# Circular Regression Applied to GNSS-R Phase Altimetry
## 5 Discussion  
The proposed receiver architecture and associated signal processing are applicable for static ground-based height measurements. In the dynamic case, the proposed method would only be valid for a constant height during the observation duration. The required observation duration for centimeter accuracy being linked with the number of observable reflected signals, very short durations could be sufficient with a multi-constellations and/or multi-frequency receiver. In the case of height variations during the measurements, the phase delay model and the associated regression estimator would have to be modified in order to be able to provide accurate, non-constant height estimation.  
One of the drawbacks of the proposed technique is that the receiver altitude is limited. The largest observable path difference is indeed equal to 300 m (which corresponds to the C/A code length). The maximum observable height is 150 m for a satellite elevation of 90\({}^{\circ}\), 300 m for a satellite elevation of 30\({}^{\circ}\) and higher for lower elevations. Assuming the height is approximately known by other means (an accuracy of tens of meters is sufficient), this limitation can be avoided by adding the corresponding approximate code delay in the replicas (7) and (8). The other height limitation is linked to the assumption that the emitting satellite can be considered at an infinite distance from the observation zone: the technique described in this article is not applicable for spaceborne GNSS-R.  
We found in the experimentation on real data the effects of the elevation variation span and observation duration that were predicted by the theoretical study. The main difference between the results on synthetic and real data is linked to the oscillations that affected the phase observations. These oscillations, which are not considered in the phase model, affected the data as an additional non-von Mises noise and altered the height estimation accuracy.  
The classical approach used to improve the robustness of the estimation toward high noise levels is to reduce the filter bandwidth of the receiver. In the digital case, this is realized by increasing the correlation integration time, at the expense of time resolution. However, for high integration time, if the local replica frequency is not equal to the frequency of the received GNSS signal, the correlation to noise ratio is reduced. This problem can appear in the GNSS-R context if the direct and reflected signals have different frequencies. In addition, from a statistical point of view, using 1-ms integration time or more provides the same accuracy for the proposed estimator, as larger integration time provides less data points.  
Larger integration times would be required for tracking the weak reflected signal in a classic lock loop. In the approach proposed in this article, the main advantage of the master/slave receiver architecture is that only the stronger direct signal has to be tracked. The interferometric GNSS-R (iGNSS-R) approach based on cross-correlating the direct and reflected received signals [(22; 41)] allows using the military encrypted components of GNSS signals, leading to better performances than the proposed approach when only one satellite reflection is considered. However, iGNSS-R requires the use of directional antennas aiming at a single satellite signal. Being able to use omnidirectional antennas  
Figure 13: Observed interferometric phase from Satellite 25.  
and to combine the data from several satellites reflections is a major advantage of the proposed method over iGNSS-R. Finally, in terms of signal processing, the linear-circular regression technique defined in this article can be used to get height estimates from datasets with large data gaps, which is something that cannot be done with the classical use of unwrapping techniques. Moreover, unwrapping can fail in the case of low \(C/N_{0}\), provoking cycle slips in the phase evolution and subsequently affecting the height estimation.