## 4 Processing and Modeling  
Besides the roughness of the reflecting surface, the attenuation of the SNR data is governed by the antenna gain pattern. For antennas of the same type, the gain patterns can be assumed to be similar, whereas it is commonly different for different antenna types. Hence, if SWH is described as a function of the damping coefficient of the attenuation, individual models must be constructed for every antenna type. For that purpose, we used the SNR data from both experiments and calculated the damping coefficient according to Equation (1) in conjunction with either Equation (2) or (3). Later, we used in situ SWH data from buoys to find individual describing functions. For all datasets, we split the satellite tracks into descending and ascending arcs with elevations above 1\({}^{\circ}\) and below 10\({}^{\circ}\). Observations from lower elevation angles were too noisy for evaluation. The higher elevation limit was chosen due to the antenna gain pattern in the case of the static antenna, while in the case of the ship's measurement, the specular point had to lay outside of the ship's wave system [16]. The elevation angles were corrected for tropospheric refraction using the refraction model \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Buoy Name** & **Latitude**\({}^{\circ}\) & **Longitude**\({}^{\circ}\) & **Water Depth m** & **Mean Difference m** & **RMS m** & **Correlation \%** \\ \hline HelgolandNorthWR & 54\({}^{\circ}\)13.164\({}^{\prime}\) & 7\({}^{\circ}\)49.109\({}^{\prime}\) & 26 & 0.027 & 0.207 & 97 \\ ElbeWR & 53\({}^{\circ}\)59.825\({}^{\prime}\) & 8\({}^{\circ}\)06.630\({}^{\prime}\) & 25 & \(-\)0.002 & 0.202 & 97 \\ BunkerHill & 54\({}^{\circ}\)47.500\({}^{\prime}\) & 8\({}^{\circ}\)16.300\({}^{\prime}\) & 6 & 0.014 & 0.148 & 96 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison statistics between the numerical wave model and wave buoy observations at three different sites. of [22]. The required pressure and temperature data were taken from DWD weather stations at Alte Weser Lighthouse and on Heligoland. The first experiment was done under ideal circumstances as the GNSS antenna was installed directly above a tide gauge station. Therefore, the tide gauge readings could be directly used to generate the height variation correction \(h_{gauge}\). The estimation of the unknown damping coefficient \(\delta\), amplitude, and phase, as well as of the unknown trend function parameters was done for each GPS and GLONASS satellite separately. To ensure a sufficient amount of data for a reliable parameter estimation, only arcs with elevation angles ranges of more than 3\({}^{\circ}\) were analyzed. The estimation was carried out as a non-linear least squares adjustment. We solved the optimization problem by applying the Levenberg-Marquardt algorithm to avoid divergence due to insufficient initial values for the damping coefficients. The reference SWH values were interpolated from ElbeWR wave buoy data. The reference epochs for interpolation were set as the mid epochs of the satellite arcs. Due to data gaps at the wave buoy, epochs with a time gap of larger than 30 min to the next buoy measurement epoch had to be excluded. Since the distance between the tide gauge station and the ElbeWR buoy is about 24 km, a correction was required. For that, the SWH from the NWM at both the position of the buoy and the tide gauge was calculated. The difference between the resulting values was used as a correction term. Because of the buoy observation resolution and the model uncertainty, the corrected SWH could become negative during low wave heights. Two percent of the corrected SWH were negative. To avoid such unrealistic values, the minimal SWH was set to 0.1 m, taking into account the data resolution. In total, 6193 (3528 GPS/2665 GLONASS) satellite arcs could be processed. In the second experiment, the 83 days of ship tracks were split into segments of deep or shallow water, while a water depth threshold of 25 m was assumed.