Journal of Geodesy

, Volume 90, Issue 1, pp 15–44 | Cite as

On the estimability of parameters in undifferenced, uncombined GNSS network and PPP-RTK user models by means of \(\mathcal {S}\)-system theory

  • Dennis Odijk
  • Baocheng Zhang
  • Amir Khodabandeh
  • Robert Odolinski
  • Peter J. G. Teunissen
Original Article

Abstract

The concept of integer ambiguity resolution-enabled Precise Point Positioning (PPP-RTK) relies on appropriate network information for the parameters that are common between the single-receiver user that applies and the network that provides this information. Most of the current methods for PPP-RTK are based on forming the ionosphere-free combination using dual-frequency Global Navigation Satellite System (GNSS) observations. These methods are therefore restrictive in the light of the development of new multi-frequency GNSS constellations, as well as from the point of view that the PPP-RTK user requires ionospheric corrections to obtain integer ambiguity resolution results based on short observation time spans. The method for PPP-RTK that is presented in this article does not have above limitations as it is based on the undifferenced, uncombined GNSS observation equations, thereby keeping all parameters in the model. Working with the undifferenced observation equations implies that the models are rank-deficient; not all parameters are unbiasedly estimable, but only combinations of them. By application of \(\mathcal {S}\)-system theory the model is made of full rank by constraining a minimum set of parameters, or S-basis. The choice of this S-basis determines the estimability and the interpretation of the parameters that are transmitted to the PPP-RTK users. As this choice is not unique, one has to be very careful when comparing network solutions in different \(\mathcal {S}\)-systems; in that case the S-transformation, which is provided by the \(\mathcal {S}\)-system method, should be used to make the comparison. Knowing the estimability and interpretation of the parameters estimated by the network is shown to be crucial for a correct interpretation of the estimable PPP-RTK user parameters, among others the essential ambiguity parameters, which have the integer property which is clearly following from the interpretation of satellite phase biases from the network. The flexibility of the \(\mathcal {S}\)-system method is furthermore demonstrated by the fact that all models in this article are derived in multi-epoch mode, allowing to incorporate dynamic model constraints on all or subsets of parameters.

Keywords

GNSS Undifferenced model Network PPP-RTK theory \(\mathcal {S}\)-system theory Rank-deficient model  Dynamic model 

