The best integer equivariant (BIE) estimator for the multivariate t-distribution was introduced in Teunissen (J Geod, 2020. https://doi.org/10.1007/s00190-020-01407-2), where it was shown that the BIE-weights will be different from that of the normal distribution. In this contribution, we analyze these BIE estimators while making use of multi global navigation satellite system (GNSS) data. The BIE-estimators are also compared to their least-squares (LS) and integer least-squares (ILS) contenders. Monte Carlo simulations are conducted so as to realize controlled performance comparisons of the different estimators for the purpose of multi-GNSS (GPS, Galileo, BDS and QZSS) single-frequency real-time kinematic positioning. The analyses are done in a qualitative sense by means of positioning scatter plots, and in a quantitative sense by means of numerical mean-squared-error (MSE) curves for the different estimators under different model strengths (receiver-satellite geometries and varying degrees of freedom). Particular attention is given to the difference in impact the multivariate t-distribution has when either only its cofactor matrix is in common with the normal distribution or its complete variance-covariance matrix. It will be shown that the BIE-estimators give better MSEs to both the LS- and ILS-estimator when the ILS success rate is different from zero and one, respectively. We also demonstrate that using the same BIE-estimator on different data distributions can give users an unrealistic sense of their solution quality, while the usage of two different BIE-estimators on the same data can have a marginal impact.@en

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.@en

With the combination of emerging GNSSs, single-frequency (SF) precise RTK positioning becomes possible. In this contribution we evaluate such low-cost ublox receiver and antenna performance when combining real data of four CDMA systems, namely L1 GPS, E1 Galileo, L1 QZSS, and B1 BDS. Comparisons are made to more expensive dual-frequency (DF) GPS receivers and antennas. The formal and empirical ambiguity success rates and positioning precisions will first be evaluated while making use of L1+E1, so as to investigate whether instantaneous SF RTK is possible without the need of B1 BDS or L1 QZSS. This is followed by an analysis of the SF 4-system model performance when the residual ionosphere can be ignored and modeled as a function of the baseline length, respectively. The analyses are conducted for a location in Dunedin, New Zealand, and compared to Perth, Australia with the better visibility of BDS and QZSS. The results indicate that successful instantaneous and precise RTK positioning is feasible while using L1 GPS and E1 Galileo data, and that the SF 4-system model is competitive to DF GPS even when residual ionospheric delays are present. We finally demonstrate that when the impact from the ionosphere increases and more than one epoch is needed for successful ambiguity resolution, the SF 4-system model performance can still remain competitive with the DF GPS receivers. This is particularly true in Perth with more satellites and when higher than customary elevation cut-off angles need to be used to avoid low-elevation multipath.@en

The concept of single-frequency, dual-system (SF-DS) real-time kinematic (RTK) positioning has become feasible since, for instance, the Chinese BeiDou Navigation Satellite System (BDS) has become operational in the Asia-Pacific region. The goal of the present contribution is to investigate the single-epoch RTK performance of such a dual-system and compare it to a dual-frequency, single-system (DF-SS). As the SF-DS we investigate the L1 GPS + B1 BDS model, and for DF-SS we take L1, L2 GPS and B1, B2 BDS, respectively. Two different locations in the Asia-Pacific region are analysed with varying visibility of the BDS constellation, namely Perth in Australia and Dunedin in New Zealand. To emphasize the benefits of such a model we also look into using low-cost ublox single-frequency receivers and compare such SF-DS RTK performance to that of a DF-SS, based on much more expensive survey-grade receivers. In this contribution a formal and empirical analysis is given. It will be shown that with the SF-DS higher elevation cut-off angles than the conventional 10 ∘ or 15 ∘ can be used. The experiment with low-cost receivers for the SF-DS reveals (for the first time) that it has the potential to achieve comparable ambiguity resolution performance to that of a DF-SS (L1, L2 GPS), based on the survey-grade receivers.@en

The emerging GNSSs make single-frequency (SF) RTK positioning possible. In this contribution two different types of low-cost (few hundred USDs) RTK receivers are analyzed, which can track L1 GPS, B1 BDS, E1 Galileo and L1 QZSS, or any combinations thereof, for a location in Dunedin, New Zealand. These SF RTK receivers can potentially give competitive ambiguity resolution and positioning performance to that of more expensive (thousands USDs) dual-frequency (DF) GPS receivers. A smartphone implementation of one of these SF receiver types is also evaluated. The least-squares variance component estimation (LS-VCE) procedure is first used to formulate a realistic stochastic model, which assures that our receivers at hand can achieve the best possible ambiguity resolution and RTK positioning performance. The best performing low-cost SF RTK receiver types are then assessed against DF GPS receivers and survey-grade antennas. Real data with ionospheric disturbances at low, medium and high levels are analyzed, while making use of the ionosphere-weighted model. It will be demonstrated that when the presence of the residual ionospheric delays increases, instantaneous RTK positioning is not possible for any of the receivers, and a multi-epoch model is necessary to use. It is finally shown that the low-cost SF RTK performance can remain competitive to that of more expensive DF GPS receivers even when the ionospheric disturbance level reaches a Kp-index of 7−, i.e. for a strong geomagnetic storm, for the baseline at hand.@en

