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

The task of this paper is to develop a Time-Domain Probe Method for thereconstruction of impenetrable scatterers. The basic idea of the method isto use pulses in the time domain and the time-dependent response of the scattererto reconstruct its location and shape. The method is based on thebasic causality principle of time-dependent scattering. The method is independentof the boundary condition and is applicable for limited aperture scattering data. In particular, we discuss the reconstruction of the shape of a rough surface inthree dimensions from time-domain measurements of the scattered field.In practise, measurement data is collected where the incident field is given bya pulse. We formulate the time-domain field reconstruction problemequivalently via frequency-domainintegral equations or via a retarded boundaryintegral equation based on results of Bamberger, Ha-Duong, Lubich. In contrastto pure frequency domain methods here we use a time-domain characterizationof the unknown shape for its reconstruction. Our paper will describe the Time-Domain Probe Method and relate it to previousfrequency-domain approaches on sampling and probe methodsby Colton, Kirsch, Ikehata, Potthast, Luke, Sylvester et al. The approachsignificantly extends recent work ofChandler-Wilde and Lines (2005) and Luke and Potthast (2006) on the time-domainpoint source method.We provide a complete convergence analysis for the method for the rough surfacescattering case and provide numerical simulations and examples.