# An Improved RAIM Availability Assessment Method Based on the Characteristic Slope
## 3 PL-Based RAIM Availability Assessment
### Deficiency of Classic- and Enhanced-PL-Based RAIM Availability Assessment
#### 3.2.3. Tangency  
In this situation, there are two kinds of positional relationships among the \(\xi_{b}-lgP(\text{HMI}|1F)\) curve, the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve and the \(lgP_{r}(\text{HMI}|1F)\) line, as shown in Figure 4.  
The \(\xi_{b}-lgP(\text{HMI}|1F)\) curve intersects with the \(lgP_{r}(\text{HMI}|1F)\) line, and is higher than the blue dashed \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve, shown as the brown \(P_{1}(\text{HMI}|1F)\) curve in Figure 4. In this situation, RAIM is unavailable, and the VPL is larger than the VAL, i.e., \(\text{VPL}\geq\text{VAL}|H_{1}\), meaning a successful detection of \"RAIM being unavailable\".  
The \(\xi_{b}-lgP(\text{HMI}|1F)\) curve is separated from the \(lgP_{r}(\text{HMI}|1F)\) line, and is lower than the blue dashed \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve, shown as the black \(P_{2}(\text{HMI}|1F)\) curve in Figure 4. In this situation, RAIM is available and the VPL is smaller than the VAL, i.e., \(\text{VPL}<\text{VAL}|H_{0}\), meaning a successful detection of \"RAIM being available\".  
Therefore, for the condition that the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve is tangent to the \(lgP_{r}(\text{HMI}|1F)\) line, both optimistic and conservative assessments can be prevented.  
Based on the above analysis, the accuracy of PL-based RAIM availability assessment depends on the positional relationship between the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve and the \(lgP_{r}(\text{HMI}|1F)\) line. There is a risk of conservative assessment when the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve is separated from the \(lgP_{r}(\text{HMI}|1F)\) line and risk of optimistic assessment when the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve intersects with the \(lgP_{r}(\text{HMI}|1F)\) line. PL-based RAIM availability assessment is accurate only if the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve is tangent to the \(lgP_{r}(\text{HMI}|1F)\) line. Because only the \(P_{a}(\text{HMI}|1F)\) value at a specific \(\xi_{b}\), i.e., \(\xi_{b}=\sigma_{m}^{2}\sqrt{\frac{\lambda_{a}}{s_{\text{max}}}}\), is determined for \(\text{VPL}_{c}\) or \(\text{VPL}_{e}\), the position relationship between the entire \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve and the \(lgP_{r}(\text{HMI}|1F)\) line is uncertain. Consequently, both optimistic and conservative assessments might happen when using \(\text{VPL}_{c}\) or \(\text{VPL}_{e}\) to assess whether vertical RAIM is available.  
Because \(VPL_{c}<\text{VPL}_{e}\), the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve for \(VPL_{c}\) is much higher than that for \(VPL_{e}\). Thus the possibility of intersection between the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve and the \(lgP_{r}(\text{HMI}|1F)\) line for \(\text{VPL}_{c}\) is much higher than that for \(\text{VPL}_{e}\), which may lead to an optimistic assessment, while the possibility of separation between the \(\xi_{b}-lgP_{a}(\text{HMI}|1F)\) curve and the \(lgP_{r}(\text{HMI}|1F)\) line for \(\text{VPL}_{e}\) is much higher than that for \(\text{PL}_{c}\), which may lead to conservative assessment. Consequently, the optimistic assessment risk of using \(\text{VPL}_{c}\) is  
Figure 4: Tangency.  
higher than that of using VPL\({}_{e}\) for vertical RAIM availability; in contrast, the conservative assessment risk of using VPL\({}_{e}\) is higher than that of using VPL\({}_{c}\).