Event Location from Noisy Time-of-Arrival Data by Total Least Squares

While it is widely known how to reduce the time-of-arrival equations to a linear system, this appears to be of limited value for noisy, over-determined problems as there is no guarantee that the standard least squares solution of this linear system will be close to the maximum likelihood solution of the errors in the original problem. This paper uses the idea of errors-in-variables to construct a total least squares problem from the linear system that is a close approximation to likelihood maximisation. The errors-in-variables problem has a non-standard form in that the same error appears in two different columns; the paper shows it can nevertheless be reduced to an equation in a single variable, and that this can be solved efficiently through use of the eigendecomposition of a quadratic matrix pencil. A sensitivity analysis relating the uncertainty in the estimate to the uncertainties in the data and an asymptotic expansion of the solution when the transmitter is far away from the receiver array are also provided.