## 2 Ionospheric Models Used for Single-Frequency GNSS Receivers
The ionospheric delay in SBAS systems has been the subject of many previous scientific studies [33; 34; 35]. Thanks to the fact that these models are created in real time (updated every 5 min), they can be applied for safety-of-life applications such as aviation, where there is a need for a high integrity of data; thus, positioning based on information only from GPS/GLONASS (Global Positioning System/GLObal Navigation Satellite System) systems [36] is not satisfactory for the aviation users. Thanks to SBAS systems, the user is able to obtain not only better accuracy, but is also able to assess the level of integrity of positioning. Beside the real-time ionospheric delay, the SBAS correction messages include GNSS satellite orbit and clock corrections as well as pseudorange fast corrections. The SBAS version utilised in Europe is EGNOS, which was developed by ESA, with SoL service available since 2 March 2011. SBAS systems use thin-shell ionosphere approximation and broadcast the vertical delay estimates at specified Ionospheric Grid Points (IGPs) situated at an altitude of 350 km [37]. The density of IGPs depends on latitude ranging from 5 by 5 degrees for low latitudes (55 degrees and less), 10 by 10 degrees for latitudes 65-75 degrees, and 10 degrees latitude and 90 degrees longitude at latitudes equal to 85 degrees [38]. Since it would be impossible to broadcast IGP delays for all possible locations, an up-to-date Ionospheric Grid Mask to define IGP locations is transmitted by SBAS systems in message 18, which is updated every 5 min and valid for a maximum of 20 min. The European system EGNOS transmits data only for IGPs covering Europe and the surroundings [39]. The predefined EGNOS IGPs are presented in Figure 2.  
Figure 1: _MODIP_ regions associated to different ionospheric characteristics [2]. MODIP: the Modified Dip Latitude. EGNOS updates ionospheric corrections every 5 min and transmits them by geostationary satellites in message 26, for each of the points of previously defined mask of IGPs grid. Thus, the receiver knows the position of these particular points and the estimated delay for each of them. To estimate the ionospheric error for each receiver/satellite line of sight, the receiver must first identify the ionospheric pierce points (IPPs) and perform a bi-linear interpolation between the values provided for the IGPs close to each IPP in order to calculate the vertical delay of each IPP. Finally, the slant ionospheric delay \(IC_{i}\)--which is added to the pseudorange measurement--is determined according to the formula [38]:  
\[IC_{i}=F_{pp}\tau_{\mathit{IPP}}\big{(}\phi_{pp},\lambda_{pp}\big{)} \tag{5}\]  
where \(F_{pp}\) is an obliquity factor, \(\tau_{\mathit{IPP}}\) is an interpolated vertical delay at the IPP, and \(\phi_{pp},\lambda_{pp}\) are coordinates of IPP. The accuracy of today's SBAS models is very high, since they are used in SoL applications. The latest study by [PERSON] et al. [14] confirmed that the ionospheric errors were reduced by more than 90% for both EGNOS and WAAS (Wide Area Augmentation System) interpolations. It is worth noticing that the GLONASS navigation message does not contain any ionospheric correction parameters. Nevertheless, many modern multi-constellation receivers can adapt the broadcast ionospheric models of GPS, BeiDou, or Galileo to mitigate the ionospheric delay to the GLONASS observations [40]. The basic characteristics of examined ionospheric models are presented in Table 1.