3D shallow ocean waveguide with an elastic seabed

Keji Liu

doi:10.3934/ipi.2024027

Article Contents

2025, Volume 19, Issue 1: 109-132. Doi: 10.3934/ipi.2024027

This issue Previous Article Denoising of sphere- and SO(3)-valued data by relaxed tikhonov regularization Next Article Stability estimates for the inverse source problem with passive measurements

Direct imaging of inhomogeneities in a 3D shallow ocean waveguide with an elastic seabed

Keji Liu

School of Mathematics, Dishui Lake Advanced Finance Institute, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China

Received: December 2023

Revised: May 2024

Early access: June 2024

Published: February 2025

The work of K. Liu was supported by [the NNSF of China under grant No. 12071275].

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Abstract

We were concerned with identifying scattering inhomogeneities in a three-dimensional shallow ocean waveguide with an elastic seabed and a pressure-release (or perfectly rigid) surface, which was motivated by applications in ocean acoustics. An extended direct imaging method (DIM) was proposed to identify the marine point sources and medium objects from the far-field data, and the key component of the DIM was an imaging functional whose indicator property was quantitatively characterized. The DIM can generate reliable initial estimates of submerged inhomogeneities, which advanced inversion methods can then be utilized to accurately determine their physical properties. As exhibited in the numerical experiments, the DIM was computationally efficient, noise-tolerant, and can identified multiple sources and scatterers of varying shapes and locations using a few observation datasets.

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Mathematics Subject Classification: Primary: 35R30, 41A27; Secondary: 76Q05.

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Figure 1. The demonstration of the 3D shallow ocean waveguide with an elastic seabed

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Figure 2. The asymptotic behavior for the wave propagation of a point source at $ (50, 50, 20) $ in a 3D shallow ocean waveguide with an elastic seabed and a pressure-release surface : (a) a slice of contour, (b) the corresponding 3D display

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Figure 3. The asymptotic behavior for the wave propagation of a point source at $ (50, 50, 25) $ in a 3D shallow ocean waveguide with an elastic seabed and a perfectly rigid surface: (a) a slice of contour, (b) the corresponding 3D display

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Figure 4. The illustration of the numerical settings

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Figure 5. Experiment 1. (a) The demonstration of the submerged point sources $ {\bf x}_s $ and the sampling region $ \mathcal{D} $; (b) the reconstruction of the sources

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Figure 6. Experiment 2. (a) The demonstration of the submerged point sources $ {\bf x}_s $ and the sampling region $ \mathcal{D} $; (b) the reconstruction of the sources

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Figure 7. Experiment 3. (a) The demonstration of the submerged medium scatterers $ \Omega $ and the sampling region $ \mathcal{D} $; (b) the reconstruction of the scatterers

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Figure 8. Experiment 4. (a) The demonstration of the submerged medium scatterers $ \Omega $ and the sampling region $ \mathcal{D} $; (b) the reconstruction of the scatterers

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Figure 9. Experiment 5. (a) The demonstration of the submerged point sources $ {\bf x}_s $, medium scatterers $ \Omega $, and the sampling region $ \mathcal{D} $; (b) the reconstruction of the sources; (c) the reconstruction of the scatterer; (d) the identification of all the inhomogeneities

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References

[1]	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, $10^{th}$ edition, New York, 1972.
[2]	J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1976.
[3]	D. Ahluwalia and J. Keller, Polynomials in finite geometries, Exact and Asymptotic Representations of the Sound Field in a Stratified Ocean, in Wave Propagation and Underwater Acoustics, Lecture Notes in Phys., 70, Springer, Berlin 14-85, 1977.
[4]	H. Ammari, Y. Chow and K. Liu, Optimal mesh size for inverse medium scattering problems, SIAM J. Num. Anal., 58 (2020), 733-756.
[5]	H. Ammari, J. Garnier and W. Jing, Passive array correlation-based imaging in a random waveguide, SIAM Multi. Model. simul., 11 (2013), 656-681.
[6]	H. Ammari, J. Garnier, W. 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.
[7]	H. Ammari, E. Iakovleva and H. Kang, Reconstruction of a small inclusion in a 2D open waveguide, SIAM J. Appl. Math., 65 (2005), 2107-2127.
[8]	T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM J. Appl. Math., 71 (2011), 753-772.
[9]	L. M. Brekhouvskikh, Waves in Layered Media, Academic Press, 1960.
[10]	J. Buchanan, R. Gilbert, A. Wirgin and Y. Xu, Marine Acoustics: Direct and Inverse Problems, SIAM, Philadelphia, 2004.
[11]	M. Cheney, The linear sampling method and the MUSIC algorithm, Inv. Prob., 17 (2001), 591-595.
[12]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inv. Prob., 12 (1996), 383-393.
[13]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2013.
[14]	Y. Deng, X. Fang and H. Liu, Gradient estimates for electric fields with multi-scale inclusions in the quasi-static regime, SIAM Multi. Model. simul., 20 (2022), 641-656.
[15]	R. Gilbert, M. Wirby and Y. Xu, Determination of a buried object in a two-layered shallow ocean, J. Comput. Acoust., 9 (2001), 1025-1037.
[16]	R. Gilbert and Y. Xu, The propagation problem and far-field pattern in a stratified finite-depth ocean, Math. Methods Appl. Sci., 12 (1990), 199-208.
[17]	R. Gilbert and Y. Xu, Acoustic imaging in a shallow ocean with a thin ice cap, Inv. Prob., 16 (2000), 1799-1811.
[18]	K. Ito, B. Jin and J. Zou, A direct sampling method for inverse electromagnetic medium scattering, Inv. Prob., 29 (2013), 095018.
[19]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, New York, 2008.
[20]	P. Li, X. Yao and Y. Zhao, Direct and inverse problems for the Schrödinger operator in a three-dimensional planar waveguide, preprint, arXiv: 2106.10597, 2021.
[21]	K. Liu, Near-field imaging of inhomogeneities in a stratified ocean waveguide, J. Comput. Phys., 398 (2019), 108901.
[22]	K. Liu, Fast imaging of sources and scatterers in a stratified ocean waveguide, SIAM J. Imag. Sci., 14 (2021), 224-245.
[23]	K. Liu and D. Xu, Simultaneous reconstruction of sources and scatterers in a three-dimensional stratified ocean waveguide, Math. Meth. Appl. Sci., 2022. doi: 10.1002/mma.8727.
[24]	K. Liu and J. Zou, A multilevel sampling algorithm for locating inhomogeneous media, Inv. Prob., 29 (2013), 095003 (19pp).
[25]	X. Liu and B. Zhang, A uniqueness result for the inverse electromagnetic scattering problem in a two-layered medium, Inv. Prob., 26 (2010), 105007.
[26]	X. Lutron, An Introduction to Underwater Acoustics, Springer Science & Business Media, 2002.
[27]	S. Meng, A sampling type method in an electromagnetic waveguide, Inv. Prob. Imag., 15 (2021), 745-762.
[28]	R. Potthast, A survey on sampling method and probe methods for inverse problems, Inv. Prob., 22 (2006), R1-R47.
[29]	Y. Xu, C. Mawata and W. Lin, Generalized dual space indicator method for underwater imaging, Inv. Prob., 16 (2000), 1761-1776.
[30]	J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inv. Prob., 28 (2012), 035004.
[31]	B. Zhang and S. N. Chandler-Wilde, Acoustic scattering by an inhomogeneous layer on a rigid plate, SIAM J. Appl. Math., 58 (1998), 1931-1950.