# An Improved RAIM Availability Assessment Method Based on the Characteristic Slope
## Appendix A Distribution of Position Error  
According to the pseudorange position principle, the linearized observation equation between the receiver and the visible satellites is as follows:  
\[\Delta\mathbf{Z}=\mathbf{H}\Delta\mathbf{X}+\varepsilon \tag{1}\]  
In (1), \(\Delta\mathbf{Z}\) is the \(K\)-dimensional pseudorange offset vector from the real receiver position to the linearization point, where \(K\) is the total number of visible satellites. \(\Delta X=\left[\Delta x,\Delta y,\Delta z,\Delta t_{1},\cdots,\Delta t_{n} \right]^{T}\), where \(\left[\Delta x,\Delta y,\Delta z\right]^{T}\) and \(\left[\Delta t_{1},\cdots,\Delta t_{n}\right]^{T}\) are respectively the offset vectors for the position and clock errors from the real receiver position to the linearization point; \(\Delta t_{n}\) is the clock error for the \(n\)-th GNSS. \(\mathbf{H}\) is a \(K\times(3+n)\) linearized observation matrix in the local Cartesian coordinate (ENU). \(\varepsilon\) is the observation error vector. The observation error includes both noise and bias. The measurement noise for the \(i\)-th visible satellite, denoted as \(VS_{i}\), is assumed to be the independent WGN with mean zero and variance \(\sigma_{i}^{2}\) (\(i=1,\cdots,K\)). \(\sigma_{i}\) is set according to the ARAIM interim report [21]. It should be mentioned that the measurement noise is modeled as non-zero mean Gaussian distribution with a nominal bias in ARAIM interim report [21]. This nominal bias is not considered here to simplify the formula derivation. It really needs to be considered in practical application. \[\sigma_{i}^{2}=\sigma_{URE,i}^{2}+\sigma_{tropo,i}^{2}+\sigma_{user,i}^{2} \tag{2}\]  
where \(\sigma_{URE,i}\) is the standard deviation of a Gaussian error distribution modelling the range component of the signal-in-space error, containing the ephemeris and satellite clock errors.\(\sigma_{tropo,i}\) is the standard deviation of the tropospheric error. \(\sigma_{user,i}\) is the standard deviation of the user error, including the multipath error and the receiver noise. \(\sigma_{URE,i}\), \(\sigma_{tropo,i}\) and \(\sigma_{user,i}\) are calculated using the equations from the ARAIM interim report [21]. The weighted least squares solution for \(X\) in (1) is as follows:  
\[\mathbf{\hat{X}}=\left(\mathbf{H}^{T}\mathbf{P}\mathbf{H}\right)^{-1}\mathbf{H}^{T}\mathbf{P}\mathbf{Z} \tag{3}\]  
In (3), \(\mathbf{P}=diag\big{[}(\sigma_{i}^{2})^{-1}\big{]}\) is the weighted matrix, where \(diag\big{[}(\sigma_{i}^{2})^{-1}\big{]}\) presents a diagonal matrix with the diagonal element \(\big{(}\sigma_{i}^{2})^{-1}\) in the \(i\)-th row. According to (1) and (3), the solution error for \(X\), represented by \(\mu\), can be expressed as follows:  
\[\mathbf{\mu}=\mathbf{\hat{X}}-\mathbf{X}=\left(\mathbf{H}^{T}\mathbf{P}\mathbf{H}\right)^{-1}\mathbf{H}^{T} \mathbf{P}\varepsilon \tag{4}\]  
Set \(\mathbf{A}=\left(\mathbf{H}^{T}\mathbf{P}\mathbf{H}\right)^{-1}\mathbf{H}^{T}\), \(\mathbf{\mu}=-\mathbf{A}\mathbf{P}\varepsilon\). Thus, the VPE can be expressed as follows:  
\[\mu_{3}=\left(\frac{a_{31}\varepsilon_{1}}{\sigma_{1}^{2}}+\frac{a_{32} \varepsilon_{1}}{\sigma_{2}^{2}}+\cdots+\frac{a_{3K\xi K}}{\sigma_{K}^{2}}\right) \tag{5}\]  
where \(a_{3i}\) is the element in the 3-th row of \(\mathbf{A}\). Under the nominal mode:  
\[\mu_{3}\sim N\Big{(}0,a_{o}^{2}\Big{)} \tag{6}\]where \(N(\mu,\sigma^{2})\) represents a normal distribution with mean \(\mu\) and variance \(\sigma^{2}\); \(a_{v}=\sqrt{\sum_{i=1}^{K}\frac{\sigma_{N}^{2}}{\sigma_{i}^{2}}}\). Under the single-satellite fault mode:  
\[\mu_{3}\sim N\bigg{(}\frac{a_{3m}\xi_{b}}{\sigma_{m}^{2}},a_{v}^{2}\bigg{)} \tag{10}\]  
where \(VS_{m}\) presents the faulty satellite and the measurement bias for \(VS_{m}\) is \(\xi_{b}\).