References

  1. Baarda W (1973) S-transformations and criterion matrices. In: Publications on geodesy, 18 (vol. 5, no. 1), Netherlands Geodetic Commission, Delft, The NetherlandsGoogle Scholar
  2. Banville S, Collins P, Zhang W, Langley RB (2014) Global and regional ionospheric corrections for faster PPP convergence. Navigation 61(2):115–124CrossRefGoogle Scholar
  3. Bar-Sever YE, Kroger PM, Borjesson JA (1998) Estimating horizontal gradients of tropospheric path delay with a single GPS receiver. J Geophys Res 103(B3):5019–5035CrossRefGoogle Scholar
  4. Boehm J, Schuh H (2004) Vienna mapping functions in VLBI analyses. Geophys Res Lett 31(1):L01603CrossRefGoogle Scholar
  5. Boehm J, Heinkelmann R, Schuh H (2007) Short note: a global model of pressure and temperature for geodetic applications. J Geodesy 81(10):679–683CrossRefGoogle Scholar
  6. Chatfield C (1994) The analysis of time series–an introduction, 4th edn. Chapman & Hall, LondonGoogle Scholar
  7. Collins P, Bisnath S, Lahaye F, Heroux P (2010) Undifferenced GPS ambiguity resolution using the decoupled clock model and ambiguity datum fixing. Navigation 57(2):123–135CrossRefGoogle Scholar
  8. de Jonge PJ (1998) A processing strategy for the application of the GPS in networks. In: Publications on geodesy, 46, Netherlands Geodetic Commission, Delft, The NetherlandsGoogle Scholar
  9. Dow JM, Neilan RE, Rizos C (2009) The international GNSS service in a changing landscape of Global Navigation Satellite Systems. J Geodesy 83(3):191–198CrossRefGoogle Scholar
  10. Ge M, Gendt G, Rothacher M, Shi C, Liu J (2008) Resolution of GPS carier-phase ambiguities in precise point positioning (PPP) with daily observations. J Geodesy 82(7):389–399CrossRefGoogle Scholar
  11. Geng J, Bock Y (2013) Triple-frequency GPS precise point positioning with rapid ambiguity resolution. J Geodesy 87(5):449–460CrossRefGoogle Scholar
  12. Geng J, Teferle FN, Meng X, Dodson AH (2011) Towards PPP-RTK: ambiguity resolution in real-time precise point positioning. Adv Space Res 47(10):1664–1673CrossRefGoogle Scholar
  13. Goad CC (1985) Precise relative position determination using Global Positioning System carrier phase measurments in a nondifference mode. In: Proceedings of 1st international symposium on precise positioning with the Global Positioning System, Rockville, MD, 15–19 April, pp 347–356Google Scholar
  14. Heroux P, Kouba J (1995) GPS precise point positioning with a difference. In: Paper presented at Geomatics ’95, Ottawa, Ontario, Canada, 13–15 JuneGoogle Scholar
  15. Herring TA, Davis JL, Shapiro II (1990) Geodesy by radio interferometry: the application of Kalman filtering to the analysis of very long baseline interferometry data. J Geophys Res 95(B8):12561–12581CrossRefGoogle Scholar
  16. Hofmann-Wellenhof B, Lichtenegger H, Wasle E (2008) GNSS–Global Navigation Satellite Systems: GPS, GLONASS. Galileo & more, Springer, Wien, New YorkGoogle Scholar
  17. Khodabandeh A, Teunissen PJG (2014) Array-based satellite phase bias sensing: theory and GPS/BeiDou/QZSS results. Meas Sci Technol 25Google Scholar
  18. Komjathy A, Sparks L, Wilson BD, Mannucci AJ (2005) Automated daily processing of more than 1000 ground-based GPS receivers for studying intense ionospheric storms. Radio Sci 40:RS6006Google Scholar
  19. Kouba J, Heroux P (2001) Precise point positioning using IGS orbit products. GPS Solut 5(2):12–28CrossRefGoogle Scholar
  20. Lannes A, Prieur J-L (2013) Calibration of the clock-phase biases of GNSS networks: the closure-ambiguity approach. J Geodesy 87(8):709–731CrossRefGoogle Scholar
  21. Lannes A, Teunissen PJG (2011) GNSS algebraic structures. J Geodesy 85(5):273–290Google Scholar
  22. Laurichesse D, Mercier F, Berthias JP (2009) Zero-difference integer ambiguity fixing on single frequency receivers. In: Proceedings of ION ITM-2009, Anaheim, 26–28 Jan 2009, pp 2460–2469Google Scholar
  23. Li X, Zhang X, Ge M (2011) Regional reference network augmented precise point positioning for instantaneous ambiguity resolution. J Geodesy 85(3):151–158CrossRefGoogle Scholar
  24. Li X, Ge M, Dousa J, Wickert J (2014) Real-time precise point positioning regional augmentation for large GPS reference networks. GPS Solut 18(1):61–71CrossRefGoogle Scholar
  25. Lindlohr W, Wells D (1985) GPS design using undifferenced carrier beat phase observations. Manuscr Geod 10:255–295Google Scholar
  26. Montenbruck O, Hauschild A (2013) Code biases in multi-GNSS positioning. In: Proceedings of ION ITM-2013, San Diego, CA, 28–30 January, pp 616–628Google Scholar
  27. Niell AE (1996) Global mapping functions for the atmosphere delay at radio wavelengths. J Geophys Res 101(B):3227–3246Google Scholar
  28. Odijk D (2002) Fast precise GPS positioning in the presence of ionospheric delays. In: Publications on geodesy, 52, Netherlands Geodetic Commission, Delft, The NetherlandsGoogle Scholar
