system
Given the following post, respond as if you're commenting on Reddit:
human
Why do GPS clocks gain 38 rather than 60 microseconds per day over clocks on the surface of the Earth?

I was under the impression that satellites could be essentially modelled as inertial reference frames in that they are in free fall and thus have no net forces, thus making time run faster for them by a factor of sqrt(1 - 2GM/r*c^2 ) as compared to an object on the surface of the Earth.  This gives me 60 microseconds per day, which seems to agree with what Wikipedia suggests (.0219 seconds per year), yet every source states that GPS satellites only gain 38 microseconds per day, not 60.  

What am I missing here?  Can small satellites not actually be modelled as inertial reference frames?  If so, why?  

edit: I do understand that by doing change in gravitational field, then Lorentz factor, one gets the standard 38 microseconds, but I do not understand why the two results are different.  

edit 2: I think I was missing the glaring fact that time is still relative, even in inertial frames.  I do not think that it is incorrect to model a satellite as inertial, as free-fall being inertial is pretty much the whole point of the equivalence principle, but closer orbits will still run slower and this is not a contradiction.
gpt
As you know, there are two effects, which go in opposite directions.

Because of the orbital motion, there is a Lorentz time dilation effect that causes time to pass more slowly GPS satellite than on Earth.

Because the gravitational field is stronger at the surface of the Earth than at the location of the satellite, there is a gravitational time dilation effect that causes time to pass more quickly at the satellite.

You cannot model the satellite as being in an inertial reference frame, because its orbit is not at constant velocity; it is accelerating due to its changing direction of motion.