Far Field Splitting for the Helmholtz Equation

Given far field data of a time-harmonic wave radiated by an ensemble of well separated acoustic or electromagnetic sources as well as a priori information on the locations of these sources, we discuss an algorithm to approximate the far field data radiated by each of these sources separately. The method is based on a Galerkin procedure considering subspaces spanned by the singular vectors of “restricted” far field operators that map local source distributions to the corresponding radiated far field patterns. We provide an error analysis for this algorithm and consider its stability. Furthermore, we exemplify a means to extract the required a priori knowledge directly from the far field data, and we show how to utilize the splitted far fields to recover information on the supports of the individual sources beyond that a priori information. Numerical results for the most important example of inverse obstacle scattering illustrate our theoretical findings. Version from September 13, 2012.