A direct imaging method for the half-space inverse scattering problem with phaseless data

Zhiming Chen; Shaofeng Fang; Guanghui Huang

doi:10.3934/ipi.2017042

Article Contents

2017, Volume 11, Issue 5: 901-916. Doi: 10.3934/ipi.2017042

This issue Previous Article An undetermined time-dependent coefficient in a fractional diffusion equation Next Article

A direct imaging method for the half-space inverse scattering problem with phaseless data

1.
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, School of Mathematical Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China
2.
School of Mathematical Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China
3.
The Rice Inversion Project, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892, USA

* Corresponding author: Guanghui Huang
* Corresponding author: Guanghui Huang

Received: November 30, 2016

Published: September 2017

The first author was supported in part by the China NSF under the grant 113211061.

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Abstract

We propose a direct imaging method based on the reverse time migration method for finding extended obstacles with phaseless total field data in the half space. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the obstacle is far away from the surface of the half-space where the measurement is taken. Numerical experiments are included to illustrate the powerful imaging quality.

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Mathematics Subject Classification: Primary: 65N30; Secondary: 78A45, 35Q60.

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Figure 1. Examples 6.1: Imaging results of penetrable obstacles with different shapes from left to right. The top results are imaged with phaseless data by our RTM algorithm, and the bottom one are imaing results with full phase data. The probe wave number is $k=4\pi$, and $N_s=512$, $N_r=512$

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Figure 2. Example 6.2: Imaging results of an elliptic obstacle with boundary conditions as sound soft, sound hard and impedance boundary with $\lambda=1$, respectively. The probe wave number $k=4\pi$, and $N_s=512$, $N_r=512$

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Figure 3. Examples 6.3 (first test): Imaging results of a penetrable obstacle with noise levels $\mu=0.1, 0.2, 0.3, 0.4$ (from left to right). The top row is imaged with single frequency data, and the bottom row is imaged with multi-frequency data

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Figure 4. Example 6.3 (second test): Imaging results of two sound soft obstacles. All the parameters are the same as that in Figure 3

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Figure 5. Example 6.4: Imaging results of two sound soft obstacles overlaid with the true obstacle model. The left column is imaged with the single frequency data, and the right one is imaged with the multi-frequency data

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