# Impact Analysis of Satellite Geometry Variation on ARAIM Integrity Risk over Exposure Interval
## 3 Integrity Risk Evaluation over the Exposure Interval
### Hazardous Misleading Information  
This section provides a description of the HMI definition for various satellite geometry. According to (2), the integrity risk of navigation application is determined by the temporal characteristics of the HMI events. In the baseline ARAIM algorithm, the HMI event under fault mode hypotheses \(k\) is described as [17]  
\[(\text{HMI}_{i}|\text{H}_{k})=\bigg{(}\Big{|}x_{0,q}^{(i)}\Big{|}>\text{AL} \cap\underset{d=1}{\overset{N}{\cap}}\Big{|}x_{d,q}^{(i)}-x_{0,q}^{(i)}\Big{|} \leq T_{d,q}^{(i)}\Big{|}\text{H}_{k}\bigg{)} \tag{3}\]  
\[\subset\Bigg{(}\Big{|}x_{k,q}^{(i)}\Big{|}>\frac{\text{AL}-T_{k,q}^{(i)}- \Big{|}\mathbf{s}_{k,q}^{(i)}\Big{|}^{T}\mathbf{b}_{\text{nom}}}{\sigma_{k,q}^ {(i)}}\Bigg{|}\text{H}_{k}\Bigg{)}=\Big{(}\Big{|}x_{k,q}^{(i)}\Big{|}>F_{k,q}^ {(i)}\Big{|}\text{H}_{k}\Big{)} \tag{4}\]  
where \(x_{0,q}^{(i)}\) is a position estimate error using all satellites in view at epoch \(i\). The subscript \(q=1\), \(2\), and \(3\) denote the east, north, and up components, respectively. \(x_{d,q}^{(i)}\) is a position estimate error using all satellites except the one(s) included in fault mode \(d\). The distribution of \(x_{d,q}^{(i)}\) in (3) and (4) can be written in detail as  
\[x_{d,q}^{(i)}\sim N\bigg{(}\mathbf{s}_{d,q}^{(i)}\mathbf{b}_{\text{nom}},\bigg{(} \sigma_{d,q}^{(i)}\bigg{)}^{2}=\mathbf{e}_{q}^{\text{T}}\Big{(}\mathbf{H}_{i}^ {\text{T}}\mathbf{R}_{c}\mathbf{H}_{i}\Big{)}^{-1}\mathbf{e}_{q}\bigg{)} \tag{5}\]  
where \(\mathbf{s}_{d,q}^{(i)}\) is a \(1\times n\) project vector to exclude the fault satellite, i.e., \(\mathbf{s}_{d,q}^{(i)}=\mathbf{e}_{q}^{\text{T}}\big{(}\mathbf{H}_{i}^{\text{ T}}\mathbf{R}_{c}\mathbf{H}_{i}\big{)}^{-1}\mathbf{H}_{i}^{\text{T}}\mathbf{R}_{c}\), \(\mathbf{R}_{c}\) is the identity matrix with the \(c\)-th diagonal element equal to 0 when the \(c\)-th satellite is included in fault mode hypothesis \(d\), and \(\mathbf{e}_{q}\) is the \(m\times 1\) vector used to extract the \(q\)-th state out of the full state vector. For example, use of \(\mathbf{e}_{q}=\left[0\ 0\ 1\ \mathbf{0}_{(m-3)\times 1}^{\text{T}}\right]^{ \text{T}}\) to extract the vertical position error. \(\mathbf{b}_{\text{nom}}\) is the \(n\times 1\) nominal bias vector. \(\sigma_{d,q}^{(i)}\) is the standard deviation of the estimated error for the \(q\)-th state. \(\mathbf{H}_{i}\) is the \(n\times m\) normalized geometry matrix in epoch \(i\) and the normalization procedure is described in [24], \(n\) is the number of all-in-view satellites, and \(m\) is the number of estimated states. In addition, the detection threshold \(T_{d,q}^{(i)}\) in (3) and (4) is set based on an allocated continuity requirement \(P_{PA,q,REQ}\) and \(\text{NES}_{\text{{scott}}}\) for continuity risk evaluation [16],  
\[T_{q,d}^{(i)}=Q^{-1}\bigg{(}\frac{P_{FA,q,REQ}}{2P_{\text{H}_{0}}\cdot N\cdot \text{NES}_{\text{{scott}}}}\bigg{)}\sigma_{\Delta_{d,q}}^{(i)} \tag{6}\]  
where \(Q^{-1}\) is the inverse tail probability function of a zero-mean unit-variance Gaussian distribution. \(\sigma_{\Delta_{d,q}}^{(i)}\) is the standard deviation of the detection statistic for the \(q\)-th state. A detail calculation of above parameters can be found in [23], and the superscript \(q\) is eliminated in the following to simplify notations. It should be noted that this section introduces two distinct definitions of HMI characterization, denoted as (3) and (4). Compared to (4), Equation (3) describes a more precise HMI event and is often utilized to precisely analyze the values of certain parameters associated with the ARAIM [21]. However, conducting an integrity risk evaluation based on this definition involves a complex challenge of determining the worst-case fault. Currently, there is a lack of appropriate strategies to determine the worst-case fault considering the variation in satellite geometry. From a navigation safety standpoint, the HMI definition based on (4) is more conservative than that based on (3).