A non-iterative sampling method for inverse elastic wave scattering by rough surfaces

Zhu Tielei; Yang Jiaqing

doi:10.3934/ipi.2022009

Article Contents

2022, Volume 16, Issue 4: 997-1017. Doi: 10.3934/ipi.2022009

This issue Previous Article A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise Next Article Quasiconformal model with CNN features for large deformation image registration

A non-iterative sampling method for inverse elastic wave scattering by rough surfaces

Zhu Tielei and
Yang Jiaqing^,

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China

^*Corresponding author: Jiaqing Yang
^*Corresponding author: Jiaqing Yang

Received: May 31, 2021

Revised: December 31, 2021

Early access: March 2022

Published: August 2022

The research of JY is supported by NNSF of China Grants No. 12122114 and 11771349

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Abstract

Consider the two-dimensional inverse elastic wave scattering by an infinite rough surface with a Dirichlet boundary condition. A non-iterative sampling method is proposed for detecting the rough surface by taking elastic field measurements on a bounded line segment above the surface, based on reconstructing a modified near-field equation associated with a special surface, which generalized our previous work for the Helmholtz equation (SIAM J. Imag. Sci. 10(3) (2017), 1579-1602) to the Navier equation. Several numerical examples are carried out to illustrate the effectiveness of the inversion algorithm.

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Mathematics Subject Classification: Primary: 35P25, 35Q74; Secondary: 74J20, 74J25.

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Figure 1. Elastic wave scattering by unbounded rough surfaces

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Figure 2. A chosen Lipschitz domain $ D_{R, h} $

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Figure 3. A chosen Lipschitz domain $ \Omega_{a, b} $

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Figure 4. Local perturbation of the plane $ x_2 = 0.5 $

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Figure 5. Reconstruction of the locally rough surface $ \Gamma $ given in Example 1 from data with no noise (a), 2% noise (b) and 5% noise (c) by the indicator function (54)

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Figure 6. Reconstruction of the locally rough surface $ \Gamma $ given in Example 2 from data with no noise (a), 2% noise (b) and 5% noise (c) by the indicator function (54)

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Figure 7. Reconstruction of the locally rough surface $ \Gamma $ given in Example 3 from data with no noise (a), 2% noise (b) and 5% noise (c) by the indicator function (54)

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Figure 8. Reconstruction of the locally rough surface $ \Gamma $ given in Example 4 from data with $ a = 5 $ (a), $ a = 7.5 $ (b) and $ a = 10 $ (c) by the indicator function (54)

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Figure 9. Reconstruction of the locally rough surface $ \Gamma $ given in Example 5 from data with no noise (a), 2% noise (b) and 5% noise (c) by the indicator function (54)

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Figure 10. Reconstruction of the locally rough surface $ \Gamma $ given in Example 6 from data with $ R = 20 $ (a), $ R = 40 $ (b) and $ R = 60 $ (c) by the indicator function (54)

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Table 1. Numerical solutions of $G^{ sc}(x, z, p;R)$ with different $R$

