Satellite transmitter position constraints for zero, one and two solutions

In a satellite navigation system such as the Global Positioning System (GPS), the receiver is required to determine its position accurately from time-difference-of-arrival (TDOA) measurements of incident signals. We will examine the receiver solution space of an estimator which is maximum-likelihood under the assumption of Gaussian measurement errors. We find the satellite position constraints which cause the receiver to have zero, one or two real solutions. We derive the equations for the boundary between the domains in which one and two solutions lie and find that there is a simple geometric interpretation.