# Circular Regression Applied to GNSS-R Phase Altimetry
## 2 GNSS-R Altimetry Using Phase Measurement
### Phase Observation  
Figure 2 presents the different blocks of the receiver architecture used to observe the phase delay \(\psi_{s,k}\). This receiver architecture is designed as a master/slave configuration as in [18]. The incoming direct and reflected signals, respectively \(S_{s}^{D}\) and \(S_{s}^{R}\), are demodulated with an internal oscillator at frequency \(f^{OL}\) and digitized with a sampling frequency \(f_{s}\). The local oscillator frequency \(f^{OL}\) is fixed so the residual frequency after down-conversion is both larger than the inverse of the integration interval and lower than half the sampling frequency \(f_{s}\). The tracking module of the direct signal provides the GNSS parameters \(f_{s}^{D}\), \(\tau_{s}^{D}\) and \(\phi_{s}^{D}\) in order to construct the replicas \(p_{s}^{LD}\) and \(p_{s}^{QD}\). The demultiplexing module described in Figure 2 provides the observations \(I_{s,k}\) and \(Q_{s,k}\).  
Let us define \(I_{s,k}\) and \(Q_{s,k}\) as the in-phase and quadrature components at instant \(k\) for the \(s\)-th satellite. \(k\) is the measurement index, the measurements being separated by a fixed period corresponding to the coherent integration \(T_{c}\) used in the tracking module. This period has a minimum value of \(T_{c}=1\) ms, which corresponds to the C/A code length. These components are obtained after the demodulation of the reflected signal with local replicas of the direct signal:  
\[I_{s,k} = \int_{k\;T_{c}}^{(k+1)\;T_{c}}S_{s}^{R}(t)p_{s}^{LD}(t)\mathrm{d}t \tag{5}\]  
\[Q_{s,k} = \int_{k\;T_{c}}^{(k+1)\;T_{c}}S_{s}^{R}(t)p_{s}^{QD}(t)\mathrm{d}t \tag{6}\]  
where \(p_{s}^{LD}(t)\) and \(p_{s}^{QD}(t)\) are the in-phase and quadrature clean replicas of the direct signal. These local replicas are constructed with the parameters provided by the tracking loop of the direct signal.  
Figure 2: Construction of the \(\psi_{s,k}\) observable.  
These parameters are the estimates of \(f_{s}^{D}\), \(\phi_{s}^{D}\) and \(\tau_{s}^{D}\). The local replicas are defined by:  
\[p_{s}^{l,D}(t)=CA_{s}\left(t-\tau_{s}^{D}(t)\right)\sin\left(2\pi f _{s}^{D}\;t-\phi_{s}^{D}(t)\right) \tag{7}\]  
\[p_{s}^{Q,D}(t)=CA_{s}\left(t-\tau_{s}^{D}(t)\right)\cos\left(2\pi f _{s}^{D}\;t-\phi_{s}^{D}(t)\right) \tag{8}\]  
Finally, \(I_{s,k}\) and \(Q_{s,k}\) can be modeled as:  
\[I_{s,k} = \frac{A_{s}^{R}}{2}\;\Lambda\left(-\Delta_{s,k}/c\right)\;\;\cos \left(-\psi_{s,k}\right)+\eta_{s,k}^{l} \tag{9}\]  
\[Q_{s,k} = \frac{A_{s}^{R}}{2}\;\Lambda\left(-\Delta_{s,k}/c\right)\;\;\sin \left(-\psi_{s,k}\right)+\eta_{s,k}^{Q} \tag{10}\]  
where \(\Lambda\left(\right)\) models the normalized correlation function of the CDMA code. For ground-based measurements, we assume that the altitude of the receiving antennas is lower than 150 m, implying that \(\Lambda\left(\Delta_{s}/c\right)\) is not null. \(\eta_{s,k}^{l}\) and \(\eta_{s,k}^{Q}\) are two independent zero-mean Gaussian noises.  
Using Relations (9) and (10), a noisy observation of the phase delay \(\psi_{s,k}\), also called interferometric phase observations [18], can be obtained every millisecond with the following model:  
\[\psi_{s,k} = -\;\mathrm{arctan}^{*}\left(\frac{Q_{s,k}}{I_{s,k}}\right) \tag{11}\] \[= \left(\frac{4\pi}{\lambda}\;h\;\sin\left(\varrho_{s,k}^{el}\right) +\eta_{s,k}^{\psi}\right)mod(2\pi)\]  
where \(\mathrm{arctan}^{*}\) is the quadrant-specific arc-tangent and \(\eta_{s,k}^{\psi}\) is a zero-mean noise. The noise is assumed to be distributed according to a von Mises distribution [27; 30] with a concentration parameter \(\kappa_{s}^{\psi}\). The noise on the observations of \(\psi_{s,k}\) is not correlated over time as there is no tracking of the reflected signal. In this model, the height \(h\) and the concentration parameter \(\kappa_{s}^{\psi}\) are the only unknown parameters.