Journal of Geodesy

, Volume 87, Issue 1, pp 1–14 | Cite as

Reliability of partial ambiguity fixing with multiple GNSS constellations

Original Article

Abstract

Reliable ambiguity resolution (AR) is essential to real-time kinematic (RTK) positioning and its applications, since incorrect ambiguity fixing can lead to largely biased positioning solutions. A partial ambiguity fixing technique is developed to improve the reliability of AR, involving partial ambiguity decorrelation (PAD) and partial ambiguity resolution (PAR). Decorrelation transformation could substantially amplify the biases in the phase measurements. The purpose of PAD is to find the optimum trade-off between decorrelation and worst-case bias amplification. The concept of PAR refers to the case where only a subset of the ambiguities can be fixed correctly to their integers in the integer least squares (ILS) estimation system at high success rates. As a result, RTK solutions can be derived from these integer-fixed phase measurements. This is meaningful provided that the number of reliably resolved phase measurements is sufficiently large for least-square estimation of RTK solutions as well. Considering the GPS constellation alone, partially fixed measurements are often insufficient for positioning. The AR reliability is usually characterised by the AR success rate. In this contribution, an AR validation decision matrix is firstly introduced to understand the impact of success rate. Moreover the AR risk probability is included into a more complete evaluation of the AR reliability. We use 16 ambiguity variance–covariance matrices with different levels of success rate to analyse the relation between success rate and AR risk probability. Next, the paper examines during the PAD process, how a bias in one measurement is propagated and amplified onto many others, leading to more than one wrong integer and to affect the success probability. Furthermore, the paper proposes a partial ambiguity fixing procedure with a predefined success rate criterion and ratio test in the ambiguity validation process. In this paper, the Galileo constellation data is tested with simulated observations. Numerical results from our experiment clearly demonstrate that only when the computed success rate is very high, the AR validation can provide decisions about the correctness of AR which are close to real world, with both low AR risk and false alarm probabilities. The results also indicate that the PAR procedure can automatically chose adequate number of ambiguities to fix at given high-success rate from the multiple constellations instead of fixing all the ambiguities. This is a benefit that multiple GNSS constellations can offer.