  29. Odijk D, Teunissen P, Zhang B (2012) Single-frequency integer ambiguity resolution enabled GPS precise point positioning. J Surv Eng 138:193–202. doi:10.1061/(ASCE)SU.1943-5428.0000085 CrossRefGoogle Scholar
  30. Ovstedal O (2002) Absolute positioning with single-frequency GPS receivers. GPS Solut 5(4):33–44CrossRefGoogle Scholar
  31. Pratt J, Axelrad P, Larson KM, Lesage B, Gerrena R, DiOrio N (2013) Satellite clock bias estimation for iGPS. GPS Solut 17(3):381–389CrossRefGoogle Scholar
  32. Rao CR (1973) Linear statistical interference and its applications, 2nd edn. Wiley, New YorkCrossRefGoogle Scholar
  33. Rocken C, Ware R, Hove TV, Solheim F, Alber C, Johnson J (1993) Sensing atmospheric water vapor with the Global Positioning System. Geophys Res Lett 20(23):2631–2634CrossRefGoogle Scholar
  34. Saastamoinen J (1972) Atmospheric correction for the troposphere and stratosphere in radio ranging of satellites. In: Henriksen SW, Mancini A, Chovitz BH (eds) The use of artificial satellites in geodesy, vol 15. AGU Geophys Monogr, Washington, pp 247–251CrossRefGoogle Scholar
  35. Sardon E, Rius A, Zarraoa A (1994) Estimation of transmitter and receiver differential biases and the ionospheric total electron content from Global Positioning System observations. Radio Sci 29(3):577–586CrossRefGoogle Scholar
  36. Schaer S (1999) Mapping and predicting the Earth’s ionosphere using the Global Positioning System. PhD thesis, Astronomical Institute, University of Berne, SwitzerlandGoogle Scholar
  37. Schönemann E, Becker M, Springer T (2011) A new approach for GNSS analysis in a multi-GNSS and multi-signal environment. J Geodetic Sci 1(3):204–214CrossRefGoogle Scholar
  38. Steigenberger P, Hugentobler U, Loyer S, Perosanz F, Prange L, Dach R, Uhlemann M, Gendt G, Montenbruck O (2015) Galileo orbit and clock quality of the IGS Multi-GNSS Experiment. Adv Space Res 55(1):269–281CrossRefGoogle Scholar
  39. Teunissen PJG (1985) Generalized inverses, adjustment, the datum problem and S-transformations. In: Sanso F, Grafarend EW (eds) Optimization of geodetic networks. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, pp 11–55CrossRefGoogle Scholar
  40. Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geodesy 70:65–82CrossRefGoogle Scholar
  41. Teunissen PJG, Khodabandeh A (2013) BLUE, BLUP and the Kalman filter: some new results. J Geodesy 87(5):461–473CrossRefGoogle Scholar
  42. Teunissen PJG, Khodabandeh A (2015) Review and principles of PPP-RTK methods. J Geodesy 89(3):217–240CrossRefGoogle Scholar
  43. Teunissen PJG, Kleusberg A (1998) GPS for geodesy. Springer-Verlag, Berlin Heidelberg New YorkCrossRefGoogle Scholar
  44. Teunissen PJG, Odijk D, Zhang B (2010) PPP-RTK: results of CORS network-based PPP with integer ambiguity resolution. J Aeronaut Astronaut Aviat Ser A 42(4):223–230Google Scholar
  45. Wen Z, Henkel P, Guenther C (2011) Reliable estimation of phase biases of GPS satellites with a local reference network. In: Proceedings of 53rd international symposium on ELMAR-2011, Zadar, Croatia, 14–16 September, pp 321–324Google Scholar
  46. Wilson BD, Mannucci AJ (1993) Instrumental biases in ionospheric measurements derived from GPS data. In: Proceedings of ION-GPS 1993, Salt Lake City, UT, 22–24 September, pp 1343–1351Google Scholar
  47. Wübbena G, Schmitz M, Bagge A (2005) PPP-RTK: precise point positioning using state-space reprentation in RTK networks. In: Proceedings of ION GNSS-2005, Long Beach, CA, 13–16 September, pp 2584–2594Google Scholar
  48. Yunck TP (1993) Coping with the atmosphere and ionosphere in precise satellite and ground positioning. In: Jones AV (ed) Environmental effects on spacecraft trajectories and positioning, vol 73. AGU Geophys Monogr Ser, Washington, pp 1–16CrossRefGoogle Scholar
  49. Zhang B, Teunissen PJG, Odijk D (2011) A novel un-differenced PPP-RTK concept. RIN J Navig 64:S180–S191CrossRefGoogle Scholar
  50. Zhong J, Lei J, Dou X, Yue X (2015) Is the long-term variation of the estimated GPS differential code biases associated with ionospheric variability? GPS Solut. doi:10.1007/s10291-015-0437-5 Google Scholar
  51. Zumberge JF, Heflin MB, Jefferson DC, Watkins MM, Webb FH (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102(3):5005–5017CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Dennis Odijk
    • 1
  • Baocheng Zhang
    • 1
  • Amir Khodabandeh
    • 1
  • Robert Odolinski
    • 1
    • 2
  • Peter J. G. Teunissen
    • 1
    • 3
  1. 1.GNSS Research CentreCurtin UniversityPerthAustralia
  2. 2.School of SurveyingUniversity of OtagoDunedinNew Zealand
  3. 3.Department of Geosciences and Remote SensingDelft University of TechnologyDelftThe Netherlands

Personalised recommendations