Journal of Geodesy

, Volume 92, Issue 11, pp 1241–1253 | Cite as

Adaptive Kalman filter based on variance component estimation for the prediction of ionospheric delay in aiding the cycle slip repair of GNSS triple-frequency signals

  • Guobin Chang
  • Tianhe XuEmail author
  • Yifei Yao
  • Qianxin Wang
Original Article


In order to incorporate the time smoothness of ionospheric delay to aid the cycle slip detection, an adaptive Kalman filter is developed based on variance component estimation. The correlations between measurements at neighboring epochs are fully considered in developing a filtering algorithm for colored measurement noise. Within this filtering framework, epoch-differenced ionospheric delays are predicted. Using this prediction, the potential cycle slips are repaired for triple-frequency signals of global navigation satellite systems. Cycle slips are repaired in a stepwise manner; i.e., for two extra wide lane combinations firstly and then for the third frequency. In the estimation for the third frequency, a stochastic model is followed in which the correlations between the ionospheric delay prediction errors and the errors in the epoch-differenced phase measurements are considered. The implementing details of the proposed method are tabulated. A real BeiDou Navigation Satellite System data set is used to check the performance of the proposed method. Most cycle slips, no matter trivial or nontrivial, can be estimated in float values with satisfactorily high accuracy and their integer values can hence be correctly obtained by simple rounding. To be more specific, all manually introduced nontrivial cycle slips are correctly repaired.


GNSS BDS Triple frequency Cycle slip Ionospheric delay Adaptive Kalman filter Variance component estimation 



This work is supported by the National Natural Science Foundation of China (41574013, 41774005) and the National Key Research and Development Program of China (2016YFB0501701). We thank three reviewers for their valuable comments which greatly improve the paper. The test data are provided by iGMAS. We thank Glenn Pennycook, MSc, from Liwen Bianji, Edanz Group China (, for editing the English text of a draft of this manuscript.


  1. Amiri-Simkooei AR (2016) Non-negative least-squares variance component estimation with application to GPS time series. J Geod 90:451–466CrossRefGoogle Scholar
  2. Chang G (2014) On Kalman filter for linear system with colored measurement noise. J Geod 88:1163–1170CrossRefGoogle Scholar
  3. Chang G, Liu M (2015) An adaptive fading Kalman filter based on Mahalanobis distance. Proc Inst Mech Eng Part G J Aerosp Eng 229:1114–1123CrossRefGoogle Scholar
  4. Cocard M, Bourgon S, Kamali O, Collins P (2008) A systematic investigation of optimal carrier-phase combinations for modernized triple-frequency GPS. J Geod 82:555–564CrossRefGoogle Scholar
  5. de Lacy MC, Reguzzoni M, Sanso F (2012) Real-time cycle slip detection in triple-frequency GNSS. GPS Solut 16:342–353Google Scholar
  6. Huang L, Lu Z, Zhai G, Ouyang Y, Huang M, Lu X, Wu T, Li K (2016) A new triple-frequency cycle slip detecting algorithm validated with BDS data. GPS Solut 20:761–769CrossRefGoogle Scholar
  7. Li B, Shen Y, Feng Y, Gao W, Yang L (2014) GNSS ambiguity resolution with controllable failure rate for long baseline network RTK. J Geod 88:99–112CrossRefGoogle Scholar
  8. Li B, Qin Y, Li Z, Lou L (2017a) Undifferenced cycle slip estimation of triple-frequency BeiDou signals with ionosphere prediction. Mar Geod 39:348–365CrossRefGoogle Scholar
  9. Li J, Yang Y, He H, Guo H (2017b) An analytical study on the carrier-phase linear combinations for triple-frequency GNSS. J Geod 91:151–166CrossRefGoogle Scholar
  10. Li W, Teunissen PJG, Zhang B, Verhagen S (2013) Precise point positioning using GPS and compass observations. In: Sun J, Jiao W, Wu H, Shi C (eds) Proceedings of China satellite navigation conference (CSNC) 2013. Springer, Wuhan, pp 367–378Google Scholar
  11. Liu Z (2011) A new automated cycle slip detection and repair method for a single dual-frequency GPS receiver. J Geod 85:171–183CrossRefGoogle Scholar
  12. Odijk D, Khodabandeh A, Nadarajah N, Choudhury M, Zhang B, Li W, Teunissen PJG (2017) PPP-RTK by means of S-system theory: Australian network and user demonstration. J Spat Sci 62:3–27CrossRefGoogle Scholar
  13. Simon D (2006) Optimal state estimation: Kalman, H\(\infty \), and nonlinear approaches. Wiley, HobokenCrossRefGoogle Scholar
  14. Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geod 70:65–82CrossRefGoogle Scholar
  15. Teunissen PJG (2002) The parameter distributions of the integer GPS model. J Geod 76:41–48CrossRefGoogle Scholar
  16. Teunissen PJG, Amiri-Simkooei AR (2008) Least-squares variance component estimation. J Geod 82:65–82CrossRefGoogle Scholar
  17. Teunissen PJG, de Bakker PF (2013) Single-receiver single-channel multi-frequency GNSS integrity: outliers, slips, and ionospheric disturbances. J Geod 87:161–177CrossRefGoogle Scholar
  18. Verhagen S, Li B, Teunissen PJG (2013) Ps-LAMBDA: ambiguity success rate evaluation software for interferometric applications. Comput Geosci 54:361–376CrossRefGoogle Scholar
  19. Wanninger L, Beer S (2015) BeiDou satellite-induced code pseudorange variations: diagnosis and therapy. GPS Solut 19:639–648CrossRefGoogle Scholar
  20. Xu P, Liu J (2014) Variance components in errors-in-variables models: estimability, stability and bias analysis. J Geod 88:719–734CrossRefGoogle Scholar
  21. Xu P, Liu Y, Shen Y, Fukuda Y (2007) Estimability analysis of variance and covariance components. J Geod 81:593–602CrossRefGoogle Scholar
  22. Yang Y, Gao W (2006) An optimal adaptive Kalman filtering. J Geod 80:177–183CrossRefGoogle Scholar
  23. Yang Y, Xu T (2003) An adaptive Kalman filter based on sage windowing weights and variance components. J Navig 56:231–240CrossRefGoogle Scholar
  24. Yang Y, He H, Xu G (2001) Adaptively robust filtering for kinematic geodetic positioning. J Geod 75:109–116CrossRefGoogle Scholar
  25. Yao Y, Gao J, Wang J, Hu H, Li Z (2016) Real time cycle slip detection and repair for BeiDou triple frequency undifferenced observations. Surv Rev 48:367–375CrossRefGoogle Scholar
  26. Zhang X, He X (2015) BDS triple-frequency carrier-phase linear combination models and their characteristics. Sci China Earth Sci 58:896–905CrossRefGoogle Scholar
  27. Zhang X, Li P (2016) Benefits of the third frequency signal on cycle slip correction. GPS Solut 20:541–560Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Guobin Chang
    • 1
    • 4
  • Tianhe Xu
    • 2
    • 4
    Email author
  • Yifei Yao
    • 3
  • Qianxin Wang
    • 1
    • 4
  1. 1.School of Environmental Science and Spatial InformaticsChina University of Mining and TechnologyXuzhouChina
  2. 2.Institute of Space ScienceShandong UniversityWeihaiChina
  3. 3.College of Water Resources and Architectural EngineeringNorthwest A&F UniversityYanglingChina
  4. 4.State Key Laboratory of Geo-Information EngineeringXi’an Research Institute of Surveying and MappingXi’anChina

Personalised recommendations