The GPS uses the flat space light propagation formula to calculate the distance from the source (the satellite) to the receiver (observer on Earth): $$ d=c \cdot \Delta t$$ where $c$ is the speed of light in Minkowski vacuum, $\Delta t$ is the difference between the times of emission and absorption of the signal (corrected for relativistic time dilations) and $d$ is the Euclidean distance. This formula is fed with the data beamed from 4 satellites to solve for the location of the receiver.
My questions are: what is the rationale for using this formula? Shouldn't the distance be calculated in the curved geometry setting, e.g. using the Schwarzschild metric? What are the errors in using the Euclidean version $ d=c \cdot \Delta t$?
N.B.: The time difference $\Delta t$ contains relativistic corrections to times. However, it is not clear to me why it is correct to use the flat space (Minkowski) formula for light propagation with just the value of $\Delta t$ amended to account for gravity.
Please, try to be as clear as possible and support your statements with calculations/derivations.
ADDENDUM: I found really good papers discussing in depth all the relativistic details and effects to GPS(-like) navigation in spacetime. They are Thomas B. Bahder's Navigation in Curved Space-Time, Clock Synchronization and Navigation in the Vicinity of the Earth and Relativity of GPS Measurement.