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GNSS pseudorange with receiver clock bias
Measured pseudorange to one satellite: the true geometric range plus the range equivalent of the receiver clock error and other biases.
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- ρ [km]
A satellite receiver measures a pseudorange: the true geometric range R plus the distance light would travel during the receiver's clock error, c δt, plus other biases b (satellite clock, atmosphere and so on). The clock term is common to every satellite, so it is solved for together with the three position unknowns, which is why at least four suitably placed satellites are needed. Measurement noise is not included. The full pseudorange in km is the output.
Measured pseudorange to one satellite: the true geometric range plus the range equivalent of the receiver clock error and other biases. The clock term is large, 300 m per microsecond, so the clock must be solved for as a fourth unknown.
Calculator
- Error from the receiver clock, c δt29,980 m
- Total pseudorange error, ρ − R29,980 m
- Timing error that is worth 1 m of range, 1/c3.336 ns
- Signal travel time over R73.38 ms
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A receiver clock only 100 µs out adds 30 km to every pseudorange, thousands of times larger than the other biases (5 m). A clock error is huge in range terms: 1 m of range is only 3.34 ns of time.
- Since the clock term is the same for all satellites, differencing two pseudoranges removes it, and a fourth satellite allows it to be solved for along with x, y and z. With three satellites the three position unknowns plus the clock make four unknowns and only three equations.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (km) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R True geometric range | 20,000 to 26,000 km | 20,030 | 26,030 | 6,000 | +10% | +9.99 % |
| δt Receiver clock bias | 0 to 1,000 µs | 22,000 | 22,300 | 299.8 | +10% | +0.0136 % |
| b Other biases (satellite clock, atmosphere) | 0 to 30 m | 22,030 | 22,030 | 0.03 | +10% | +2.27e-6 % |
Sweep of Receiver clock bias
| Receiver clock bias (µs) | Pseudorange (km) |
|---|---|
| 0 | 22,000 |
| 125 | 22,040 |
| 250 | 22,070 |
| 375 | 22,110 |
| 500 | 22,150 |
| 625 | 22,190 |
| 750 | 22,220 |
| 875 | 22,260 |
| 1,000 | 22,300 |
Sweep one variable
Variables and how each one contributes
Pseudorange
What the equation gives you. Everything else feeds it.
True geometric range
The true geometric range from the receiver to the satellite, in km. GNSS satellites are typically 20 000 to 26 000 km away. The pseudorange follows the true range almost exactly, because the error terms are small beside it. Raising it by 10 % (to 24,200) moves the result by +9.99 %, from 22,030 to 24,230 km.
Receiver clock bias
The receiver clock bias in microseconds. Each microsecond adds 299.8 m of apparent range. Each microsecond of clock error adds 299.8 m to the pseudorange. Raising it by 100 µs (to 200 µs) moves the result by +0.136 %, from 22,030 to 22,060 km.
Other biases (satellite clock, atmosphere)
Other biases in metres: satellite clock error, ionospheric and tropospheric delay and similar effects. Adds one-for-one in metres, but it is tiny next to the range, so the percentage effect is negligible. Raising it by 5 m (to 10 m) moves the result by +2.27×10−5 %, from 22,030 to 22,030 km.
Worked example with the values above
- Clock term: c δt = 299 792 458 × 100 × 10⁻⁶ = 29,980 m.
- Total error: ρ − R = 29,980 + 5 = 29,980 m.
- ρ = 22,000 km + 29.98 km = 22,030 km.
Assumptions and limits
- Receiver and satellite positions in the same Earth-fixed frame, and a single satellite considered with the clock bias common to all.
- Measurement noise omitted. b groups satellite-clock, atmospheric and other corrections.
Common mistakes
- Treating the pseudorange as the true range. It includes the receiver clock term and so can be off by many kilometres.
- Mixing units: δt here is in microseconds and the speed of light in m/s. One microsecond is 299.8 m.
Statistics and uncertainty
The clock term is common to all satellites, so it does not degrade the final position once it is solved for. The accuracy is set by the biases bi that differ from satellite to satellite, and by the satellite geometry, which scales how pseudorange errors turn into position errors. Poor geometry, with satellites bunched together, magnifies the error.
Where it is used
- Understanding why four satellites are needed for a GNSS fix.
- Judging the size of timing errors and the quality of oscillator a receiver needs.
Maths you need
- Speed of light: c = 299 792 458 m/s
- Range equals speed times time: distance = c × delay