Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Araujo-Pradere EA, Redmon R, Fedrizzi M, Viereck R, Fuller-Rowell TJ (2011) Some characteristics of the ionospheric behavior during the solar cycle 23–24 minimum. Sol Phys 274(1–2):439–456. https://doi.org/10.1007/s11207-011-9728-3
Basu S (2013) The peculiar solar cycle 24—where do we stand? J Phys Conf Ser 440:012001. https://doi.org/10.1088/1742-6596/440/1/012001
Brčić D (2015) A model of non-specific daily pattern of the satellite positioning signal ionospheric delay. Ph.D. thesis, Faculty of Maritime Studies, University of Rijeka, Rijeka, Croatia
CSNO (2018) BeiDou navigation satellite system signal in space interface control document—open service signal B3I (Version 1.0). China Satellite Navigation Office, Beijing
Filjar R, Kos T, Kos S (2009) Klobuchar-like local model of quiet space weather GPS ionospheric delay for northern Adriatic. J Navig 62(03):543–554. https://doi.org/10.1017/s0373463309005281
Goswami DY (2015) Principles of solar engineering, 3rd edn. Taylor & Francis Group, Boca Raton, FL
Guo J, Li W, Liu X, Kong Q, Zhao C, Guo B (2015) Temporal–spatial variation of global GPS-derived total electron content 1999–2013. PLoS ONE 10:e0133378. https://doi.org/10.1371/journal.pone.0133378
Hartigan JA, Wong MA (1979) Algorithm AS 136: a k-means clustering algorithm. J R Stat Soc Ser C (Appl Stat) 28(1):100–108. https://doi.org/10.2307/2346830
Hernández-Pajares M, Roma-Dollase D, Krankowski A, García-Rigo A, Orús-Pérez R (2017) Methodology and consistency of slant and vertical assessments for ionospheric electron content models. J Geodesy 91(12):1405–1414. https://doi.org/10.1007/s00190-017-1032-z
Hochegger G, Nava B, Radicella S, Leitinger R (2000) A family of ionospheric models for different uses. Phys Chem Earth Part C 25(4):307–310. https://doi.org/10.1016/S1464-1917(00)00022-2
Hofmeister SJ, Veronig A, Reiss MA, Temmer M, Vennerstrom S, Vršnak B, Heber B (2017) Characteristics of low-latitude coronal holes near the maximum of solar cycle 24. Astrophys J 835:268. https://doi.org/10.3847/1538-4357/835/2/268
Hoque MM, Jakowski N (2015) An alternative ionospheric correction model for global navigation satellite systems. J Geodesy 89(4):391–406. https://doi.org/10.1007/s00190-014-0783-z
Hoque MM, Jakowski N, Berdermann J (2017) Ionospheric correction using NTCM driven by GPS Klobuchar coefficients for GNSS applications. GPS Solut 21(4):1563–1572. https://doi.org/10.1007/s10291-017-0632-7
Hoque MM, Jakowski N, Orús-Pérez R (2019) Fast ionospheric correction using Galileo Az coefficients and the NTCM model. GPS Solut 23(2):41. https://doi.org/10.1007/s10291-019-0833-3
Jakowski N, Hoque MM, Mayer C (2011) A new global TEC model for estimating transionospheric radio wave propagation errors. J Geodesy 85(12):965–974. https://doi.org/10.1007/s00190-011-0455-1
Jin S, Cho J-H, Park J-U (2007) Ionospheric slab thickness and its seasonal variations observed by GPS. J Atmos Solar Terr Phys 69(15):1864–1870. https://doi.org/10.1016/j.jastp.2007.07.008
Jongsintawee S, Runraengwajiake S, Supnithi P, Panachart C (2016) Improvement of GPS positioning accuracy when utilizing Klobuchar model with ionospheric conditions in Thailand. In: 2016 13th international conference on electrical engineering/electronics, computer, telecommunications and information technology (ECTI-CON), June 28–July 1, pp 1–5. https://doi.org/10.1109/ecticon.2016.7561391
Klobuchar JA (1975) A first-order, worldwide, ionospheric, time-delay algorithm. AFCRL-TR-75-0502; air force surveys in geophysics: 324. Ionospheric Physics Laboratory, Air Force Cambridge Research Laboratories, Air Force Systems Command (USAF)
Klobuchar JA (1987) Ionospheric time-delay algorithm for single-frequency GPS users. IEEE Trans Aerosp Electron Syst AES 23(3):325–331. https://doi.org/10.1109/taes.1987.310829
MacQueen JB (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, Berkeley, 1967, vol 1: statistics. University of California Press, Berkeley, pp 281–297
Mosert M, Buresova D, Magdaleno S, de la Morena B, Altadill D, Ezquer RG, Scida L (2012) An analysis of the scale height at the F2-layer peak over three middle-latitude stations in the European sector. Earth Planets Space 64(6):493–503. https://doi.org/10.5047/eps.2011.04.013
Nava B, Coïsson P, Radicella SM (2008) A new version of the NeQuick ionosphere electron density model. J Atmos Solar Terr Phys 70(15):1856–1862. https://doi.org/10.1016/j.jastp.2008.01.015
Ratnam DV, Dabbakuti JRKK, Lakshmi NVVNJS (2018) Improvement of Indian-regional Klobuchar ionospheric model parameters for single-frequency GNSS users. IEEE Geosci Remote Sens Lett 15(7):971–975. https://doi.org/10.1109/lgrs.2018.2827081
Romma-Dollase D et al (2018) Consistency of seven different GNSS global ionospheric mapping techniques during one solar cycle. J Geodesy 92(6):691–706. https://doi.org/10.1007/s00190-017-1088-9
Rousseeuw P (1987) Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J Comput Appl Math 20:53–65. https://doi.org/10.1016/0377-0427(87)90125-7
Schaer S (1999) Mapping and predicting the Earth’s ionosphere using the global positioning system. Ph.D. dissertation, Astronomical Institute, University of Berne, Switzerland
Schaer S, Gurtner W, Feltens J (1998) IONEX: the IONosphere Map EXchange Format Version 1. Astronomical Institute, University of Bern, Bern
Stepniak K, Wielgosz P, Paziewski J (2014) Accuracy analysis of the Klobuchar ionosphere model transmitted by the GPS system. In: The 9th international conference “environmental engineering”, selected papers, article number: enviro.2014.246, Vilinus, Lithuania, May 22–23, 2014, pp 1–6. https://doi.org/10.3846/enviro.2014.246
Tibshirani R, Walther G, Hastie T (2001) Estimating the number of clusters in a data set via the gap statistic. J R Stat Soc Ser B (Stat Methodol) 63(2):411–423. https://doi.org/10.1111/1467-9868.00293
Wang N, Yuan Y, Li Z, Huo X (2016) Improvement of Klobuchar model for GNSS single-frequency ionospheric delay corrections. Adv Space Res 57(7):1555–1569. https://doi.org/10.1016/j.asr.2016.01.010
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Pongracic, B., Wu, F., Fathollahi, L. et al. Midlatitude Klobuchar correction model based on the k-means clustering of ionospheric daily variations. GPS Solut 23, 80 (2019). https://doi.org/10.1007/s10291-019-0871-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10291-019-0871-x