Acta Geodaetica et Geophysica

, Volume 54, Issue 3, pp 333–357 | Cite as

Model assessment of GNSS-based regional TEC modeling: polynomial, trigonometric series, spherical harmonic and multi-surface function

  • Bofeng LiEmail author
  • Miaomiao Wang
  • Yueliang Wang
  • Haijing Guo
Original Study


The ground-based GNSS VTEC model can adequately capture the spatiotemporal characteristics of ionosphere activities. However, it is difficult to precisely model VTECs with a unified mathematical model. We dedicate to comprehensively assess the performance of four mathematical models for VTEC modeling, i.e., polynomial, trigonometric series, spherical harmonic and multi-surface function. To capture the varying ionosphere situations, three typical regions in Western Europe, Southeast China, and North America are chosen. To reflect the precision and accuracy from different aspects, four evaluation measures are defined based on the comparison with CODE GIM products, the poster-fitting residuals of VTEC modeling, the comparison with high-precision quasi-VTEC computed with L1–L2 phase observations as well as the analysis of precision and stability of receiver and satellite DCB by-products. The results show that, in small regions, the performances of all models are insensitive to the model orders but sensitive to the ionosphere activity. On the whole, the polynomial and spherical harmonic function are comparable in terms of their performance and computation efficiency; Trigonometric series is instable with systematic biases in large region due to its incapability of describing the spatial ionosphere variations; Multi-surface function outperforms the others thanks to its epoch-wise solution with low computation efficiency. In addition, the annual solutions of satellite and receiver DCBs indicate their sufficient stability, which can then be applied as known for single-epoch ionosphere modeling.


Ionosphere modeling Polynomial Trigonometric series Spherical harmonic Multi-surface function Differential code bias 



The study is sponsored by the National Key Research and Development Program of China (2016YFB0501802, 2017YFA0603102), the National Natural Science Funds of China (41874030, 41622401 and 41730102), the Scientific and Technological Innovation Plan from Shanghai Science and Technology Committee (18511101801) and the Fundamental Research Funds for the Central Universities. The second author is financially supported by the China Scholarship Council (CSC) for his study at the German Research Centre for Geosciences GFZ.


