Source splitting via the point source method

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 u j , j = 1, . . . , n of n ∈ N sound sources supported in different bounded domains G 1 , . . . , G n in R 3 from measurements of the field on some microphone array-mathematically speaking from the knowledge of the sum of the fieldson some open subset of a plane. The main idea of the scheme is to calculate filter functions g 1 , . . . , g n , n ∈ N, to construct u for = 1, . . . , n from u| in the form u (x) = g ,x (y)u(y) ds(y), = 1, . . . , n.(1)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.

Uncertainty Principles for Inverse Source Problems, Far Field Splitting, and Data Completion

“…The novel mathematical contribution of the present work is the stability analysis for these algorithms based on new uncertainty principles, and their application to data completion. For alternate approaches to far field splitting that however, so far, lack a rigorous stability analysis we refer to [12,22] (see also [11] for a method to separate time-dependent wave fields due to multiple sources). This paper is organized as follows.…”

Time-dependent wave splitting and source separation

“…In pratice, due to the two dividing characteristics at θ 1 = 0, π, we actually solve four partial differential equations on four quarter-circles independently of each other. Since (22) and (27) both involve only partial derivatives but no integrals, our approach is local in time and in space.…”

“…Note that in the special case when u 1 or u 2 corresponds to a purely radial wave generated by a point source, the corresponding partial differential equation (22) or (27) reduces to an ordinary differential equation, which can be solved for any particular angle independently, regardless of the shape of Ω.…”

“…In [15,17], nonreflecting boundary conditions for time-dependent multiple scattering were derived, which avoid the space-time integrals involved in standard integral-based formulations. By Fourier transform in the frequency domain, Potthast, Fazi and Nelson [27] devised a filter via the point source method for time-dependent source separation.…”

On wave splitting, source separation and echo removal with absorbing boundary conditions

“…By combining the inverse Radon approximation with a Galerkin ansatz, Griesmaier, Hanke and Sylvester determine the convex scattering support of individual far-field components separately [7]. For time-dependent source separation, Potthast, Fazi and Nelson recently devised a filter using the point source method via Fourier transform in the frequency domain [15]. Recently, Grote, Kray, Nataf and Assous devised a method to split a time-dependent, scattered, total wave field into its distinct outgoing components induced by separate sources or obstacles [9].…”