## 2 Materials and Methods
Scientific publications show better positioning results when using a model based on data transmitted by EGNOS. The ionospheric slant delay \(RC_{iono}\) determined according to the model used in EGNOS (based on four-point interpolation), can be determined by the following formula [30]:  
\[RC_{iono}=-F_{pp}\tau_{cpp}\big{(}\lambda_{pp},\phi_{pp}\big{)} \tag{8}\]  
where  
\(F_{pp}\) is a coefficient that changes vertical delay to slant;  
\(\tau_{cpp}\) is the vertical ionospheric delay for IPP;  
\(\lambda_{pp},\phi_{pp}\) are IPP coordinates. \(F_{pp}\) and \(\tau_{cpp}\) can be determined from the following formulae:  
\[F_{pp}=\left[1-\left(\frac{R_{e}cosE}{R_{e}+h_{I}}\right)^{2}\right]^{-\frac{ 1}{2}} \tag{9}\]  
\[\tau_{cpp}=\sum_{i=1}^{4}W_{i}\big{(}x_{pp},y_{pp}\big{)}\tau_{vi} \tag{10}\]  
where  
\(R_{e}\) is the Earth's radius;  
\(E\) is the elevation of the satellite;  
\(h_{I}\) is the altitude at which the highest electron density occurs (350 km);  
\(\tau_{vi}\) are vertical delays for four gridpoints, transmitted by EGNOS;  
\(W_{i}\big{(}x_{pp},y_{pp}\big{)}\) is the weighing function. SBAS systems use averaged and seasonal atmospheric data determined for a specific day and user position to determine the tropospheric correction. The model is based on tropospheric delay values towards the zenith, determined based on five meteorological parameters: pressure, temperature, water vapour pressure, rate of change of temperature, and rate of change of water vapour pressure. Their values are made available from meteorological data mainly from North American centres. The influence of tropospheric refraction on GPS/EGNOS positioning results is determined individually for the receiver location (without using EGNOS transmitted data). According to [30], the tropospheric delay \(TC_{i}\) for satellite \(i\) takes the following form:  
\[RC_{fast}(t)=PRC(1)+RRC(t_{1})(t-t_{1}) \tag{2}\]0  
where  
\(d_{hyd},\ d_{wet}(\mathrm{m})\) are estimated signal delays for a satellite at zenith, determined from receiver altitude, pressure, temperature, water vapour pressure, rate of change of temperature, and rate of change of water vapour pressure,  
\(m(El_{i})\) is the mapping function that transforms the delay based on the satellite's current horizontal altitude \((El_{i})\). Previous studies on the quality of the tropospheric EGNOS model have shown very good correlation with the Saastamoinen model, which is considered to accurately represent the local state of the troposphere [37]. The use of smoothing according to the Hutch filter of carrier-phase smoothed code observations in GPS/EGNOS positioning before applying EGNOS corrections can reduce the noise level [30; 38]:  
\[ \begin{split} PR_{k}^{S}=\alpha PR_{k-1}^{C}+(1-\alpha)\left[\!\left[PR _{k-1}^{p}+\frac{\lambda}{2\pi}(PR_{k}^{p}-PR_{k-1}^{p})\right]\!\right]\\ \\ \alpha=\frac{dt}{T}\end{split}  \tag{12}\]  
where  
\(PR_{k}^{S}\) is the smoothed pseudorange code;  
\(PR_{k-1}^{C}\) is the code pseudorange in the epoch \(k\);  
\(PR_{k}^{P}\) is the pseudorange distance in the epoch \(k\);  
\(\lambda\) is the wavelength of GPS carrier frequency at L\({}_{1}\);  
\(T\) is the smoothing constant;  
\(dt\) is the sampling interval. Data acquired with a Xiaomi Mi 8 smartphone (Xiaomi Corporation, Beijing, China) operating in duty cycling off mode were used for the study. Satellite observations were recorded on 25 April 2020 between 7:00 and 12:00 GPS time with an interval of 1 s in the vicinity of the city of Olsztyn in the north-eastern part of Poland. Measurements were taken at a fixed point over which a Xiaomi Mi8 smartphone was mounted on a tripod. At a distance of approximately 3 m, data were collected simultaneously using a Septentrio AsteRx2 receiver (Septentrio, Leuven, Belgium), which was equipped with a geodesic-grade antenna that has a uniform quasi-hemispherical gain pattern, right-handed circular polarisation, and a stable phase centre (Figure 1).