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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

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Abstract

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.

References

  • Aggrey J, Bisnath S, Naciri N, Shinghal G, Yang S (2020) Multi-GNSS precise point positioning with next-generation smartphone measurements. J Spat Sci 65(1):79–98. https://doi.org/10.1080/14498596.2019.1664944

    Article  Google Scholar 

  • Amiri-Simkooei AR, Tiberius CCJM (2007) Assessing receiver noise using GPS short baseline time series. GPS Solut 11(1):21–35

    Article  Google Scholar 

  • Axelrad P, Larson K, Jones B (2005) Use of the correct satellite repeat period to characterize and reduce site-specific multipath errors. In: Proceedings of ION GNSS 18th international technical meeting of the satellite division, Long Beach, CA

  • Brack A (2015) On reliable data-driven partial GNSS ambiguity resolution. GPS Solut 19(4):411–422

    Article  Google Scholar 

  • Brack A (2017) Reliable GPS + BDS RTK positioning with partial ambiguity resolution. GPS Solut 21:1083–1092

    Article  Google Scholar 

  • Brack A (2019) Partial carrier-phase integer ambiguity resolution for high accuracy GNSS positioning. PhD dissertation, Lehrstuhl fur Kommunikation und Navigation Technische Universitat Munchen

  • Brack A, Gunther C (2014) Generalized integer aperture estimation for partial GNSS ambiguity fixing. J Geod 88(5):479–490

    Article  Google Scholar 

  • Brack A, Henkel P, Gunther C (2014) Sequential best integer-equivariant estimation for GNSS. Navigation 61(2):149–158

    Article  Google Scholar 

  • De Jonge PJ, Tiberius CCJM (1996) The LAMBDA method for integer ambiguity estimation: implementation aspects. LGR-Series, technical report, Delft University of Technology (12)

  • Euler HJ, Goad C (1991) On optimal filtering of GPS dual frequency observations without using orbit information. Bull Geod 65:130–143

    Article  Google Scholar 

  • Krietemeyer A, van der Marel H, van de Giesen N, ten Veldhuis MC (2020) High quality zenith tropospheric delay estimation using a low-cost dual-frequency receiver and relative antenna calibration. Remote Sens 12:1393

    Article  Google Scholar 

  • Laurichesse D, Banville S (2018) Instantaneous centimeter-level multi-frequency precise point positioning. GPS World, Innovation column 4 July

  • Mongredien C, Doyen JP, Strom M, Ammann D (2016) Centimeter-level positioning for UAVs and other mass-market applications. In: ION GNSS, Portland, Oregon

  • Nie Z, Liu F, Gao Y (2020) Real-time precise point positioning with a low-cost dual-frequency GNSS device. GPS Solut. https://doi.org/10.1007/s10291-019-0922-3

    Article  Google Scholar 

  • Odijk D, Teunissen PJG (2013) Characterization of between-receiver GPS–Galileo inter-system biases and their effect on mixed ambiguity resolution. GPS Solut 17(4):521–533

    Article  Google Scholar 

  • Odijk D, Khodabandeh D, Nadarajah N, Choudhury M, Zhang B, Li W, Teunissen PJG (2017) PPP-RTK by means of Ssystem theory: Australian network and user demonstration. J Spat Sci 62(1):3–27. https://doi.org/10.1080/14498596.2016.1261373

    Article  Google Scholar 

  • Odolinski R, Teunissen PJG (2016) Single-frequency, dual-GNSS versus dual-frequency, single-GNSS: a low-cost and high-grade receivers GPS-BDS RTK analysis. J Geod 90(11):1255–1278. https://doi.org/10.1007/s00190-016-0921-x

    Article  Google Scholar 

  • Odolinski R, Teunissen PJG (2017a) Low-cost, 4-system, precise GNSS positioning: a GPS, Galileo, BDS and QZSS ionosphere-weighted RTK analysis. Meas Sci Technol. https://doi.org/10.1088/1361-6501/aa92eb

    Article  Google Scholar 

  • Odolinski R, Teunissen PJG (2017b) Low-cost, high-precision, single-frequency GPS-BDS RTK positioning. GPS Solut 21(3):1315–1330. https://doi.org/10.1007/s10291-017-0613-x

    Article  Google Scholar 

  • Odolinski R, Teunissen PJG (2019) An assessment of smartphone and low-cost multi-GNSS single-frequency RTK positioning for low, medium and high ionospheric disturbance periods. J Geod 93:701–722. https://doi.org/10.1007/s00190-018-1192-5

