## 5 Validation  
The linear functions calculated in the previous section describe the SWH as linear functions of the damping coefficients \(\delta\) individually for the antenna types and can now be used to calculate SWH from \(\delta\). For real applications, it can be assumed that the SWH in a particular area is constant in space and over longer time spans since the SWH is an average measure of roughness and, commonly, changes slowly over time. Therefore, the individually-derived \(\delta\) values can be averaged within a reasonable time span. This should increase the quality of the derived SWH. The calculation of a mean \(\delta\) should take into account a correct stochastic model that adopts the standard deviations derived from the non-linear robust least squares adjustment. In this way, negative impacts of outliers and false reflections will be down-weighted, as well. The \(\delta\) values from the static antenna were sampled into hourly time slots according to their reference epoch. For the moving antenna scenario, all satellites from a segment were used to produce a mean value because the segments were always shorter than one hour. Finally, the SWH was calculated from these mean damping coefficients. For an independent validation, SWH data that were not used in the previous modeling section are required. Although many altimeter satellite ground tracks cross the German Bight, altimetry data are not useful for validation of this experiments. Due to the long revisit time, too few observations are available. Because of their distance to the measuring sites, data from other wave buoys could not be used either. Even if high frequency radar systems can be used for wave estimation [23], the system that is observing the German Bight [24] was only used for surface current calculation during both experiments. Due to the lack of other suitable data sources, the validation was done in comparison with NWM data. It should be emphasized that the wave model is independent of the buoy measurements used in the previous section because no in situ data were assimilated into the model. Therefore, these data can be used to validate the results obtained from the ship's measurement without any concern. In the case of the static experiment, differences of SWM estimates from the NWM were used \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Static Antenna m** & **Moving Antenna m** \\ \hline \(s_{0}\) & 0.126 & 0.085 \\ \(a_{0}\) & \(-1.161\) & \(-0.157\) \\ \(a_{1}\) & 5.300 & 3.104 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Standard deviation of unit weight and estimated parameters of the linear models. Figure 5: Damping coefficients \(\delta\) from SNR analysis plotted against in situ SWH from wave buoys. Shading with respect to their final weights in the robust adjustment. Estimated first degree polynomial (red). (**a**) Data from the static antenna. (**b**) Data from the ship-based antenna. to correct the buoy data. A systematic bias that would have been canceled out by this procedure would still be in the validation data. Hence, the datasets are not completely independent, which may result in an underestimation of uncertainty. Reference values were interpolated from the NWM results for each hourly slot and ship segment. In total, 733 time slots of the static antenna could be compared. The time series of the reference and of the empirical calculated SWH for the static antenna are shown in Figure 6. The reference series was less noisy, and both series matched well. On average, the differences between SWH from wave model and SWH from the empirical model was \(-0.022\) m. The RMS was \(0.146\) m. Seventeen percent of all time slots showed differences above \(0.20\) m. In Figure 7a, both series are plotted against each other. The correlation coefficient was \(0.92\), and both high SWH and low SWH values matched well. The average \(\sigma\) of the estimates SWH was \(0.048\) m. The \(\sigma\) increased for higher SWH values, and a maximum of \(0.253\) m was found.