# An Improved RAIM Availability Assessment Method Based on the Characteristic Slope
## 4 Slope-Based RAIM Availability Assessment
### Slope-Based RAIM Availability Assessment Method  
Slope-based RAIM availability assessment should be implemented in the worst case to fully prevent the integrity risk. Using slope-based method, RAIM is considered available if the maximum slope is less than the ideal threshold \(T_{Slope}\); otherwise, it is considered unavailable. The specific execution process for slope-based RAIM availability assessment is presented in Figure 7.  
As shown in Figure 7, the inputs are the observation matrix \(\mathbf{H}\) and the weighted matrix \(\mathbf{P}\). The first step is an assessment based on the total number of visible satellites \(K\). RAIM is considered unavailable if \(K\leq 4\) because fault detection cannot be executed with less than 5 visible satellites. For \(K>4\), the second step is an assessment based on the \(a_{v}\) value. RAIM is considered to be unavailable if \(a_{v}\geq Ta_{v}\) because only the sum of \(P(\mathrm{HMI},0F)\) and \(P(\mathrm{HMI},2F)\) has exceed \(P_{r}(\mathrm{HMI})\) when \(a_{v}\) exceeds \(Ta_{v}\). For \(a_{v}<Ta_{v}\), the last step is an assessment based on the maximum slope \(Slope_{max}\). RAIM is finally considered avail  
Figure 6: Practical meaning of \(T_{Slope}\). _Ta\({}_{v}\)_ can be calculated online according to Equation (6) with given VAL, \(P_{sat,A}\), \(P_{r}(\text{HMI})\) and \(P_{r}(\text{HMI}|2F)\). \(T_{Slope}\) should be calculated offline with discrete \(a_{v}\) with a small step size for different numbers of visible satellites and saved in the GNSS receiver or the onboard computer. Assuming that the total number of visible satellites is \(K\) and \(a_{v}\) is in the interval of \((a_{v1},a_{v2})\), where \(a_{v1}\) and \(a_{v2}\) are the indexes of two adjacent discrete points \(\left[a_{v1},T_{Slope_{1}}\right]\) and \(\left[a_{v2},T_{Slope_{2}}\right]\) presaved for \(K\) visible satellites, \(T_{Slope}\) corresponding to \(a_{v}\) should be assigned as:  
\[T_{Slope}=\frac{T_{Slope1}-T_{Slope2}}{a_{v1}-a_{v2}}(a_{v}-a_{v1})+T_{Slope1} \tag{21}\]  
This value is calculated according to a linear fit for \(T_{Slope}\) in the interval of \((a_{v1},a_{v2})\). Because the \(a_{v}-T_{Slope}\) curve is convex as shown in Figure 5, the assigned \(T_{Slope}\) value is smaller than the real \(T_{Slope}\) value, which may cause a small conservative assessment risk. However, if the \(a_{v}\) step size is small enough, the assigned \(T_{Slope}\) value would be nearly equal to the true value, thereby preventing the small conservative assessment risk. A simulation is designed to find the desirable range \(a_{v}\) step. \(1.2\times 10^{6}\) times of \(a_{v}\) and \(Slope_{max}\) for different location and epochs are collected. As shown in Table 2, the times of conservative assessment increases with step size widen. Taking into account both the amount of calculation and conservative assessment rate, 0.01-0.02 is the desirable range \(a_{v}\) step because it is the maximum step size with 0 time of conservative assessment. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline Step & 0.001 & 0.01 & 0.02 & 0.03 & 0.04 & 0.05 & 0.1 & 0.2 & 0.5 \\ \hline Times of conservative assessments & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 2 & 4 \\ \hline \end{tabular}
\end{table} 
Table 2: Step and times of conservative assessment. Figure 7: Slope-based RAIM availability assessment process. It should be mentioned that the calculation amount of the ideal-PL and \(T_{Slope}\) is exactly the same. The ideal PL procedure begins with an improbably large VPL of 2000 m and halves the search step by checking if the corresponding integrity risk exceeds the requirement [3]. For each step of the iteration, the integrity risk needs to be calculated for different bias values with a fixed step. This process is computationally intensive. Similarly, the \(T_{Slope}\) procedure begins with an improbably large slope value of 15 and halves the search step.