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Precise modeling of slant total electron contents with multidimensional free interpolation

DOI 10.18127/j20700784-201812-18


M.M. Valikhanov – Ph.D. (Eng.), Associate Professor, Siberian Federal University (Krasnoyarsk)
V.V. Denisenko – Dr.Sc. (Phys.-Math.), Professor, Siberian Federal University (Krasnoyarsk); Institute of computational modeling SB RAS (Krasnoyarsk)
S.P. Tsarev – Dr.Sc. (Phys.-Math.), Professor, Siberian Federal University (Krasnoyarsk)

Standard models of ionospheric delays widely used for practical processing of navigation signals from global navigation satellite systems (GNSS) usually approximate the complicated three-dimensional electron density distribution in the Earth's ionosphere and plasmasphere with one- or two-layer distributions and have errors of order 1−8 TECU (total electron content units). Publicly available modern three-dimensional ionospheric models IRI-2016 and NeQuick2 are very complex, they are not easy to use in GNSS applications and have similar large deviations from real ionospheric data. We propose a new interpolation model of ionospheric delays which is much more precise than the one- or two-layer models but simple enough to be used in radiophysics and GNSS applications. It is based on our recent free interpolation framework used previously in finding positions of GNSS satellites from SP3 data with very high precision. The free interpolation framework is not limited to polynomial, trigonometric or spherical interpolating functions and uses a simple machine learning approach for definition of the interpolation coefficients. Our model approximates the slant TEC angular distributions obtained from modern three-dimensional ionospheric models with RMS error 0,02 TECU. The new model of ionospheric delays is universal, stable and have weak dependence on the position of the GNSS ground receiver.
Experimental verification of our model was performed for two IGS stations: АМС4 (B=38,80312°, L= -104,524594°, H=1912,5 м) and PIE1 (B=34,301505°, L= -108,118927°, H = 2347,7 м) for STEC distributions calculated using both ionospheric models IRI-2016 and NeQuick2 for the years 2008 and 2017. We show that interpolation coefficients may be chosen equal for both stations and both years. Investigating deeper one may observe a small seasonal variations of the residuals of our model: RMS = 0,03 TECU for winter months and RMS = 0,05 TECU for summer months for the smallest number of parameters in our model N=7. For N=10 parameters one obtains RMS < 0,02 TECU.
Further research is planned to test the free interpolation ionospheric STEC model on real measurements from GNSS stations. Integration of our model and methods for differential code biases (DCBs in satellite transmitters and ground receivers) separation is obviously necessary for this test. Possible inclusion of higher order ionospheric effects is under study.

  1. Продукты IGS. URL:
  2. Hernández-Pajares M., Juan J.M., Sanz, J. et al. The ionosphere: effects, GPS modeling and the benefits for space geodetic techniques. J. of Geodesy. 2011. № 85(12). P. 887–907.
  3. Roma-Dollase D., Hernández-Pajares M., Krankowski A. et al. Consistency of seven different GNSS global ionospheric mapping techniques during one solar cycle // Journal of Geodesy. 2018. Т. 92. №. 6. P. 691–706.
  4. Lyu H., Hernández-Pajares M., Nohutcu M. et al. The Barcelona ionospheric mapping function (BIMF) and its application to northern mid-latitudes // GPS Solutions. 2018. Т. 22. С. 1–13.
  5. Myl'nikova A.A., Yasyukevich Yu.V., Dem'yanov V.V. Opredelenie absolyutnogo vertikal'nogo polnogo elektronnogo soderzhaniya v ionosfere po dannym GLONASS/GPS // Solnechno-zemnaya fizika. 2013. №. 24. S. 70–77.
  6. Pustoshilov A., Tsarev S. Universal Coefficients for Precise Interpolation of GNSS Orbits from Final IGS SP3 Data // International Siberian Conference on Control and Communications, SIBCON 2017 – Proceedings. URL: document/7998463
  7. Pustoshilov A., Carev S.P. Vysokotochnoe vosstanovlenie orbit sputnikov GNSS metodom obucheniya po rasshirennym SP3–dannym // Uspehi sovremennoj radioelektroniki. 2017. № 12. S. 48–52.
  8. Bilitza D., Altadill D., Truhlik V., Shubin V., Galkin I., Reinisch B., Huang X. International Reference Ionosphere 2016: From ionospheric climate to real-time weather predictions, Space Weather. № 15. P. 418–429. URL:
  9. Radicella S. The NeQuick model genesis, uses and evolution // Annals of Geophysics. 2010. № 52(3–4). URL:
  10. Banville S. et al. On the estimation of higher-order ionospheric effects in precise point positioning // GPS Solutions. 2017. Т. 21. №. 4. P. 1817–1828.
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