# An Improved RAIM Availability Assessment Method Based on the Characteristic Slope
## 2 Technical Background
### Integrity Risk Requirement for the Single-Satellite Fault Mode  
The integrity risk \(P(\mathrm{HMI})\)[19] is the probability of undetected faults causing unacceptably large errors in the estimated position [20]. HMI is short for hazardous misleading information (HMI). \(P(\mathrm{HMI})\) can be divided into three fault modes, the nominal mode, the single-satellite fault mode and the multiple-satellite fault mode, expressed as:  
\[P(\mathrm{HMI})=\sum_{i=0}^{2}P(\mathrm{HMI},iF)=\sum_{i=0}^{2}P(\mathrm{HMI}| iF)P(iF) \tag{1}\]  
In Equation (1), \(0F\), \(1F\) and \(2F\) respectively represent the nominal, the single-satellite fault and the multiple-satellite fault modes, where \(P(iF)\) is the prior probability of the fault mode \(iF\) and \(P(\mathrm{HMI}|iF)\) is the probability of HMI under the \(iF\) fault mode. \(P(0F)=\left(1-P_{sat}\right)^{K}\) and \(P(1F)=C_{K}^{1}P_{sat}\left(1-P_{sat}\right)^{K}\), where \(C_{K}^{1}\) means the number of combinations for choosing one element from K elements; \(P_{sat}\) is the prior fault probability of a satellite, and \(K\) is the total visible satellite number. Taking vertical plane for example, \(P(\mathrm{HMI}|iF)\) is calculated with the following equation [21]:  
\[P(\mathrm{HMI}|iF)=P(|\mathrm{VPE}|\geq\mathrm{VAL}|iF\,)P(T\mathrm{s}<Td|iF) \tag{2}\]  
In Equation (2), VPE is the vertical position error; \(\mathrm{VAL}\) is the vertical \(\mathrm{AL}\); \(T\mathrm{s}\) and \(Td\) are the fault detection test statistic and threshold, respectively.  
To ensure the integrity of the navigation system, \(P(\mathrm{HMI})\) should be less than its requirement, denoted as \(P_{r}(\mathrm{HMI})\).  
\[P(\mathrm{HMI},0F)+P(\mathrm{HMI},1F)+P(\mathrm{HMI},2F)<P_{r}(\mathrm{HMI}) \tag{3}\]  
Given a geometry between the user and all-in-view satellites, \(\mathrm{VAL}\), \(\alpha\) and \(P_{sat}\), \(P(\mathrm{HMI},0F)\) can be calculated following:  
\[P(\mathrm{HMI},0F)=(1-\alpha)\bigg{[}1-\int_{-\mathrm{VAL}}^{\mathrm{VAL}}g \Big{(}x;0,a_{v}^{2}\Big{)}dx\bigg{]}\Big{(}1-P_{sat}\Big{)}^{K} \tag{4}\]  
In Equation (4), \(\alpha\) is the allowable false alarm probability under the nominal fault mode, satisfying \(P(T\mathrm{s}<Td|0F)=1-\alpha\); \(g\big{(}x;0,a_{v}^{2}\big{)}\) is the probability density function (PDF) of the normal-distributed VPE under the nominal fault mode with mean value 0 and standard deviation value \(a_{v}\), where \(a_{v}\) is explained in Appendix A.  
According to GNSS Evolutionary Architecture Study (GEAS) report [22], the integrity risk requirement allocated on the multiple satellites fault, denoted as \(P_{r}(\mathrm{HMI},2F)\), can be set \(1.3\times 10^{-8}\)[22].  
To ensure Equation (3) to be true, the integrity risk for single fault mode should satisfy:  
\[P(\mathrm{HMI},1F)<P_{r}(\mathrm{HMI}|1F) \tag{5}\]  
where \(P_{r}(\mathrm{HMI}|1F)=\left[P_{r}(\mathrm{HMI})-P_{r}(\mathrm{HMI},2F)-P( \mathrm{HMI},0F)\right]/P(1F)\). Thus, the problem of \(P(\mathrm{HMI})<P_{r}(\mathrm{HMI})\) has evolved into the problem of \(P(\mathrm{HMI}|1F)<P_{r}(\mathrm{HMI}|1F)\).  
Moreover, to ensure that \(P_{r}(\text{HMI},1F)\) is nonnegative, the threshold for \(a_{v}\), denoted as \(Ta_{v}\), can be derived from the inequality \(P_{r}(\text{HMI})-P_{r}(\text{HMI},2F)-P(\text{HMI},0F)\geq 0\).  
\[Ta_{v}=-\text{VAL}/\Phi^{-1}\bigg{(}\frac{P_{r}(\text{HMI})-P_{r}(\text{HMI},2F )}{2(1-\alpha)(1-P_{sat})^{K}}\bigg{)} \tag{6}\]  
where \(\Phi^{-1}(\cdot)\) represents the inverse function of the cumulative distribution function (CDF) for the standard normal distribution. \(a_{v}\geq Ta_{v}\) indicates that only the integrity risk of the nominal and multiple-satellite fault modes have exceeded the total requirement, i.e., \(R_{0}+P_{r}(\text{HMI},2)\geq P_{r}(\text{HMI})\).