# Circular Regression Applied to GNSS-R Phase Altimetry
## 4 Experimentation
### Assessment with Real Data
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **15 h 25 min** & **15 h 38 min** & **15 h 51 min** \\
**Satellite PRN** & \(h_{ref}=\) **12.60 m** & \(h_{ref}=\) **12.60 m** & \(h_{ref}=\) **12.60 m** \\ \cline{2-4}  & **Estimated Height \(\tilde{h}\) (m)** & **Estimated Height \(\tilde{h}\) (m)** & **Estimated Height \(\tilde{h}\) (m)** \\ \hline
18 & 12.68 & 12.87 & 13.02 \\
21 & 12.65 & 12.49 & 12.36 \\ \hline
18 + 21 & 12.61 \({}^{*}\) & 12.61 \({}^{*}\) & 12.51 \({}^{*}\) \\ \hline
18 + 21 & & 12.60 ** & \\ \hline \hline \end{tabular}  
* Results obtained using the proposed fusion operation on both satellites. ** Results obtained using the proposed fusion operation on both satellites and all recordings.  
\end{table} 
Table 3: Estimated heights 30 June 2016 for 600 seconds of observation, Satellites PRN 18 and 21. In addition to merging data obtained from several satellites, it is also possible to merge data corresponding to different observation instants using the proposed estimator. On the fourth line of Table 3, we display the estimated height corresponding to the fusion of all of the available datasets, for both satellites and all three dates. The result of 12.60 m corresponds to the tape measured height. The second experiment took place on 6 July 2016 between 13 h 52 min and 14 h 10 min. The reference height between the antennas and the water surface was 11.27 m (constant height during the experimentation). The GPS satellites in view and the associated footprints are displayed Figure 12. The aim of this experiment is to show the feasibility of the proposed technique in the case of a single satellite signal observation including data losses. We considered the footprint of Satellite PRN 25 for this experimentation. We acquired five 13-s sets of data distributed over a 20-min span. Figure 13 shows the obtained interferometric phases as a function of the satellite elevation sine. We used all of the observations of phase for the height estimation, so the total amount of data corresponds to 65 s of observation (65,000 data points), with a satellite elevation varying over 20 min (in this case, over 7.5\({}^{\circ}\)). The theoretical study thus predicts a good accuracy. The estimated height is 11.30 cm, three centimeters away from the tape measured height. This experimentation shows that the amount of data can be highly reduced if the span of variation of the satellite elevation is large. It also shows that the proposed linear-circular regression approach can be applied to sets with missing data, which could not be done in an estimation process including phase unwrapping. Such a case could correspond to signal losses or attenuations yielding to the selection of limited data intervals. \begin{table}
\begin{tabular}{l c c c c c c c} \hline
**Sat.: 18 + 21** & **15 s** & **30 s** & **50 s** & **100 s** & **150 s** & **300 s** & **600 s** \\ \hline
15 h 25 min & 8.80 & 12.63 & 12.62 & 12.62 & 12.61 & 12.62 & 12.61 \\
15 h 38 min & 7.54 & 12.62 & 12.61 & 12.61 & 12.62 & 12.62 & 12.61 \\
15 h 51 min & 16.01 & 12.56 & 12.53 & 12.54 & 12.55 & 12.55 & 12.51 \\ \hline \end{tabular}
\end{table} 
Table 4: Estimated height (m) as a function of the observation duration, obtained by fusion of Satellite 18 and 21 data at 15 h 25 min, 15 h 38 min and 15 h 51 min, 30 June 2016 (reference height: \(h_{ref}=12.60\) m.)  
Figure 12: GPS footprints on the canal basin, 6 July 2016 at 13 h 59 min.