# An Improved RAIM Availability Assessment Method Based on the Characteristic Slope
## 4 Slope-Based RAIM Availability Assessment
### Derivation of the Ideal Threshold for the Slope  
The ideal slope threshold derivation begins with searching for a condition satisfying \(P(\text{HMI}|1F)<P_{r}(\text{HMI}|1F)\) for all possible measurement bias values of an arbitrary faulty satellite. To ensure \(P(\text{HMI}|1F)<P_{r}(\text{HMI}|1F)\) constantly true, the maximum value of \(P(\text{HMI}|1F)\) should be less than \(P_{r}(\text{HMI}|1F)\),  
\[max[P(\text{HMI}|1F)]<P_{r}(\text{HMI}|1F) \tag{14}\]  
where:  
\[\begin{array}{ll}P(HMI|1F|&=P(|\text{VPE}|\geq VAL|1F)P(\text{Ts}<Td|1F)\\ &=\left[1-\int_{-\text{VAL}}^{\text{VAL}}g\left(x;\frac{a_{3m}\xi_{b}}{\sigma_ {m}^{2}},a_{v}^{2}\right)dx\right]\int_{0}^{Td}f\left(y;\left(\frac{\xi_{b}}{ \sigma_{m}^{2}}\right)^{2}s_{mm},K-4\right)dy\\ \end{array} \tag{15}\]  
Substituting Equation (7) into Equation (15),  
\[\begin{array}{ll}P(\text{HMI}|1F)=\left[1-\int_{-\text{VAL}}^{\text{VAL}}g \left(x;\frac{a_{3m}\xi_{b}}{\sigma_{m}^{2}},a_{v}^{2}\right)dx\right]\int_{ 0}^{Td}f\left(y;\left(\frac{\xi_{b}}{\sigma_{m}^{2}}\right)^{2}\left(\frac{a_ {3m}}{Slope_{m}}\right)^{2},K-4\right)dy\right)\\ \end{array} \tag{16}\]  
Setting \(t=\frac{x-a_{3m}\xi_{b}/\sigma_{m}^{2}}{a_{v}}\), Equation (16) can be transformed into:  
\[\begin{array}{ll}P(\text{HMI}|1F)=\left[1-\int_{-\frac{\text{VAL}-\frac{a_ {3m}\xi_{b}}{\sigma_{m}^{2}}}{\sigma_{m}^{2}}}^{\frac{\text{VAL}-\frac{a_{3m} \xi_{b}}{\sigma_{m}^{2}}}{\sigma_{m}^{2}}}g\left(t;0,1\right)dt\right]\int_{ 0}^{Td}f\left(y;\left(\frac{a_{3m}\xi_{b}}{\sigma_{m}^{2}}\right)^{2}\left( \frac{1}{Slope_{m}}\right)^{2},K-4\right)dy\right)\\ \end{array} \tag{17}\]  
Setting \(\mu_{s}=\frac{a_{3m}\xi_{b}}{\sigma_{m}^{2}}\), \(P(\text{HMI}|1F)\) can be taken as a function of \(\mu_{s}\) as follows:  
\[\begin{array}{ll}\Phi(\mu_{s})=P(\text{HMI}|1F)=\left[1-\int_{-\frac{\text{VAL}-\mu_{s}}{ \sigma_{m}^{2}}}^{\frac{\text{VAL}-\mu_{s}}{\sigma_{m}^{2}}}g\left(t;0,1\right) dt\right]\int_{0}^{Td}f\left(y;\left(\frac{\mu_{s}}{Slope_{m}}\right)^{2},K-4 \right)dy\end{array} \tag{18}\]When substituting Equation (18) into Equation (14), an ideal threshold must exist for \(Slope_{m}\) with the given \(P_{r}(\text{HMI}|1F)\), VAL, \(a_{v}\), \(Td\) and \(K\), denoted as \(T_{Slope}\), which satisfies the limit situation  
\[max\Big{\{}\Phi(\mu_{s})\Big{|}Slope_{m}=T_{Slope}\Big{\}}=P_{r}(\text{HMI}|1F) \tag{19}\]  
According to Equations (18) and (19), once \(P_{r}(\text{HMI}|1F)\), VAL, \(Td\) and \(K\) are given, \(T_{Slope}\) is determined by only one parameter \(a_{v}\).  
Figure 5 presents the numerically solved \(T_{Slope}\) for \(a_{v}\) values with a step of 0.001 at \(P_{r}(\text{HMI})=1\times 10^{-7}\), \(P_{r}(\text{HMI},2)=1.3\times 10^{-8}\), \(\text{VAL}=50\) m, \(P_{sat}=1\times 10^{-5}\), \(\alpha=1\times 10^{-6}\) and \(K=7,\cdots,11\). As shown in Figure 5, \(T_{Slope}\) decreases with the increase of \(a_{v}\). Each \(a_{v}-T_{Slope}\) curve exhibits a nearly constant segment at the beginning and a sharply decreasing segment at the end. The \(a_{v}-T_{Slope}\) curve ends when \(a_{v}\) reaches \(T_{av}\), and the spacing between two adjacent curves obviously decreases as \(K\) increases.