# Circular Regression Applied to GNSS-R Phase Altimetry
## 4 Experimentation
### Assessment with Real Data  
This experimentation on real data is an application of the experimental setup presented in the previous section, the receiver architecture described Section 2.2 and the circular estimator described Section 3.3. The proposed system is tested using the civil component of GPS-L1 signals reflected on an artificial canal basin in Calais. As shown Figure 8, the receiving antennas system, composed of an up-looking RHCP antenna and a down-looking LHCP (Left Hand Circular Polarization)antenna, is located at the top of a telescopic mast. The mast can rise up to a height of 10 m and is mounted on an experimental vehicle. The reference height between the reflecting water surface and the receiving antennas is obtained using a measurement tape. Figure 6: Experimental setup. Figure 7: Example of delay-Doppler map obtained with the proposed receiver. The direct and reflected signals are separated in time by a delay \(\tau^{L}=5.12\) us (in addition to the path delay). The first experiment took place on 30 June 2016. The reference height between the antennas and the water surface was 12.60 m. This height did not change during the experimentations, as the basin gates were locked. In Figure 9, an example of interferometric phase evolution in the time frame of the sine of the satellite elevation is shown. Each measure is obtained with one millisecond of coherent integration. In this figure, we observe, in accordance with Relation (4), that the phase difference \(\psi_{s}\) evolves linearly as a function of the elevation sine. However, the linear slope is affected by oscillations that are not considered in our model. This effect can be due to the oscillations of the different elements of the system such as the mast or the surface of reflection. The GPS satellites in view and their associated footprints on the basin are displayed in Figure 10. Only two footprints, corresponding to satellites PRN (Pseudo-Random Number) 18 and 21, can be used, as the footprint for satellite PRN 20 was partially out of sight during the experimentations. We acquired three successive sets of data at 15 h 25 min, 15 h 38 min and 15 h 51 min. For each dataset, the recording duration was of 600 s. We report in Table 2 the elevation and the carrier-to-noise ratio of the interferometric signal for each satellite. Figure 8: Experimental vehicle carrying the GNSS-R altimeter. The observed GPS signals are reflected on an artificial basin in Calais, France (50.962689\({}^{\circ}\)N, 1.857305\({}^{\circ}\)E). Figure 9: Example of interferometric phase observations obtained for a GPS-L1 reflection on the artificial basin. Table 3 shows the estimated heights obtained for Satellites 18 and 21. On the first two lines of this table, the heights estimated independently for Satellites 18 and 21 at 15 h 25 min, 15 h 38 min and 15 h 51 min are presented. The estimated heights are within 10 cm of accuracy except for Satellite 21 at 15 h 51 min, with a difference with the tape measurement of 25 cm, and for Satellite 18 at 15 h 38 min and 15 h 51 min with a difference of 27 and 42 cm. For Satellite 21 at 15 h 51 min, the higher inaccuracy can be explained by the low \(C/N_{0}\) of the signal. For Satellite 18 at 15 h 38 min and 15 h 51 min, the higher inaccuracy is explained by the low elevation rates (in absolute value), in accordance with the theoretical study. For observation durations of 600 s, the accuracy was expected to be of at least 5 cm according to this study. However, the additional oscillations, due to the waves being not taken into account in the model, affect the estimation precision. Over 600 s, the oscillations are