GPS Solutions

, 23:80 | Cite as

Midlatitude Klobuchar correction model based on the k-means clustering of ionospheric daily variations

  • Barbara Pongracic
  • Falin WuEmail author
  • Loghman Fathollahi
  • David Brčić
Original Article


The ionosphere influences GNSS radio waves and causes errors in measurements. The majority of GNSS users employ single-frequency receivers that mitigate ionospheric effects by utilizing various models. The GPS system corrects for ionospheric errors through the Klobuchar model, which successfully mitigates approximately 50% of the delay on the global scale; this model estimates the ionospheric delay by using one daily peak value at 14:00 local time (LT) with constant nighttime values. However, the daily ionospheric distribution shows a deviation from the Klobuchar model regarding a secondary peak during periods with higher incoming solar radiation and the occurrence of a nighttime peak. We propose a model, namely the midlatitude Klobuchar correction (ML-KC) model, to correct the Klobuchar model for midlatitude users. The proposed model is a function of the day of the year and the LT of the user adjusted to the local solar time. The dependency on the day of the year is modeled by using the k-means algorithm, thereby producing three clusters based on the correlation between daily modeling coefficients, which are expressed as the ratio between the delay from ionospheric maps and the delay estimated by the Klobuchar model. Furthermore, the time dependency is modeled with three harmonic components. The ML-KC was modeled from ionospheric maps over Europe during the period from 2005 to 2016. The performance of the ML-KC model was tested not only on the same dataset with one additional year of data from 2017 but also in two larger regions different from the modeling area to avoid model overfitting. The performance of the ML-KC model was better than that of the Klobuchar model during all assessed years and areas with the most significant improvements in RMS; during 2011, which demonstrated high solar activity, the RMS improvement reached 36.24%. The proposed model, which can be easily implemented in single-frequency GNSS receivers, offers a simple improvement to the Klobuchar model.


Ionospheric delay Klobuchar model improvement Midlatitudes k-means clustering  GNSS 



We would like to express gratitude to the efforts of the International GNSS Service (IGS) for creating and making publicly available scientific data by CDDIS. We are thankful to the NOAA for publishing the continuously operating reference station (CORS) data and to NASA OmniWeb for making the historical solar data available. We would also like to thank all the reviewers on their time and insightful comments which improved our manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Satellite Navigation and Remote Sensing (SNARS) Research Group, School of Instrumentation and Optoelectronics EngineeringBeihang UniversityBeijingChina
  2. 2.iOLAP Inc.RijekaCroatia
  3. 3.Faculty of Maritime StudiesUniversity of RijekaRijekaCroatia

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