## 3 Results
### ISB Estimates Derived from Smartphones
For the two smartphones of the same brand, the daily ISBs are roughly consistent, and the differences are 1.597-2.956, 0.220-0.326, 3.413-5.367, and 0.230-4.316 ns for GLONASS, Galileo, BDS-2, and BDS-3 with respect to GPS, respectively. Figure 15: Two smartphones for the SPP experiment. To investigate the enhanced performance for the SPP solutions considering a priori ISB constraints with the use of smartphones, the smartphone-based four-system integrated SPP processing with and without a priori ISB constraints is compared. The daily ISBs on DOY 319 of 2020 are introduced into the SPP processing on DOY 320 of 2020 as a priori constraints. For completeness, in addition to the SPP processing under open sky with an elevation mask of 10\({}^{\circ}\), the cutoff elevations are increased to 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), and 50\({}^{\circ}\) to perform the SPP processing in the constrained visibility environments. Figure 17 illustrates the epochwise positioning errors of four-system integrated SPP for two Xiaomi Mi8 smartphones with and without a priori ISB constraints under a cutoff elevation of 40\({}^{\circ}\) on DOY 320 of 2020. When ignoring the a priori ISBs, the epochwise position errors vary within 20, 20, and 40 m for most of the time in the east, north, and up directions, respectively, but can exceed 60 m after 14:00 (GPS Time) in all the three directions, except for the north direction for Xiaomi Mi8 (2). After considering the a priori ISB constraints, the epochwise position errors are reduced in all the three directions, especially for those after 14:00 (GPS Time) with an accuracy improvement of tens of meters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Devices**} & \multirow{2}{*}{**DOY**} & \multirow{2}{*}{**Items**} & \multicolumn{4}{c}{**Statistics of Epochwise ISBs (ns)**} \\ \cline{3-6}  & & & **GPS-** & **GPS-** & **GPS-** & **GPS-BDS-2** & **GPS-BDS-3** \\ \hline \multirow{3}{*}{Xiaomi Mi8 (1)} & \multirow{3}{*}{319} & Average & 30.285 & 0.064 & 35.957 & 39.839 \\  & & STD & 19.357 & 7.038 & 6.754 & 10.697 \\ \cline{2-6}  & \multirow{3}{*}{320} & Average & 23.642 & \(-\)0.747 & 32.568 & 36.107 \\  & & STD & 17.949 & 5.994 & 6.670 & 7.897 \\ \hline \multirow{3}{*}{Xiaomi Mi8 (2)} & \multirow{3}{*}{319} & Average & 27.329 & 0.390 & 30.590 & 35.523 \\  & & STD & 17.135 & 6.365 & 7.966 & 6.768 \\ \cline{1-1} \cline{2-6}  & \multirow{3}{*}{320} & Average & 25.239 & \(-\)0.967 & 29.155 & 36.337 \\ \cline{1-1}  & & STD & 18.599 & 7.896 & 7.039 & 12.185 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average value and STD of epochwise ISBs for two smartphones. Figure 16: Epochwise ISB estimates for two smartphones on DOY 319 and 320 of 2020. For further analysis, Table 8 provides the positioning accuracy (i.e., RMS statistics of epochwise position errors) in (m) of four-system integrated SPP for two smartphones with and without a priori ISB constraints under different elevation masks. For the smartphone-based four-system integrated SPP considering a priori ISB constraints, the positioning accuracy can be about 2.5 m in the east and north directions, and about 5-7 m in the up direction, respectively, when the cutoff elevations are lower than or equal to 20\({}^{\circ}\). The positioning accuracy degrades with the increasing elevation masks, and can be approximately 5-6, 4-5 and 23.5-24.5 m at an elevation mask of 50\({}^{\circ}\) in the three directions, respectively. Compared with the SPP solutions without a priori ISB constraints, the positioning accuracy for those with a priori ISB constraints can be improved, and the corresponding improvement rates are listed in Table 9. Different from the results of improvement rates with the geodetic receivers (see Table 5), the accuracy improvement coming with a priori ISB constraints is also significant for the smartphone-based SPP applications when the cutoff elevations are no more than 30\({}^{\circ}\).