$R$	$G^{ sc}(x^{(1)}, z, p;R)$	$G^{sc}(x^{(2)}, z, p;R)$	$G^{sc}(x^{(3)}, z, p;R)$
$10^2$	$1.0{\text{e}}-2\cdot(1.3750+0.7988{\rm i}) \\ 1.0{\text{e}}-3\cdot(-9.2804+4.6422{\rm i})$	$1.0{\text{e}}-2\cdot(1.5892+0.9119{\rm i}) \\ 1.0{\text{e}}-3\cdot(-5.6177-3.3897{\rm i})$	$1.0{\text{e}}-2\cdot(1.5368+0.8773{\rm i}) \\ 1.0{\text{e}}-2\cdot(-0.1188-1.2583{\rm i})$
$10^3$	$1.0{\text{e}}-5\cdot(-4.6964+3.1245{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.8867-7.7026{\rm i})$	$1.0{\text{e}}-5\cdot(-3.6553+2.5430{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.8972-7.7034{\rm i})$	$1.0{\text{e}}-5\cdot(-2.6095+1.9516{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9051-7.7043{\rm i})$
$10^4$	$1.0{\text{e}}-7\cdot(0.0569+1.3741{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2793+9.0659{\rm i})$	$1.0{\text{e}}-7\cdot(0.0848+1.1093{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2786+9.0665{\rm i})$	$1.0{\text{e}}-8\cdot(1.1240+8.4444{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2780+9.0669{\rm i})$
$10^5$	$1.0{\text{e}}-9\cdot(-0.4012-1.0360{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1725+1.0492{\rm i})$	$1.0{\text{e}}-10\cdot(-1.5029-7.2349{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1725+1.0492{\rm i})$	$1.0{\text{e}}-10\cdot(1.0061-4.1102{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$	$1.0{\text{e}}-11\cdot(0.0181+1.2834{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$	$1.0{\text{e}}-12\cdot(0.0330-7.2675{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495 - 0.4940{\rm i})$	$1.0{\text{e}}-12\cdot(-0.1152+1.7009{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$
$R$	$G^{ sc}(x^{(4)}, z, p;R)$	$G^{sc}(x^{(5)}, z, p;R) $	$G^{sc}(x^{(6)}, z, p;R) $
$10^2$	$1.0{\text{e}}-2\cdot(1.2156-0.6900{\rm i})\\ 1.0{\text{e}}-2\cdot(0.3021-2.1052{\rm i})$	$1.0{\text{e}}-3\cdot(6.7691+3.7704{\rm i})\\ 1.0{\text{e}}-2\cdot(0.6058-2.7001{\rm i})$	$1.0{\text{e}}-4\cdot(1.6784-0.7840{\rm i})\\ 1.0{\text{e}}-2\cdot(0.7233-2.9150{\rm i})$
$10^3$	$1.0{\text{e}}-5\cdot(-1.5604+1.3531{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9101-7.7055{\rm i})$	$1.0{\text{e}}-6\cdot(-5.0901+7.5026{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9124-7.7068{\rm i})$	$1.0{\text{e}}-6\cdot(5.4321+1.4604{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9120-7.7083{\rm i})$
$10^4$	$1.0{\text{e}}-8\cdot(1.3992+5.7956{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2777+9.0671{\rm i})$	$1.0{\text{e}}-8\cdot(1.6737+3.1466{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2776+9.0672{\rm i})$	$1.0{\text{e}}-8\cdot(1.9479+0.4977{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2777+9.0671{\rm i})$
$10^5$	$1.0{\text{e}}-10\cdot(3.5152-0.9855{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$	$1.0{\text{e}}-10\cdot(6.0243+2.1393{\rm i})\\ 1.0{\text{e}}-5\cdot( 1.1726+1.0492{\rm i})$	$1.0{\text{e}}-10\cdot(8.5334+5.2640{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$	$1.0{\text{e}}-12\cdot(-0.2635-3.8658{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-4.9404{\rm i})$	$1.0{\text{e}}-12\cdot(-0.4117-9.4324{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$	$1.0{\text{e}}-11\cdot(-0.0560-1.4999{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4941{\rm i})$
$R$	$G^{ sc}(x^{(7)}, z, p;R)$	$G^{sc}(x^{(8)}, z, p;R)$	$G^{sc}(x^{(9)}, z, p;R)$
$10^2$	$1.0{\text{e}}-3\cdot(-6.4395-3.9529{\rm i}) \\ 1.0{\text{e}}-2\cdot(0.6266-2.7036 {\rm i})$	$1.0{\text{e}}-2\cdot(-1.1845-0.7152{\rm i}) \\ 1.0{\text{e}}-2\cdot(0.3364-2.1128{\rm i})$	$1.0{\text{e}}-2\cdot(-1.5090-0.9115{\rm i}) \\ 1.0{\text{e}}-2\cdot(-0.0834-1.27712{\rm i})$
$10^3$	$1.0{\text{e}}-5\cdot(1.5950-0.4567{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.9087-7.7099{\rm i})$	$1.0{\text{e}}-5\cdot(2.6452-1.0550{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.9028-7.7118{\rm i})$	$1.0{\text{e}}-5\cdot(3.6925-1.6461{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.8941-7.7138{\rm i})$
$10^4$	$1.0{\text{e}}-8\cdot(2.2226-2.1510{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2781+9.0669{\rm i})$	$1.0{\text{e}}-8\cdot(2.4981-4.7993{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2786+9.0666{\rm i})$	$1.0{\text{e}}-8\cdot(2.7749-7.4469{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2794+9.0660{\rm i})$
$10^5$	$1.0{\text{e}}-9\cdot(1.1042+0.8839{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$	$1.0{\text{e}}-9\cdot(1.3551+1.1514{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$	$1.0{\text{e}}-9\cdot(1.6060+1.4639{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$	$1.0{\text{e}}-11\cdot(-0.0708-2.0566{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$	$1.0{\text{e}}-11\cdot(-0.0856-2.6132{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$	$1.0{\text{e}}-11\cdot(-0.1004-3.1699{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$

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