The integration of the Chinese BDS with other systems, such as the American GPS, makes precise RTK positioning possible with low-cost receivers. We investigate the performance of low-cost ublox receivers, which cost a few hundred USDs, while making use of L1 GPS + B1 BDS data in Dunedin, New Zealand. Comparisons will be made to L1 + L2 GPS and survey-grade receivers which cost several thousand USDs. The least-squares variance component estimation procedure is used to determine the code and phase variances and covariances of the receivers and thus formulate a realistic stochastic model. Otherwise, the ambiguity resolution and hence positioning performance would deteriorate. For the same reasons, the existence of receiver-induced time correlation is also investigated. The low-cost RTK performance is then evaluated by formal and empirical ambiguity success rates and positioning precisions. It will be shown that the code and phase precision of the low-cost receivers can be significantly improved by using survey-grade antennas, since they have better signal reception and multipath suppression abilities in comparison with low-cost patch antennas. It will also be demonstrated that the low-cost receivers can achieve competitive ambiguity resolution and positioning performance to survey-grade dual-frequency GPS receivers.@en

When the GPS was the only satellite constellation in orbit, instantaneous single-frequency precise (millimeter-level) positioning was not possible and more expensive survey-grade multiple-frequency receivers had to be used. With the advent of the GNSSs, such as the Chinese BDS, low-cost receiver precise positioning will potentially become feasible. In this contribution we investigate the performance of such a low-cost single-frequency GPS+BDS model, making use of ublox EVK-M8T receivers, and compare its performance to a survey-grade dual-frequency GPS receiver solution, in Dunedin New Zealand. The least-squares variance-component estimation (LS-VCE) procedure is used to investigate the precision of the receiver code and phase observables of the low-cost receivers. The estimated (co)variances are needed so as to formulate a realistic stochastic model for precise RTK positioning. The performance of ambiguity resolution would otherwise deteriorate and hence the achievable positioning precisions as well. The low-cost RTK performance is then evaluated by formal and empirical ambiguity success-rates and positioning precisions. Our results indicate that the quality of the antennas used plays an important role, especially in suppressing multipath. We will demonstrate that the low-cost solution, which costs a few hundred USDs, can give a competitive ambiguity resolution and positioning performance to the survey-grade receivers, which cost several thousand USDs.@en

Carrier phase integer ambiguity resolution with a high success rate is the key to precise Global Navigation Satellite System (GNSS) positioning. When the success rate is too low the user will normally prefer the float solution, whereas the alternative can be to use the Best Integer Equivariant (BIE) estimator. Low-cost receiver real-time kinematic (RTK) precise positioning has become possible through combining signals from several GNSSs, such as BDS, Galileo, QZSS, and GPS. In this contribution we will use single-frequency (SF) low-cost receiver multi-GNSS data to compare the performance of the BIE estimator and the standard method of Integer Least Squares (ILS). The GNSS data is evaluated in Dunedin, New Zealand, with short baselines so that the relative atmospheric delays can be neglected. We show, with real multi-GNSS data and when the success rate is at low to medium levels, that the positioning performance of the BIE estimator will resemble or be better than that of the float solution, and always be better in the minimum mean squared error (MMSE) sense than the ILS fixed solutions. Whereas for very high success rates we get a BIE performance similar to that of the ILS estimator and much better than the float solution.@en

In this contribution we will focus on instantaneous (single-epoch) single-baseline Real- Time Kinematic (RTK) combining four CDMA satellite systems. We will combine the Chinese BeiDou, the European Galileo, the American GPS and the Japanese QZSS system. To further strengthen the underlying model and maximize the redundancy, attention will be given to overlapping frequencies between the systems. With calibrated Inter System Biases (ISBs), it enables one to use a common pivot satellite between the respective systems when parameterizing the double-differenced ambiguities. We make use of the LAMBDA method for ambiguity resolution, and the performance is evaluated by ambiguity success-rates and by comparing the estimated positions to very precise benchmark coordinates. This will be based on various elevation cut-off angles so as to mimic conditions with obstructed satellite visibility (such as in urban canyons). It will be shown by how much the increased strength of the combined models allow for improved ambiguity resolution performance and positioning robustness over the single-systems.@en