Built onFacts: Time and Navigation

Matt gives a brief summary of time and navigation. There’s one point that he glosses over, and it’s something that a lot of GPS summaries gloss over, to the point that they are misleading.

All a GPS satellite does is eternally broadcast two continuously updated pieces of information: its position and the time on its atomic clock. Knowing that light travels at about 1 foot per nanosecond, we can calculate how far we are from the satellite to the foot, as long as the GPS clock is accurate to the nanosecond and we have a receiver that can handle such a precise signal.

Actually you can’t do this unless you have a synchronized clock, and unless you’ve done this already, in order to synchronize the clocks properly you have to know … [wait for it] … the distance to the satellite. Many of the explanations of GPS completely miss this little tidbit. If you haven’t got a synchronized clock, and all you have are the GPS signals, you need *four* satellites to find your position. In practice four may not be necessary, because if you know your approximate position on the earth and have a topographic map, you can get the elevation from that, in which case three satellites is sufficient to get your position, to some level of uncertainty.

One satellite puts you on the surface of a sphere. Two intersect spheres into a circle. Three intersect two circles to allow two points. One point is generally impossible, locating you by default. Four satellites is an absolute fix.

That’s the common explanation, but it’s wrong. You don’t know the distance to the satellite. The equation for one satellite is not d = ct, it’s d = c(t2-t1) and has two unknowns — you only know the time the signal was sent, not when it was received.