# A Method to Reduce Non-Nominal Troposphere Error
## 5 Method to Reduce the Non-Nominal Troposphere Error
### Simulation Analysis of the Proposed Method  
To improve the GBAS integrity under abnormal troposphere, this paper selects the optimal satellite subset that minimizes the airborne VPL based on the brute method for every sample epoch. After selecting the final optimal subset, the elevation angles of the optimal satellites are calculated and the stable distribution proportions in different elevation ranges are analyzed for every sample epoch at one location. To improve the accuracy of the statistical results and eliminate the impact of temporal factors, the distribution characteristics at all sample epochs during 2015 and 2016 are analyzed. Finally, the mean value of the distribution proportions in different elevation ranges during 2015 and 2016 are obtained. To eliminate the influence of geographical factor and determine whether the distribution proportions of the optimal satellite subsets are applicable worldwide, this paper analyzes the distribution characteristics at different locations in detail using a \(1^{\circ}\times 1^{\circ}\) grid. Thus, the distribution characteristics fully account for the temporal and geographical characteristics. The distribution characteristics of optimal satellite subsets under non-nominal troposphere are shown in detail in Figures 14 and 15 using a \(1^{\circ}\times 1^{\circ}\) grid. Figure 14: Global distribution ratios of the optimal satellite subsets with the brute method. Figure 14 shows the stable distribution proportions of the satellites in the optimal subsets obtained by brute method in the different elevation ranges during 2015 and 2016 globally. The coloring indicates the distribution ratios (%) of the optimal satellites in different elevation ranges. Figure 15 shows the number of visible satellites in different elevation ranges worldwide. The coloring indicates the number of visible satellites distributed in different elevation ranges. The results shown in Figures 14 and 15 indicate that the numbers of optimal satellite subsets are stably distributed in elevation ranges of (\(5^{\circ}\), \(30^{\circ}\)), (\(30^{\circ}\), \(60^{\circ}\)) and (\(60^{\circ}\), \(90^{\circ}\)) when the airborne VPL is minimized under the non-nominal troposphere. Based on the above analysis, the proposed method divides the satellite elevation into three categories: low elevation range (\(5^{\circ}\), \(30^{\circ}\)), mid-elevation range (\(30^{\circ}\), \(60^{\circ}\)) and high elevation range (\(60^{\circ}\), \(90^{\circ}\)). Based on the results shown in Figure 9, the change rates of the non-nominal troposphere error in different elevation ranges are shown in Table 3.  
The change rate of the non-nominal troposphere error is calculated as:  
\[rate=\left|\frac{error_{elev_{k+1}}-error_{elev_{k}}}{error_{elev_{k}}}\right| \tag{17}\]  
where \(i\) indicates the \(i\)th elevation range and \(elev_{k}\) is the starting elevation of the \(k\)th elevation range. Figure 9 and Table 3 show that, when the satellite elevation angle is within (\(60^{\circ}\), \(90^{\circ}\)), the error change rate is the smallest. The mean value of the non-nominal troposphere error is 0.21 m, indicating \begin{table}
\begin{tabular}{c c} \hline \hline
**Elevation Range** & **Change Rate of Error** \\ \hline (\(5^{\circ}\), \(30^{\circ}\)) & 74.25\% \\ (\(30^{\circ}\), \(60^{\circ}\)) & 48.76\% \\ (\(60^{\circ}\), \(90^{\circ}\)) & 14.95\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Change rates of the non-nominal troposphere error for different elevation ranges. Figure 15: The number of visible satellites in different elevation ranges. that the satellites with elevation angles in (60\({}^{\circ}\), 90\({}^{\circ}\)) can contribute to reducing the non-nominal troposphere error in the constellation. In contrast, when the satellite elevation angle is in the range of (5\({}^{\circ}\), 30\({}^{\circ}\)), the error change rate and non-nominal troposphere error are the largest.