# Fault-Free Protection Level Equation for CLAS PPP-RTK and Experimental Evaluations
## 3 Proposed Fault-Free Protection Level Equations for CLAS PPP-RTK Service
### Overview of Least-Squares Ambiguity Decorrelation Adjustment (LAMBDA)  
The PPP-RTK service allows the use of the GNSS carrier phase for ranging measurements in a manner similar to a VRS-RTK process. A VRS-RTK process typically resolves the integer ambiguities in carrier phase measurements using the LAMBDA algorithm [22; 23], and its upper bound probability of correctly fixing integer ambiguities is assessed using the integer bootstrapping method [24]. With GNSS measurements of the two receivers at a short baseline, the basic observation measurements are as follows:  
\[\Delta\
abla\rho=r+\varepsilon_{\rho}, \tag{8}\]  
\[\Delta\
abla\Phi=r+\lambda_{f}N+\varepsilon_{\Phi}, \tag{9}\]  
where \(\Delta\
abla\rho\) and \(\Delta\
abla\Phi\) are the double-difference code and carrier-phase measurements, respectively. \(r\) is the geometric range from a user to a satellite. \(\lambda_{f}\) is the \(f\) frequency wavelength of a GNSS carrier. \(N\) is integer ambiguity. \(\varepsilon\) represents the uncorrected range measurements and noise. The linearized observation equation for a set of double-difference codes and carrier-phase measurements of visible satellites is  
\[\mathbf{y}=\mathbf{A}\mathbf{a}+\mathbf{B}\mathbf{b}+\varepsilon, \tag{10}\]  
where \(\mathbf{y}\) is a vector of the double-difference code and carrier-phase measurements. \(\mathbf{a}\) is a vector of integer ambiguities and \(\mathbf{A}\) is a corresponding matrix with wavelengths as elements. \(\mathbf{B}\) is a satellite geometry matrix, and \(\mathbf{b}\) is the relative position vector from a VRS reference position to a GNSS receiver. The popular LAMBDA method resolves \(\mathbf{a}\) and computes the precise \(\mathbf{b}\) in two steps. First, an unconstrained solution of Equation (10) is computed as  
\[\left[ \begin{array}{c}\hat{\mathbf{b}}\\ \hat{\mathbf{a}}\end{array} \right]=\left(\left[ \begin{array}{c}\mathbf{B}^{T}\\ \mathbf{A}^{T}\end{array} \right]\mathbf{Q}_{y}^{-1}\left[ \begin{array}{cc}\mathbf{B}&\bm {A}\end{array} \right]\right)^{-1}\left[ \begin{array}{c}\mathbf{B}^{T}\\ \mathbf{A}^{T}\end{array} \right]\mathbf{Q}_{y}^{-1}\mathbf{y}, \tag{11}\]  
where \(\hat{\mathbf{b}}\) and \(\hat{\mathbf{a}}\) are the estimated baseline and float-integer ambiguities, respectively. \(\mathbf{Q}_{y}\) is the covariance matrix of \(\varepsilon\) in Equation (10). The covariance matrix of \(\hat{\mathbf{b}}\) and \(\hat{\mathbf{a}}\) is  
\[\mathbf{Q}=\left[ \begin{array}{cc}\mathbf{Q}_{\hat{b}}&\mathbf{Q}_{\hat{a}\hat{b}}\\ \mathbf{Q}_{\hat{b}\hat{a}}&\mathbf{Q}_{\hat{a}}\end{array} \right]. \tag{12}\]  
The second procedure of LAMBDA consists of the re-parameterization and search for \(\mathbf{a}\). The re-parameterization of the integer vector is implemented as follows:  
\[\mathbf{z}=\mathbf{Z}\mathbf{a},\hat{\mathbf{z}}=\mathbf{Z}\mathbf{\hat{a}},\mathbf{Q}_{\hat{z}}=\mathbf{Z}^{ T}\mathbf{Q}_{\hat{a}}\mathbf{Z}, \tag{13}\]  
where \(\mathbf{Z}\) is the decorrelation transformation matrix. Then, the re-parameterized integer ambiguity is searched with respect to the following objective function:  
\[\underset{\mathbf{z}}{min}(\hat{\mathbf{z}}-\mathbf{z})^{T}\mathbf{Q}_{\hat{z}}^{-1}(\hat\mathbf{{z}}-\mathbf{z}) \tag{14}\]  
Once an optimal integer ambiguity vector, \(\hat{\mathbf{z}}\), is obtained from Equation (14), the original integer ambiguity vector estimate is obtained using the inverse of the transformation such that  
\[\hat{\mathbf{a}}=\mathbf{Z}^{-1}\hat{\mathbf{z}}. \tag{15}\]  
The presumed fixed baseline vector, \(\hat{\mathbf{b}}\) is  
\[\hat{\mathbf{b}}=\hat{\mathbf{b}}-\mathbf{Q}_{\hat{b}\hat{a}}\mathbf{Q}_{\hat{a}}^{-1}(\hat{ \mathbf{a}}-\hat{\mathbf{a}}) \tag{16}\]In addition, the covariance matrix of \(\tilde{\mathbf{b}}\) is as follows:  
\[ \begin{array}{ll}\mathbf{Q}_{\tilde{b}}&=\mathbf{Q}_{\tilde{b}}-\mathbf{Q}_{\tilde{b}\mathbf{ \alpha}}\mathbf{Q}_{\tilde{a}}^{-1}\mathbf{Q}_{\tilde{a}\tilde{b}}\\ &=\left(\mathbf{B}^{T}\Big{(}\mathbf{Q}_{\Phi}^{-1}+\mathbf{Q}_{\rho}^{-1}\Big{)}\mathbf{B} \right)^{-1}\,\end{array}