Geolocation using TOA, FOA, and altitude information at singular geometries

We discuss singularities that arise when geolocating an radio frequency (RF) emitter using time-of-arrival (TOA) and altitude (ALT) information. When TOA is measured at only three satellites, it has been found that the system regularly passes through singularities where the errors predicted through linearization go to infinity. We derive the conditions for such a system to be singular and show that it is generic in a one-parameter family of problems and, hence, likely that we will encounter this singularity as time varies. We give a simple heuristic argument to show that when the system passes through the singularity, the actual solution errors do not go to infinity and are proportional to the square root of the noise, rather than linear in the noise, as predicted by linearization. Although the errors are much larger than in the nonsingular case, they are not as large as predicted by the linear analysis. We show that errors near the singularity can be reduced by including frequency of arrival (FOA) information. This gives an overdetermined system of nonlinear equations.We evaluate several techniques for finding good initial guesses to start Gauss-Newton, which gives the minimum variance solution. We show that several of the initialization schemes can perform as well as beginning the iterations with the true device location. We also analyze FOA system singularities and discuss the complementary problem of augmenting a singular FOA-ALT solution with TOA data. We note that the principles discussed could apply to satellite navigation, but those systems usually have more than three satellites in view.