## 2 Methodology  
According to the full model [12], the SNR is a combination of the direct and reflected power related to the noise power. For a plane reflecting surface, the SNR at time \(t_{i}\) can be decomposed into a trend and interference fringes that are attenuated with respect to the elevation angle:  
\[SNR =trend+attenuation\cdot oscillation \tag{1}\] \[trend =c_{0}+c_{1}t_{i}+c_{2}t_{i}^{2}+\cdots+c_{n}t_{i}^{n}\] \[attenuation =e^{-ak^{2}\beta^{2}\sin^{2}\epsilon},k=\pi/\lambda\] \[oscillation =A\cos\left(\frac{4\pi}{\lambda}h_{ref,t_{i}}\sin\epsilon+\phi_{0}\right)\]  
Here, the trend is described by a low-order polynomial function of the \(n\) unknown parameters \(c\). The attenuation depends on the wave number \(k\), the GNSS signal wavelength \(\lambda\), the elevation angle \(\epsilon\), and the unknown damping coefficient \(\delta\). \(A\) and \(\phi_{0}\) are the unknown amplitude and phase offset of the oscillation. The key parameter describing the oscillation frequency is the reflector height \(h_{ref,t_{i}}\). If this parameter is known, all remaining parameters can be estimated by non-linear least squares estimation for each satellite separately. Therefore, \(h_{ref,t_{i}}\) has to be described as well as possible. In the case of a static antenna, measuring the reflections from a tidally-influenced water surface, this can be done by:  
\[h_{ref,t_{i}}=h_{APC}-h_{gauge,t_{i}}+dh_{sphere,t_{i}} \tag{2}\]  
The height of the antenna phase center above tide gauge zero \(h_{APC}\) can be assumed to be stable for a static antenna. It can either be measured by leveling techniques or estimated using GNSS reflectometry [13]. The tides \(h_{gauge}\) cause a height variation with respect to tide gauge zero at epoch \(t_{i}\). If no tidal data are recorded in the vicinity of the measurement site, tidal interpolation may be necessary [14]. Such a procedure will reduce the quality of the correction term and, therefore, the quality of the analysis. In particular, for observations from low elevation angles, the specular point can be several hundred meters away, and a correction \(dh_{sphere}\) due to the curvature of the spherically-approximated Earth's surface is required. A simplified geometry is presented in Figure 2a. For moving antennas, the formula has to be extended for height variations of the antenna. Antennas aboard moving ships are a special case, as the tide-dependent height variations of the water surface can be ignored, but additional hydrodynamic and hydrostatic effects have to be taken into account. The reflector height can be expressed as:  
\[h_{ref,t_{i}}=h_{SRF}-h_{static}+dh_{att,t_{i}}+leave_{t_{i}}-squat_{UKC_{t_{i} },STW_{t_{i}}}+dh_{sphere,t_{i}} \tag{3}\]  
Here, \(h_{SRF}\) is the antenna height above the ship's heel. The hydrostatic draft \(h_{static}\) can be assumed to be stable for short journeys. For longer journeys, the temporal behavior has to be modeled based on information about the loading condition or from measurements [15]. The hydrodynamic squat is an apparent change of trim and draft while the ship moves through the surrounding water. This effect is caused by the changed pressure conditions around the ship's hull. The main impact parameters are the shape of the ship's hull, the under keel clearance (UKC), and the speed at which the ship is moving relative to the water (speed through water (STW)). Speed- and depth-dependent empirical squat models might be applied if available or a method based on GNSS reflectometry can be used [16]. The ship's attitude \(dh_{att}\) and heave have an impact on the reflector height and must be taken into account. Both corrections can be obtained by GNSS analysis [16] or from inertial measurement unit (IMU). data [17]. A simplified geometry showing the connection between the parameters in Equation (3) is presented in Figure 2b. In order to keep the illustration simple the parameters \(leave_{t_{i}}\) and \(dh_{sphere,t_{i}}\) are not shown.