# A Method to Reduce Non-Nominal Troposphere Error
## 5 Method to Reduce the Non-Nominal Troposphere Error
### Simulation Results  
Figure 17 presents the \(VPL_{C}\) of different methods under non-nominal troposphere. The figure presents the following results:  
* \(VPL_{C-all}\) is the \(VPL_{C}\) of all visible satellites
* \(VPL_{C-Brute}\) is the \(VPL_{C}\) of the brute method. * \(VPL_{C-Greedy}\) is the \(VPL_{C}\) of the greedy method. * \(VPL_{C-elevation}\) is the \(VPL_{C}\) of the elevation method. * \(VPL_{C-GDOP}\) is the \(VPL_{C}\) of the GDOP method. Figure 18 presents the \(VPL_{non}\) of different methods under non-nominal troposphere. The figure presents the following results:  
* \(VPL_{non-all}\) is the \(VPL_{non}\) of all visible satellites. * \(VPL_{non-Brute}\) is the \(VPL_{non}\) of the brute method. * \(VPL_{non-Greedy}\) is the \(VPL_{non}\) of the greedy method. * \(VPL_{non-elevation}\) is the \(VPL_{non}\) of the elevation method. * \(VPL_{non-GDOP}\) is the \(VPL_{non}\) of the GDOP method. From Figures 17 and 18, we can derive the following inequalities:  
\[VPL_{C\_Brute}<VPL_{C\_Greedy}<VPL_{C\_GDOP}<VPL_{C\_all}<VPL_{C\_elevation} \tag{21}\] \[VPL_{non\_all}<VPL_{non\_Brute}<VPL_{non\_Greedy}<VPL_{non\_ GDOP}<VPL_{non\_elevation} \tag{22}\]  
Figure 17: \(VPL_{C}\) of different methods under non-nominal troposphere. From Inequalities (21) and (22), we can infer that all visible satellites will increase the non-nominal troposphere error, which in turn increases \(VPL_{C}\). Compared with the \(VPL_{non}\) of all visible satellites, both \(VPL_{non\_Brute}\) and \(VPL_{non\_Gready}\) increase. However, the non-nominal troposphere errors in the selected satellite subsets decrease, and as a result, \(VPL_{C\_Brute}\) and \(VPL_{C\_Gready}\) are smaller than \(VPL_{C\_all}\). The above results verify the accuracy of the inference that the non-nominal troposphere error in the constellation can be reduced using an appropriate satellite selection method. The average satellite number \(M\), \(VPL_{non}\) and \(VPL_{bias}\) of the optimal satellite subsets obtained by different methods are summarized in Table 4.  
According to Table 4, the \(VPL_{non}\) of the brute method and the greedy method increase by less than 5% compared with the VPL of all visible satellites, which meets the GBAS integrity requirements [6]. Thus, the number of satellites in the optimal subset is set to 16. Table 5 shows the average distribution of the optimal satellite subsets obtained by the brute method in different elevation ranges for the Linzhi airport during 2015 and 2016. Table 6 presents the average distribution of the optimal satellite subsets obtained by the greedy method in different elevation ranges for the Linzhi airport during 2015 and 2016. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Satellite Subset** & **M** & \(VPL_{non}\) **(m)** & \(VPL_{bias}\) **(m)** \\ \hline All visible satellites & 31.6 & 1.97 & 2.32 \\ Brute method & 16.2 & 2.04 & 0.94 \\ Greedy method & 15.9 & 2.07 & 1.12 \\ Elevation method & 19.8 & 2.34 & 2.07 \\ GDOP method & 17.4 & 2.11 & 1.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The average satellite number M, \(VPL_{non}\) and \(VPL_{bias}\) of different satellite subsets. Figure 18: \(VPL_{non}\) of different methods under non-nominal troposphere. \begin{table}
\begin{tabular}{c c} \hline \hline
**Elevation Range** & **Distribution Ratio** \\ \hline \((5^{\circ},30^{\circ})\) & 24.6\% \\ \((30^{\circ},60^{\circ})\) & 50.3\% \\ \((60^{\circ},90^{\circ})\) & 25.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average distribution of the optimal satellite subsets under the brute method. According to the results shown in Tables 5 and 6, the distribution ratios of the optimal satellite subsets obtained by the brute and greedy methods in different elevation ranges are 24.6:50.3:25.1 and 25.7:49.5:24.8, respectively. For convenience, in this paper, the ratio is set to 1:2:1.