# Noises in Double-Differenced GNSS Observations
## 4 Results  
### Autocorrelation Function  
The ACF was determined to be able to find a temporal correlation in the ZB DD time series. Figures 4-6 present examples from one daily solution (2019 DOY:055) sample autocorrelation coefficient for all of the analyzed GNSS signals calculated as an average for all satellites. Each plot shows the ACF for a 0-3600 sec time lag and an enlarged function curve for the time lag range 0-60 s for the code and 0-30 s for the carrier phase observations. Additionally, the \(\rho_{t}\) values for selected \(\tau\) values are listed. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**System/Observation Type**} & \multicolumn{2}{c}{**Code (m)**} & \multicolumn{2}{c}{**Phase (mm)**} \\ \cline{3-6}  & & **Mean** & **Std.** & **Mean** & **Std.** \\ \hline \multirow{4}{*}{GPS} & 1C & 0.002 & \(\pm\)0.370 & 0.00 & \(\pm\)1.50 \\  & 2L & \(-\)0.001 & \(\pm\)0.243 & 0.00 & \(\pm\)1.69 \\  & 2W & \(-\)0.002 & \(\pm\)0.229 & 0.00 & \(\pm\)3.41 \\  & 5Q & \(-\)0.001 & \(\pm\)0.095 & 0.00 & \(\pm\)1.95 \\ \hline \multirow{4}{*}{GLONASS} & 1C & 0.001 & \(\pm\)0.891 & 0.00 & \(\pm\)2.41 \\  & 1P & \(-\)0.002 & \(\pm\)0.151 & 0.00 & \(\pm\)2.50 \\  & 2C & 0.002 & \(\pm\)0.444 & 0.03 & \(\pm\)3.03 \\  & 2P & \(-\)0.001 & \(\pm\)0.133 & 0.01 & \(\pm\)2.92 \\ \hline \multirow{4}{*}{Galileo} & 1C & 0.000 & \(\pm\)0.182 & 0.00 & \(\pm\)1.47 \\  & 5Q & 0.003 & \(\pm\)0.121 & 0.00 & \(\pm\)1.81 \\ \cline{1-1}  & 7Q & 0.000 & \(\pm\)0.064 & 0.00 & \(\pm\)1.66 \\ \cline{1-1}  & 8Q & 0.000 & \(\pm\)0.020 & 0.00 & \(\pm\)1.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistics for ZB DD code and carrier phase noise for the weekly solution. Figure 4: Sample autocorrelation coefficients for GPS observations (2019 DOY: 055). For all of the investigated code signals, it is easy to notice a high temporal correlation for a time lag equal to a few seconds. Especially for the GPS observations, the temporal correlation coefficient maintains a significant value that exceeds 0.2 for the time lag that is in the range of 10-20 s (except for C1C observations, for which the correlation time is only 2 s). For the GLONASS and Galileo systems, the obtained values of the time correlation are at a very similar level of 2-4 s for each code signal. Only slightly higher correlation time values (up to 6 s) are obtained for the Galileo C5Q and C8Q observations. For the carrier phase measurements, the correlation time is much smaller and, in most cases, does not exceed 1 s. Non-zero temporal correlation coefficient values for a time lag of at least 1 s can only be seen for the GPS L2W signal and GLONASS L2C and L2P signals. However, for the GPS L2W observations, it immediately drops to zero for \(\tau=2\) s, while for the GLONASS observations, the decrease in the \(\rho_{t}\) value is very  
Figure 5: Sample autocorrelation coefficients for GLONASS observations (2019 DOY: 055). Figure 6: Sample autocorrelation coefficients for Galileo observations (2019 DOY: 055). slow and exceeds 2000 s, which may be due to the influence of the residual value of the frequency-dependent part of the IFB when using the FDMA technique for the GLONASS signals. In that case, the assumption IFB elimination using a homogenous receiver pair [55] may not be accurate for the L2 signals, but this requires additional confirmation. The results for the weekly solution, which was determined as the average of the daily solutions, are presented in Table 3 in the form of an autocorrelation time lag, for which the temporal correlation coefficient drops below 0.2. The time lag values are calculated as an average for all of the satellites for the analyzed signals while rejecting outliers with a confidence level of about 99%. For the 1 Hz data, the smallest possible autocorrelation time lag is denoted as \(\leq\)1 s.  
Figure 7 shows the corresponding autocorrelation time lag results for a weekly solution in the form of a box plot.