How does GPS find your location?
A Socratic walk-through of GPS positioning — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #How does GPS find your location?
Suppose your phone receives a radio message from a satellite saying, in effect, "I was here when I sent this." What could your phone learn if it also knew exactly how long that message traveled?
Because radio waves move at a known speed, travel time can become distance. Yet one distance does not identify one place. Where might you stand if you were exactly 20,000 kilometers from a particular satellite?
Reasoning it through
REASONING #You could be anywhere on a vast sphere around that satellite. A second satellite gives another sphere, and two spheres meet in a circle. A third cuts that circle down to two points, one of which is usually absurd — far out in space, or deep inside the Earth — so you can throw it away. Which suggests three satellites should be enough. Why, then, does every receiver want four?
Because of a problem hiding inside the phrase "how long that message traveled." Light covers about thirty centimeters in a nanosecond, so a quartz watch good to a thousandth of a second — excellent, by watch standards — would misjudge every distance by some three hundred kilometers. The measurement you actually have, then, is not a distance. It is a distance plus an unknown offset, which is why engineers call it a pseudorange.
Now notice the feature that rescues everything: if the receiver samples all the satellites at once, its clock is wrong by the same amount in every measurement. One unknown, shared. And an unknown appearing in every equation is not an error to be lived with — it is a variable to be solved for. Four measurements, four unknowns: latitude, longitude, altitude, and the clock offset itself. Do you see what that buys? A receiver with a cheap oscillator ends up knowing the time to within billionths of a second, as a by-product of asking where it is.
A fifth satellite helps differently: it is redundant, good for averaging and for catching a signal that disagrees with the rest. What matters more is where the satellites sit. Picture four bunched into one patch of sky — the spheres meet at shallow angles, and a small ranging error smears the intersection out along the line of sight. Same clocks, far worse fix. That sensitivity is called dilution of precision, and it is why a phone in a narrow street is not merely losing satellites but keeping a badly shaped selection of them — and why altitude is reliably the worst of the three coordinates.
One assumption is still worth poking. Light travels at c in vacuum, and the last few hundred kilometers are not vacuum: free electrons in the ionosphere delay the signal by an amount that swings with solar activity and can reach tens of meters. How do you remove a delay you cannot see? It depends on frequency, so a receiver listening on two frequencies can solve for it. Or put a second receiver on a surveyed point nearby, let it compute how wrong it is, and broadcast the difference — which works because two receivers tens of kilometers apart share much the same sky and much the same error. That is differential GPS.
The analogy
THE ANALOGY #Imagine hearing four tower bells whose locations and ringing times are known. The different arrival times at your ears let you reason backward to where you must be, while also correcting a wristwatch that is slightly wrong. GPS performs that same kind of reasoning with radio signals and much more precise clocks.
Bells stand still, and sound travels slowly enough to feel arriving. GPS satellites are moving thousands of miles an hour, and their clocks must be corrected for relativity — both their speed and their weaker gravity — or the fix would drift kilometers off within a day.
Clarifying the model
THE MODEL #The phone usually does not ask the satellites where it is. It listens to their broadcasts and performs the calculation locally. Nothing goes up, which is why the system serves any number of receivers at once and why it cannot, by itself, track you. Nor does GPS provide the street map; the calculated coordinates are later placed onto map data by another part of the device.
Two more things are commonly run together. Assisted GPS does not make your position more accurate: the satellites broadcast their orbital data at fifty bits per second, so reading it cold takes tens of seconds, and assisted GPS simply hands the phone that data over the cellular network. It fixes the wait, not the error. Differential GPS is the one that fixes the error.
Then the correction that most surprises people. The satellites sit about 20,200 kilometers up and move at roughly four kilometers a second, and both facts bend their clocks. Their speed makes them tick slow, by about 7 microseconds a day; their weaker gravity makes them tick fast, by about 45. The second effect wins, so a satellite clock gains some 38 microseconds a day on a clock at your feet. Small? Multiply by the speed of light: uncorrected, positional error would accumulate at roughly ten kilometers a day. It is not a frantic ongoing repair, though — the oscillators were built to run slightly slow before launch, so that from the ground they read correctly. And the familiar claim that GPS "proves" relativity is a shade too strong: the drift could in principle have been calibrated out by engineers who did not know its cause. What is true is that the effect is large, and was doubted enough at the time that the first satellite carried its relativistic offset as a switch, to be turned on once the prediction had been checked in orbit.
A picture of it
THE PICTURE #How to readTime runs downward. Each satellite only ever talks; nothing is sent back, which is why an unlimited number of receivers can use the system at once. The two self-directed arrows are the receiver working alone. Read the notes in order to find the real puzzle: three satellites are enough to fix a point in space, but only if you already know the exact time, and your receiver does not. The fourth satellite is not a spare for accuracy — it is what lets the receiver solve for its own clock error as a fourth unknown, which is why a device with no atomic clock inside it can still behave as though it had one.
What became clearer
WHAT CLEARED #GPS finds location by turning signal travel times into distances and combining several distance constraints. The fourth satellite is what lets an ordinary receiver correct its imperfect clock — and once you see that, the rest of the system falls into place as a long argument about time. Satellite clocks corrected for motion and altitude; atmospheric delay measured out or differenced away; satellite geometry deciding how many meters a leftover nanosecond costs. A receiver is not really a position finder. It is a clock that happens to report where it must be standing.
Where to go next
ONWARD #- How multipath — a signal arriving twice, once bounced off a wall — corrupts a fix in a city.
- Why GPS, Galileo, GLONASS and BeiDou can be combined, and what a receiver gains by doing so.
- How carrier-phase measurement reaches centimeters where code timing reaches meters.
Key terms
TERMS #| Term | What it means |
|---|---|
| Trilateration | finding a position from measured distances to known points. |
| Atomic clock | an extremely precise clock based on atomic transitions. |
| Clock bias | the receiver clock's difference from GPS system time. |
| Pseudorange | a measured satellite distance still containing the receiver's clock bias, which is |
| Dilution of precision | the factor by which satellite geometry magnifies a ranging error into a |
| Ephemeris | the precise orbital description each satellite broadcasts about itself. |
| Ionospheric delay | the slowing of the signal by charged particles in the upper atmosphere, |
Every term the collection defines is gathered in the glossary.