## 2 Performance Analysis of WAAS  
The availability of WAAS is determined by the confidence bounds on position errors and the satellite geometry. The computation of confidence estimates for the corrections to various error sources are defined in the WAAS Minimum Operational Performance Standard (MOPS) [7]. The error due to ionospheric delay and satellite errors will be corrected according to the WAAS MOPS as well, and then the local errors such as error due to tropospheric delay and user receiver noise and multipath errors will be removed by a standard model [7]. The corrected range measurements are used to compute 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, \(\sigma_{i}^{z}\) which is calculated in Equation (2). \[\sigma_{i}^{2}=\sigma_{i,th}^{2}+\sigma_{i,lURE}^{2}+\sigma_{i,air}^{2}+\sigma _{i,prop}^{2} \tag{2}\]  
where  
\(\sigma_{i,th}^{2}\) is the fast and long-term degradation confidence, which is the confidence bound on satellite clock and ephemeris corrections [7],  
\(\sigma_{i,lURE}^{2}\) is the user ionospheric range error confidence, which is the confidence bound on ionospheric delay corrections [7],  
\(\sigma_{i,air}^{2}\) is the airborne receiver error confidence, which is the confidence bound on aircraft user receiver error [7], and \(\sigma_{i,n_{\text{propo}}}^{2}\) is the tropospheric error confidence, which is the confidence bound on residual tropospheric error [7]. As a result, the inverse of \(W\) can be written in Equation (3)  
\[W^{-1}= \begin{bmatrix}\sigma_{1}^{2}&0&\cdots&0\\ 0&\sigma_{2}^{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&&\sigma_{N}^{2}\end{bmatrix}  \tag{3}\]  
The variance of the vertical position estimate is the third diagonal element of the position estimate covariance matrix,  
\[d_{3,3}= \begin{bmatrix}\left(G^{T}WG\right)^{-1}\end{bmatrix} _{3,3} \tag{4}\]  
where  
\(d_{3,3}\) is the variance of the vertical position estimate. The VPL (Vertical Protection Level) is  
\[VPL_{W,AS}=K_{V,P,A}d_{3,3} \tag{5}\]  
where, \(K_{V,P,A}\) equals 5.33. This is a multiplier on the standard deviation of the vertical error such that the VPL is only exceeded at most one time in ten million (\(10^{-7}\)), the tolerable probability of HMI (Hazardously Misleading Information), provided that the error distribution is a zero mean Gaussian [8]. The protection level calculation is specified in the WAAS MOPS Appendices A and J [7]. The VPL is very important and will be used to determine the availability of WAAS. The simulation tool in this paper is the MATLAB(r) Algorithm Availability Simulation Tool (MAAST) [9], a publicly available software tool co-developed by the author, which can be customized to simulate the WAAS confidence estimation algorithms and evaluate the effect of service availability with algorithm changes. Note that MAAST is available in [10]. MAAST assumes 100% asset reliability (_i.e._, no satellite, reference station, or communication failures), and is therefore somewhat optimistic, and one would expect actual performance to be slightly worse. As indicated in [11] and Figure 1, the simulation result of MAAST (the left plot of Figure 1) is very similar to the actual performance of the WAAS (right plot of Figure 1).