Shapiro Time Delay within the Geolocation Problem by TDOA

By means of Synge's world function and one Ho and Chan method a covariant description of the time difference of arrival (TDOA) geolocation problem is derived to provide two related exact locations of a radio transmitter placed on the surface of a spherical Earth. These locations correspond to the two models of the Earth surrounding space-time currently used in the navigation problem by GPS. Hence, one location corresponds to the Newtonian approximation of the Earth surrounding space-time, and the other, to its post-Newtonian approximation, which is where Shapiro time delay appears as the only post-Newtonian correction that modifies the classical transmitter location. Both locations are provided by means of two linear systems which involve three equatorial Earth satellites. The dependence of Shapiro time delay on the satellites distribution and on the transmitter classical location is shown by means of numerical solutions of these linear systems.