# Impact Analysis of Satellite Geometry Variation on ARAIM Integrity Risk over Exposure Interval
## 3 Integrity Risk Evaluation over the Exposure Interval
### Evaluation of \(P_{\text{HMI}}\)  
With the definition of HMI, this section applies it to evaluate the integrity risk over the exposure interval. Figure 2 illustrates the epochs that are essential to consider for the integrity risk evaluation under a specific fault mode. Figure 2 indicates that an HMI event will have an impact on navigation safety only when its duration exceeds the time length of \(t_{\text{TTA}}\). Additionally, HMI events can potentially occur at any point within the exposure interval. Therefore, a rigorous integrity risk evaluation requires the incorporation of geometry information from all epochs across the exposure interval. The integrity risk within the exposure interval can be derived from (2) by summing up the probabilities of \(\mathrm{HMI_{TTA,j}}\) across all fault modes and epochs. For clarity, we will only present the expression for the integrity risk in a single \(t_{\mathrm{TTA}}\) under a specific fault event \(k\),  
\[P\Big{\{}\mathrm{HMI_{TTA,j}}|H_{k}\Big{\}}=P\bigg{\{} \begin{pmatrix}j+n_{aa} \\ \cap\limits_{i=j}^{i}\mathrm{HMI_{i}}\end{pmatrix} \bigg{|}H_{k}\Big{\}}=1-P \bigg{\{} \begin{pmatrix}j+n_{aa}\\ \cup\limits_{i=j}^{i}\mathrm{HMI_{i}}\end{pmatrix} \bigg{|}H_{k}\bigg{\}} \tag{7}\]  
where \(\overline{\mathrm{HMI_{i}}}=\Big{\{}\Big{|}x_{k}^{(i)}\Big{|}\leq F_{k}^{(i)} \Big{\}}\). Due to the positioning error \(x_{k}^{(i)}\) being highly time-correlated within the allowable time-to-alert, we introduce the PIE to accurately deal with the impact of time-correlation and satellite geometry variation on the integrity risk evaluation simultaneously. The probabilistic PIE is a method used to calculate the probability of unions of events [12]. Based on the PIE, the probability of the \(\overline{\mathrm{HMI_{i}}}\) occurring under fault mode hypothesis \(H_{k}\) can be described as follows:  
\[\begin{array}{ll}P\bigg{\{} \begin{pmatrix}j+n_{aa}\\ \cup\limits_{i=j}^{i}\mathrm{HMI_{i}}\end{pmatrix} \bigg{|}H_{k}\bigg{\}}&= \sum\limits_{a=-j}^{j+n_{aa}}P(C_{a}|H_{k})-\sum\limits_{j\leq a<b\leq j+n_{aa }}P(C_{a,b,c}|H_{k})\\ &+\cdots+(-1)^{n_{aa}-1}\sum\limits_{j\leq l_{1}<l_{2}\cdots<l_{aa}\leq j+n_{ aa}}P\Big{(}C_{l_{1},l_{2}\cdots l_{aa}}|H_{k}\Big{)}+(-1)^{n_{aa}}P\Big{(}C_{j,(j+1) \cdots(j+n_{aa})}|H_{k}\Big{)}\\ &\text{where the intersection of }\overline{\mathrm{HMI_{i}}}\) is descried as \(C_{l_{1},l_{2}\cdots l_{n_{aa}}}=\Big{\{}\overline{\mathrm{HMI_{i}}}\cap \overline{\mathrm{HMI_{i_{2}}}}\cap\cdots\cap\overline{\mathrm{HMI_{i_{aa}}}} \Big{\}}\) to simplify notations. Given the threshold \(F_{k}^{(i)}\), the probability of the \(C_{l_{1},l_{2}\cdots l_{n_{aa}}}\) can be expressed as follows,  
\[ \begin{array}{ll}P\Big{(}C_{l_{1},l_{2}\cdots l_{aa}}|H_{k}\Big{)}&=P\Big{[} \begin{subarray}{c}l_{aa}\\ \cup\limits_{i=1}^{l_{aa}}\end{subarray}\Big{(}-F_{k}^{(i)}<x_{k}^{(i)}<F_{k}^ {(i)}\Big{)}\Big{|}H_{k}\Big{]}\\ &=\frac{(2\pi)^{-(n_{aa})/2}}{\sqrt{|\Sigma|}}f_{-F_{k}^{(i)}}^{F_{k}^{(i_{aa} )}}\cdots f_{-F_{k}^{(i_{aa})}}^{F_{k}^{(i_{aa})}}\exp\Bigl{[}-\frac{1}{2} \Big{(}X_{k}^{\top}\Sigma^{-1}X_{k}\Big{)}\Bigr{]}\mathrm{d}x_{k}^{(i_{aa})} \cdots\mathrm{d}x_{k}^{(l_{1})}\end{array}  \tag{9}\]  
where the sequence of random vectors \(\mathbf{X}_{k}\) composed of the positioning errors, i.e., \(\mathbf{X}_{k}=\left(x_{k}^{(l_{1})},\ldots,x_{k}^{(l_{aa})}\right)^{\top}\). Equation (9) involves a multidimensional-integration calculation. In this paper, the MATLAB function _mvncdf_ was used to implement the integral calculation. Moreover, the normalized positioning errors covariance matrix of adjacent epochs can be expressed as  
\[\mathbf{\Sigma}=\left[ \begin{array}{cccc}1&&e^{-\beta At}&&\cdots&e^{-(n_{aa}-1 )\beta At}\\ e^{-\beta\Delta At}&&1&&\cdots&e^{-(n_{aa}-2)\beta At}\\ \vdots&&\vdots&&\ddots&\vdots\\ e^{-(n_{aa}-1)\beta At}&&e^{-(n_{aa}-2)\beta At}&&\cdots&1\end{array} \right] \tag{10}\]  
Figure 2: A conceptual illustration of the integrity risk evaluation within exposure interval.