Introduction
RTK positioning depends on resolving the integer ambiguities in carrier-phase measurements. The difference between a float solution and a fixed solution - and the process of getting from one to the other - is one of the most important concepts in precision GNSS engineering. Understanding ambiguity resolution explains why RTK receivers sometimes take seconds, sometimes minutes, to reach centimetre accuracy, and why the fix/float distinction matters so much in practice.
The Integer Ambiguity Problem
When a receiver begins tracking a satellite, it can measure the fractional phase of the carrier wave precisely, but it cannot directly determine how many complete wavelengths lie between the satellite and the antenna. This unknown integer count - the integer ambiguity N - is constant as long as lock is maintained. The mathematical relationship is:
Carrier Phase Observable = Geometric Range / wavelength + N + atmospheric and clock errors / wavelength
If N were known precisely as an integer, the carrier phase would effectively become a very precise distance measurement. Resolving N to its correct integer value is the goal of ambiguity resolution. Until this is achieved, the ambiguity must be treated as a real-valued unknown - the float solution.
Float Solution
A float RTK solution is one in which the integer ambiguities have not yet been resolved. The receiver is processing carrier-phase measurements and using them to improve the position estimate, but the ambiguity parameters are being estimated as continuous real numbers rather than fixed integers. This is mathematically valid - the Kalman filter or least-squares estimator converges toward the correct position over time - but it is less precise than a fixed solution because the ambiguities carry uncertainty that propagates into position error.
Float accuracy is typically 10 cm to 50 cm horizontally, depending on geometry, baseline length, and the number of satellites and frequencies tracked. In challenging environments where ambiguity resolution cannot be completed - heavy multipath, insufficient satellites, long baseline - the receiver may remain in float indefinitely.
Fixed Solution
A fixed RTK solution is one in which the integer ambiguities have been correctly resolved to their true integer values with high statistical confidence. Once fixed, the carrier phase measurements effectively become the equivalent of very precise distance measurements anchored to the true integer count. Horizontal accuracy of 1–3 cm is typical for a well-designed RTK system with a short baseline.
The quality of the fix is assessed by the ratio test - the ratio of the residual of the second-best ambiguity set to that of the best set. A ratio above a threshold (commonly 3.0) indicates sufficient separation between the correct and next-best candidates to validate the fix. A low ratio indicates ambiguous candidates and signals that the fix should not be trusted.
Convergence Time
| Factor | Effect on Convergence |
|---|---|
| Number of satellites | More satellites = faster convergence |
| Number of frequencies | Dual/triple-frequency converges much faster than single-frequency |
| Baseline length | Longer baselines have larger atmospheric residuals, slowing convergence |
| Multipath level | Contaminated measurements slow or prevent convergence |
| Satellite geometry (DOP) | Poor geometry increases convergence time significantly |
With a short baseline (under 10 km), multi-constellation dual-frequency receivers typically achieve fixed solutions within 5–30 seconds in open sky. Single-frequency receivers on the same baseline may take several minutes. In PPP without a nearby base station, convergence to sub-decimetre accuracy typically takes 15–30 minutes with dual-frequency, reduced to under a minute with PPP-RTK atmospheric corrections.
Maintaining the Fix
Once fixed, the solution remains fixed as long as the receiver maintains continuous lock on a sufficient set of satellites. A cycle slip on any satellite used in the fixed solution invalidates that satellite's ambiguity and may force a partial or full re-initialisation. RTK engines try to maintain fix through partial disruptions by holding the ambiguities of unaffected satellites and re-resolving only those that slipped. In severe environments - urban canyons, tree canopy, tunnels - maintaining fix continuously is difficult, and the engineer must plan for the possibility of intermittent float periods.
Vital Points
- Float RTK uses carrier phase but with real-valued (non-integer) ambiguities - accuracy is decimetre level, not centimetre.
- Fixed RTK resolves ambiguities to exact integers - full carrier-phase precision of 1–3 cm is achieved.
- Convergence time depends primarily on the number of satellites, frequencies, baseline length, and multipath environment.
- The ratio test validates the fix but is not infallible; wrong fixes can occur in multipath-dominated environments.