Keywords

Reliability Partial ambiguity resolution Partial ambiguity decorrelation Multiple GNSS 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cao W, O’Keefe K, Cannon M (2007) Partial ambiguity fixing within multiple frequencies and systems. In: Proceedings of ION GNSS07, The Satellite Division of the Institute of Navigation 20th International Technical Meeting, Fort Worth, TX, September 25–28 2007, pp 312–323Google Scholar
  2. Cao W, O’Keefe K, Cannon M (2008a) Performance evaluation of GPS/Galileo multiple-frequency RTK positioning using a single-difference processor. In: Proceedings of the 21st international technical meeting of the satellite division of the institute of navigation, Savannah, GA, September 2008, pp 2841–2849Google Scholar
  3. Cao W, O’Keefe K, Petovello M, Cannon M (2008b) Simulated performance of multiple-signal and multiple-system positioning for land vehicle navigation. In: Proceedings of the 2008 national technical meeting of the institute of navigation, San Diego, CA, January 2008, pp 603–613Google Scholar
  4. Cao W (2009) Multi-frequency GPS and Galileo kinematic positioning with partial ambiguity fixing. Master thesis, University of CalgaryGoogle Scholar
  5. Euler HJ, Schaffrin B (1990) On a measure for the discernibility between different ambiguity solutions in the static-kinematic GPS-mode. In: Paper presented at the IAG Symposia no. 107, Kinematic Systems in Geodesy, Surveying, and Remote Sensing. Springer, New York, pp 285–295Google Scholar
  6. Feng Y (2005) Future GNSS performance predictions using GPS with a virtual Galileo constellation. GPS World 16(3): 46–52Google Scholar
  7. Feng Y, Wang J (2011) Computed success rates of various carrier phase integer estimation solutions and their comparison with statistical success rates. J Geod 85(2): 93–103. doi:10.1007/s00190-010-0418-y CrossRefGoogle Scholar
  8. Grafarend EW (2000) Mixed integer-real valued adjustment (IRA) problems: GPS initial cycle ambiguity resolution by means of the LLL algorithm. GPS Solut 4(2): 31–44CrossRefGoogle Scholar
  9. Han S, Rizos C (1996) Integrated methods for instantaneous ambiguity resolution using new-generation GPS receivers. In: Proceedings of IEEE PLANS’96, Atlanta, GA, pp 245–261Google Scholar
  10. Hassibi A, Boyd S (1998) Integer parameter estimation in linear models with applications to GPS. IEEE Trans Signal Process 46(11): 2938–2952CrossRefGoogle Scholar
  11. Henkel P, Günther C (2009) Partial integer decorrelation: optimum trade-off between variance reduction and bias amplification. J Geod 84(1): 51–63CrossRefGoogle Scholar
  12. Henkel P, Günther C (2012) Reliable integer ambiguity resolution: multi-frequency code carrier linear combinations and statistical a priori knowledge of attitude. Navigation 59(1): 61–75CrossRefGoogle Scholar
  13. Kovach K, Maquet H, Davis D (1995) PPS RAIM algorithms and their performance. Navigation (Washington, DC) 42(3): 515–529Google Scholar
  14. Leick A (2004) GPS satellite surveying, 3rd edn. Wiley, New YorkGoogle Scholar
  15. Mowlam AP, Collier PA (2004) Fast ambiguity resolution performance using partially-fixed multi-GNSS phase observations. In: Paper presented at the the 2004 International Symposium on GNSS/GPS, Sydney, Australia, 6–8 DecemberGoogle Scholar
  16. O’Keefe K (2001) Availability and reliability advantages of GPS/Galileo integration. In: Proceedings of the 14th international technical meeting of the satellite division of the institute of navigation (ION GPS 2001), Salt Lake City, UT, September 2001, pp 2096–2104Google Scholar
  17. O’Keefe K, Petovello M, Lachapelle G, Cannon ME (2006) Assessing probability of correct ambiguity resolution in the presence of time-correlated errors. Navigation (Washington, DC) 53(4): 269–282Google Scholar
  18. Ong RB, Petovello MG, Lachapelle G (2009) Assessment of GPS/GLONASS RTK under a variety of operational conditions. In: Proceedings of the 22nd international technical meeting of the satellite division of the institute of navigation, Savannah, GA, September 2009, pp 3297–3308Google Scholar
  19. Parkins A (2011) Increasing GNSS RTK availability with a new single-epoch batch partial ambiguity resolution algorithm. GPS Solut 15(4): 391–402CrossRefGoogle Scholar
  20. Teunissen PJG (1993) Least-squares estimation of the integer GPS ambiguities. Invited lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing, China, August, also in Delft Geodetic Computing Centre LGR series, No. 6, pp 16Google Scholar
  21. Teunissen PJG (1998) Success probability of integer GPS ambiguity rounding and bootstrapping. J Geod 72(10): 606–612CrossRefGoogle Scholar
  22. Teunissen PJG (1999) An optimality property of the integer least-squares estimator. J Geod 73(11): 587–593CrossRefGoogle Scholar
  23. Teunissen PJG (2000) The success rate and precision of GPS ambiguities. J Geod 74(3): 321–326CrossRefGoogle Scholar
  24. Teunissen PJG (2001) Integer estimation in the presence of biases. J Geod 75(7): 399–407CrossRefGoogle Scholar
  25. Teunissen PJG (2003) An invariant upperbound for the GNSS bootstrapped ambiguity success rate. JGPS 2(1): 13–17CrossRefGoogle Scholar
  26. Teunissen PJG, Odijk D (1997) Ambiguity dilution of precision: definition, properties and application. In: Proceedings of the 10th international technical meeting of the satellite division of the institute of navigation, Kansas City, MO, September 16–19, pp 891– 899Google Scholar
  27. Teunissen PJG, Verhagen S (2008) GNSS ambiguity resolution: when and how to fix or not to fix?. In: VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy: Challenge and Role of Modern Geodesy, Wuhan, China, 29 May–2 June 2006, Series: International Association of Geodesy Symposia, vol 132, pp 143–148Google Scholar
  28. Teunissen PJG, Joosten P, Odijk D (1999) The reliability of GPS ambiguity resolution. GPS Solut 2(3): 63–69CrossRefGoogle Scholar
  29. Teunissen PJG, Joosten P, Tiberius C (1999b) Geometry-free ambiguity success rates in case of partial fixing. In: Proceedings of the 1999 National Technical Meeting of The Institute of Navigation, San Diego, CA, January 25–27 1999, pp 201–207Google Scholar
  30. Verhagen S (2003) On the approximation of the integer least-squares success rate: which lower or upper bound to use. JGPS 2(2): 117–124CrossRefGoogle Scholar
  31. Verhagen S (2005) On the reliability of integer ambiguity resolution. Navigation (Washington, DC) 52(2): 99–110Google Scholar
  32. Verhagen S, Odijk D, Teunissen PJG, Huisman L (2010) Performance improvement with low-cost multi-GNSS receivers. In: Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (NAVITEC), 2010 5th ESA Workshop on, 8–10 December 2010, pp 1–8Google Scholar
  33. Wei M, Schwarz KP (1995) Fast ambiguity resolution using an integer nonlinear programming method. In: Proceedings of the 8th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 1995), Palm Springs, CA, September 1995, pp 1101–1110Google Scholar
  34. Xu P (2001) Random simulation and GPS decorrelation. J Geod 75(7): 408–423CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Faculty of Science and TechnologyQueensland University of TechnologyBrisbaneAustralia

Personalised recommendations