American Institute of Mathematical Sciences

May  2013, 7(2): 377-396. doi: 10.3934/ipi.2013.7.377

Near-field imaging of the surface displacement on an infinite ground plane

 1 Department of Mathematics, Zhejiang University, Hangzhou, China 2 Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, United States

Received  November 2011 Revised  June 2012 Published  May 2013

This paper is concerned with the inverse diffraction problem for an unbounded obstacle which is a ground plane with some local disturbance. The data is collected in the near-field regime with a distance above the surface displacement that is smaller than the wavelength. In this regime, the evanescent modes carried by the scattered wave are significant, which makes it different from the far-field measurement. We formulate explicitly the connection between the evanescent wave modes and the high frequency components of the surface displacement, and present a new numerical scheme to reconstruct the surface displacement from the boundary measurements. By extracting the information carried by the evanescent modes effectively, it is shown that the resolution of the reconstructed image is significantly improved in the near field. Numerical examples show that images with a resolution of $\lambda/10$ are obtained.
Citation: 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
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References:
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