Journal of Geodesy

, Volume 93, Issue 6, pp 877–888 | Cite as

Helmert-VCE-aided fast-WTLS approach for global ionospheric VTEC modelling using data from GNSS, satellite altimetry and radio occultation

  • Andong HuEmail author
  • Zishen Li
  • Brett Carter
  • Suqin Wu
  • Xiaoming Wang
  • Robert Norman
  • Kefei ZhangEmail author
Original Article


Vertical total electron content (VTEC) global ionospheric maps (GIM) are commonly used to correct the ionospheric delay of global navigation satellite system (GNSS) signals for single-frequency positioning and other ionospheric studies. The measurements observed by inhomogeneously distributed ground reference stations are the only data used to generate the GIMs. Thus the accuracy of the GIMs over ocean and polar regions is relatively poor due to the lack of measurements over these regions. In this study, space-borne VTECs obtained from ocean-altimetry and GNSS radio occultation measurements are incorporated into the modelling process. Since the three types of VTEC data have different qualities, the weight for each type of data is determined using the Helmert-variance component estimation (Helmert-VCE) method. In addition, unlike the traditional weighted least squares (WLS) estimation method in which the design matrix of observation equations is fixed, in this study, the design matrix, especially those elements in design matrix that are derived from the coordinates of either tangent point or ionospheric pierce point, are considered to be inaccurate. Thus they are adjusted together with the unknown coefficient parameters of the fitting model using the fast-weighted total least squares (fast-WTLS) technique. The proposed approach, named Helmert-WTLS, was tested using the data in the period of day of year (DOY) 217–224, 2016 and validated using GIMs produced by the research team for ionosphere and precise positioning based on BDS/GNSS (GIPP) at the Academy of Opto-Electronics, Chinese Academy of Sciences (CAS). Comparison results showed that the GIMs (with a 2 h temporal resolution) generated using the new approach can improve the determination of ionospheric TEC by 0.28 TEC units (TECU) over those from the Helmert-VCE-aided WLS approach (w.r.t CAS references, respectively) and by 1.61 TECU better than those from WLS, in terms of the mean of all root-mean-squares errors of all 2 h time slots in the 8-day testing period. In addition, in comparison with out-of-sample Jason-3 observations, results from the proposed method also outperformed Helmert-VCE-aided WLS, CAS and CODE models by 1.5, 2.4 and 2.4 TECU, respectively.


Global navigation satellite system (GNSS) Radio occultation Satellite altimetry VTEC GIM WTLS Helmert-VCE 



This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDA17010304) and Australian Research Council project (LP160100561). This work was also funded by the Jiangsu dual creative talents and teams programme projects awarded in 2017. We also would like to acknowledge the support of the Natural Science Foundation of China (41730109, 41674043, 41704038, 41874040), Beijing Nova programme (xx2017042), Beijing Youth Talent Support program (2017000021223ZK13), Pioneer Hundred Talents Program, National Key Research and Development Plan (2016YFB0501405) of the Chinese Academy of Sciences and Beijing Natural Science Foundation (8184092). Andong Hu acknowledges the China Scholarship Council for the provision of a scholarship for his study at the SPACE Research Centre, RMIT University. Zishen Li acknowledges the Australian Dept. of Education and Training for the provision of an Endeavour scholarship for his research at the SPACE Research Centre. We thank IGS for providing GNSS data and precise DCB values for all transmitters and receivers, UCAR/CDAAC for providing ionPrf data, and CNES and NOAA for providing Jason-3 IGDR data.