    Article  Google Scholar 

  • Odolinski R, Teunissen PJG, Odijk D (2013) Quality analysis of a combined COMPASS/BeiDou-2 and GPS RTK positioning model. In: IGNSS symposium, Golden Coast, Australia

  • Odolinski R, Teunissen PJG, Odijk D (2014) Combined GPS + BDS + Galileo + QZSS for long baseline RTK positioning. In: ION GNSS, Tampa, Florida, USA

  • Odolinski R, Teunissen PJG, Odijk D (2015) Combined GPS + BDS for short to long baseline RTK positioning. Meas Sci Technol 26:045801. https://doi.org/10.1088/0957-0233/26/4/045801

    Article  Google Scholar 

  • Paziewski J, Sieradzki R, Radoslaw B (2019) Signal characterization and assessment of code GNSS positioning with low-power consumption smartphones. GPS Solut. https://doi.org/10.1007/s10291-019-0892-5

    Article  Google Scholar 

  • Riley S, Lentz W, Clare A (2017) On the path to precision—observations with android GNSS observables. In: Proceedings of ION GNSS, Portland, Oregon

  • Teunissen PJG (1990) An integrity and quality control procedure for use in multi sensor integration. In: Proceedings of the 3rd international technical meeting of the satellite division of the institute of navigation (ION GPS 1990), Colorado Spring, CO, pp 513–522, also published in: volume VII of the GPS red book: integrated systems, ION Navigation, 2012

  • Teunissen PJG (1995) The least squares ambiguity decorrelation adjustment: a method for fast GPS integer estimation. J Geod 70:65–82

    Article  Google Scholar 

  • Teunissen PJG (1999a) An optimality property of the integer least-squares estimator. J Geod 73:587–593

    Article  Google Scholar 

  • Teunissen PJG (1999b) The probability distribution of the GPS baseline for a class of integer ambiguity estimators. J Geod 73(5):275–284

    Article  Google Scholar 

  • Teunissen PJG (2003a) Adjustment theory—an introduction. Series on mathematical geodesy and positioning. Delft University Press, Delft

    Google Scholar 

  • Teunissen PJG (2003b) Theory of integer equivariant estimation with application to GNSS. J Geod 77:402–410. https://doi.org/10.1007/s00190-003-0344-3

    Article  Google Scholar 

  • Teunissen PJG (2005) On the computation of the best integer equivariant estimator. Artif Satell 40(3):161–171

    Google Scholar 

  • Teunissen PJG, Amiri-Simkooei AR (2008) Least-squares variance component estimation. J Geod 82(2):65–82

    Article  Google Scholar 

  • Teunissen PJG, Verhagen S (2009) The GNSS ambiguity ratio-test revisited: a better way of using it. Surv Rev 41(312):138–151

    Article  Google Scholar 

  • Teunissen PJG, Joosten P, Tiberius CCJM (1999) Geometry-free ambiguity success rates in case of partial fixing. In: Proceedings of ION ITM, San Diego, CA, USA, pp 201–207

  • Verhagen S (2005) The GNSS integer ambiguities: estimation and validation. PhD dissertation, Netherlands Geodetic Commission, Publications on Geodesy, 58

  • Verhagen S, Teunissen PJG (2005) Performance comparison of the BIE estimator with the float and fixed GNSS ambiguity estimators. In: A window on the future of geodesy, international association of geodesy symposia, vol 128. Springer, Berlin, Heidelberg, pp 428–433

  • Wand MP, Jones M (eds) (1995) Kernel smoothing. Chapman and Hall, London

    Google Scholar 

  • Yang Y, Li J, Wang A, Xu J, He H, Guo H, Shen J, Dai X (2014) Preliminary assessment of the navigation and positioning performance of BeiDou regional navigation satellite system. Sci China Earth Sci 57:144–152

    Article  Google Scholar 

  • Yang Y, Gao W, Guo S, Mao Y, Yang Y (2019) Introduction to BeiDou-3 navigation satellite system. Navigation 66:7–18

    Article  Google Scholar 

  • Zhang X, Tao X, Zhu F, Shi X, Wang F (2018) Quality assessment of GNSS observations from an Android N smartphone and positioning performance analysis using time differenced filtering approach. GPS Solut. https://doi.org/10.1007/s10291-018-0736-8

    Article  Google Scholar 

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Acknowledgements

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). https://doi.org/10.1007/s00190-020-01423-2

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