## Appendix A  
This section describes the methodology of analyzing statistics of the SQM metrics. Under nominal conditions, in low multipath open sky environments, calm ionospheric condition and the absence of other GNSS signal degradation errors, the output of each SQM metric is a random process whose statistics (e.g., mean and variance) is determined based on the location of the constituent correlators and receiver noise. Recall the in-phase output of the \(i\)th correlator in code-delay domain given by Equation (4) and repeated here for convenience:  
\[I_{c_{i}}=\sqrt{C}R_{\tau}(c_{i}T_{c})+\eta_{c_{i}}^{I}\]  
where \(T_{c}\) is the chip duration and \(c_{i}T_{c}\) denotes the spacing of the \(i\)th early or late correlator from the reference prompt correlator. \(C\) is the power of the received signal and \(\eta_{c_{i}}^{I}\) is the corresponding in-phase zero-mean Gaussian noise whose variance is \(\sigma_{0}^{2}=N_{0}/2T_{I}\). \(T_{I}\) is the coherent integration time and \(N_{0}\) is the power spectral density of the receiver noise. \(R_{\tau}(\cdot)\) is the correlation function related to the choice of the GNSS signaling scheme [22]. With this definition, it can be shown that \(I_{c_{i}}\) is a random process with the following statistics [22]. \[\mu_{i}=E\Big{\{}I_{c_{i}}\Big{\}}=\sqrt{C}R_{\tau}(c_{i}T_{c}) \tag{1}\]  
\[\sigma_{i}^{2}=E\bigg{\{}\Big{(}I_{c_{i}}-E\Big{\{}I_{c_{i}}\Big{\}}^{2}\Big{)} ^{2}\bigg{\}}=\sigma_{0}^{2}=\frac{N_{0}}{2T_{I}} \tag{2}\]where \(E\{\cdot\}\) denotes the statistical expectation. The covariance of the two \(I_{c_{i}}\) and \(I_{c_{j}}\) correlators is also calculated as follows [22]:  
\[\sigma_{ij}=\sigma_{ji}=E\Big{\{}\Big{(}I_{c_{i}}-E\Big{\{}I_{c_{i}}\Big{\}} \Big{)}\Big{(}I_{c_{j}}-E\Big{\{}I_{c_{j}}\Big{\}}\Big{)}\Big{\}}=\sigma_{0}^{2 }R_{\tau}\big{(}d_{ij}T_{c}\big{)}\qquad for\ \ \forall\ i\ \ and\ j \tag{11}\]  
where \(d_{ij}\) is the code distance (spacing) between the \(I_{c_{i}}\) and \(I_{c_{j}}\) correlators in the unit of chips. If the SQM metric \(m\) is considered as a general function of different correlators:  
\[m=f(I_{c_{i}},I_{c_{j}},\ldots) \tag{12}\]  
the variance of the metric can be calculated based on the error propagation law [10; 24]:  
\[\sigma_{m}^{2}=\mathbf{J}\,\mathbf{D}\,\mathbf{J}^{T} \tag{13}\]  
where \(\mathbf{J}\) is the Jacobian matrix containing the partial derivatives of the metric \(m\) with respect to the constituent correlators:  
\[\mathbf{J}=\Big{[} \begin{array}{cc}\frac{\partial m}{\partial I_{c_{i}}}& \frac{\partial m}{\partial I_{c_{j}}}&\cdots\end{array} \Big{]} \tag{14}\]  
\(\mathbf{D}\) is the symmetrical covariance matrix of the correlators with the following structure:  
\[\mathbf{D}=\left[ \begin{array}{ccc}\sigma_{i}^{2}&\sigma_{ij}&\cdots\\ \sigma_{ji}&\sigma_{j}^{2}&\cdots\\ \vdots&\vdots&\ddots\end{array} \right]=\sigma_{0}^{2}\left[ \begin{array}{ cccc}1&R_{\tau}(d_{ij}T_{c})&\cdots\\ R_{\tau}(d_{ji}T_{c})&1&\cdots\\ \vdots&\vdots&\ddots\end{array} \right]=\frac{N_{0}}{2T_{1}}\left[ \begin{array} []{ccc}1&R_{\tau}(d_{ij}T_{c})&\cdots\\ R_{\tau}(d_{ij}T_{c})&1&\cdots\\ \vdots&\vdots&\ddots\end{array} \right] \tag{15}\]  
By defining the SQM metrics as a linear combination of different early and late correlators normalized by the prompt value, Equation (13) can be rewritten as:  
\[\sigma_{m}^{2}=\mathbf{J}\,\mathbf{D}\,\mathbf{J}^{T}=\Big{(}\frac{1}{\sqrt{C }}\Big{[} \begin{array}{cc}g_{1}&g_{2}&\cdots\end{array} \Big{]}\Big{)} \Bigg{(}\begin{array}{ccc}1&R_{\tau}(d_{ij}T_{c})&\cdots\\ \frac{N_{0}}{2T_{1}}\Bigg{[} \begin{array}{ccc}R_{\tau}(d_{ij}T_{c})&1&\cdots \\ \vdots&\vdots&\ddots\end{array} \Bigg{]}\Bigg{)}\Bigg{(} \begin{array}{c}g_{1} \\ \frac{1}{\sqrt{C}}\Big{[}\begin{array}{c}g_{2}\\ \vdots\end{array}\end{array}