📍How GPS Works - Pseudorange Positioning, DOP, Relativity, RTK

How to use How GPS Works

1. 1On Pseudorange fix, drag satellites or the receiver, set the clock error and noise, and compare solving or ignoring the clock error
2. 2Switch between the good and bad geometry presets and watch HDOP and the expected error
3. 3On Constellation & DOP, set latitude, longitude and elevation mask, play the constellation and read the sky plot and DOP
4. 4On Relativistic clocks and Errors & RTK, change the orbit altitude, positioning mode and baseline

FAQ

Why does GPS need at least four satellites?

The receiver has a cheap quartz clock with an unknown offset δt from the satellites' atomic clocks. Each satellite gives one pseudorange equation ρ = |s − x| + c·δt, and the unknowns are x, y, z and δt: four in all, so four satellites are the minimum. One microsecond of clock error is 300 m of range, so the fix is useless unless the clock error is solved. As a bonus the receiver gets the time to within tens of nanoseconds, which is how GPS time transfer works.

What is DOP, and why are clustered satellites bad?

Dilution of precision says how much the geometry amplifies range errors: position error ≈ DOP × pseudorange error. It comes from G = (HᵀH)⁻¹, where each row of H is the unit vector to a satellite plus a column of ones. Satellites spread across the sky constrain every direction and give a small DOP. Satellites bunched in one patch of sky make the range circles cross at shallow angles, so one direction is poorly determined and DOP grows. An HDOP around 1 is good; above 5 is poor.

How big is the relativistic effect on GPS?

GPS satellites move at about 3.9 km/s, so special relativity slows their clocks by about 7 μs per day. They sit higher in Earth's gravity well, so general relativity speeds them up by about 45 μs per day. The net is about +38 μs per day; at 300 m per microsecond that would add roughly 11 km of position error each day. That is why the satellite clocks are set to about 10.22999999543 MHz instead of 10.23 MHz before launch, and receivers correct the periodic eccentricity term.

Why is phone GPS good to a few metres but RTK to centimetres?

For a single-frequency civilian receiver the main errors are ionospheric delay, multipath, satellite clock and ephemeris errors, about 5 m of pseudorange error in total; multiplied by DOP that is a few metres horizontally. Dual-frequency receivers remove most of the ionosphere. RTK uses the carrier phase with its 19 cm wavelength instead of the code, measures phase to about 1% of a cycle, and differences against a nearby base station to cancel the shared satellite and atmospheric errors. Once the integer ambiguity is fixed, 1–2 cm is routine.

Are the constellation and error values real?

The constellation is idealised: six planes at 55° inclination, four evenly spaced satellites per plane, period half a sidereal day. The real constellation has about 31 unevenly spaced satellites, so the visible count and DOP here are representative only. The error budget uses typical textbook 1σ values; real values vary with solar activity, surroundings and receiver. The physics itself (pseudorange equations, DOP, relativity) is exact.