# An Improved RAIM Availability Assessment Method Based on the Characteristic Slope
## 5 Simulation
### Specific Example Analysis  
Three specific examples for a single grid point and a simulation epoch are chosen to show the RAIM availability assessment using the four methods in detail. Table 3 records the VPLs, \(Slope_{max}\) and \(T_{Slope}\) values for these specific examples. Figures 8-11 present the real \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve for \(Slope_{max}\), the \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve for \(T_{Slope}\), and the \(\mu_{s}-lgP_{u}(\mathrm{HMI}|1F)\) curves for VPL\({}_{c}\), VPL\({}_{e}\) and VPL\({}_{d}\) in these examples, respectively. In Figures 8-11, the \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve for \(T_{Slope}\) and the \(\mu_{s}-lgP_{u}(\mathrm{HMI}|1F)\) curve for \(VPL_{d}\) are tangent to the \(lgP_{r}(\mathrm{HMI}|1F)\) line, while the \(\xi_{b}-lg_{a}P(\mathrm{HMI}|1F)\) curve for \(VPL_{e}\) is separated from the \(lgP_{r}(\mathrm{HMI}|1F)\) line. The positional relationship between the \(\mu_{s}-lg_{a}P(\mathrm{HMI}|1F)\) curve for \(VPL_{e}\) and the \(lgP_{r}(\mathrm{HMI}|1F)\) line could be either intersection or separation. As shown in Figure 8, the real \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve for \(Slope_{max}\) intersects with the \(lgP_{r}(\mathrm{HMI}|1F)\) line, which means that RAIM is actually unavailable in the first example. The \(\mu_{s}-lgP_{u}(\mathrm{HMI}|1F)\) curve for VPL\({}_{c}\) is higher than the real \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve, i.e., VPL\({}_{c}<\mathrm{VAL}\), which means that RAIM is assessed to be available when using VPL\({}_{c}\). All of the \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve for \(T_{Slope}\), the \(\mu_{s}-lgP_{u}(\mathrm{HMI}|1F)\) curves for VPL\({}_{e}\) and VPL\({}_{d}\) are lower than the real \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve, i.e., \(Slope\geq Slope_{max}\), VPL\({}_{e}\geq\mathrm{VAL}\) and VPL\({}_{d}\geq\mathrm{VAL}\), which means that RAIM is assessed to be unavailable when using \(Slope_{max}\), VPL\({}_{e}\) or VPL\({}_{d}\). Therefore, the RAIM availability is optimistically assessed when using VPL\({}_{c}\) and successfully assessed using when using \(Slope_{max}\), VPL\({}_{e}\) or VPL\({}_{d}\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Num** & **Location** & **UTC Time** & \(K\) & \(a_{v}\) & \(Slope_{max}\) & VPL\({}_{c}\) & VPL\({}_{e}\) & VPL\({}_{d}\) & \(T_{Slope}\) \\ \hline
1 & \(8^{\circ}\) N, \(82^{\circ}\) W & 03:55:00 & 8 & 6.346 & 5.232 & 45.287 & 66.168 & 50.424 & 5.172 \\ \hline
2 & \(45^{\circ}\) S, \(132^{\circ}\) W & 00:00:00 & 8 & 2.704 & 5.330 & 46.139 & 55.037 & 45.475 & 5.887 \\ \hline
3 & \(53^{\circ}\) N, \(122^{\circ}\) E & 08:35:00 & 10 & 1.358 & 5.666 & 50.656 & 55.305 & 49.683 & 5.702 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Specific examples for RAIM availability assessment. Figure 8: RAIM availability judgment for the first specific example. Figure 10: RAIM availability assessment for the third specific example. Figure 9: RAIM availability assessment for the second specific example. As shown in Figure 9, the real \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve for \(Slope_{max}\) is separated from the \(lgP_{r}(\mathrm{HMI}|1F)\) line, which means that RAIM is actually available in the second example. The \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve for \(T_{Slope}\), and the \(\mu_{s}-lgP_{a}(\mathrm{HMI}|1F)\) curves for \(\mathrm{VPL}_{c}\) and \(\mathrm{VPL}_{d}\) are higher than the real \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve, i.e., \(Slope<Slope_{max}\), \(\mathrm{VPL}_{c}<\mathrm{VAL}\) and \(\mathrm{VPL}_{d}<\mathrm{VAL}\), which means that RAIM is assessed to be available when using \(Slope_{max}\), \(\mathrm{VPL}_{c}\) or \(\mathrm{VPL}_{d}\). The \(\mu_{s}-lg_{a}P(\mathrm{HMI}|1F)\) curve for \(\mathrm{VPL}_{e}\) is lower than the real \(\mu_{s}-lgP(\mathrm{HMI}|1F)\) curve, i.e., \(\mathrm{VPL}_{e}\geq\mathrm{VAL}\), which means that RAIM is assessed to be unavailable when using \(\mathrm{VPL}_{e}\).