The satellite positioning system, commonly known as GNSS (Global Navigation Satellite System), lets us determine precise geographic positions anywhere on Earth 🌍. Today, we’re breaking down how it works in broad strokes, even though it can get pretty complex!

What we call GPS (Global Positioning System) is just one type of GNSS operated by the United States since the 1970s. There are others, like Galileo in Europe, GLONASS in Russia, or BeiDou in China.
So when people say “GPS,” they’re really referring to the generic term GNSS regardless of which system is in play.
To form a GNSS constellation, about twenty satellites orbit Earth at roughly 20000km in the medium Earth orbit.

These satellites continuously broadcast radio waves, and since their positions are known, you can use triangulation to figure out the distance between each satellite and your receiver on Earth.
In theory, 3 satellites could provide your latitude and longitude, but in practice, at least 4 are needed. The fourth one helps correct the clock offset, more on that later.

A reference frame is simply a point of view, an origin from which we measure things. For GNSS calculations, we use the geocentric reference frame (or ECEF in English), which takes the center of the Earth as its origin.

This system uses a Cartesian framework with three mutually perpendicular axes (designated X, Y, and Z) to pinpoint positions. These coordinates (x, y, z) let you precisely locate points relative to Earth’s center. Since raw (x,y,z)’s aren’t very user-friendly, we convert them into the much simpler latitude and longitude format.

All the satellites’ clocks are synchronized to a source known as GPS Time. When your GNSS receiver (for example, your smartphone) picks up signals from at least 4 satellites, it can calculate your position by measuring 4 pseudo-ranges.

We call them “pseudo” distances because they include not just the true distance, but also various errors such as:

To correct these issues, each satellite transmits a unique random code. Your receiver generates an identical code in sync with the satellite’s transmission and compares the time shift between sending and receiving. This time offset is then used in the formula:
P = c * Δt
where P is the pseudo-range in meters, c is the speed of light in m/s, and Δt is the time offset in seconds.

By solving an equation with 4 unknowns, your device can determine its (x,y,z) coordinates on Earth as well as the clock offset relative to the satellites.

These equations essentially calculate the real distance between each satellite and your receiver using the 3D version of the Pythagorean theorem, with X, Y, and Z being the satellite’s coordinates and U representing your unknown location. Then, the clock error (expressed as c times the time difference) is added to account for the time discrepancy. We won’t dive into the nitty-gritty of solving these equations, it gets pretty involved, but there are two primary methods: (Least Squares Method) and (Kalman Filter). If you’re curious, check them out!
Once solved, you obtain the (x,y,z) coordinates of U.

Ultimately, we want to know exactly where we are on Earth, and the raw x,y,z coordinates aren’t very intuitive. Luckily, we can convert them into latitude and longitude. Keep in mind that latitude and longitude are based on an ellipsoidal approximation of Earth. It makes sense: you’re converting 3D data into a 2D map. The best way to represent our blue planet (which is roughly spherical) is by using a reference ellipsoid defined by the WGS 84 model. We won’t get into all the details here, the Wikipedia page is there for the curious, but with a bit of trigonometry, the conversion happens.
And there you have it, now you know exactly where you are on the map 🗺️! You could even go further and compute your altitude, but that’s a whole other level of complexity.