## 2 Materials and Methods  
Positioning using SBAS is achieved by implementing corrections to the mathematical model of positioning. In SBAS systems, data related to long-term correction, fast correction, ionospheric delay and tropospheric delay, among others, are transmitted to improve positioning quality. The corrected pseudorange determined in SBAS systems can be written with the general formula [28; 29]:  
\[P_{corr}=P_{meas}+RC_{fast}+RC_{clock}-RC_{iono}+RC_{tropo} \tag{1}\]  
where  
\(P_{corr}\) is the corrected pseudorange based on SBAS data;  
\(P_{meas}\) is the pseudorange as measured by the receiver;  
\(RC_{fast}\) is the fast correction;  
\(RC_{clock}\) is the clock correction;\(RC_{iono}\) is the ionospheric correction;  
\(RC_{tropo}\) is the tropospheric correction. Short-term corrections are used to correct rapid changes in ephemeris errors and clock errors [30]. They are applied in all positioning modes of air operations using SBAS with an interval of no more than 60 s. Fast corrections for a given epoch can be calculated according to the formula:  
\[RC_{fast}(t)=PRC(1)+RRC(t_{1})(t-t_{1}) \tag{2}\]  
where  
\(RC_{fast}\) is the total fast correction,  
\(PRC\) is the pseudorange correction,  
\(RRC\) is the range rate correction,  
\(t\) is the user time,  
\(t_{1}\) is the application time of the most up-to-date fast correction. Range rate corrections are not transmitted by SBAS systems, but calculated at receiver algorithm level:  
\[RRC(t_{1})=\frac{PRC(t_{1})-PRC(t_{0})}{t_{1}-t_{0}} \tag{3}\]  
where  
\(t_{0}\) is the time of application of the recently received fast correction. The use of raw GNSS data in smartphone-based positioning puts some strain on the computing resources of Android-based devices. The use of fast corrections in the GNSS receiver algorithm significantly consumes the bandwidth of transmitted messages [24]. Long-term corrections are used to correct infrequent changes in ephemeris errors and clock errors of GPS satellites [30]. Data on the parameters of the satellites' orbits and information on the coefficients are transmitted to determine the values of the clock corrections. Some of the correction data are applied to the calculated satellite positions and velocity, while the remaining data are applied to the pseudorange measurements. Corrections to satellite positions are calculated using linear interpolation and applied to satellite positions according to the following formulas:  
\[DX(t) =DX+DX_{ROC}(t-t_{V}) \tag{4}\] \[DY(t) =DY+DY_{ROC}(t-t_{V})\] (5) \[DZ(t) =DZ+DZ_{ROC}(t-t_{V}) \tag{6}\]  
where  
\(DX(t)\), \(DY(t)\), \(DZ(t)\) are the current corrections for satellite position;  
\(DX\), \(DY\), \(DZ\) are the corrections to satellite positions transmitted as part of slow corrections;  
\(DX_{ROC}\), \(DY_{ROC}\), \(DZ_{ROC}\) are transmitted adjustments to velocity;  
\(t\) is the user time;  
\(t_{V}\) is the application time of transmitted corrections to velocity. Slow corrections related to satellite clocks are determined by the following formula:  
\[RC_{clock}=\Big{[}\delta a_{f0}+\delta a_{f1}(t-t_{0})\Big{]}c \tag{7}\]  
where  
\(RC_{clock}\) is the resulting correction of the satellite clock;  
\(\delta a_{f0}\), \(\delta a_{f1}\) are clock correction parameters provided in EGNOS system messages;  
\(t\) is the user time;  
\(t_{0}\) is the time of application of clock correction parameters;  
\(c\) is the speed of propagation of an electromagnetic wave in a vacuum. Standard satellite clock corrections determined for absolute GNSS positioning and additional clock corrections determined based on the SBAS model are calculated independently. Applying them to GNSS/SBAS positioning is also carried out independently. The impact of the ionospheric correction on GNSS/SBAS positioning has been the subject of many scientific studies [31; 32; 33; 34; 35; 36]. According to [30], its value is determined by either an SBAS model or a classical model based on the Klobuchar algorithm.