## 3 Empirical Stochastic Modeling Methodology
The cross-correlation coefficients of measurements were determined based on the covariance of two time series \(y_{1}\) and \(y_{2}\) according to the formula:  
Figure 4:4  
The correlation coefficient is then calculated as:  
\[\rho_{y_{1},y_{2}}=\frac{\text{cov}_{y_{1},y_{2}}}{\sigma_{y_{1}}\sigma_{y_Figure 4:5 Sample autocorrelation coefficient for ZB DD GPS code measurements for signals: C1C (**a**), C1W (**b**), C2L (**c**), C2W (**d**), C5Q (**e**). Figure 4:6 Sample autocorrelation coefficient for ZB DD GPS carrier-phase measurements for signals: L1C (**a**), L2L (**b**), L2W (**c**), L5Q (**d**). Figure 4:7 Sample autocorrelation coefficient for ZB DD GLONASS code measurements for signals: C1C (**a**), C1P (**b**), C2C (**c**), C2P (**d**). Figure 4:8 Sample autocorrelation coefficient for ZB DD GLONASS carrier-phase measurements for signals: L1C (**a**), L1P (**b**), L2C (**c**), L2P (**d**). Figure 4:9 Sample autocorrelation coefficient for ZB DD Galileo carrier-phase measurements for signals: L1C (**a**), L5Q (**b**), L7Q (**c**), L8Q (**d**), L6C (**e**). \[\sigma=a_{1}\cdot 10^{-C/N_{0}/20}+a_{2}\cdot 10^{-C/N_{0}/10} \tag{6}\]0 Sample autocorrelation coefficient for ZB DD Galileo code measurements for signals: C1C (**a**), C5Q (**b**), C7Q (**c**), C8Q (**d**), C6C (**e**). The cross-correlation coefficients were determined between corresponding ZB DD time series of the observations to the same satellites for different types of signals and frequencies within one GNSS system, and the average for all satellites was taken as the final value. Figure 21 shows the cross-correlation coefficients of observations presented in a form of a correlation matrix for individual observation types for 1 Hz daily data. This Figure shows that only for the GPS C1C/C1W and Galileo L5Q/L7Q/L8Q measurements, we observe a significant cross-correlation between the observations. In the first case, it is due to a use of the Z-tracking technique of encrypted P(Y) code in which the difference measurements C1C-C1W are acquired instead of the direct observations of C1W [83,84]; this leads to a cross-correlation of observations reaching 0.95. In the second case, high correlation (up to 0.55) is a result of the structure of the Galileo E5 signal, which can be tracked independently for the E5a and E5b components or as a single wide bandwidth E5 ab signal. In other cases, slight correlations between GPS L1C/L2W, GLONASS C1P/C2P and L2C/L2P, Galileo CSQ/C8Q measurements reaching 0.15 can be observed. No correlation was found between code and carrier-phase observations to the same satellites for all systems, except for the GLONASS system. However, in that case, when correlation coefficients are lower than 0.25, it can be concluded that the correlation is very weak and insignificant. Summing up, it can be stated that the used combinations of observation residuals and the adopted fitting function allow for precise modeling of the dependence of the code and carrier-phase observation variance on the C/N\({}_{0}\) parameter. The determined time- and cross-correlation coefficients may reach non-negligible values and significantly differ depending on the analyzed signal. \[\sigma=a_{1}\cdot 10^{-C/N_{0}/20}+a_{2}\cdot 10^{-C/N_{0}/10} \tag{6}\]1 Sample autocorrelation coefficient for ZB DD BeiDou carrier-phase measurements for signals: L2I (a), L7I (b), L6I (c). \[\sigma=a_{1}\cdot 10^{-C/N_{0}/20}+a_{2}\cdot 10^{-C/N_{0}/10} \tag{6}\]2 Sample autocorrelation coefficient for ZB DD BeiDou code measurements for signals: C2I (a), C7I (b), C6I (c).