# Stochastic Modeling of Smartphones GNSS Observations Using LS-VCE and Application to Samsung S20
## 4 Experimental Results
### SF-PPP Performance Employing Reliable Stochastic Model  
The performance of the SF-PPP model when using the reliable stochastic model is investigated in kinematic mode using a dataset from the same dual-frequency Samsung S20 device as the previous Section 4.1. A kinematic test was carried out on 22 November 2022 with a duration of nearly 1 h in a mostly open-sky environment with overpasses, in Calgary, Alberta, Canada. Figure 9 displays the kinematic test configuration and the reference vehicle's path in this experiment. Figure 8: Estimated STD for GPS and GLONASS code observations on L1 frequency (in meters). Figure 9: A kinematic experiment conducted on 22 April 2022: (**a**) test configuration and (**b**) reference trajectory [33]. For the kinematic experiment, we used three geodetic receivers, two U-blox F9 Ps, and one Septentrio AsteRx-m2, as indicated by the three pick arrows in Figure 9. Additionally, we placed six smartphones on the vehicle roof to be used for future research. However, for this particular study, we only utilized the dataset from the Samsung S20 Black. The reference trajectory of the vehicle during the kinematic experiment was obtained by the RTK fixed solutions from the three geodetic receivers (shown by the three pick arrows in Figure 9) as the rover receivers and a geodetic receiver on a geodetic pillar (with true position) on the rooftop of the civil engineering building at the University of Calgary as the base receiver. The offsets between all units were measured and applied prior to comparison. Table 6 provides GNSS data information and processing setting. Considering \(\hat{a}^{GPS}_{code}=0.35\times 10^{5}\), \(\hat{a}^{GPS}_{code}=0.04\times 10^{5}\) for GPS and \(\hat{a}^{GLO}_{code}=1.08\times 10^{5}\), \(\hat{a}^{GLO}_{code}=0.04\times 10^{5}\) for GLONASS as reported in Table 5, one can obtain the C/N0/elevation-based stochastic model of GNSS observations (\(\sigma^{2,sys}_{[i],type}\)) as follows:  
\[\sigma^{2,sys}_{[i],type}=\hat{a}^{sys}_{type}\frac{10^{-\frac{C/N0_{i}}{10}}}{ \sin E_{i}} \tag{18}\]  
The horizontal positioning error in the cases of the SF-PPP model using the nominal stochastic model (i.e., \(a^{GPS}_{code}=10^{5}\), \(a^{GPS}_{phase}=10^{3}\) and \(a^{GLO}_{type}=2\times a^{GPS}_{type}=2\times 10^{5}\)) and the reliable stochastic model (considering the estimated scale factors \(\hat{a}^{sys}_{type}\)) are provided in Figure 10. Please note that the values displayed in Figure 10 were calculated using all positioning solutions, including the convergence period. Figure 11 depicts the cumulative distribution error plot of the horizontal positioning error for both cases. The results indicate that incorporating a more reliable stochastic model resulted in an improvement in positioning performance, specifically a 25.1% decrease in horizontal root mean square (RMS) and a 32.7% decrease in the 50 th percentile error. Furthermore, the maximum error decreased from approximately 7 m to about 5 m when the reliable stochastic model \begin{table}
\begin{tabular}{c c} \hline \hline
**Device** & Samsung S20 Black \\ \hline App logger & GnssLogger (v3.0.5.6) \\ \hline RINEX converter & UofC CSV2 RINEX tool [32] \\ \hline Measurements used & GPS (L1), GLONASS (L1) \\ \hline Mode & Kinematic \\ \hline Date & 22 November 2022 \\ \hline Duration & 1 h \\ \hline Sampling interval & 1 s \\ \hline
**Toposphere model** & Saastamoinen model \\ \hline Ionosphere model & Global ionospheric maps (GIM) \\ \hline Functional model & SF PPP model \\ \hline
**Stochastic model** & C/N0 and elevation weighting function (nominal and estimated stochastic model) \\ \hline Elevation mask angle & 10 deg \\ \hline C/N0 mask & 20 dB-Hz \\ \hline Satellite orbit & CODE MGEX precise ephemerides (5 min interval) \\ \hline Clock error & CODE MGEX precise clock (1 s interval) \\ \hline Satellite DCB correction & CAS DCBs in Bias SINEX (BSX) format \\ \hline \hline \end{tabular}
\end{table} 
Table 6: GNSS data information and processing setting. was introduced.