The two-dimensional source location problem for time differences of arrival at minimal element monitoring arrays.

The time difference of arrival (TDOA) source localization inverse problem is analyzed for two-dimensional signal propagation detected by a small number of sensor elements in a monitoring array. Nonlinear least-squares solutions are found based on the assumptions of geodesic rays propagating at constant speed. The two-dimensional (2D) TDOA source location problem is shown in the case of three sensors to have dual possible solutions for some combinations of arrival time differences. In the case of four non-collinear sensors, there are unique solutions for all physically possible combinations of time differences. Dual solutions to the three-sensor problem are associated with a small range of arrival time differences but large regions in physical space. The locations of the dual solutions are separated by a wide variety of distances, which in some cases prevent the use of alternative reasoning to remove the ambiguity. Three-sensor TDOA cannot be reliably used for 2D source location unless the source is a priori known to be within either the spatial region spanned by the sensor array or the external zones of unique solution. Determining the minimum number of sensors necessary to unambiguously solve the source location problem assists in cost-effective design of sensor arrays.