GPS and GNSS

A GNSS receiver, such as GPS, measures its distance to several satellites from signal travel time and finds the one point that fits all the distances. Range errors of a few meters are normal, and the satellite layout decides how much they grow into position error.

Satellite spread across the sky Number of satellites Range error per satellite RTK corrections (centimeter level) Buildings block the low satellites

Solving for position

Each satellite $i$ at known position $\mathbf{s}_i$ gives a measured range $\rho_i$. The receiver solves for the position $\mathbf{p}$ that best fits all of them, usually by least squares. A real receiver also solves for its own clock error, which is why it needs at least four satellites in 3D.

$$\rho_i = \lVert \mathbf{s}_i - \mathbf{p} \rVert + \varepsilon_i \qquad \sigma_{\text{position}} \approx \text{DOP} \times \sigma_{\text{range}}$$

DOP, the dilution of precision, depends only on geometry. Satellites spread across the sky give a low DOP. Satellites bunched together give a high one.

What to notice

Bunch the satellites together and the cloud of fixes stretches into a long streak, since the range lines cross at a shallow angle. Spread them out and the cloud shrinks. Turn on "Buildings block" to see a city street, where only the overhead satellites are visible and the sideways error gets large. RTK uses corrections from a nearby base station to cut the error from meters to centimeters, which is how farm robots and survey rovers drive straight rows.