## 3 Empirical Stochastic Modeling Methodology
Formulas are presented in a generalized form (omitting the indexes of satellites and receivers) for intra-system differences (no differences are computed between two GNSS systems). It is assumed that two receivers of the same type are used to create differences, thus the measurement noise affecting observations from both receivers have the same magnitude. This prerequisition allows to eliminate the inter-frequency bias occurring in the differences computed using observation on a different frequency (as in GLOASS system, which uses Frequency Division Multiple Access (FDMA)). The dispersion of observation residuals of linear combinations represents the measurement noise with variance \(\sigma_{L}^{2}\) for phase observation and takes into account two types ofcorrelations: time-correlation \(\rho_{t}\) which affects the TD dispersion when time differencing is performed and cross-correlation \(\rho_{\Delta}\) between observations from two receivers [21; 22]. For the SB setup \(\rho_{\Delta}\) can be neglected; however, for the ZB, when two receivers use the same antenna, the large part of the noise is the same due to the common antenna Low-Noise Amplifier (LNA) [45; 82], which results in a significant cross-correlation of measurement from the receivers. When the observations from both receivers are burdened with common LNA noise, it is eliminated in the observation difference, significantly reducing the dispersion. Figure 4 presents the time series of phase residuals for four observation combinations listed in Table 2, for exemplary one of the processed signal--Galileo L1C from the Sv8 satellite. The satellite with the highest elevation for a given epoch was used as a reference satellite, so it can be assumed that its noise was constant and its impact on the presented results was minimized. The SB length was 4.75 m. Based on these series, one can see an apparent multipath effect in the DD combination for SB, which is eliminated for the TD combination and ZB observations. There is also a clear dependence on the satellite elevation (marked on top graphs for DD) and the C/N\({}_{0}\) parameter (marked on bottom graphs for TD). The magnitude of the measurement noise clearly decreases for observations from the ZB, which is confirmed by the standard deviation of the time series, calculated for a uniform undifferenced (UD) level, amounting to \(\pm\)0.76 mm, \(\pm\)0.32 mm and \(\pm\)0.31 mm, respectively for SB TD, ZB DD and ZB TD combinations. This confirms the existence of significant correlations that must be taken into account to correctly determine the measurement noise variance based on the presented combinations of observations. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Obs.** & **Formula** & \(\mathbf{E\{\cdot\}}\) & \(\mathbf{D\{\cdot\}}\) \\ \hline SB:DD & \(\
abla\Delta\hat{L}(t_{1})=\) & \(\
abla\Delta L(t_{1})-\
abla\Delta R(t_{1})-\lambda\
abla\Delta N(t_{1})\) & \(\
abla\Delta\delta M_{L}(t_{2})-\
abla\Delta\delta M_{L}(t_{1})\approx\) & \(4\sigma_{L}^{2}\) \\ SB:TD & \(\
abla\Delta L(t_{12})=\
abla\Delta L(t_{2})-\
abla\Delta L(t_{1})\) & \(\
abla\Delta\delta M_{L}(t_{2})-\
abla\Delta\delta M_{L}(t_{1})\approx\) & \(8(1-\rho_{t})\sigma_{L}^{2}\) \\  & & 0 & \(4(1-\rho_{\Delta})\sigma_{L}^{2}\) \\ \hline ZB:DD & \(\
abla\Delta\hat{L}(t_{1})=\
abla\Delta L(t_{1})-\lambda\
abla\Delta N(t_{1})\) & 0 & \(4(1-\rho_{\Delta})\sigma_{L}^{2}\) \\ ZB:TD & \(\
abla\Delta\hat{L}(t_{2})=\
abla\Delta\hat{L}(t_{2})-\
abla\Delta\hat{L}(t_{1})\) & 0 & \(8(1-\rho_{t})(1-\rho_{\Delta})\sigma_{L}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Expectation and dispersion values of DD and TD carrier-phase combination residuals for SB and ZB setup. Figure 4: Time series of carrier-phase residuals of Galileo L1C signal from Sv8 for combinations: SB DD (**a**), ZB DD (**b**), SB TD (**c**), ZB TD (**d**). In order to determine the variance of carrier-phase and code observations, the SB TD combination was used.