## 3 Results
In the case of the analysis performed for the Septentrio AsteRx2 receiver, similar conclusions were also drawn (for the no-smoothing variant, \(\text{HPE}_{\text{mean}}=0.44\) m, \(\text{HPE}_{\text{std}}=0.23\) m, \(\text{HPE}_{\text{max}}=1.89\) m, \(\text{VPE}_{\text{mean}}=0.58\) m, \(\text{VPE}_{\text{std}}=0.50\) m, \(\text{VPE}_{\text{max}}=3.77\) m). The next stage of the research was to analyse the use of different observation weighting strategies in the observation model. The Xiaomi Mi 8 device is equipped with a Broadcom chipset, BCM4775, GLL ver. 107.20.22 414037, characterised by dual-frequency (E1/L1 + E5/L5) satellite observation capabilities. The quality of GNSS positioning using smartphones is highly dependent on the antenna design these devices are equipped with [26; 41; 42; 43]. Smartphone GNSS antennas are characterised by low gain and low multipath resistance [44; 45]. As a rule, mobile devices are equipped with either omnidirectional linearly or elliptically polarised antennas due to the theoretically differently oriented smartphone performing the positioning [46]. When using more advanced GNSS receivers and their antennas, the signal-to-noise ratio increases with increasing elevation, whereas this relationship cannot be observed for smartphone devices [47]. Therefore, for calculations related to data acquired with a smartphone, a signal-to-noise ratio-dependent weighting model is preferred to one related to satellite elevation [48]. The study analysed the impact of the type of observation weighting within the autonomous positioning model using a Xiaomi Mi8 smartphone (Formula (14)). Values of standard deviation are equal to 1 (\(\sigma=1\)), dependent on satellite elevation and dependent on the value of the signal-to-noise ratio. \[W=\frac{1}{\sigma^{2}} \tag{14}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Smoothing Window [s]** & **Steady State [s]** & **HPE\({}_{\text{mean}}\) [m]** & **HPE\({}_{\text{std}}\) [m]** & **HPE\({}_{\text{max}}\) [m]** & **VPE\({}_{\text{mean}}\) [m]** & **VPE\({}_{\text{std}}\) [m]** & **VPE\({}_{\text{max}}\) [m]** \\ \hline \multirow{4}{*}{10} & 1 & 3.16 & 2.55 & 26.56 & 4.67 & 4.41 & 60.46 \\  & 60 & 3.23 & 2.85 & 57.05 & 4.52 & 4.22 & 48.37 \\  & 180 & 4.01 & 3.87 & 64.24 & 5.73 & 5.58 & 51.58 \\  & 360 & 4.74 & 5.42 & 86.87 & 22.21 & 46.78 & 171.00 \\ \hline \multirow{4}{*}{20} & 1 & 2.84 & 2.46 & 26.25 & 4.25 & 4.23 & 59.31 \\  & 60 & 2.24 & 1.95 & 30.62 & 3.27 & 2.78 & 24.48 \\  & 180 & 3.12 & 2.84 & 40.65 & 4.47 & 4.27 & 39.29 \\  & 360 & 4.06 & 4.20 & 60.89 & 6.27 & 7.12 & 106.42 \\ \hline \multirow{4}{*}{30} & 1 & 2.68 & 2.45 & 26.03 & 4.05 & 4.16 & 58.94 \\  & 60 & 2.59 & 2.22 & 33.49 & 3.70 & 3.25 & 30.84 \\ \cline{1-1}  & 180 & 2.66 & 2.39 & 27.02 & 3.81 & 3.58 & 32.71 \\ \cline{1-1}  & 360 & 3.48 & 3.66 & 50.00 & 5.41 & 6.21 & 86.42 \\ \hline \multirow{2}{*}{No smoothing} & - & 4.47 & 3.01 & 43.04 & 6.38 & 5.32 & 64.95 \\ \cline{1-1}  & & & & & & & \\ \end{tabular}
\end{table} 
Table 1: Numerical values of the accuracy analysis performed for the Xiaomi Mi8 smartphone with varying steady states and smoothing windows. In the case of the option of adopting a satellite elevation-dependent standard deviation value, it takes the following form:  
\[\sigma=a+b\cdot e^{(-E/c)} \tag{15}\]  
where  
\(E\) is the satellite elevation. According to [30], \(a=0.13\) m, \(b=0.53\) m, \(c=10^{\circ}\). A variant taking the value of the standard deviation depending on the signal-to-noise ratio's standard deviation takes the following form:  
\[\sigma=a+b\cdot 10^{(-SNR/10)} \tag{16}\]  
where  
\(SNR\) is the signal-to-noise ratio for pseudorange measurement.