**3. Results**

*3.1. Impacts in the IBI Wave Solution Related to the Use of Surface Current–Wave Coupling*

For the assessment of the surface current–wave coupling, model outputs from the control run (IBI-CO; run with no currents coupling and no data assimilation) and the no data assimilation run with current–wave coupling (IBI-CU) were compared with in-situ observations, corresponding to the year 2018, focusing the model validation on the wave parameters of significant wave height, *Hs*, and mean wave period, *Tm*02.

The error metrics from both model configuration test runs are summarized in Tables 2 and 3. Table 2 shows results of a comparison between the wave hindcast best estimate and in-situ observations, and Table 3 shows a similar comparison with altimeter data for the *Hs* parameter, with skill metrics differentiated between CMEMS L3 altimeters and HY-2A. The general trend of validation for Tables 2 and 3 is very similar due to the good correlation ( ≥0.98) between buoys measurements and altimeters, as the comparison performed by [42] reveals.

The results shown in both tables do not show substantial improvement between the coupled and uncoupled model, but differences are definitively evident in terms of mean period. Coupling effects are stronger for coastal buoys locations, in some cases with worse metrics due to overestimation in the peaks of *Tm*02 time series. On the other hand, the systematic negative bias in *Hs*, caused by the tendency of the IBI wave system to underpredict, is reduced with the current surface implementation.

Metrics in Strait of Gibraltar (CADIZ an GIBST subregions) have different solutions for coastal and deep-water buoys. In general, results are slightly better for the coupled model IBI-CU. This is due to current coupling increasing values of wave height, but in some coastal locations, when the Hs is slight, an unrealistic ocean current overestimates the *Tm*02.

Figure 3 depicts the pattern of agreemen<sup>t</sup> between the IBI wave system with wave– current coupling (IBI-CU) and no coupling (IBI-CO). As Table 2 shows, the western part of the IBI region (WIBSH, GOBIS and IRISH subregions) has analogous solutions: slightly better results for significant wave height and better metrics for the mean period, in the case of the coupled model (IBI-CU). At buoy 6200083, located on the western coast of Galicia (depth of 386 m), the comparison of model results serves as an explicit example of the general IBI performance. The effect of the current coupling is not very sensitive in the Hs, but bias for the *Tm02* drops from 0.53 to 0.46 and the RMSD from 0.72 to 0.60 (note the time series agreemen<sup>t</sup> except for the mean period on 9 December 2018). Similar error reductions are found throughout the year.

**Table 2.** Error metrics for the test runs with and without current–wave coupling (IBI-CU (CU) and IBI-CO (CO) test runs, respectively) computed with hourly observations at mooring buoys. Time period: 2018 Variables: Significant wave height (SWH) and mean wave period (TM02). Metrics computed for the whole IBI service domain and for the 9 Validation regions (i.e., IRISH, ECHAN, GOBIS, NIBSH, WIBSH, GIBST, CADIZ, WSMED, ICANA) used by the Copernicus Marine IBI-MFC service. Metrics are gathered using all the available buoys in each region, and also using exclusively Coastal and Deep-water Buoys (CB and DB, respectively). Each error metric (Bias, Root-Mean-Square differences (RMSD) Correlation (CCOR)) provided for each model solution. N counts the size of the sample. Bold numbers highlight the best performing dataset. Mooring buoys in the English Channel area give the zero-crossing wave period (Tz) instead of *Tm*02, so mean period measurements for the ECHAN region are not provided for this validation.


**Table 3.** Error metrics for the test runs with and without current–wave coupling (IBI-CU and IBI-CO test runs, respectively) computed with satellite observations. Time period: 2018. Variable: Significant Wave Height (SWH). Metrics computed for the whole IBI service domain and for the 9 Validation regions (i.e., IRISH, ECHAN, GOBIS, NIBSH, WIBSH, GIBST, CADIZ, WSMED, ICANA) used by the Copernicus Marine IBI-MFC service. Metrics are computed using the available L3 CMEMS altimeter data (Janson-2, Janson-3, Saral, Cryosat-2 and Sentinel3) and HY-2A satellite data. Each error metric (Bias, Root-Mean-Square differences (RMSD) Correlation (CCOR) and Scatter Index (SI2)) provided for each model solution. N counts the size of available sample after the SWH data pre-process. Bold numbers highlight the best performing dataset.


Predictably, current coupling has no influence in severe storms, where strong wind forcings control the wave model output, with good accuracy in both cases: IBI-CU and IBI-CO solutions. Figure 4 shows the time series at locations for the three biggest storms in western IBI area in 2018: Carmen, the 1 January in the Cantabrian Sea, Emma, the 28 February in the Gulf of Cadiz and Ali, the 19 September on the Irish coast. In these storms, current refraction has a limited impact on wave height patterns, with results (IBI-CU and IBI-CO) more similar than the usual state.

**Figure 3.** Typical time series of significant wave height (**a**) and mean period (**b**) at the buoy 6200083 for a period of 2 weeks (5–12 December 2018). The observed values are represented by the black dots. Two model results are shown, one including current coupling (red line, IBI-CU) and the other without currents (blue, IBI-CO). On the bottom, example of the modeled situation at 18:00 UTC 9 December 2018 for the western coast of Galicia for the coupled model, IBI-CU (**d**) and no coupled model, IBI-CO (**c**). Point B14 is the location of the buoy 6200083.

**Figure 4.** Comparison of coupled model IBI-CU, (red line) and no coupled, IBI-CO (blue line) against mooring buoys for different Storms: Ali ((**a**) Hs and (**b**) *Tm02* at buoy 620092 on Irish coast), Carmen ((**c**) *Hs*, (**d**)*Tm02* at the buoy 6200082, Gijon coastal buoy at Cantabrian Sea) and Emma ((**e**) *Hs*, (**f**) *Tm02* at buoy 620085 in Gulf of Cadiz). The observed values are represented by the black dots.

