## 2 Signal Analysis
### Elevation, C/N\({}_{0}\) and Biases  
To show the changes of the DD residuals with elevation angles, the elevation angle \(\theta^{1s}\) is computed based on the elevation-dependent weighting function \(w(\theta^{s})=1/\left(1+10\times\exp(-\theta^{s}/10^{\circ})\right)^{2}\) (see also Appendix A), so that:  
\[(\sigma_{j}^{1s})^{2} = 2\sigma_{j,0}^{2}\times\left(1+10\times\exp\bigg{(}-\frac{ \theta^{1s}}{10^{\circ}}\bigg{)}\right)^{2} \tag{9}\] \[= \sigma_{j,0}^{2}\times\left(\left(1+10\times\exp\Big{(}-\frac{ \theta^{1}}{10^{\circ}}\Big{)}\right)^{2}+\left(1+10\times\exp\Big{(}-\frac{ \theta^{s}}{10^{\circ}}\Big{)}\right)^{2}\right)\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Receiver** & **Function** & **Receiver Type** & **Antenna Type** \\ \hline UWA0 & Reference receiver & SEPT POLARX5 & JAVRINGANT\_DM SCIS \\ \hline CUT0 & Rover (Open-sky scenario) & TRIMBLE NetR9 & TRM59800.00 SCIS \\ \hline SUB0 & Rover (Suburban scenario) & TRIMBLE R10 & TRMR10 NONE \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Receiver and antenna types of the two baselines used in data analysis. Figure 1: Rover receivers in: (**left**) the open-sky scenario and (**middle** and **right**) the suburban scenario. where \(\sigma_{j,0}\) represents the between-receiver zenith-referenced standard deviation for code or phase observations on frequency \(j\), and \(\sigma_{j}^{1s}\) represents that on the DD level, mapped to the slant direction. Based on Equation (9):  
\[\left(1+10\times\exp\!\bigg{(}-\frac{\theta^{1s}}{10^{\circ}}\bigg{)}\right)^{2} =\frac{\left(1+10\times\exp\!\Big{(}-\frac{\theta^{1}}{10^{\circ}}\Big{)} \right)^{2}+\left(1+10\times\exp\!\Big{(}-\frac{\theta^{s}}{10^{\circ}}\Big{)} \right)^{2}}{2} \tag{10}\]  
so that \(\theta^{1s}\) can be obtained with:  
\[\theta^{1s}=-10^{\circ}\times\log\!\left(0.1\times\sqrt{\frac{\left(1+10 \times\exp\!\Big{(}-\frac{\theta^{1}}{10^{\circ}}\Big{)}\right)^{2}+\left(1+1 0\times\exp\!\Big{(}-\frac{\theta^{s}}{10^{\circ}}\Big{)}\right)^{2}}{2}}-0.1\right) \tag{11}\]  
where \(\log(\cdot)\) is the natural logarithm of the element in (\(\cdot\)). The DD phase (\(\delta\phi_{DDj}^{1s}\)) and code (\(\delta p_{DDj}^{1s}\)) residuals are grouped for the elevation intervals of \(\theta^{1s}\) for every one degree, and the root mean square (RMS) of the residuals are plotted in the top panels of Figure 2 for the open-sky and suburban scenarios. The code RMS is divided by a factor of 100 for the purpose of illustration. Together with the RMS, the total number of observations used for computing the RMS in each elevation interval is also shown in the bottom panels of Figure 2. In general, the RMS of all observation types decrease with increasing elevation angles. Compared to the open-sky scenario, larger increase in the phase RMS can be observed for the suburban scenario when decreasing the low elevation angles (the red and blue lines in the top panel of Figure 2b). This should be related to the larger phase multipath effects in the suburban scenario. For the code RMS, compared with their smooth behaviours in the open-sky scenario, greater randomness in RMS is visible under the suburban scenario due to the fact that the residuals become less sensitive to the elevation angles in the more complicated environment (the green and magenta lines in the top panel of Figure 2b). In addition to the relationship between the DD residuals and the elevation angles, the changes of the residuals with the C/N\({}_{0}\) were also analysed. Similar to Equation (9), the C/N\({}_{0}\)\(S_{j}^{1s}\) are computed  
Figure 2: Root Mean Squares (RMS) of the double-difference (DD) residuals with respect to the elevation angles (\(\theta^{1s}\)) in: (**a**) the open-sky scenario and (**b**) the suburban scenario. Note that the RMS of the code residuals is divided by a factor of 100. In the bottom panels, the blue, the red and the green lines are almost overwritten by the magenta lines.