## 2 Ionospheric Models Used for Single-Frequency GNSS Receivers
The NeQuick is a three-dimensional and time-dependent ionospheric electron density model developed at the former Aeronomy and Radiopropagation Laboratory of The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, and the Institute for Geophysics, Astrophysics and Meteorology (IGAM) of the University of Graz, Austria. It is based on the DGR (Di Giovanni and Radicella) \"profiler\" proposed by [PERSON] and [PERSON] [28] and subsequently modified by [PERSON] and [PERSON] [29]. It is a quick-run model that is particularly tailored for transionospheric propagation applications, and it is based on an empirical climatological representation of the ionosphere, which predicts monthly mean electron density from analytical profiles, depending on the solar activity related input values including: sunspot number or solar flux, month, geographic latitude and longitude, height, and universal time (UT). The original version of the NeQuick model has been used by the European Geostationary Navigation Overlay Service (EGNOS) of the European Space Agency (ESA) for system assessment analysis. For such an objective, a series of ionospheric scenarios have been created under ESA contracts to simulate realistic ionospheric conditions under disturbed conditions [30]. A specific version of the NeQuick model named NeQuick G has been developed by the ESA involving the original authors and other European ionospheric scientists under various ESA contracts, and it has been adopted for the real-time Galileo Single-Frequency Ionospheric Correction algorithm. The tests performed during the Galileo In-Orbit-Validation phase proved that the algorithm is capable of correcting more than 70% of the ionospheric group delay error under nominal ionospheric conditions [31]. Recently, the International Committee on Global Navigation Satellite Systems Working Group B has recommended to distribute the document providing the detailed description of the NeQuick algorithm implemented in Galileo [2] and to assess the performance and usability of a NeQuick ionospheric correction algorithm for the single-frequency users, similar to the one adopted by Galileo. The NeQuick G model used by Galileo for real-time predictions of ionospheric corrections is based on a single input parameter--the Effective Ionisation Level--\(Az\), which is determined using three coefficients (\(a_{i0}\), \(a_{i1}\), \(a_{i2}\)) broadcast in the navigation message and \(MODIP\)--the Modified Dip Latitude at the location of the user receiver:  
\[Az=a_{i0}+a_{i1}\cdot MODIP+a_{i2}\cdot(MODIP)^{2}. \tag{4}\]  
The coefficients (\(a_{i0}\), \(a_{i1}\), \(a_{i2}\)) are updated at least once a day. \(MODIP\) is expressed in degrees, and a table grid of \(MODIP\) values versus geographical location is provided together with the NeQuick G model. Depending on the severity and general characteristics of ionospheric effects, five \(MODIP\) regions (associated with the geomagnetic field) have been defined and presented in Figure 1[2]. Once the _Az_ is determined, the receiver then calculates the integrated slant total electron content along the path using the NeQuick G model and by means of Equation (1) converts it to slant delay. The step-by-step procedure to implement the ionospheric algorithm for Galileo single frequency receivers is given in [32]. The latest study by [PERSON] et al. [14] demonstrates that on the global scale, the NeQuick G was found to be only around 6% more accurate than the computationally less expensive Klobuchar model. The other research by [PERSON] [32], which was focused on GNSS single-frequency quality at the 2014 solar maximum, confirmed that on the global scale, NeQuick G presents a better RMS (Root Mean Square) than the GPS Klobuchar model by approximately 20% and 11% for the horizontal and vertical navigation error components, respectively. A somewhat different genesis of creation, in relation to the global Klobuchar and NeQuick models, is characterised by the local ionosphere models used by SBAS (Satellite-Based Augmentation System) systems.