# Research on Satellite Selection Strategy for Receiver Autonomous Integrity Monitoring Applications
## 2 Problem Statement
### Accuracy and Integrity Models in Weighted Slope-Based RAIM Algorithm  
#### 2.1.1 Accuracy Model  
The linearized form of pseudo-range measurements model for GNSS can be described as [36]:  
\[\Delta y_{n\times 1}=G_{n\times p}\Delta x_{p\times 1}+\varepsilon_{n\times 1} \tag{1}\]  
where \(\Delta y_{n\times 1}\) is the measurement vector in the form of increment to the pseudo-range between satellites and receiver. \(n\) is the number of satellites in use and \(p\) is the constellation number plus three. \(\Delta x_{p\times 1}\) is the states vector which involves increments to three components of receiver position and the clock offsets between constellations and the receiver. \(G_{n\times p}\) is the observation matrix in the local navigation frame and \(\varepsilon_{n\times 1}\) is the measurement error vector produced by receiver noise, wave propagation, satellite clock offsets, and possibly unexpected errors caused by failures to the assumption of normal-distributed noises or a satellite malfunction. The model of measurement error corresponding to each satellite is described later in Section 4.  
When the accuracy of the pseudo-range measurements is known to differ, for example, due to variation in the carrier power to noise density or the residual ionosphere and troposphere propagation errors, which mainly depend on the elevation angle, a weighted least-squares estimate can be computed iteratively:  
\[\Delta\hat{x}_{p\times 1}=\left(G_{p\times n}^{T}W_{n\times n}G_{n\times p} \right)^{-1}G_{p\times n}^{T}W_{n\times n}\Delta y_{n\times 1}=K_{p\times n} \Delta y_{n\times 1} \tag{2}\]  
where \(W_{n\times n}\) is the inverse of the covariance matrix of the measurement error vector. For simplification, the error sources for each satellite are assumed to be uncorrelated with that for any other satellite. Therefore, all off-diagonal elements are set to zero and the diagonal elements are the inverses of the variances corresponding to each satellite.  
\[W=\left[\begin{array}{cccc}\sigma_{1}^{-2}&0&\ldots&0\\ 0&\sigma_{2}^{-2}&\ddots&\vdots\\ \vdots&\ddots&\ddots&0\\ 0&\ldots&0&\sigma_{n}^{-2}\end{array}\right] \tag{3}\]  
The covariance of the estimated states vector can be expressed as:  
\[cov(\Delta\hat{x})=\left(G^{T}WG\right)^{-1} \tag{4}\]  
Thus, the accuracy of that least-squares solution is mainly decided by two factors: the quality of the measurements, i.e., \(\sigma_{i}^{2}\), and the satellite geometry which is described by observation matrix \(G\).  
The horizontal accuracy of the navigation solution can be represented by the radius of the circle assured to contain the indicated horizontal position with a 95% probability under fault-free conditions [37]. It can be approximated by the two times distance root mean square (2 drms) [10]  
\[2 drms\approx 2\sqrt{\left[\left(G^{T}WG\right)^{-1}\right]_{11}+\left[\left(G^{ T}WG\right)^{-1}\right]_{22}} \tag{5}\]If the measurement errors are assumed to be independent and identically distributed with zero mean and variance \(\sigma_{0}^{2}\), the horizontal accuracy of the navigation solution can be simplified as:  
\[2 drms\approx 2*HDOP*\sigma_{0} \tag{6}\]  
where the horizontal dilution of precision (_HDOP_) describes the satellite geometry factor of the horizontal accuracy and is defined as:  
\[HDOP=\sqrt{\left[\left(G^{T}G\right)^{-1}\right]_{11}+\left[\left(G^{T}G\right) ^{-1}\right]_{22}} \tag{7}\]