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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.

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Data availability

The broadcast ephemerides were used for satellite orbits and clocks. The low-cost and survey-grade receiver observation data are stored at University of Otago and the School of Surveying data facilities, and can be made available upon request by contacting the corresponding author R. Odolinski by email.


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Kade Phillips, School of Surveying, collected the GNSS data. The National Institute of Water & Atmospheric Research (NIWA) in Alexandra, NZ, allowed us to conduct a GNSS experiment at their property. The second author is the recipient of an Australian Research Council (ARC) Federation Fellowship (Project Number FF0883188). All this support is gratefully acknowledged.

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The first author R. Odolinski performed the research, wrote the manuscript and did the data analysis. P. J. G. Teunissen gave feedback on the written manuscript and wrote some of the theoretical parts of the paper.

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Correspondence to Robert Odolinski.

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Odolinski, R., Teunissen, P.J.G. 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. J Geod 94, 91 (2020).

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  • Best integer equivariant (BIE) estimator
  • Integer least squares (ILS) estimator
  • Low-cost receiver
  • Low-cost antenna
  • Single-frequency (SF)
  • Dual-frequency (DF)
  • Real-time kinematic (RTK) positioning