How Satellite Navigation Finds Your Position

A Not long ago, finding the way across an unfamiliar city meant unfolding a paper map on the passenger seat and hoping for the best. Today a small device in almost every pocket can announce a person's position to within a few metres, anywhere on the surface of the Earth, by day or night and in almost any weather. This everyday marvel depends on a fleet of satellites circling far overhead and on clocks of extraordinary precision. Although we use it without a second thought, the idea behind it is a remarkably elegant piece of applied physics, and understanding it reveals just how much careful engineering lies behind a simple blue dot on a screen.

B The heart of the system is a group of satellites, usually numbering around thirty, that orbit the Earth at a height of roughly twenty thousand kilometres. They are arranged in their paths so carefully that, from any point on the ground, several of them are always above the horizon at once. Each satellite does nothing more complicated than broadcast a steady radio signal, and that signal carries three pieces of information: which satellite is sending it, exactly where that satellite is at the moment of sending, and the precise instant at which the signal left. Everything the receiver does is built upon these three simple facts arriving from the sky.

C A receiver on the ground works out its position by measuring time. Because radio waves travel at the speed of light, a known and constant speed, the receiver can multiply that speed by the tiny interval the signal took to arrive and so calculate its distance from that particular satellite. One such measurement is not enough, for it shows only that the receiver lies somewhere on an imaginary sphere drawn around the satellite. But by measuring its distance from several satellites at once, and finding the single point where all those spheres cross, the receiver can fix its location. This method of pinning down a position from several known distances is called trilateration, and it is the mathematical core of the whole system.

D In principle the distances to three satellites would be enough to place a receiver in three dimensions, yet in practice a fourth is needed. The reason lies in the receiver's own clock. The satellites keep near-perfect time, but the cheap clock inside a phone or a car does not, and even a small error in its timekeeping would ruin the calculation. By taking a fourth measurement, the receiver gains just enough extra information to solve for its own clock error as well as its position, correcting the two together and so escaping the need for an expensive clock of its own.

E Everything, then, turns on time, and the demands are severe. Light travels about three hundred thousand kilometres every second, so an error of a single millionth of a second in a satellite's clock would translate into an error of some three hundred metres on the ground. For this reason each satellite carries an atomic clock, a device so stable that it would lose or gain no more than a second over many thousands of years. Stranger still, the engineers must allow for the predictions of Einstein's theories, since a clock in orbit runs at a very slightly different rate from one on the ground, and without this correction the whole system would soon drift into uselessness.

F No system is perfect, and several things conspire to blur the result. As the signals pass through the upper atmosphere they are slowed by a small and variable amount, introducing an error that must be estimated and removed. In cities the signals bounce off tall glass buildings and reach the receiver by more than one path, confusing the measurement. To keep the satellites honest, a network of ground stations continually tracks them, checks their clocks and their orbits, and sends up corrections. Newer additions to the system broadcast extra reference signals that allow a well-designed receiver to narrow its error from several metres down to a few centimetres.

G From guiding aircraft and ships to stamping an exact time on financial transactions, satellite navigation has quietly woven itself into the fabric of modern life, and much of what we now take for granted would stop working without it. Yet its principle remains as simple as it is powerful: listen to several distant clocks, measure how long their signals took to arrive, and let the unforgiving speed of light do the rest.