← Tolerance · exhibit 03 of 06
03

Where Am I

GPS pseudoranging · side view, squashed to fit

Each GPS satellite carries an atomic clock and broadcasts the time. Your receiver notes when each broadcast arrives, multiplies the delay by the speed of light, and draws a circle. The catch is that your phone has a cheap quartz clock, and one microsecond of error is 300 m of range. None of these "ranges" are true. They're all wrong by the same unknown amount.

So GPS doesn't solve for where you are. It solves for where you are and what time it is, together, which is why you need one more satellite than you have dimensions. Slide the clock-error guess until the circles meet at a single point. That point is the fix.

circles miss by
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fix error
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horizontal DOP
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vertical DOP
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Drag the receiver or any satellite. Tap a satellite to switch it off. The faint cloud is 150 fixes with fresh noise, showing where the answer could land with this geometry.

Try this

  1. Slide the clock guess by hand until the circles close. Then hit solve it to watch Gauss–Newton do it in four or five steps.
  2. Look at the noise cloud. It's taller than it is wide, and the vertical DOP is worse than the horizontal. Every satellite is above you, so nothing pins you from below. That's why GPS altitude is always the sloppiest number.
  3. Drag all the satellites into a tight cluster. The cloud stretches into a long streak. Good satellites are spread out, not close.
  4. Switch off satellites until only two are left. Two circles and an unknown clock is three unknowns and two equations, so there's no unique answer.
  5. Press all sat clocks +10 µs. The fix doesn't move, and only the solved time changes. Then try one sat clock.

The deep cut

Satellite clocks tick faster than ground clocks by about 38 µs a day: weaker gravity up there speeds them up by about 45 µs, and orbital speed slows them by about 7 µs. They're built to run slow on the ground, at 10.22999999543 MHz instead of 10.23 MHz, so they're right in orbit.

You'll often read that without relativity "GPS would drift 10 km per day." The last button shows why that's only half right. An error that's the same on every satellite lands in the clock term you were already solving for, so position barely notices. What actually hurts is error that differs between satellites. Orbits are slightly eccentric, so each satellite's relativistic offset wobbles on its own schedule, and receivers correct for that term by term.