Some info from the USGS
* Survey grade GPS receivers typically have horizontal accuracies around 1cm. Vertical accuracy is twice that (2cm).
* Differential grade GPS receivers from 0.3 to 1.0 meter horizontal. Vertical accuracy is 2 to 3 times that.
* Consumer grade GPS receivers are between 3 and 10 meters horizontal. "This type of GPS handheld unit provides elevation data with poor accuracy."
So figure at best
a consumer grade GPS is going to have 20 to 30 meter vertical accuracy. Cell phone GPS is probably also at the low end of consumer devices. And if we have stacked roads, we also are going to have signal multi-path issues. So that will make that number even larger.
Plus, GPS signals will tell us how far we are from the satellite. It will not directly tell us how high off the ground we are. There are mathematical models to estimate the surface of the earth and that is used to give an indication of where you are in relation to the ground. Different devices may use a different model or at least interpret the same model differently.
I am by no means saying this is a bad idea, I'm just saying there is a big signal to noise ratio so it isn't going to be easy.