  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 Geod 85(12):975–987CrossRefGoogle Scholar
  2. Alves D, Monico J (2011) GPS/VRS positioning using atmospheric modeling. GPS Solut 15(3):253–261CrossRefGoogle Scholar
  3. An J, Wang Z, Ning X (2014) GPS-based regional ionospheric models and their suitability in Antarctica. Adv Polar Sci 25(1):32–37Google Scholar
  4. Bidaine B, Warnant R (2010) Assessment of the NeQuick model at mid-latitudes using GNSS TEC and ionosonde data. Adv Space Res 45(9):1122–1128CrossRefGoogle Scholar
  5. Bilitza D, Reinisch BW (2008) International reference ionosphere 2007: improvements and new parameter. Adv Space Res 42(4):599–609CrossRefGoogle Scholar
  6. Bilitza D, McKinnell L, Reinisch B, Fuller-Rowell T (2011) The international reference ionosphere today and in the future. J Geod 85(12):909–920CrossRefGoogle Scholar
  7. Camargo P (2009) Quality of TEC estimated with Mod_Ion using GPS and GLONASS data. Math Probl Eng 16(4):266–287Google Scholar
  8. Camargo P, Monico J, Ferreira L (2000) Application of ionospheric corrections in the equatorial region for L1 GPS users. Earth Planets Space 52(11):1083–1089CrossRefGoogle Scholar
  9. Cander L (2008) Ionospheric research and space weather services. J Atmos Solar Terr Phys 70(15):1870–1878CrossRefGoogle Scholar
  10. Ciraolo L, Azpilicueta F, Brunini C, Meza A, Radicella S (2007) Calibration errors on experimental slant total electron content (TEC) determined with GPS. J Geod 81(2):111–120CrossRefGoogle Scholar
  11. Coster A, Gaposchkin E, Thornton L (1992) Real-time ionospheric monitoring system using GPS. Navigation 39(2):191–204CrossRefGoogle Scholar
  12. Georgiadiou Y (1994) Modeling the ionosphere for an active control network of GPS stations. Delft Geodetic Computing Centre LGR Series No. 7, Delft University of TechnologyGoogle Scholar
  13. Hardy R (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915CrossRefGoogle Scholar
  14. Huang L, Tao B, Zhao C (1993) The application of fitting method of multiquadric functions in research on crustal vertical movement. Acta Geod Cartogr Sin 22(1):25–32Google Scholar
  15. Huba J, Joyce G, Fedder J (2000) SAMI2 (Sami2 is another model of the ionosphere): a new low-latitude ionosphere model. J Geophys Res: Space Phys 105(10):035–053Google Scholar
  16. Jakowski N, Mayer C, Hoque M, Wilken V (2011) Total electron content models and their use in ionosphere monitoring. Radio Sci 46(6):1. CrossRefGoogle Scholar
  17. Jin R, Jin S, Feng G (2012) M_DCB: matlab code for estimating GNSS satellite and receiver differential code biases. GPS Solut 16(4):541–548CrossRefGoogle Scholar
  18. Klobuchar JA (1987) Ionospheric time-delay algorithm for single-frequency GPS users. IEEE Trans Aerosp Electr Syst 23(3):325–331CrossRefGoogle Scholar
  19. Komjathy A, Sparks L, Wilson B, Mannucci AJ (2005) automated daily processing of more than 1000 ground-based GPS receivers to study intense ionospheric storms. Radio Sci 40:RS6006. CrossRefGoogle Scholar
  20. Lanyi GE, Roth TA (1988) A comparison of mapped and measured total ionospheric electron content using global positioning system and beacon satellite observations. Radio Sci 23(4):483–492CrossRefGoogle Scholar
  21. Liu J, Wang Z, Zhang H, Zhu W (2008) Comparison and consistency research of regional ionospheric TEC models based on GPS measurements. Geom Inform Sci Wuhan Univ 33(5):479–482Google Scholar
  22. Liu J, Chen R, Wang Z, Zhang H (2011) Spherical cap harmonic model for mapping and predicting regional TEC. GPS Solut 15(2):109–119CrossRefGoogle Scholar
  23. Memarzadeh Y (2009) Ionospheric modeling for precise GNSS applications. Ph.D. dissertation, Aerospace Engineering, Delft University of TechnologyGoogle Scholar
  24. Nava B, Coïsson P, Radicella S (2008) A new version of the NeQuick ionosphere electron density model. J Atmos Sol Terr Phys 70(15):1856–1862CrossRefGoogle Scholar
  25. Prasad R, Kumar S, Jayachandran PT (2016) Receiver DCB estimation and GPS vTEC study at a low latitude station in the South Pacific. J Atmos Sol Terr Phys 149:120–130CrossRefGoogle Scholar
  26. Qian L, Burns A, Emery B, Foster B, Lu G, Maute A, Richmond A, Roble R, Solomon S, Wangm W (2013) The NCAR TIE-GCM: a community model of the coupled thermosphere/ionosphere system. Geophys Monogr 201(2):73–83Google Scholar
  27. Sardón E, Zarraoa N (1997) Estimation of total electron content using GPS data: how stable are the differential satellite and receiver instrumental biases? Radio Sci 32(5):1899–1910CrossRefGoogle Scholar
  28. Schaer S (1999) Mapping and predicting the earth’s ionosphere using the global positioning system. Ph.D. dissertation, Bern, Switzerland, University of BernGoogle Scholar
  29. Tao A, Jan S (2016) Wide-area ionospheric delay model for GNSS users in middle- and low-magnetic-latitude regions. GPS Solut 20(1):9–21CrossRefGoogle Scholar
  30. Wilson B, Mannucci A (1993) Instrumental biases in ionospheric measurements derived from GPS data. In: Proceedings of international technical meeting of the Satellite Division of the Institute of Navigation, Salt Lake City, UtahGoogle Scholar
  31. Yuan Y, Ou J (2004) A generalized trigonometric series function model for determining ionospheric delay. Prog Nat Sci 14(11):1010–1014CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó 2019

Authors and Affiliations

  1. 1.College of Surveying and Geo-InformaticsTongji UniversityShanghaiChina
  2. 2.Department of GeodesyGeoForschungsZentrum (GFZ)PotsdamGermany
  3. 3.Surveying and Mapping Institute, Lands and Resource Department of Guangdong ProvinceGuangzhouChina

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