## 2 Ionospheric Models Used for Single-Frequency GNSS Receivers  
Single-frequency GNSS users often relay on standard ionosphere models broadcast in navigation message. The parameters of ionospheric models are determined by ground segments and transmitted to users. Nowadays, the models used in GNSS systems include the Klobuchar model (GPS, BeiDou) and NeQuick G model (Galileo). For demanding safety-of-life applications, single-frequency receivers can also use augmentation systems that use their own real-time ionosphere models; in the case of Europe, this is EGNOS. The Klobuchar model, also known as the Ionospheric Correction Algorithm (ICA), is widely used because of its simple structure and straightforward calculation [20; 21]. It assumes the ionosphere as a single layer model (SLM) at the height of 350 km or 375 km for GPS and BeiDou, respectively. In order to calculate the delay, the satellites of each constellation transmit eight ionospheric correction coefficients in the navigation message, which are typically updated once per day for GPS and every 2 h for BeiDou, as the driving parameter set for the Klobuchar model. The Klobuchar model gives a representation of the mean vertical delay at the GPS L1 and BeiDou BII frequency as a half-cosine function with varying amplitude and period. The Klobuchar model used by GPS removes the ionospheric error by 50% to 70% on the basis of TEC on a global scale. Its performance is contingent on the solar activity, time of day, and user's position. There are two main reasons that limit the accuracy of the Klobuchar model used by GPS: firstly, the Klobuchar coefficients are determined based on the day of the year (DOY) and the average values of the solar flux from the previous five days, so the coefficients cannot be updated more often than once a day; secondly, only eight coefficients have been introduced to describe the global variability of the ionosphere due to the limitation of communication in the GNSS navigation message [22]. The Klobuchar model adopted in BeiDou resembles the GPS one, but it is more local than global. It is formulated in a geographic coordinate system instead of a geomagnetic coordinate system, and the coefficients of model are calculated only from the regional BeiDou monitoring stations located in the mainland of China with an update interval of 2 h [23]. The algorithm of Klobuchar model is presented as follows [24]:  
\[I(t)=N+P\cdot cos\left(\frac{2\pi(t-14)}{T}\right) \tag{2}\]  
where \(I(t)\) is the vertical ionosphere delay, \(t\) is the local time of the ionospheric pierce point, \(N\) is a night-time constant, \(P\) is the amplitude of the cosine term, and \(T\) is the period of the cosine term. Parameters \(P\) and \(T\) are determined on the basis of coefficients broadcasted by satellites. The vertical ionospheric delay can be converted to slant delay with a mapping function \(MF(e)\) depending on the satellite elevation angle \(e\) (in semi-circles), with the formula:  
\[MF(e)=1.0+16.0\cdot(0.53-e)^{3}. \tag{3}\]  
The exact computation process based on this model for GPS and BeiDou can be found in [25] and [26], respectively. The latest research by [PERSON] et al. [27] performed on GNSS stations located on the territory of China shows that for a long period test (DOY 001-180, 2015), the Klobuchar broadcast ionospheric models can correct the ionospheric delay by 64.8% for GPS and 65.4% for BeiDou. Another study by [PERSON] et al. [23] confirms significant improvement of the three-dimension, single-frequency positioning accuracy by using the BeiDou model compared with the GPS one in the northern hemisphere. At the same time, the positioning accuracy in the southern hemisphere is degraded by almost the same amount, which is apparently due to the lack of BeiDou monitoring stations there.