## 2 Performance Analysis of the WADGPS System  
The Protection Level (PL) calculation [8] and Stanford Chart [9, 10] are used as the metrics to evaluate the performance of a GPS-based aircraft approach and landing system. The performance criterion chosen for this paper is the comparison of the LNAV availability. The PL is the computation of the confidence bounds on the post-correction position errors, and the PL calculation is defined in the Minimum Operational Performance Standards for GPS/WAAS Airborne Equipment (WAAS MOPS, RTCA DO-229D) [8]. The ionospheric delay error and satellite ephemeris and clock errors will be corrected according to the WAAS MOPS, and then the local errors such as the tropospheric delay error \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
**Phase of Flight** & **Accuracy (95\% error)** & \multicolumn{2}{c|}{**Integrity**} & \multicolumn{2}{c|}{**Alert Limit**} & \multicolumn{1}{c|}{**Continuuit**} & \multicolumn{1}{c|}{**Availability**} \\ \cline{3-6} \cline{6-6} \multicolumn{1}{|c|}{} & **Accuracy** & **Time to** & **Pr(HMI)** & **(H: Horizontal** & **y** & **Availability** \\ \hline En route & H: 740 m & 15 s & \(1\times 10^{-7}\)/h & H: 3,704 m & \(1\times 10^{-5}\)/hr & 0.99 to \\ (continental) & V: NA & 15 s & \(1\times 10^{-7}\)/h & V: NA & \(1\times 10^{-5}\)/h & 0.99999 \\ Terminal & H: 220 m & 15 s & \(1\times 10^{-7}\)/h & H: 1,852 m & \(1\times 10^{-5}\)/h & 0.99 to \\ LNAV(NPA) & H: 220 m & 10 s & \(1\times 10^{-7}\)/h & H: 556 m & \(1\times 10^{-5}\)/h & 0.99999 \\ LNAV/VAV & H: 220 m & 10 s & \(2\times 10^{-7}\)/approach & H: 556 m & \(5.5\times 10^{-5}\)/ approach & 0.9999 \\ LPV & H: 16 m & 6 s & \(2\times 10^{-7}\) & H: 40 m & \(5.5\times 10^{-5}\)/ & 0.99 to \\ V: 20 m & 6 s & /approach & V: 50 m & approach & 0.9999 \\ \hline \end{tabular}
\end{table} 
Table 1: Required Navigation Performance (RNP) [6, 7]. and user receiver noise and multipath errors will be removed by a standard model [11]. The corrected range measurements are used to compute the GPS position and receiver clock errors using weighted least squares as follows:  
\[\hat{x}=\left(G^{T}WG\right)^{-1}G^{T}Wy \tag{1}\]  
where:  
\(\hat{x}\) is the position and clock errors,  
\(G\) is the observation matrix,  
\(W\) is the weighting matrix for the measurement, and  
\(y\) is the corrected range residual vector. The weighting matrix, \(W\), is a diagonal matrix and the inverse of the \(i^{th}\) diagonal element is given by the variance for the corresponding satellite, as depicted in Equation (2). \[W^{-1}=\left[ \begin{array}{cccc}\sigma_{1}^{2}&0&\cdots&0\\ 0&\sigma_{2}^{2}&\cdots&0\\ \vdots&\vdots&\ddots&\cdots\\ 0&0&\cdots&\sigma_{i}^{2}\end{array} \right] \tag{2}\]  
where, \(\sigma_{i}^{2}\) is calculated in Equation (3). \[\sigma_{i}^{2}=\sigma_{i,th}^{2}+\sigma_{i,UWE}^{2}+\sigma_{i,air}^{2}+\sigma _{i,prop}^{2} \tag{3}\]  
where:  
\(\sigma_{i,th}^{2}\) is the fast and long-term degradation confidence, which is the confidence bound on satellite clock and ephemeris corrections [8, 12],  
\(\sigma_{i,UWE}^{2}\) is the user ionospheric range error confidence, which is the confidence bound on ionospheric delay corrections [8, 11],  
\(\sigma_{i,air}^{2}\) is the airborne receiver error confidence, which is the confidence bound on aircraft user receiver error [8], and  
\(\sigma_{i,prop}^{2}\) is the tropospheric error confidence, which is the confidence bound on residual tropospheric error [8]. The position error is proportional to the measurement errors and the satellite geometry through the matrix \(\left(G^{T}WG\right)^{-1}\). This matrix is composed of the variance from each direction as indicated in Equation (4).