Best integer equivariant estimation: performance analysis using real data collected by low-cost, single- and dual-frequency, multi-GNSS receivers for short- to long-baseline RTK positioning

The key to precise global navigation satellite system (GNSS) positioning is carrier phase integer ambiguity resolution with a high success rate. On the other hand when the success rate is too low, the user will normally prefer the float solution. The alternative can be to use the best integer equivariant (BIE) estimator, since it is optimal in the minimum mean squared error (MMSE) sense. Low-cost receiver real-time kinematic precise positioning has become possible through the many signals that can be obtained by combining several GNSSs, such as BDS, Galileo, QZSS and GPS. In this contribution, we will use both simulations and such low-cost multi-GNSS data to compare the performance of the BIE and integer least squares (ILS) estimator, based on full ambiguity resolution. The GNSS data are evaluated in Dunedin, New Zealand, with a short- (670 m) and long-baseline (112.9 km) where the relative atmospheric delays can be neglected and need to be estimated, respectively. We compare the BIE and ILS results by using both single-frequency and dual-frequency (DF) low-cost and survey-grade receivers and antennas. We demonstrate, for the first time, the distributional properties of BIE positioning, where it will be shown that a ‘star-like’ pattern reveals itself once the model gets stronger and the ILS success rate increases. It will further be shown that the DF low-cost receivers give a very good positioning performance, but still not yet competitive to the survey-grade counterparts for the long-baseline. We will also demonstrate that the positioning performance of the BIE estimator will always equal or be better than that of the float solutions. It will finally be shown that BIE will always be better in the MMSE sense than the ILS solution when the success rate is at low to medium levels, whereas for high success rates we get a similar performance to ILS.