## 4 Results and Discussion
### Occluded Environment  
The distributions and counts of observed satellites in the occluded environment are depicted in Figure 9. It is evident that satellites predominantly cluster in the southwest direction. The combined count of GPS/BDS satellites observed in each epoch is never less \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Device**} & \multirow{2}{*}{**Direction**} & \multicolumn{2}{c}{**C/N0-Dependent**} & \multicolumn{2}{c}{**DDPR-Dependent**} & \multicolumn{2}{c}{**Accuracy**} & \multicolumn{2}{c}{**C/N0-Dependent**} \\  & & **Dependent** & **Dependent** & **Improvement** & **(Oipital Subset)** \\  & & **(m)** & **(m)** & **(m)** \\ \hline \multirow{3}{*}{Mate40} & N & 3.36 & 3.14 & 6\% & 3.17 \\  & E & 1.93 & 1.80 & 7\% & 2.19 \\  & U & 5.85 & 5.72 & 2\% & 6.28 \\ \hline \multirow{3}{*}{P40} & N & 1.95 & 2.24 & \(-15\%\) & 2.91 \\  & E & 2.58 & 2.56 & 1\% & 3.06 \\ \cline{1-1}  & U & 6.61 & 6.12 & 7\% & 6.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: RMS values of positioning results with different stochastic models in an open environment. Figure 8: Time series of the positioning errors with different stochastic models in the open environment. than 14, with the count consistently hovering between 21 and 24 for the majority of the time. The surrounding buildings' occlusion resulted in a reduction of 4-6 satellites on average for the smartphone, compared to the open environment, leading to significant fluctuations in satellite counts across epochs. Single-frequency code pseudorange observations were employed to calculate the relative positioning of the Huawei Mate40 and Huawei P40 smartphones per epoch, which were then juxtaposed against known benchmark coordinates. Figure 10 illustrates the time series of positioning errors using different stochastic models. In the occluded environment, smartphones were susceptible to errors like multipath ones and NLOSs, degrading the quality of observations. The relative positioning accuracy per epoch is diminished, with errors in the N and E directions primarily spanning the [\(-\)10 m, \(+\)10 m] range. However, errors in the U direction are more pronounced, clustering within the [\(-\)20 m, \(+\)20 m] range. A comparison of the two stochastic models, as visualized in Figure 10, revealed that the error fluctuation range in the N, E, and U directions for the DDPR-dependent model (based on the optimal subset) was markedly reduced compared to the C/N0-dependent stochastic mode, especially during specific periods. For instance, during the initial 2000 epochs for the Huawei P40 smartphone in the N direction, the C/N0-dependent mode's positioning accuracy frequently exceeded 10 m. In contrast, the DDPR-dependent model proposed in this study managed to maintain accuracy within 5 m.  
In Figure 11, the cumulative distribution function (CDF) for positioning error indicated that for the Huawei Mate40 and Huawei P40 smartphones, the proportion of horizontal positioning accuracy within 5 m using the DDPR-dependent model stood at 57% and 60%, respectively. This is a significant improvement over the C/N0-dependent mode's respective 23% and 42%. Similarly, in the vertical direction, the DDPR-dependent model achieved 41% and 40% accuracy within 5 m, respectively, outperforming the C/N0-dependent mode's respective 27% and 30%. It is evident that the positioning accuracy comparisons between the two smartphones using the two stochastic models are consistent. In the occlusion environment, the C/N0 of certain satellites failed to accurately represent the impact of multipath errors and NLOSs in observations, revealing the C/N0-dependent model's inherent limitations. The proposed DDPR-dependent model, based on the optimal satellite subset's terminal approximate position, can identify significant gross errors caused by multipath and NLOS issues. By reducing the weight of corresponding observations, positioning accuracy is enhanced to a degree. Figure 9: Distributions and counts of the observed satellites in the occluded environment.