## 3 Results
### 3D-Var Results  
The presentation of DA results underscores the persistent challenge in refining wind speed forecasts, despite advancements in assimilation techniques. Wind speed stands out as a critical variable for subsequent wave and hydrodynamic models. A test case was examined, incorporating all available data sources (I, II, and III, Section 2.1) whenever they were available, augmenting the NOAA METAR and POSEIDON Buoys dataset with approximately 30 meteorological stations situated across the island of Crete. The inclusion of additional observation locations was anticipated to significantly enhance accuracy, demonstrating a path towards improved forecasting precision. Notably, not all stations seen in Figure 1 always have availability of data to be assimilated, and hence we only included in each run the stations that had data availability closer to the analysis time, which is the time of the simulation when the functional J from Equation (1) is minimized (here we used the 00:00 of the hindcast day). In Figure 3, the plot shows the absolute bias for wind speed for both the case of the control simulation (WRF without any DA) and the corrected one with the 3D-Var, under the dynamically downscaled 3 km spatial resolution. The predicted values at each case were compared to the corresponding observations by the six METAR stations of Dataset II, and these biases were plotted with representative values across 3 months of daily simulations for each forecast hour in 3-hourly temporal resolution that most METAR data showcased, with their mean values shown in stars. Additionally, their 90 th percentiles over the 92 simulations are shown with the shaded same color areas. Those simulations fall within mostly summer period in Greece, performed during the months of mid-July to mid-October 2024, and hence the covariance matrix that was calculated for the summer period was used. As the lower value translates to improved model in absolute bias terms, we observed that between forecast hours 26 and 72, the mean values as well as the percentiles for the 3D-Var case were consistently lower than the control case. Evidently, calculating the average percentagewise improvement from the control simulations to the 3D-Var case, for all simulations in the wind speed variable compared to the observational data, we found a 5.6% improvement, whereas the overall improvement for all forecast hours was calculated at 4.7%. This finding is consistent with [20], which features a 3D-Var study with dynamical downscaling in the southern region of China, which also demonstrated bigger contributions of the 3D-Var around the middle of the forecast horizon period. In comparison with their study, our case study showcased comparable improvement percentages, although in our case, we had considerably less observational data coverage. Specifically, in our wider D01 region shown in Figure 1, there were approximately 30 METAR stations provided by the NOAA ISD/ISH dataset, which were mostly consistently available throughout the simulation period, the nearly 40 stations of Data Source IV as well as 4 buoys from Data Source III (Section 2.1), compared to the study of [20] that featured more than 300 METAR stations and resulted in an average improvement in 10 m wind speeds of about 4% throughout their entire forecast horizon. Additionally, in Figure 4, a bar chart is presented depicting absolute biases for each station utilized in the validation process from the METAR data that were used for validation purposes. Note that some METAR stations that were used for DA were not used for validation, since the specific 3D-Var technique implemented requires data availability only during the analysis time, whereas for validation, we used the entire forecasting horizon. Furthermore, some stations had available data with a coarser 6 h temporal resolution instead of the 3 hourly that we calculated for the errors of Figure 3. Hence, for consistency, the six stations shown in Table 1 and Figure 4 are retained as validation stations.