## 3 Empirical Stochastic Modeling Methodology
The low variability of the C/N0 for geostationary satellites makes precise modeling difficult--for GEO L2I and L6I signals, the differences of C/N0 for the determined carrier-phase variances are only 4 dB-Hz. Figure 5: Empirical variance model for SB TD GPS carrier-phase measurements for signals: L1C (**a**), L2L (**b**), L2W (**c**), L5Q (**d**). Figure 6: Empirical variance model for SB TD GPS code measurements for signals: C1C (**a**), C1W (**b**), C2L (**c**), C2W (**d**), C5Q (**e**). Figure 8: Empirical variance model for SB TD GLONASS code measurements for signals: C1C (**a**), C1P (**b**), C2C (**c**), C2P (**d**). Figure 7: Empirical variance model for SB TD GLONASS carrier-phase measurements for signals: L1C (**a**), L1P (**b**), L2C (**c**), L2P (**d**). Figure 10: Empirical variance model for SB TD Galileo code measurements for signals: C1C (**a**), C5Q (**b**), C7Q (**c**), C8Q (**d**), C6C (**e**). Figure 9: Empirical variance model for SB TD Galileo carrier-phase measurements for signals: L1C (**a**), L5Q (**b**), L7Q (**c**), L8Q (**d**), L6C (**e**). Figure 4:0 Empirical variance model for SB TD BeiDou carrier-phase measurements for signals: L2I (**a**), L7I (**b**), L6I (**c**) for all satellite blocks except for GEO satellite; L2I (**d**), L7I (**e**), L6I (**f**) for GEO satellites. Figure 4:1 Empirical variance model for SB TD BeiDou code measurements for signals: C2I (**a**), C7I (**b**), C6I (**c**) for all satellite blocks except for GEO satellite; C2I (**d**), C7I (**e**), C6I (**f**) for GEO satellites. The time- and cross-correlation were determined based on the ZB DD observation time series analysis. To determine the time correlation, the autocorrelation function (ACF) for time series was used that measures the correlation between \(y(t)\) and \(y(t+\tau)\) of \(y(t)\) stochastic process given by [18]:  
Figure 4:2  
where \(\sigma^{2}(\tau)\) is the sample covariance of time series \(y(t)\) with itself at time lag \(\tau\), \(\bar{y}\) denotes the mean of the time series \(y\) and \(T\) is the effective sample size of \(y\). The time-correlation coefficient for lag \(\tau\) is computed as:  
Figure 4:3  
where \(\sigma_{0}^{2}\) is a sample variance of the time series for \(\tau=0\). For the ZB DD combination, the variance of the time series is burdened with the influence of correlation \(\rho_{\Delta}\), but assuming that the LNA noise is constant over time, it will burden the \(\sigma^{2}(\tau)\) and \(\sigma_{0}^{2}\) equally, so as a result, it does not affect the \(\rho_{t}(\tau)\). Figures 13-20 show the sample autocorrelation coefficient for code and phase observations for all analyzed GNSS signals based on daily 1 Hz measurements. As for the variance model analysis presented previously, the presented ACFs did not distinguish between satellite blocks and were calculated as averages for all satellites. The average \(\rho_{t}\) for the selected lags (1, 5, 10 and 30 s) were also listed in the graphs. As can be easily seen, the time-correlation of the code observations reaches significant values for \(\tau<10\) s for all signals. There is also a noticeable high temporal correlation of pseudo-observation of encrypted GPS L2W and GLONASS C1P and C2P codes, for which this coefficient has significant values for the entire range of analyzed lags (\(\tau\leq 60\) s). For carrier-phase measurements, the time-correlation coefficient is much smaller, especially for new systems Galileo and BeiDou as well as for the GPS L5Q signal, for which for \(\tau=1\) s it immediately drops off to about zero. For the remaining carrier-phase measurements, the time-correlation is significant only for \(\tau\leq 3\) s except for GLONASS L2C and L2P observations where the \(\rho_{t}\) remains constant regardless of the lag. These examples lead to the conclusion that for the analyzed signals, there is a significant time-correlation of observations, especially for 1 Hz measurements, and its neglect may cause essential simplification in the realistic definition of the Multi-GNSS observation stochastic model.