# A Method to Reduce Non-Nominal Troposphere Error
## 4 VPL under Non-Nominal troposphere
### Simulation Results
#### 4.3.2 Results Analysis  
Figure 10 presents the VPL of all visible satellites based on the current GPS constellation and the future global BDS constellation under non-nominal troposphere. The figure presents the following results:  
* \(VPL_{non-all}\) is the VPL component to bound the zero-mean Gaussian error of all visible satellites. * \(VPL_{bias-all}\) is the VPL component to bound the non-nominal troposphere error of all visible satellites. According to Figure 10, for all visible satellites, the average \(VPL_{non}\) is 1.97 m and the average \(VPL_{bias}\) is 2.32 m.  
In 2016, [PERSON] of German Aerospace Center found that when the number of randomly selected satellites is set to 14, the average increase is between 1.5 and 12 cm and stays below 5 cm for medium latitudes. A significant influence on the GBAS availability and integrity is therefore very unlikely to occur when the number of satellites is limited to 14. In other words, the VPL of 14 randomly selected visible satellites increased by less than 5% compared with that of all visible satellites, which indicates that the selection of a subset of 14 satellites can maintain 95% of the initial accuracy and meet the GBAS integrity requirements [6]. Figure 11 presents the VPL of a satellite subset containing 14 randomly selected visible satellites. The figure presents the following results:  
* \(VPL_{non-14}\) is the VPL component to bound the zero-mean Gaussian error of 14 visible satellites. * \(VPL_{bias-14}\) is the VPL component to bound the non-nominal troposphere error of 14 visible satellites. Figure 11 shows that, when 14 visible satellites are included in a satellite subset, the average \(VPL_{non}\) and \(VPL_{bias}\) is 2.09 m and 1.89 m, respectively. According to the results shown in Figures 10 and 11, compared with the VPL of all visible satellites, the average \(VPL_{non}\) increase is 0.12 m, and the average \(VPL_{bias}\) decrement is 0.43 m.  
Figure 12 presents the \(VPL_{non}\) of satellite subsets containing all, 13, 14, 15, 16 and 17 visible satellites. The 13, 14, 15, 16 and 17 satellites are randomly selected from all visible satellites. This figure presents the following results:  
* \(VPL_{non-13}\) is the VPL component to bound the zero-mean Gaussian error of 13 visible satellites. * \(VPL_{non-15}\) is the VPL component to bound the zero-mean Gaussian error of 15 visible satellites. * \(VPL_{non-16}\) is the VPL component to bound the zero-mean Gaussian error of 16 visible satellites. * \(VPL_{non-17}\) is the VPL component to bound the zero-mean Gaussian error of 17 visible satellites. Figure 10: VPL of all visible satellites under non-nominal troposphere. From Figure 12, we can derive the following inequality:  
\[VPL_{non\_all}<VPL_{non\_17}<VPL_{non\_16}<VPL_{non\_15}<VPL_{non\_14}<VPL_{non\_13} \tag{13}\]  
From Inequality (13), we can conclude that more visible satellites improves satellite geometry, which, in turn, increases \(VPL_{non}\). Figure 13 presents the \(VPL_{bias}\) of satellite subsets containing all, 13, 14, 15, 16 and 17 visible satellites. The satellites are also randomly selected from all visible satellites. This figure presents the following results:  
* \(VPL_{bias-13}\) is the VPL component to bind the non-nominal troposphere error of 13 visible satellites. * \(VPL_{bias-15}\) is the VPL component to bind the non-nominal troposphere error of 15 visible satellites. * \(VPL_{bias-16}\) is the VPL component to bind the non-nominal troposphere error of 16 visible satellites. * \(VPL_{bias-17}\) is the VPL component to bind the non-nominal troposphere error of 17 visible satellites. From Figure 13, we can derive the following inequality:  
\[VPL_{bias\_16}<VPL_{bias\_14}<VPL_{bias\_13}<VPL_{bias\_15}<VPL_{bias\_17}<VPL_{ bias\_all} \tag{14}\]  
Figure 12: \(VPL_{non}\) of different satellite subsets under non-nominal troposphere. Figure 13: \(VPL_{bias}\) of different satellite subsets under non-nominal troposphere.