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Vector ellipsoidal harmonics and neuronal current decomposition in the brain
A timedomain probe method for threedimensional rough surface reconstructions
1.  Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Berkshire, RG6 6AX, United Kingdom, United Kingdom 
In particular, we discuss the reconstruction of the shape of a rough surface in three dimensions from timedomain measurements of the scattered field. In practise, measurement data is collected where the incident field is given by a pulse. We formulate the timedomain field reconstruction problem equivalently via frequencydomain integral equations or via a retarded boundary integral equation based on results of Bamberger, HaDuong, Lubich. In contrast to pure frequency domain methods here we use a timedomain characterization of the unknown shape for its reconstruction.
Our paper will describe the TimeDomain Probe Method and relate it to previous frequencydomain approaches on sampling and probe methods by Colton, Kirsch, Ikehata, Potthast, Luke, Sylvester et al. The approach significantly extends recent work of ChandlerWilde and Lines (2005) and Luke and Potthast (2006) on the timedomain point source method. We provide a complete convergence analysis for the method for the rough surface scattering case and provide numerical simulations and examples.
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