"Relativity in the Global Positioning System" by Neil Ashby of the University of Colorado is excellent. Chapter 6 discusses relativistic corrections for satellites using GPS.
Quote from the paper:
“Relativistic principles and effects which must be considered include the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts, and relativity of synchronization.“
So many factors to unpack yet the solution is right there.
Interestingly while all those points factor in it would be possible to have a working GPS system without "understanding" the error or having any grasp of relativity.
To expand on that, consider if the satellites went up (with ideas of basic triangulation from beacons orbiting) and then the drifting error for a supposedly fixed ground position was noticed what could be done?
The 'unknown cause' error function over time twixt fixed position and uncorrected GPS calculation can be fed into a Kalman filter to derive weights that eliminate the error.
Typically what happens in many instrumentation applications is models are created to derive functions to emit answers, errors are noticed, people think hard to add extra terms to account for errors and eventually either all errors are accounted for or some residual 'wobble' remains which can be smoothed away by an adaptive error model.
To this day in high precision GPS applications post processing runs are used to improve accuracy that account for relativity factors, atmospheric twinkle factors, (other factors I'd have to look up), and still there's a bit or error left over that can be sweep away (for a time) with a Kalman filter.
