# Circular Regression Applied to GNSS-R Phase Altimetry
## 4 Experimentation
### Assessment with Real Data
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**15 h 25 min**} \\ \hline \multirow{2}{*}{**Satellite PRN**} & \multirow{2}{*}{**Interferometric Signal \(C/N_{0}\) (dB-Hz)**} & \multicolumn{4}{c}{**Elevation**} \\ \cline{3-6}  & & **min. (\({}^{\circ}\))** & **max. (\({}^{\circ}\))** & **mean (\({}^{\circ}\))** & **rate (\({}^{\circ}\)/s)** \\ \hline
18 & 36.5 & 36.44 & 39.21 & 37.85 & 0.0046 \\
21 & 34 & 53.73 & 57.56 & 55.65 & \(-\)0.0064 \\ \hline \multicolumn{6}{c}{**15 h 38 min**} \\ \hline \multirow{2}{*}{**Satellite PRN**} & \multirow{2}{*}{**Interferometric Signal \(C/N_{0}\) (dB-Hz)**} & \multicolumn{4}{c}{**Elevation**} \\ \cline{3-6}  & & **min. (\({}^{\circ}\))** & **max. (\({}^{\circ}\))** & **mean (\({}^{\circ}\))** & **rate (\({}^{\circ}\)/s)** \\ \hline
18 & 37 & 39.99 & 42.23 & 41.15 & 0.0037 \\
21 & 34.1 & 48.55 & 52.51 & 50.53 & \(-\)0.0066 \\ \hline \multicolumn{6}{c}{**15 h 51 min**} \\ \hline \multirow{2}{*}{**Satellite PRN**} & \multirow{2}{*}{**Interferometric Signal \(C/N_{0}\) (dB-Hz)**} & \multicolumn{4}{c}{**Elevation**} \\ \cline{3-6}  & & **min. (\({}^{\circ}\))** & **max. (\({}^{\circ}\))** & **mean (\({}^{\circ}\))** & **rate (\({}^{\circ}\)/s)** \\ \hline
18 & 35 & 42.80 & 44.38 & 43.63 & 0.0026 \\
21 & 32.3 & 43.39 & 47.38 & 45.38 & \(-\)0.0066 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters for satellites PRN 18 and 21, 30 June 2016, during 600 s of observation at 15 h 25 min, 15 h 38 min and 15 h 51 min. Figure 10: GPS satellites in view and corresponding footprints on the basin 30 June 2016, 15 h 25 min. averaged in the estimation process with some residuals. The loss of accuracy in the estimated height as compared to the theoretical study is supposed to be due to these residuals. In order to improve the accuracy of the estimated height, we used the fusion operation defined in Equation (25) on the interferometric phase delays obtained for the two satellites. We present on the third line of Table 3 the estimated heights obtained at 15 h 25 min, 15 h 38 min and 15 h 51 min using this technique. The advantage is both to increase the amount of data and to increase the elevation sine span. At 15 h 25 min and 15 h 38 min, the estimated heights are both 1 cm close to the reference height. At 15 h 51 min, the estimated height is 9 cm away from the reference height. This can be explained looking at Figure 11, which shows the observed interferometric phases for Satellites 18 and 21 at 15 h 38 min and 15 h 51 min. In this figure, we observe that the elevation sine span is over two-times smaller at 15 h 51 min than at 15 h 38 min, explaining (through Equation (28)) the accuracy loss. We present in Table 4 the estimated heights obtained with the fusion of the observations for Satellites 18 and 21 at 15 h 25 min, 15 h 38 min and 15 h 51 min, as a function of the considered observation duration. We show that for various elevation spans (of the order of 20\({}^{\circ}\) at 15 h 25 min, 10\({}^{\circ}\) at 15 h 38 min and a few degrees at 15 h 51 min), centimeter accuracy can be reached for any duration over 30 s (60,000 data points). This shows that fusing the data from several satellites can highly reduce the observation duration required for reaching a given accuracy. The main explanation is the increase of the elevation sine span. For a 15-s duration, the amount of data is not sufficient. This is in accordance with Figure 4: the equivalent number of data points (30,000) corresponds to the fast increase of the height estimation error. Figure 11: (**Left**) Observed interferometric phase for Satellites 18 and 21 at 15 h 38 min, 30 June 2016.