Source splitting via the point source method

Potthast, Roland, Fazi, Filippo and Nelson, Philip A. (2010) Source splitting via the point source method. Inverse Problems, 26 (4), 45002. (doi:10.1088/0266-5611/26/4/045002).

Abstract

We introduce a new algorithm for source identification and field splitting based on the point source method (Potthast 1998 A point-source method for inverse acoustic and electromagnetic obstacle scattering problems IMA J. Appl. Math. 61 119–40, Potthast R 1996 A fast new method to solve inverse scattering problems Inverse Problems 12 731–42). The task is to separate the sound fields uj, j = 1, ..., n of n \in \mathbb {N} sound sources supported in different bounded domains G1, ..., Gn in \mathbb {R}^3 from measurements of the field on some microphone array—mathematically speaking from the knowledge of the sum of the fields u = u1 + sdotsdotsdot + un on some open subset ? of a plane. The main idea of the scheme is to calculate filter functions g_1, \ldots, g_n, n\in \mathbb {N} , to construct uell for ell = 1, ..., n from u|? in the form

u_{\ell }(x) = \int _{\Lambda } g_{\ell,x}(y) u(y) {\,\rm d}s(y), \qquad \ell =1,\ldots, n.

We will provide the complete mathematical theory for the field splitting via the point source method. In particular, we describe uniqueness, solvability of the problem and convergence and stability of the algorithm. In the second part we describe the practical realization of the splitting for real data measurements carried out at the Institute for Sound and Vibration Research at Southampton, UK. A practical demonstration of the original recording and the splitting results for real data is available online.

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Contributors

Author: Filippo Fazi

Author: Philip A. Nelson

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