## 5 Triple-Frequency Medium RTK Performance Analysis
### Convergence Performance  
It is crucial for real-time users to have fast convergence, i.e., a 3D positioning error less than 10 cm, when satellite signals are obstructed. To evaluate such performance, we re-initialize the medium baseline RTK every 30 min. For each baseline, there are 48 sessions per day. The probability distribution of the convergence time for 48 sessions of 6 baselines is shown Figure 9.  
It can be observed that, within 10 min, the probability of convergence for all six baselines using Scheme 2 and Scheme 3 exceeds 90%, whereas the baselines HZDQ-HZLA and WHJX-WHJA, which adopt Scheme 1, fall below 90%. Taking the baseline HZXS-HZLA, the convergence distribution for Scheme 2 and Scheme 3 is similar, with approximately 92% achieving convergence within 60 s, while, for Scheme 1, this value is only about 75%. For the baseline HZDQ-HZXS, the probability of achieving convergence within 60 s for Scheme 2 exceeds 95%, while the corresponding probability for Scheme 3 is approximately 82%, which is primarily attributed to the slower convergence of satellite ambiguities due to the absence of triple-frequency observations. We calculated the probability of convergence time within 180 s for three baselines using the three different schemes, as shown in Table 2. The probability of convergence within 180 s is calculated to be approximately 82.6% for Scheme 1, 90.6% for Scheme 2, and 92.1% for Scheme 3. Compared to Scheme 1, both Scheme 2 and Scheme 3 show an approximate improvement of 8.0% and 9.5%, respectively, with both surpassing 90%. Figure 8: (**a**) The positioning error series of Scheme 1 for WHJX-WHJA; (**b**) The positioning error series of Scheme 2 for WHJX-WHJA. The blue point represents the float solution and the red point represents the fixed solution. Furthermore, compared to other baselines, the baseline WHJX-WHJA exhibits a lower probability of achieving convergence within 180 s. This is primarily attributed to the longer length of this baseline, which results in greater atmospheric spatial variations between 15:30 and 21:30. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Baseline** & **Scheme 1** & **Scheme 2** & **Scheme 3** \\ \hline HZDQ-HZLA & 81.3\% & 87.5\% & 87.5\% \\ HZDQ-HZXS & 83.0\% & 100.0\% & 92.0\% \\ HZXS-HZLA & 81.3\% & 98.0\% & 96.0\% \\ WHJA-EZEC & 87.5\% & 89.6\% & 95.8\% \\ EZEC-WHJX & 89.6\% & 97.9\% & 100.0\% \\ WHJX-WHJA & 72.9\% & 70.8\% & 81.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The probability of convergence time within 180 s for three baselines. Figure 9: The probability distribution of the convergence time for 48 sessions of HZDQ-HZLA, HZDQ-HZLA, HZXS-HZLA, WHJA-EZEC, EZEC-WHJX, and WHJX-WHJA. Using the baselines HZDQ-HZLA and WHJX-WHJA as examples, with resets occurring every 30 min, Figure 10 illustrates the positioning errors in the Up direction for three different schemes. It can be observed that Scheme 1, which only utilizes dual-frequency observations, has more float solutions and fixed solution ratios of only 84.27% and 80.00%, indicating slower convergence. However, Scheme 2 and Scheme 3, which incorporate triple-frequency observations, significantly reduce the number of float solutions, with a mean fixed solution ratio of approximately 91.7% and 92.7%. Additionally, it is worth noting that positioning convergence between 15 h and 20 h is relatively slow, primarily due to the high activity of the ionosphere in the day, as illustrated in Figure 11.  
Figure 10: The positioning error in the Up direction for HZDQ-HZLA and WHJX-WHJA when using Scheme 1, Scheme 2, and Scheme 3. The red dots represent fixed solutions, while the blue dots represent float solutions. Furthermore, in the dual-frequency scheme (i.e., Scheme 1), BDS uses B1/B2a, and, during the transitional phase, BDS2 satellites lack the B2a signal, rendering them unusable.