# Fault-Free Protection Level Equation for CLAS PPP-RTK and Experimental Evaluations
## 3 Proposed Fault-Free Protection Level Equations for CLAS PPP-RTK Service
### Proposed Fault-Free PPP-RTK Protection Level Equation  
The integer ambiguity resolution through LAMBDA is a stochastic search process, and based on the bias-free estimates of float ambiguities, its upper bound probability of correct integer fixes is [24]  
\[\text{Prob}(\tilde{\mathbf{a}}=\mathbf{a})=\prod\
olimits_{i=1}^{n}\left(CDF\left( \frac{1}{2\sigma_{\tilde{a}_{i|I}}}\right)-1\right) \tag{18}\]  
where \(\tilde{\mathbf{a}}_{i|I}\) is the conditional least-squares estimate of integer ambiguity. Because the correctness of the fixed integer ambiguities may significantly affect the position errors, the probability of Equation (18) must be considered in a protection-level equation. A protection level (XPL) and given fault-free risk probability (\(I_{H0 req}\)) can be expressed as follows:  
\[\text{Prob}\left\{\left|\overset{\mathbf{\wedge}}{\mathbf{\mathbf{x}}}-\mathbf{\mathbf{x}} \right|>\text{XPL}_{H0}\right\}=I_{H0 req}, \tag{19}\]  
where \(\overset{\mathbf{\wedge}}{\mathbf{\mathbf{x}}}\) and \(\mathbf{\mathbf{x}}\) denote the estimated position and true position, respectively, in the horizontal or vertical directions. \(\text{XPL}_{H0}\) refers to the horizontal or vertical protection levels. Because the position error may exceed \(\text{XPL}_{H0}\) with correctly fixed (CF) or incorrectly fixed (IF) integer ambiguities, Equation (19) can be expanded to two conditional probabilities, as follows [9]:  
\[\begin{array}{ll}\text{Prob}\left\{\left|\overset{\mathbf{\wedge}}{\mathbf{\mathbf{ x}}}-\mathbf{\mathbf{x}}\right|>\text{XPL}_{H0}\right\}&=\text{Prob}\left\{\left| \overset{\mathbf{\wedge}}{\mathbf{\mathbf{x}}}-\mathbf{\mathbf{x}}\right|>\text{XPL}_{H0} \right|\text{CF}\right\}PCF\\ &+\text{Prob}\left\{\left|\overset{\mathbf{\wedge}}{\mathbf{\mathbf{x}}}-\mathbf{\mathbf{ x}}\right|>\text{XPL}_{H0}\right|\text{IF}\right\}PIF\end{array} \tag{20}\]  
where \(PCF\) is the probability of the correct fix, which can be estimated from Equation (18). \(PIF\) is the probability of incorrect fix and \(PCF\) equals to \(1-PIF\).  
\[\text{Considering Prob}\left\{\left|\overset{\mathbf{\wedge}}{\mathbf{\mathbf{x}}}-\mathbf{ \mathbf{x}}\right|>\text{XPL}_{H0}\left|\text{IF}\right\}=1\text{ as a conservative approach, the fault-free risk probability presuming that integer ambiguities are correctly fixed is  
\[\text{Prob}\left\{\left|\overset{\mathbf{\wedge}}{\mathbf{\mathbf{x}}}-\mathbf{\mathbf{x}} \right|>\text{XPL}_{H0}\left|\text{CF}\right\}=\frac{I_{H0 req}-PIF}{1-PIF}. \tag{21}\]  
Assuming that the position error follows a zero-mean Gaussian distribution, the corresponding tail value of Equation (21) was used as the \(K_{ffmd}\) factor in Equation (7).  
Using the total range error of CLAS, \(\sigma_{i}\), in Equation (2), a weighting matrix for a single-differenced ranging source can be similarly constructed as in Equation (4). Because VRS-RTK uses double-difference code and carrier phase measurements, \(\mathbf{Q}_{\rho}\) and \(\mathbf{Q}_{\Phi}\) in Equation (17) can be expressed as  
\[\mathbf{Q}_{\rho}=\mathbf{D}\mathbf{W}_{\rho}^{-1}\mathbf{D}^{T}\text{ and }\mathbf{Q}_{\Phi}=\mathbf{D} \mathbf{W}_{\Phi}^{-1}\mathbf{D}^{T}, \tag{22}\]where \(\mathbf{D}\) is the double-difference matrix. Using the \(K_{ffmd}\) derived from Equation (21) and substituting Equation (23) into Equation (17), the fault-free horizontal and vertical protection level equations were computed as follows:  
\[\begin{array}{rl}\text{HPL}&=K_{ffmd,\text{HPL}}\sigma_{H}\\ &=K_{ffmd,\text{HPL}}\sqrt{\frac{\mathbf{\alpha}_{k_{(1,1)}+\mathbf{\alpha}_{k_{(2,2)}} }}{2}+\sqrt{\left(\frac{\mathbf{\alpha}_{k_{(1,1)}}-\mathbf{\alpha}_{k_{(2,2)}}}{2} \right)^{2}+\mathbf{Q}_{k(1,2)}^{2}}}\\ &\qquad\qquad\qquad\text{VPL}=K_{ffmd,\text{VPL}}\sigma_{V}=K_{ffmd,\text{VPL} }\sqrt{\mathbf{\alpha}_{k_{(3,3)}}}.\end{array} \tag{23}\]  
In Equation (23), \(\mathbf{Q}_{k}\) is evaluated in ENU coordinates.