## 3 Data Processing and Analysis for Smartphone Real-Time Kinematic (RTK) Positioning
### Evaluation of Data Processing  
First, a combination of GPS and Galileo observations were used in RTK positioning to verify the effectiveness of the data processing strategies, and real-time differential analysis was also used to verify the stochastic models. Except for the stochastic model and ambiguity-resolution strategy, the other specifications are listed in Table 2. We consider RTK positioning convergence when the errors along the north and east directions are below 0.1 m. In addition, the fixing rate is the number of fixed epochs divided by the total number of epochs, while the error-fixing rate is the proportion of positioning error greater than 0.1 m at fixed ambiguity. Table 3 lists the real-time differential root-mean-square (RMS) errors of different stochastic models using GPS L1/L5 + Galileo E1/E5a observations for the parameters of  
the unified-CNM and individual-CNM models described in Section 3.1.1. The RMS values along the three directions for the individual-CNM model are the smallest, followed by those for the elevation model. However, the results of the SIGMA-\(\varepsilon\) and unified-CNM models are all worse than those of the elevation model in the real-time differential analysis. Thus, the proposed individual-CNM model provides optimal results. Next, we used RTK positioning to evaluate the stochastic models. Table 4 lists the RTK positioning solutions of different stochastic models using GPS L1/L5 + Galileo E1/E5a observations. The positioning accuracy of every C/N0 model is better than that of the elevation model. However, in the C/N0 model, the SIGMA-\(\varepsilon\) and unified-CNM models take longer to converge. On the other hand, the accuracy of the individual-CNM model is the highest, and its convergence time is similar to that of the elevation model. Therefore, the individual-CNM model achieves the highest positioning performance. For ambiguity resolution, Figures 7 and 8 show the fully fixed ambiguity and partially fixed ambiguity solutions for GPS L1 + Galileo E1 observations, respectively. Considering the results after 1500 s (around 9:30), most unfixed ambiguities can be fixed when using the partial resolution, effectively improving RTK positioning. A comparison of the fixed and float-only solutions is shown in Figure 9. Ambiguity resolution can substantially improve the positioning accuracy. Table 5 lists the statistical results of the float-only, full, and partial resolution. Overall, the accuracy of fixed solutions is higher than that of float-only solutions, and partial resolution can considerably improve the fixing rate, positioning accuracy, and convergence speed. However, there are also some error-fixed solutions. In the parameter settings, five epochs (i.e., minimum lock) should converge before participating in ambiguity resolution when a cycle slip occurs, aiming to mitigate the influence of cycle slips. To further weaken the influence of cycle sips, the fix-and-hold mode [29] is used to impose a tight constraint on fixed ambiguities. This mode uses the fixed ambiguity solutions as virtual observations and performs secondary filtering to improve the filtering covariance matrix, aiming to reduce the influence of state initialisation on the covariance. Figure 8: Results of partial ambiguity resolution for GPS L1 + Galileo E1 observations. Figure 9: Fixed and float-only solutions for GPS L1 + Galileo E1 observations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Fixed Mode**} & \multirow{2}{*}{**Fixing Rate (\%)**} & \multirow{2}{*}{**Error-Fixing Rate (\%)**} & \multirow{2}{*}{**Convergence Time (min)**} & \multicolumn{2}{c}{**RMS Error after Convergence**} \\ \cline{3-5}  & & & & **East (m)** & **North (m)** & **Up (m)** \\ \hline Float ambiguity & – & – & 35.0 & 0.035 & 0.020 & 0.043 \\ Full resolution & 63.4 & 0 & 35.0 & 0.018 & 0.014 & 0.031 \\ Partial resolution & 90.4 & 5.3 & 25.1 & 0.012 & 0.012 & 0.026 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical results for GPS L1 + Galileo E1 observations.