With ground based hyperbolic navigation one doesn't need absolute time and time error cancels out because one looks at time differences between the RX of stations only (3 stations two equations for 2D). Now you can treat GPS as a hyperbolic problem as well but the bad news is that you need time anyhow to determine the satellites position from its ephemeris data in the first place. So time needs to be synced nevertheless. This is a non issue when the transmitter stations are fixed ground based as with Decca, Omega and similar systems.
A good clock is also important for the first fix in order to synchronize on a specific satellites pseudo random gold code. This is one reason why cold fixes could take longer than hot fixes besides the fact that ephemeris and almanac data need upating. Quality recievers therefore feature thermo compensated quartz clocks which are approx. 20 times more accurate than regular quartz clocks. Mobile phones rely on network time synchronization instead.
Typically data from as many satellites as the reciever can track and are visible in the sky are fed into an extended Kalman filter which will provide a least mean square error solution for position and time offset.
For four sattelites a closed algebraic solution for the set of four nonlinear equation does exist.
Last edited by BDAttitude; 11th Jun 2020 at 12:56.