# Circular Regression Applied to GNSS-R Phase Altimetry
## 2 GNSS-R Altimetry Using Phase Measurement  
### Height Retrieval Using GNSS Phase Signals  
Figure 1 describes the geometry of the GNSS-R altimeter. In this figure, \(T\) is the transmitting GNSS satellite and \(S\) is the specular reflection point. In our approach, the surface of reflection is assumed to be the plane, and the incidence and reflection angles are the same. \(R\) is the position of the receiving antennas, at an altitude \(h\) over the reflecting surface. Two antennas are considered: an up-looking  
antenna, with right-hand circular polarization, and a down-looking antenna, with left-hand circular polarization. These two antennas respectively sense the direct and the reflected GNSS signal. The direct and reflected signals of the \(s\)-th satellite are respectively noted \(S_{s}^{D}\) and \(S_{s}^{R}\). The elevation angle is \(\theta_{s}^{el}\). For a low altitude \(h\), the satellite is assumed to be at an infinite distance in regards to the position of the receiver. Consequently, the elevation angle \(\theta_{s}^{el}\) is equal to the incidence and the reflection angle.  
The height \(h\) between the antennas and the reflecting surface is the parameter to estimate. The height is derived from the path difference between the direct and the reflected signals. According to the system geometry presented in Figure 1, the path difference for satellite \(s\) at instant \(t\) is given by the following equation:  
\[\Delta_{s}(t)=2h\sin\left(\theta_{s}^{el}(t)\right) \tag{1}\]  
where the satellite elevation \(\theta_{s}^{el}(t)\) can be derived from the current GNSS satellites ephemeris and from the GNSS positioning of the receiving antennas.  
In the case of the civil component of a GPS-L1 signal received from the satellite \(s\), the direct signal \(S_{s}^{D}\) and the reflected signal \(S_{s}^{R}\) can be defined as follows:  
\[S_{s}^{D}(t)=A_{s}^{D}\ CA_{s}(t-\tau_{s}^{D}(t))\sin(2\pi f_{s}^{D}\ t-\phi_{s }^{D}(t))+\eta_{s}^{D}(t) \tag{2}\]  
\[S_{s}^{R}(t)=A_{s}^{R}\ CA_{s}(t-\tau_{s}^{D}(t)-\Delta_{s}(t)/c)\sin(2\pi f_{ s}^{D}\ t-\phi_{s}^{D}(t)-\psi_{s}(t))+\eta_{s}^{R}(t) \tag{3}\]  
where \(A_{s}^{D}\) and \(A_{s}^{R}\) are the amplitudes of the direct and reflected signals. \(CA_{s}(t)\) is the CDMA code broadcast by the satellite, at a period of one millisecond. \(\tau_{s}^{D}\) is the code delay of the received signal at the RHCP (Right Hand Circular Polarization)antenna (in seconds). \(f_{s}^{D}\) is the frequency of the direct signal, and \(\phi_{s}^{D}\) is its phase delay. \(\eta_{s}^{D}\) and \(\eta_{s}^{R}\) are additive zero-mean white Gaussian noises [26], respectively, on the direct and reflected signals. \(\Delta_{s}\) is the path difference between the direct and the reflected signals for the satellite \(s\). \(c\) is the speed of light. The term \(\psi_{s}\) is the difference of phase between the direct and the reflected signals for satellite \(s\).  
Figure 1: Geometry of a ground-based GNSS-Reflectometry (R) altimeter.  
The phase \(\psi_{s}\) corresponding to the path difference observed in the phase delay can be derived from Equation (1) as:  
\[\psi_{s}(t) = \left(\frac{2\pi}{\lambda}\;\Delta_{s}(t)\right)\;mod(2\pi) \tag{4}\] \[= \left(\frac{4\pi}{\lambda}\;h\;\sin\left(\theta_{s}^{el}(t)\right) \right)\;mod(2\pi)\]  
\(\psi_{s}\) is an angular datum, defined in an interval of length \(2\pi\), which corresponds to \(\lambda\approx 19.05\) cm in range for a GPS-L1 signal. We notice, from Equation (4) that \(\psi_{s}\) evolves linearly with \(\sin\left(\theta_{s}^{el}(t)\right)\), the sine of the \(s\)-th satellite elevation. The slope of \(\psi_{s}\) in the elevation sine time frame is proportional to the height \(h\) between the receiving antennas and the reflecting surface, with a factor of \(4\pi/\lambda\).