I am pretty sure that the Edge uses only the barometer to calibrate itself. If it used the GPS I would think it would be a lot more accurate at startup than it is at times.
There are pitfalls with GPS elevation too. The military can disable this feature when the ever they want. I have read some posts about the accuracy of GPS elevation that say it is actually less accurate than barometer.
Regarded, I wish the Edge had a screen where we could tell it both the correct elevation and barometric pressure. This would solve most if not all of our problems.