Direct estimation of wave source location from weighted integral measurements on finite boundary

We deal with the problem of direction and distance estimate of wave emission sources in 3-D space. We first introduce a partial differential equation (PDE) what we call the location constraint PDE (LC-PDE), which provides a necessary and sufficient description of wave-field generated by a source at direction vector n and distance R. To remove differentials and obtain algebraic relations involving unknown direction and distance, we integrate the LC-PDE in finite two-dimensional (2-D) region or 1-D curvilinear path with sufficient number of different weight functions. By using Cartesian 2-D complex sinusoidal weight functions in 2-D measurement case and genera 1-D measurement case, we show an exact finite set of algebraic equations can be obtained including n, R as the unknown variables. Some additional techniques are introduced to eliminate the integral boundary terms. In 2-D measurement case, they involve small number of 2-D discrete Fourier transform (DFT) components of wave-field as the measurements. In case of 1-D measurement on a circle, we show sinusoidal weight functions on the circle can be used to obtain the algebraic equations. Owing to frequency domain separation by DFT, it can be applied to multiple sources with sufficient angle difference. We examine this method with numerical simulation assuming noisy measurements and multiple source conditions.