Irregular Webcomic! #2679

In our reality, GPS stands for Global Positioning System, which is a system that uses multiple artificial satellites to provide navigational position information to Earth-based receiver units.

Essentially, the system works by allowing you to measure your distance precisely from a number of satellites:

If you have the distance from one satellite, you can locate your position as being somewhere on the surface of a sphere centred on that satellite.
If you also have the distance from a second satellite, you can locate your position as somewhere on the surface of a sphere centred on the second satellite. This will (in general) intersect the first sphere along a circle - so you could be located at any point on that circle.
If you also have the distance from a third satellite, you can locate your position as being somewhere on the surface of a sphere centred on the third satellite. This will intersect the circle from the first two satellites at (in general) two points.

But how do you get the distance to the satellites? Well, each of the approximately 30 satellites broadcasts a precise timing signal. This timing signal takes a different amount of time to reach you depending on how far away the satellite is, because of the finite speed of light. So if you know the time at your own location very precisely, you can simply subtract the satellite's broadcast time from that and multiply by the speed of light to give you the distance.

In practice, however, a GPS receiver doesn't know what the time is to enough precision. This means you have an extra unknown in your mathematical calculations. As you know, if you've remembered how to solve simultaneous equations from school, an extra unknown means you need an extra equation to solve the system. Which means if you can detect timing signals from four satellites, you can solve for the precise time and the three spatial coordinates that define your location as being one of the two possible points mentioned above.

This is actually enough to give you your location, on the assumption that you are close to the surface of the Earth. By close, I mean within a dozen or so kilometres - so this also works if you're in a plane, and it would also work if you were underground, except for the small detail that the GPS radio signals don't penetrate the ground, so you wouldn't actually be able to detect them in the first place. The second possible point you could be at, given your position relative to the GPS satellites, is almost certainly in deep space somewhere - and obviously you're not likely to be there.

In practice there are a lot of fiddly details and corrections for things like clock drift, atmospheric effects, and so on, which means that your calculated position carries with it a certain uncertainty. The more satellites you can see above the horizon and free of blocking objects (like mountains or buildings), the more information your GPS receiver can use and the more accurately your position can be figured out.

Interestingly, although GPS receivers are capable of calculating two possible locations from three satellite signals, civilian GPS receivers capable of reporting to the user the non-Earth-based second possible location are not allowed to be exported from the USA, under the International Traffic in Arms Regulations. (Which means, effectively, that no civilian GPS manufacturer will make such devices.) Firstly, you don't need to know that solution, since obviously you're not going to be in deep space. But then again, what's so harmful about knowing that position that the US government have seen fit to make it illegal export GPS receiver capable of telling you what it is?

Well it's also illegal to export GPS receivers to report your location if you're at an altitude higher than 18 km and travelling at a speed of over 515 metres per second (or 1000 knots). The reason is that if you are both of these things, then you're probably a ballistic missile... Obviously the US government doesn't want anyone to be using readily available GPS receivers for guiding home-made (or terrorist made) ballistic missiles.

Another interesting thing about GPS is that, despite calculating your position being quite tricky and requiring multiple satellite fixes and solving a system of simultaneous equations, determining your velocity is pretty easy. A signal from a single satellite gives you your velocity component in a direction along the sightline, because the radio frequency of the timing signal is Doppler shifted, and that shift can be measured easily and turned into a velocity.