#### *3.2. Evaluation of Data Assimilation Performance: Validation of New IBI Wave Analysis*

Data available for the year 2018 were assimilated to produce the hindcast. Analyses were performed every hour with a data window of 3 h (i.e., data 1.5 h either side of the analysis time were assimilated). A consequence of this arrangemen<sup>t</sup> is that the data are not all independent between analyses; however, in testing, we found that this performed better than insisting on the strict independence of the data. This is because with such a short analysis time step, a narrower window does not always contain enough data for an effective analysis. It also leads to a kind of smoothing of the data in time, which could potentially benefit the analysis because, if the data are too different between analyses, and lead to too big a disparity between observation and model in an ensuing analysis step, the data could be rejected. These are not general principles in data assimilation and depend on the assimilation algorithm used. Because optimal interpolation treats all data assimilated for a given analysis as being observed at the same time as the analysis, the correlation of data between analyses is not a concern, as it would be for a time-dependent assimilation algorithm, such as the Kalman filter.

For comparison, the 2018 model runs were also performed without data assimilation. We refer to the runs with data assimilation and no currents as the IBI-DA runs, and those without as the reference runs (IBI-CO).

Bulk statistics for the entire 2018 study period show that the model is closer to the observations with data assimilated than without. These results are summarized in Table 4. Interestingly, the bias, while shrinking in magnitude, changes from being negative to being positive with the assimilation of data. On the other hand, the scatter indices are reduced by DA in the whole IBI region, particularly in CADIZ subregion, where SI drops by 1.8%. In this area, however, the impact of DA on the model during storm Emma (26 February–7 March 2018) is more like the whole IBI region.


**Table 4.** Biases and scatter indices for control run (IBI-CO) and Data Assimilation simulation (IBI-DA) for the whole IBI area and the CADIZ subregion. In the last case, metrics computed not only for the year 2018, but specifically for the period of the Emma storm (26 February–7 March 2018).

The scatter plots in Figure 5 gather all the observation–model data pairs. They were generated for the entire two-year period and allow us to examine the validation in more fine-grained detail. The IBI-DA run is visually more concentrated about the center of mass, the black line representing a one-to-one correspondence between the observations and model. The colored squares on that axis also appear hotter in color, indicating that more data pairs are concentrated on it. Finally, the linear regression of the IBI-DA run (the red line) has a gradient closer to one than that of the reference run, though again, for very low Hs, the regression line of the reference run is closer to the center of mass.

**Figure 5.** Density scatter plots comparing observed Hs against modelled Hs for all of 2018. The control run (IBI-CO, (**b**)) is on the left, the IBI-DA ((**a**)) run on the right. Each box represents the points found within the range of Hs it covers, and the color scale indicates the number of validated points within the box. The red line is a linear regression, with the black line representing an ideal 1-1 correspondence between model and data.

The quantile–quantile plots in Figure 6 allow us to compare the statistical distributions of the observed and modelled *Hs*, without and with data assimilation. These plots were produced using only the model points that correspond to observations. Assuming reliable data, a straight line of gradient equal to one would imply that the model produces an

identical distribution of *Hs* to the observations. Comparing the plot for the IBI-DA run with that of the control run, we see that data assimilation helps bring the model's Hs distribution closer to that of the observations for most of the range of *Hs*. This is especially true for *Hs* between 6 and 9 m. For low *Hs*, up to around 2 m, the IBI-DA run's representativity is slightly worse, but the difference is small, and these wave heights are of much less interest to seafarers, so the inaccuracy here can be forgiven. At extremely high Hs, the distribution of the IBI-DA run is skewed high. There are not many data in this extreme regime to begin with, and the control run already suggests this by oscillating around the ideal unit gradient. Furthermore, the modelling of extremely high waves is even less reliable, though we should be cautious, given the paucity of data in the regime. One possible explanation for this over-correction could be because the reference model is under-estimating moderately high *Hs*, which are grea<sup>t</sup> in number, so when the analysis corrects these upward, it inadvertently increases the extremely high Hs as well. In other words, the moderately high *Hs*, because of their larger number, are weighting the analysis more than the extreme *Hs*. In the simple OI data assimilation scheme implemented here, where model errors are constant and covariances are defined solely based on the distance between points, there is no way for it to selectively apply the correction, in such a way as to avoid incorrectly increasing these extreme wave heights.

**Figure 6.** Quantile–quantile plots of observed Significant Wave Height (SWH), *Hs*, against modelled *Hs* for all of 2018. On the left is the IBI-CO run (**a**), on the right the IBI-DA run (**b**). The red line is the unit gradient line.

The bar plots for the monthly means in Figure 7 show, briefly, a consistent reduction in scatter index with data assimilated; it is most reduced in December, March and April, and least in January and September. The bar plots for bias reflect the shift from negative overall bias to positive overall bias, but in some instances, the absolute bias is greater with data assimilation—especially so in July, where the control run's bias is already positive. The fact that the data assimilation always results in the bias tending positive suggests that a bias remains in the data. The monthly diagnostics reveals a moderate seasonal signature in scatter index, with higher scatter indices in the summer months, conserved in the IBI-DA run. The seasonal signature for the bias is almost inverted in the DA run, with higher absolute biases in summer than in winter (apart from November and February). With all this said, the highest absolute bias for all months in the IBI-DA run is only just over 5 cm, which, to put it into perspective, is about the same as the bias for the control run for the whole year.

**Figure 7.** Model (IBI-DA and IBI-CO) Bias (**a**) and Scatter Index (**b**) compared with HY-2A altimeter Hs observations.
