# A Method to Reduce Non-Nominal Troposphere Error
## 5 Method to Reduce the Non-Nominal Troposphere Error
### Simulation Results
\begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Satellite Number of Removed Satellites** \\ \hline All visible satellites & Non \\ Brute method & 11,17,27,52,58,70,71,72 \\ Greedy method & 11,17,27,52,55,58,71,72 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Satellites removed from the optimal satellite subset by the greedy and brute methods. Figure 20: Skylpot of the optimal satellite subset obtained by the greedy method. By examining Inequality (23), it is clear that \(VPL_{C\_new}\) decreases compared to \(VPL_{C\_all}\), \(VPL_{C\_elevation}\) and \(VPL_{C\_GDOP}\). Figure 23 presents the \(VPL_{non}\) of the new method under non-nominal troposphere, with the following results. * \(VPL_{non\_new}\) is the \(VPL_{non}\) of the new method. From Figure 23, we can derive the following inequality:  
\[VPL_{non\_all}<VPL_{non\_Brute}<VPL_{non\_Greedy}<VPL_{non\_new}<VPL_{non\_GDOP}< VPL_{C\_elevation} \tag{24}\]  
By examining Inequality (24), it is clear that \(VPL_{non\_new}\) decreases compared to \(VPL_{non\_elevation}\) and \(VPL_{non\_GDOP}\). The average \(VPL_{non}\), \(VPL_{bias}\) and \(VPL_{C}\) for the optimal satellite subsets obtained by the new method are shown in Table 9.  
Table 9 reveals that compared to all visible satellites, with the new method the average \(VPL_{bias}\) increase is 1.15 m, which corresponds to a decrease of 50.4%, and the average \(VPL_{non}\) increase of the new method is 0.12 m, which corresponds to an increase of 6%. These results show that the proposed  
Figure 21: Skylplot of the optimal satellite subset obtained by the new method. Figure 22: \(VPL_{C}\) of the new method under non-nominal troposphere. method can reduce the non-nominal troposphere error in the satellite subset. Although \(VPL_{\text{C\_new}}\) increases by 0.26 m compared to \(VPL_{\text{C\_Brute}}\), the running rate is greatly improved. Professor [PERSON] of Stanford University suggested that the running rate of different methods can be characterized by the program running time (the duration from the start to the end of the program) [7]. Based on the Matlab function combatns, the program traverses all possible satellite subsets and records the running time of different methods. Table 10 shows the running time of different methods obtained on the same day. Table 11 presents the percentage improvement in the running rate of the new method compared to those of other methods. As seen in Tables 10 and 11, the running rate is greatly improved compared to those of the brute method and the greedy method. \begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Running Time (min)** \\ \hline Brute method & 537.31 \\ Greedy method & 231.65 \\ GDOP method & 197.35 \\ Elevation method & 87.45 \\ New method & 109.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Running time of different methods. Figure 23: \(VPL_{non}\) of the new method under non-nominal troposphere. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Satellite Subset** & \(VPL_{non}\)**(m)** & \(VPL_{bias}\)**(m)** & \(VPL_{\text{C}}\)**(m)** \\ \hline All visible satellites & 1.97 & 2.32 & 4.29 \\ Brute method & 2.04 & 0.94 & 2.98 \\ Greedy method & 2.07 & 1.12 & 3.19 \\ Elevation method & 2.34 & 2.07 & 4.41 \\ GDOP method & 2.11 & 1.57 & 3.86 \\ New method & 2.09 & 1.15 & 3.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Average \(VPL_{non}\), \(VPL_{bias}\) and \(VPL_{\text{C}}\) for the optimal satellite subsets obtained by the new method. According to the above analysis, it can be concluded that the new method proposed in this paper can achieve a balance between the satellite geometry and the non-nominal troposphere error, which greatly decreases non-nominal troposphere error in the positioning solution. The new method not only minimizes the airborne VPL under severe troposphere conditions but also improves both the GBAS integrity and running rate.