A direct imaging method for the exterior and interior inverse scattering problems

Deyue Zhang; Yue Wu; Yinglin Wang; Yukun Guo

doi:10.3934/ipi.2022025

Article Contents

2022, Volume 16, Issue 5: 1299-1323. Doi: 10.3934/ipi.2022025

This issue Previous Article Well-posedness of an inverse problem for two- and three-dimensional convective Brinkman-Forchheimer equations with the final overdetermination Next Article Uniqueness of the partial travel time representation of a compact Riemannian manifold with strictly convex boundary

A direct imaging method for the exterior and interior inverse scattering problems

1.
School of Mathematics, Jilin University, Changchun 130012, China
2.
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

^*Corresponding author: Yukun Guo
^*Corresponding author: Yukun Guo

Received: January 2022

Revised: April 2022

Early access: May 2022

Published: October 2022

The work of D. Zhang, Y. Wu and Y. Wang are supported by NSFC grant 12171200. The work of Y. Guo is supported by NSFC grant 11971133 and the Fundamental Research Funds for the Central Universities

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Abstract

This paper is concerned with the inverse acoustic scattering problems by an obstacle or a cavity with a sound-soft or a sound-hard boundary. A direct imaging method relying on the boundary conditions is proposed for reconstructing the shape of the obstacle or cavity. First, the scattered fields are approximated by the Fourier-Bessel functions with the measurements on a closed curve. Then, the indicator functions are established by the superposition of the total fields or their derivatives to the incident point sources. We prove that the indicator functions vanish only on the boundary of the obstacle or cavity. Numerical examples are also included to demonstrate the effectiveness of the method.

Keywords:

Mathematics Subject Classification: Primary: 78A46, 65N21; Secondary: 35R30.

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Figure 1. An illustration of the inverse obstacle scattering problem

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Figure 2. An illustration of the interior inverse scattering problem

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Figure 3. The model scatterers for testing the reconstructions. (a) circle: $ (\cos t, \sin t) $; (b) kite: $ (\cos t+0.6\cos 2t-0.3, 1.3\sin t) $; (c) starfish: $ (1+0.2\cos 5t)(\cos t, \sin t) $

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Figure 4. Reconstruction of sound-soft obstacles with 5% noise. Top row: $ k = 3 $; Bottom row: $ k = 6 $

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Figure 5. Reconstruction of sound-hard obstacles with 2% noise. Top row: $ k = 4 $; Middle row: $ k = 5 $; Bottom row: superposition with multiple frequencies $ k = 3, 3.5, \cdots, 6 $

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Figure 6. Reconstruction of sound-soft cavities with 5% noise. Top row: $ k = 3 $; Bottom row: $ k = 5 $

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Figure 7. Reconstruction of sound-hard cavities with 2% noise. Top row: $ k = 4 $; Middle row: $ k = 5 $; Bottom row: superposition with multiple frequencies $ k = 3, 3.5, \cdots, 6 $

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