Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

Isaac Harris; Dinh-Liem Nguyen; Thi-Phong Nguyen

doi:10.3934/ipi.2022015

Article Contents

2022, Volume 16, Issue 5: 1137-1162. Doi: 10.3934/ipi.2022015

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Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

1.
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
2.
Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA

^*Corresponding author: Isaac Harris
^*Corresponding author: Isaac Harris

Received: July 2021

Revised: January 2022

Early access: April 2022

Published: October 2022

The author I. Harris is supported in part by the NSF DMS grant 2107891 and the author D.-L. Nguyen is supported in part by the NSF DMS grant 1812693.

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Abstract

In this paper, we consider the inverse scattering problem for recovering either an isotropic or anisotropic scatterer from the measured scattered field initiated by a point source. We propose two new imaging functionals for solving the inverse problem. The first one employs a 'far-field' transform to the data which we then use to derive and provide an explicit decay rate for the imaging functional. In order to analyze the behavior of this imaging functional we use the factorization of the near field operator as well as the Funk-Hecke integral identity. For the second imaging functional the Cauchy data is used to define the functional and its behavior is analyzed using the Green's identities. Numerical experiments are given in two dimensions for both isotropic and anisotropic scatterers.

Keywords:

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

Citation:

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Figure 1. The exact geometry of the scatterer(s)

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Figure 2. The reconstruction of the kite-shaped scatterer $ D $ where the boundary $ \partial D = $kite by the direct sampling method with 'Far-Field' transformation

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Figure 3. The reconstruction of the kite-shaped scatterer $ D $ where the boundary $ \partial D = $kite by the direct sampling method with Cauchy data

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Figure 4. The reconstruction of the scatterer with two disjoint components $ \partial D = {\rm{disk}} \cup {\rm{rectangle}} $ by the direct sampling method with 'Far-Field' transformation

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Figure 5. The reconstruction of the scatterer with two disjoint components $ \partial D = {\rm{disk}} \cup {\rm{rectangle}} $ by the direct sampling method with Cauchy data

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Figure 6. The reconstruction of the scatterer with three disjoint components $ \partial D = {\rm{kite}} \cup {\rm{ellipse}} \cup {\rm{peanut}} $ by the direct sampling method with 'Far-Field' transformation

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Figure 7. The reconstruction of the scatterer with two disjoint components $ \partial D = {\rm{kite}} \cup {\rm{ellipse}} \cup {\rm{peanut}} $ by the direct sampling method with Cauchy data

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Figure 8. The reconstruction of the kite shaped scatterer by the direct sampling method with 'Far-Field' transformation for different amounts of measurements

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Figure 9. The reconstruction of the kite shaped scatterer by the direct sampling method with Cauchy data for different amounts of measurements

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Figure 10. The reconstruction of the kite shaped scatterer when there are more receivers than sources with 25 receivers

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Figure 11. The reconstruction of the kite shaped scatterer by the direct sampling method with 'Far-Field' transformation for different wave numbers $ k $

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Figure 12. The reconstruction of the kite shaped scatterer by the direct sampling method with Cauchy data for different wave numbers $ k $

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Figure 13. The reconstruction of the kite shaped scatterer by the direct sampling method with Cauchy data for different values of parameter $ \rho $

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