# American Institute of Mathematical Sciences

August  2019, 13(4): 721-744. doi: 10.3934/ipi.2019033

## Inverse elastic surface scattering with far-field data

 1 School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China 2 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA

* Corresponding author

Received  April 2018 Revised  December 2018 Published  May 2019

Fund Project: The research of H.-A. Diao was supported in part by the Fundamental Research Funds for the Central Universities under the grant 2412017FZ007

A rigorous mathematical model and an efficient computational method are proposed to solving the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. We demonstrate how an enhanced resolution can be achieved by using more easily measurable far-field data. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangular slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near (FtN) field data conversion and utilizes the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem. Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction. Results show that the proposed method is capable of stably reconstructing surfaces with resolution controlled by the slab's density.

Citation: Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033
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##### References:
The problem geometry
Example 1: the reconstructed surface (dashed line) is plotted against the exact surface (solid line). (a) $\rho_1 = 1$; (b) $\rho_1 = 2$; (c) $\rho_1 = 4$; (d) 1 step of nonlinear correction when $\rho_1 = 4$; (e) 2 steps of nonlinear correction when $\rho_1 = 4$; (f) 3 steps of nonlinear correction when $\rho_1 = 4$
Example 2: the reconstructed surface (dashed line) is plotted against the exact surface (solid line). (a) $\rho_1 = 1$; (b) $\rho_1 = 2$; (c) $\rho_1 = 4$; (d) 1 step of nonlinear correction when $\rho_1 = 4$; (e) 2 steps of nonlinear correction when $\rho_1 = 4$; (f) 3 steps of nonlinear correction when $\rho_1 = 4$
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