# American Institute of Mathematical Sciences

May  2009, 3(2): 259-274. doi: 10.3934/ipi.2009.3.259

## A time-domain probe method for three-dimensional rough surface reconstructions

 1 Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Berkshire, RG6 6AX, United Kingdom, United Kingdom

Received  November 2008 Revised  March 2009 Published  May 2009

The task of this paper is to develop a Time-Domain Probe Method for the reconstruction of impenetrable scatterers. The basic idea of the method is to use pulses in the time domain and the time-dependent response of the scatterer to reconstruct its location and shape. The method is based on the basic causality principle of time-dependent scattering. The method is independent of the boundary condition and is applicable for limited aperture scattering data.
In particular, we discuss the reconstruction of the shape of a rough surface in three dimensions from time-domain measurements of the scattered field. In practise, measurement data is collected where the incident field is given by a pulse. We formulate the time-domain field reconstruction problem equivalently via frequency-domain integral equations or via a retarded boundary integral equation based on results of Bamberger, Ha-Duong, Lubich. In contrast to pure frequency domain methods here we use a time-domain characterization of the unknown shape for its reconstruction.
Our paper will describe the Time-Domain Probe Method and relate it to previous frequency-domain approaches on sampling and probe methods by Colton, Kirsch, Ikehata, Potthast, Luke, Sylvester et al. The approach significantly extends recent work of Chandler-Wilde and Lines (2005) and Luke and Potthast (2006) on the time-domain point source method. We provide a complete convergence analysis for the method for the rough surface scattering case and provide numerical simulations and examples.
Citation: Corinna Burkard, Roland Potthast. A time-domain probe method for three-dimensional rough surface reconstructions. Inverse Problems and Imaging, 2009, 3 (2) : 259-274. doi: 10.3934/ipi.2009.3.259
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