Evaluation of Direct and Iterative Methods for Overdetermined Systems of TOA Geolocation Equations

We analyze and rigorously test over a wide range of noise levels some of the most popular algorithms for solving overdetermined systems of the time of arrival (TOA) or pseudorange geolocation equations. Four criteria are given for evaluating these methods, one of which is that the method should achieve the minimum Cramer-Rao lower bound (CRLB) solution error variance. We discuss some straightforward techniques to determine if a method is minimum variance (MV) and apply this analysis to several solution methods. We consider two popular iterative algorithms, Newton-Raphson and Gauss-Newton, applied to both the primitive and squared TOA equations, both without and with explicit differencing (TDOA). We prove that each of these formulations is MV and examine the robustness of several initialization methods. We also consider three direct (noniterative) methods by Bancroft, by Chan and Ho, and by Abel and Chaffee and show when these methods achieve MV. In particular we show how the performance of each of these three algorithms depends on whether or not the large equal radius (LER) conditions are satisfied. Finally we give a new direct method that we prove is MV and show that it is robust in simulated geometries where the other direct methods are not.