## 1 Introduction  
The electromagnetic wave propagation through the atmosphere has a significant impact on radio signals transmitted by satellites towards the Earth. This effect can cause errors in the range from a few meters to several dozen meters depending on numerous factors [1]. The main concern is the ionosphere, which is an area of weakly ionised gas in the Earth's atmosphere, lying between approximately 50 kilometres and several thousand kilometres from the Earth's surface. Radio signals crossing the ionosphere suffer several ionospheric effects such as phase advance, group delay, refraction, absorption, Faraday rotation, or amplitude and phase scintillation. Most of those effects depend on the ionospheric total electron content (TEC), which is defined as the electron density in a cross-section of 1 m\({}^{2}\), integrated along a slant (or vertical) path between two points (e.g., a satellite and a receiver); it is expressed in TEC units (TECU), where 1 TECU equals 10\({}^{16}\) electrons/m\({}^{2}\)Ionosphere electron density and ionospheric effects depend on various factors such as the time of day, location, season, solar activity, and interaction between solar activity and the Earth's magnetic field or the level of ionosphere disturbance during geomagnetic storms. On a large time-scale, the Sun undergoes a cyclic variation with an average period of 11 years [2]. The ionosphere group delay along the path \(s\) from satellite to receiver, neglecting higher-order terms, for code measurements leads to a pseudorange error:  
\[I_{g}=\frac{40.3}{f^{2}}\int_{sat.}^{rec.}N_{e}ds=\frac{40.3}{f^{2}}sTEC \tag{1}\]  
where \(I_{g}\) is the group delay [m], \(f\) is frequency [Hz], \(N_{e}\) is the electron density [electrons/m\({}^{3}\)] and \(sTEC\) is the Slant Total Electron Content [electrons/m\({}^{2}\)]. The ionospheric effect has a significant impact on the accuracy of GNSS positioning. In the past, there were many scientific investigations concerning the creation and modification of ionospheric models [3; 4] as well as the influence of ionosphere storms on global navigation satellite system (GNSS) positioning [5; 6; 7]. There are various approaches to mitigate ionospheric effects on GNSS positioning. They depend on desirable accuracy and on the availability of signal frequencies. It was reported by [PERSON] and [PERSON] [8] that for RTK (Real Time Kinematic) positioning the error caused by the impact of geomagnetic storm was in the range of cm-dm level. In the case of multi-frequency receivers, the ionospheric delay can be mitigated by means of specific combinations of measurements, using ionospheric dispersion properties [9; 10]. On the contrary, single-frequency navigation requires an ionospheric model estimation of the actual state of the ionosphere to make a correction [11; 12]. High-precision approximation of the ionosphere is obtained by high-accuracy global ionosphere models (GIMs) from different International GNSS Service (IGS) analysis centres [13], but these often come with a delay and can be only used for post-processing purposes. In recent years, the GIMs with shorter refreshing time are available: for example, the IGS rapid model updated every 2 h\({}_{j}\) CODE (available since 2013) with a resolution of 1 h; or the models of the Technical University of Catalonia (UPC) providing GIM every 15 min or even every 5 min [14]. There is a number of analysis centres that offer real-time products, mainly precise satellite orbit and clock parameters. Currently, only one of the individual centres, the Centre National d'Etudes Spatiales (CNES), is broadcasting the ionospheric VTEC message to mitigate ionospheric delay [15]. Nevertheless, for real-time GNSS single-frequency users, the broadcast ionospheric models are the most effective approach to mitigate the ionospheric time delay. Thus, this article compares the quality of popular ionospheric models, which were applied into Global Positioning System/European Geostationary Navigation Overlay Service (GPS/EGNOS) single-frequency positioning.