Since Sir Edmund Hillary climbed Mt. Everest in 1953, I hardly think that he used GPS. Hence the low content warning.

Sorry, my mistake:

Bradford Washburn 1999

Mounted a GPS atop Mt Everest.

" A rock head elevation of 8,850 m (29,035 ft), and a snow/ice elevation 1 m (3 ft) higher, were obtained via this device".

Wikipedia.

If GPS antennas above your head can provide your position in three dimensions (xy movement across the earth + z altitude), then why can't GPS antennas on the same plane as you provide your position in three dimensions?

Because the time it would take the signals from each tower to reach the receiver wouldn't

be very different, and commercial GPS receivers have very slow processors so their

batteries can last all day.

You're going to have to clarify how you arrived at those calculations. GPS calculations are carried out via triangulation of coordinates between a large set of satellites, I mean towers; often 5 or more. I also don't see how sensitivity to height would be any less of an issue with satellites, remember that we don't have to assume that all towers are at the same height, so sensitivity to low angles isn't necessary.

Same as above. The further away the transmitters, the longer it takes a signal to get to the receiver,

then, the more accurate the result of the program in the receiver (on a strict time budget)

in timing the distance covered by the signal from each transmitter.

GPS uses triliteration, not triangulation. It looks at the intersection of spheres

of the radius of the calculated distance travelled from each transmitter.

Many of the bonus results can be discarded because they are below sea level where GPS doesn't work.

EDIT: Thinking about other devices that use EMR to measure distance, even a cheap unit would probably have the resolution to measure this difference, but I did make it quite large. If I had used something like a 20ft change at 100km, it would have been a much smaller difference that needed to be measured. Having satellites practically overhead definitely makes it heaps easier to get an accurate measure of altitude, no matter how far away from the receiver the satellite is.

No matter whether or not a ground based positioning system has been demonstrated

to work on Earth, resolution is always improved if the transmitter is further away.

The transmitter has a peripheral atomic clock so it can send the time and it's location

to the receiver. The receiver has a temperature compensated clock (the best we can do

at the other end, to sell it to the masses).

The receiver measures the distance travelled by comparing the transmission time from

each transmitter's clock to the receive time from the receiver's clock.

In your close to Earth GPS theory, the points of intersection in those spheres are close together,

the spheres are smaller, the erroneous results don't sound as stupid as the erroneous

results from much larger spheres, and you have several plausible locations to choose from.

This is alleviated if you don't have to think about altitude, and just use all information to calculate a 2D horizontal position.