# Impact Analysis of Satellite Geometry Variation on ARAIM Integrity Risk over Exposure Interval
## 2 Problem Definition  
Since the integrity risk requirement is defined as the probability over the specific exposure interval, the integrity risk is required to evaluate over multiple detection epochs. Without losing generality, the integrity risk over the exposure interval can be expressed as [21],  
\[P_{\text{HMI,actual}}=\sum\limits_{k=0}^{N}P\left\{\overset{n_{\text{s}}-n_{ \text{sa}}\wedge j+n_{\text{sa}}}{\underset{i=j}{\cap}}(\text{HMI}_{i},\text{ H}_{k,i})\right\}+P_{\text{NM}}\big{(}t_{\text{exp}}\big{)} \tag{1}\]  
where HMI represents the hazardous misleading information, which is the situation when an undetected positioning error exceeds a predefined alert limit (AL). There are three different time metrics for integrity risk evaluation, i.e., the exposure interval time (\(t_{\text{exp}}\)), the time-to-alert (\(t_{\text{TTA}}\)) defined by required navigation performance, and the single detection time (\(t_{\text{s}}\)). Based on these time metrics, two parameters required for integrity risk evaluation, i.e., \(n_{\text{s}}\) and \(n_{\text{sa}}\), can be determined. The \(n_{\text{s}}\) is the number of epochs in the \(t_{\text{exp}}\), i.e., \(t_{\text{exp}}/t_{\text{s}}\). The \(n_{\text{sa}}\) is the number of epochs in the \(t_{\text{TTA}}\), i.e., \(t_{\text{TTA}}/t_{\text{s}}\). As seen from Figure 1, we usually have \(t_{\text{exp}}>t_{\text{TTA}}>t_{\text{s}}\). Furthermore, \(H_{\text{k}}\) in (1) represents the fault mode hypotheses for \(k=0\ldots N\), which account for all fault satellite combinations including \(H_{0}\), single satellite fault, multi-satellite faults, and constellation fault [22; 23; 24; 25]. \(N\) is the number of fault modes to be monitored. \(P_{\text{NM}}(t_{\text{exp}})\) is the probability of the fault modes that need not be monitored in the exposure interval. The calculation method of \(P_{\text{NM}}(t_{\text{exp}})\) and the fault modes number \(N\) can be found in [16]. \(i\) and \(j\) represent the epoch time within the exposure interval. It should be noted that we will denote the event of continuous HMI occurrence within \(t_{\text{TTA}}\) under fault mode \(k\), i.e., \(\cap_{i=j}^{i+n_{\text{sa}}}(\text{HMI}_{i},\text{H}_{k,i})\), as \(\text{HM}_{\text{TTA},j}^{k}\) in the following to simplify the notation. Figure 1: Epochs over the exposure interval. The integrity risk over the exposure interval can be accurately evaluated based on (1) [13]. However, the evaluation of integrity risk based on (1) is computationally challenged by the presence of time-correlation among multiple epochs [8]. A conservative but computationally efficient risk-evaluation model is usually used to calculate the integrity risk, instead of (1) [23],  
\[ \begin{array}{ll}\overline{P}_{\text{HMI,actual}}&=\sum\limits_{k=0}^{N}\sum \limits_{j=1}^{n_{\text{s}}-n_{\text{eq}}}P\Big{\{}\text{HMI}_{\text{TTA}j}^{k} \Big{\}}+P_{\text{NM}}\big{(}t_{\text{exp}}\big{)}\\ &=\sum\limits_{k=0}^{N}\sum\limits_{j=1}^{n_{\text{s}}-n_{\text{eq}}}P\Big{\{} \text{HMI}_{\text{TTA}j}|\text{H}_{k}\Big{\}}P(\text{H}_{k})+P_{\text{NM}}\big{(} t_{\text{exp}}\big{)}\end{array}  \tag{2}\]  
where \(P(\text{H}_{k})\) is the prior probability of the fault mode H\({}_{k}\). It should be noted that the subscript \(i\) of \(H_{k,i}\) is eliminated in (2) because we assume that the fault mode occurring at the first epoch in the exposure interval will always persist in this interval. This assumption is reasonable, because according to analysis in [1], the mean fault duration of GPS and Galileo is 1 h, which is usually much longer than the exposure interval. Equation (2) strictly considers all epochs' geometry in the exposure interval and is referred to as the geometry-sensitive risk-evaluation model. Provided the statistical distribution of HMI and the satellite geometry from all epochs across the exposure interval, the integrity risk can be obtained based on the geometry-sensitive risk-evaluation model.