# Circular Regression Applied to GNSS-R Phase Altimetry
## 4 Experimentation  
### Assessment with Synthetic Data  
We first assess the linear-circular regression on synthetic GNSS-R data. In this section, we consider synthetic data corresponding to a realistic GPS constellation scenario. The height is fixed at 100 m (\(\beta~{}\approx\) 6603.63), and the observation rate is fixed at 1000 Hz (one measure per millisecond, the length of the GPS-L1 civil code). The noise of the interferometric phase observations is defined for different values of carrier to noise ratio \(C/N_{0}\). The carrier to noise ratio \(C/N_{0}\) and the associated concentration parameter \(\kappa_{s}\) are described in Table 1. In common conditions, the carrier to noise ratio associated with the interferometric signal is close to 40 dB-Hz for GPS-L1 signals, so we assess the estimator for \(C/N_{0}\) values of 30, 35, 40 and 45 dB-Hz.  
The interferometric phase observations are obtained with an initial elevation fixed at 75\({}^{\circ}\) and an elevation rate of 0.006\({}^{\circ}\)/s. This corresponds to a usual case as observed over the city of Calais [7; 19]. We compute the RMSE (Root Mean Square Error) of the estimator over 300 realizations of the observations model. Figure 4 presents the obtained RMSE as a function of the integration time along with the theoretical standard deviation of the estimator, calculated using Equation (28).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(C/N_{0}\) (dB-Hz) & 30 & 35 & 40 & 45 \\ \hline \(\kappa_{s}\) & 1.35 & 2.96 & 9.34 & 30.82 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Concentration parameter \(\kappa_{s}\) corresponding to various \(C/N_{0}\) values of the interferometric signal.  
Figure 4: RMSE and theoretical standard deviation of the estimated height as a function of the integration time, for various values of \(C/N_{0}\).  
We observe from Figure 4 that the RMSE fits the theoretical standard deviation. We first note that the linear-circular regression height estimator is unbiased (the mean error is indeed null, and the RMSE thus corresponds to the standard deviation of the estimator). We also can conclude that the theoretical formulation for the standard deviation is correct. Furthermore, we observe that the linear-circular regression height estimate reaches an accuracy of at least 5 cm for a period of integration of 100 s for any carrier to noise ratio \(C/N_{0}\) superior to 35 dB-Hz. We also observe, as expected, that when the carrier to noise ratio increases, the accuracy of the estimator increases.  
We show in Figure 5 the standard deviation of the proposed estimator as a function of the satellite elevation rate and for various initial elevations. For these simulations, the considered observation period is fixed at 100 s, and the \(C/N_{0}\) is 35 dB-Hz. We observe that the standard deviation of the estimated height decreases when the elevation rate increases, as expected according to Equation (28). Figure 5 also depicts the influence of the initial elevation: the estimator is less precise for high initial elevations, with especially highly increasing precision losses for initial elevations increasing over 70\({}^{\circ}\). High values of the initial elevation indeed provide smaller \((\sin(\theta_{s,k}^{el})-\overline{\sin(\theta_{s,k}^{el})})\) spans as the sine gets very close to one for any elevation over 70\({}^{\circ}\).