# 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
##### References:

show all references

##### 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 $\rho_1 = 4$
 [1] Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757 [2] Peijun Li, Yuliang Wang. Near-field imaging of obstacles. Inverse Problems & Imaging, 2015, 9 (1) : 189-210. doi: 10.3934/ipi.2015.9.189 [3] Heping Dong, Deyue Zhang, Yukun Guo. A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data. Inverse Problems & Imaging, 2019, 13 (1) : 177-195. doi: 10.3934/ipi.2019010 [4] Ming Li, Ruming Zhang. Near-field imaging of sound-soft obstacles in periodic waveguides. Inverse Problems & Imaging, 2017, 11 (6) : 1091-1105. doi: 10.3934/ipi.2017050 [5] Gang Bao, Junshan Lin. Near-field imaging of the surface displacement on an infinite ground plane. Inverse Problems & Imaging, 2013, 7 (2) : 377-396. doi: 10.3934/ipi.2013.7.377 [6] Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems & Imaging, 2015, 9 (2) : 591-600. doi: 10.3934/ipi.2015.9.591 [7] Maria Schonbek, Tomas Schonbek. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1277-1304. doi: 10.3934/dcds.2005.13.1277 [8] Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems & Imaging, 2013, 7 (2) : 341-351. doi: 10.3934/ipi.2013.7.341 [9] Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 415-442. doi: 10.3934/dcdsb.2019188 [10] Frank Natterer. Incomplete data problems in wave equation imaging. Inverse Problems & Imaging, 2010, 4 (4) : 685-691. doi: 10.3934/ipi.2010.4.685 [11] Josselin Garnier, George Papanicolaou. Resolution enhancement from scattering in passive sensor imaging with cross correlations. Inverse Problems & Imaging, 2014, 8 (3) : 645-683. doi: 10.3934/ipi.2014.8.645 [12] Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari. A note on analyticity properties of far field patterns. Inverse Problems & Imaging, 2013, 7 (2) : 491-498. doi: 10.3934/ipi.2013.7.491 [13] Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042 [14] Peter Monk, Virginia Selgas. Near field sampling type methods for the inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2011, 5 (2) : 465-483. doi: 10.3934/ipi.2011.5.465 [15] Peijun Li, Xiaokai Yuan. Inverse obstacle scattering for elastic waves in three dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 545-573. doi: 10.3934/ipi.2019026 [16] Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343 [17] Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015 [18] Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems & Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609 [19] Huey-Er Lin, Jian-Guo Liu, Wen-Qing Xu. Effects of small viscosity and far field boundary conditions for hyperbolic systems. Communications on Pure & Applied Analysis, 2004, 3 (2) : 267-290. doi: 10.3934/cpaa.2004.3.267 [20] Jiakun Liu, Neil S. Trudinger. On the classical solvability of near field reflector problems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 895-916. doi: 10.3934/dcds.2016.36.895

2018 Impact Factor: 1.469