  1. Alizadeh M, Schuh H, Todorova S, Schmidt M (2011) Global ionosphere maps of VTEC from GNSS, satellite altimetry, and formosat-3/COSMIC data. J Geodesy 85(12):975–987CrossRefGoogle Scholar
  2. Anthes RA et al (2008) The COSMIC/FORMOST-3 mission: early results. Am Meteorol Soc 89:313–333CrossRefGoogle Scholar
  3. Azpilicueta F, Brunini C (2009) Analysis of the bias between TOPEX and GPS vTEC determinations. J Geodesy 83(2):121–127CrossRefGoogle Scholar
  4. Bilitza D (1997) International reference ionosphere—status 1995/96. Adv Space Res 20(9):1751–1754CrossRefGoogle Scholar
  5. Bilitza D (2001) International reference ionosphere 2000. Radio Sci 36(2):261–275CrossRefGoogle Scholar
  6. Bilitza D, Reinisch BW (2008) International reference ionosphere 2007: improvements and new parameters. Adv Space Res 42(4):599–609CrossRefGoogle Scholar
  7. Bilitza D, Rawer K, Bossy L, Kutiev I, Oyama K-I, Leitinger R, Kazimirovsky E (1990) International reference ionosphereGoogle Scholar
  8. Chen P, Yao W, Zhu X (2015) Combination of ground-and space-based data to establish a global ionospheric grid model. IEEE Trans Geosci Remote Sens 53(2):1073–1081CrossRefGoogle Scholar
  9. Ciraolo L, Azpilicueta F, Brunini C, Meza A, Radicella S (2007) Calibration errors on experimental slant total electron content (TEC) determined with GPS. J Geodesy 81(2):111–120CrossRefGoogle Scholar
  10. Dettmering D, Schmidt M, Heinkelmann R, Seitz M (2011) Combination of different space-geodetic observations for regional ionosphere modeling. J Geodesy 85(12):989–998CrossRefGoogle Scholar
  11. Dumont J, Rosmorduc V, Carrere L, Picot N, Bronner E, Couhert A, Guillot A, Desai S, Bonekamp H, Figa J (2016) Jason-3 products handbook. Rep SALP-MU-M-OP-16118-CN Google ScholarGoogle Scholar
  12. Golub GH, Van Loan CF (1980) An analysis of the total least squares problem. SIAM J Numer Anal 17(6):883–893CrossRefGoogle Scholar
  13. Hajj GA, Ibanez-Meier R, Kursinski ER, Romans LJ (1994) Imaging the ionosphere with the global positioing system. Int J Imaging Syst Technol 5:174–184CrossRefGoogle Scholar
  14. Henderson HV, Searle SR (1981) The vec-permutation matrix, the vec operator and Kronecker products: a review. Linear Multilinear Algebra 9(4):271–288CrossRefGoogle Scholar
  15. Hernández-Pajares M, Juan J, Sanz J (1999) New approaches in global ionospheric determination using ground GPS data. J Atmos Solar Terr Phys 61(16):1237–1247CrossRefGoogle Scholar
  16. Hernández-Pajares M, Juan JM, Sanz J, Orus R, Garcia-Rigo A, Feltens J, Komjathy A, Schaer SC, Krankowski A (2009) The IGS VTEC maps: a reliable source of ionospheric information since 1998. J Geodesy 83(3):263–275. CrossRefGoogle Scholar
  17. Hernández Pajares M, Roma Dollase D, Krankowski A, García Rigo A, Orús Pérez R (2016) Comparing performances of seven different global VTEC ionospheric models in the IGS context. In: Paper presented at international GNSS service workshop (IGS 2016): Sydney, Australia: 8–12 Feb 2016, International GNSS Service (IGS)Google Scholar
  18. Hochegger G, Nava B, Radicella S, Leitinger R (2000) A family of ionospheric models for different uses. Phys Chem Earth Part C Solar Terr Planet Sci 25(4):307–310Google Scholar
  19. Imel DA (1994) Evaluation of the TOPEX/poseidon dual-frequency ionosphere correction. J Geophys Res Oceans 99(C12):24895–24906CrossRefGoogle Scholar
  20. Li Z, Yuan Y, Li H, Ou J, Huo X (2012) Two-step method for the determination of the differential code biases of COMPASS satellites. J Geodesy 86(11):1059–1076CrossRefGoogle Scholar
  21. Li Z, Yuan Y, Wang N, Hernandez-Pajares M, Huo X (2015) SHPTS: towards a new method for generating precise global ionospheric TEC map based on spherical harmonic and generalized trigonometric series functions. J Geodesy 89(4):331–345CrossRefGoogle Scholar
  22. Mannucci A, Wilson B, Yuan D, Ho C, Lindqwister U, Runge T (1998) A global mapping technique for GPS-derived ionospheric total electron content measurements. Radio Sci 33(3):565–582CrossRefGoogle Scholar
  23. Radicella S, Leitinger R (2001) The evolution of the DGR approach to model electron density profiles. Adv Space Res 27(1):35–40CrossRefGoogle Scholar
  24. Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geodesy 82(7):415–421CrossRefGoogle Scholar
  25. Scherliess L, Schunk RW, Sojka JJ, Thompson DC (2004) Development of a physics‐based reduced state Kalman filter for the ionosphere. Radio Sci 39(1):RS1S04CrossRefGoogle Scholar
  26. Schreiner W, Rocken C, Sokolovskiy S, Syndergaard S, Hunt D (2007) Estimates of the precision of GPS radio occultation from the COSMIC/FORMOSAT-3 mission. Geophys Res Lett 34(4):L04808CrossRefGoogle Scholar
  27. Schunk RW, Scherliess L, Sojka JJ, Thompson DC, Anderson DN, Codrescu M, Minter C, Fuller-Rowell TJ, Heelis RA, Hairston M (2004) Global assimilation of ionospheric measurements (GAIM). Radio Sci 39(1):RS1S04CrossRefGoogle Scholar
  28. Shen Y, Li B, Chen Y (2011) An iterative solution of weighted total least-squares adjustment. J Geodesy 85(4):229–238CrossRefGoogle Scholar
  29. Sokolovskiy S, Rocken C (2006) Algorithms for inverting radio occultation signals in the ionosphere. University Corporation for Atmospheric ResearchGoogle Scholar
  30. Todorova S, Hobiger T, Schuh H (2008) Using the global navigation satellite system and satellite altimetry for combined global ionosphere maps. Adv Space Res 42(4):727–736CrossRefGoogle Scholar
  31. Wang N, Yuan Y, Li Z, Montenbruck O, Tan B (2016) Determination of differential code biases with multi-GNSS observations. J Geodesy 90(3):209–228CrossRefGoogle Scholar
  32. Xu P, Shen Y, Fukuda Y, Liu Y (2006) Variance component estimation in linear inverse ill-posed models. J Geodesy 80(2):69–81CrossRefGoogle Scholar
  33. Yuan Y, Li Z, Wang N, Zhang B, Li H, Li M, Huo X, Ou J (2015) Monitoring the ionosphere based on the crustal movement observation network of China. Geodesy Geodyn 6(2):73–80CrossRefGoogle Scholar
  34. Zehfuss G (1858) Über eine gewisse determinante. Z Math Phys 3(1858):298–301Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.SPACE Research Centre, School of SciencesRMIT UniversityMelbourneAustralia
  2. 2.School of Environmental Science and Spatial InformaticsChina University of Mining and TechnologyXuzhouChina
  3. 3.Academy of Opto-ElectronicsChinese Academy of SciencesBeijingChina

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