The approach used for the analysis of the skills of the data-reconstruction methods is based on the use of a realistic numerical simulation as a true ocean, that provides both the emulated observations and the 3D reference field (hereinafter “reference field”) that will be used to assess the results of the 3D reconstruction. This is a well-established methodology inspired by the techniques used in observing system simulation experiments (OSSEs), and it is the only approach that allows us to quantify the skills of the data-reconstruction methods in the entire 3D domain considered for the reconstruction. The assessment approach consisted of three main steps (Fig. 2). First, the observations that emulate the data obtained from the EuskOOS platforms were extracted from a numerical simulation (for simplicity these “emulated observations” are called “observations” from here on). The extracted simulation data emulate the two vertical current profiles of the ADCPs located in the Matxitxako and Donostia moorings and the surface current fields of the HFR (see locations and coverage in Fig. 1b). Second, the two data-reconstruction methods were applied to the observations to compute the 3D reconstructed fields. Note that the ROOI method also uses historical data from a simulation to estimate the spatial covariances of the currents in the study area needed for the reconstruction. Finally, the 3D reconstructed fields (outputs of the methods) were compared to the reference field to assess the performances of the data-reconstruction methods.

Since the current regime is seasonally modulated, the performances of the two data-reconstruction methods were tested for winter and summer periods: November–December–January–February (2010–2011) and June–July–August–September (2011), respectively. The data-reconstruction methods were also analysed in a reduced grid case to evaluate the performance of the reconstructions in areas where the surface currents are highly correlated with the currents at the mooring locations (hereinafter called “well-sampled areas”). Since the moorings are located along the Spanish slope, where the zonal current velocity component (U) prevails over the meridional component (V), the reduced grid was determined only by the correlations obtained for this component (cross-correlation ≥0.8). This reduced grid mainly covers the Spanish slope area and slightly differs for the winter and summer periods (black and orange lines, respectively, in Fig. 1b).

A second scenario with two additional current vertical profiles along the French slope (see Fig. 1b) was also considered in order to assess the sensitivity of the data-reconstruction methods to different observational configurations (hereinafter called the “4-mooring scenario”).

The Atlantic–Iberian Biscay Irish simulation, and particularly the IBI_REANALYSIS_PHYS_005_002 product (hereinafter IBI), provided by the Copernicus Marine Environment Monitoring Service (CMEMS), was used to obtain the true ocean from which the observations and the reference field were extracted, as explained in Sect. 2.1. The IBI reanalysis is based on a realistic configuration of the NEMO model for the Iberian Biscay Irish region (Fig. 1a), which assimilates in situ and satellite data. For more details, see Table 1; a complete description of the product and its validation can be found in Sotillo et al. (2015) and the links shown in Table 1. In Sect. 3, the realism of IBI simulations is assessed based on previous knowledge of the circulation in the area and used to provide an overview of the dynamical characteristics of the study area to support the discussion of the results.

The spatial covariances required for the ROOI have been obtained from IBI and two additional numerical simulations (see Fig. 2) with daily outputs from 1993 to 2009, with the objective of exploring the impact on the reconstruction of an imperfect definition of the covariances. The two additional numerical simulations used for this purpose were the GLORYS high-resolution (GLOBAL_REANALYSIS_PHY_001_030 product, hereinafter called ”GLORYS-HR”) and the low-resolution (GLOBAL_REANALYSIS_ PHY_ 001_025 product, hereinafter called “GLORYS-LR”) reanalyses. The general details of these products are listed in Table 1, along with links to additional information about the products and their validation. Thus, the ROOI method was tested both in an optimal configuration, where the covariance matrix was obtained from the same numerical simulation used as the reference field (i.e. IBI), and in two suboptimal configurations: one in which the covariances were obtained from a high-resolution numerical simulation (i.e. GLORYS-HR), which is supposed to capture the same range of processes as IBI although not exactly, and another one from a low-resolution numerical simulation (i.e. GLORYS-LR), which differs from IBI in the numerical code and also in the resolvable spatial scales.

The same 3D grid was considered for the reference field, the covariance matrices, and to extract the observations at the surface layer or in the vertical profiles at the grid points closest to the mooring locations (Fig. 1b). The horizontal grid spacing was given by the native horizontal grid of IBI and GLORYS-HR (1/12∘) (Fig. 1b). Thus, for the computation of the covariance matrices with GLORYS-LR, the data were linearly interpolated to the IBI grid points. The vertical configuration was adapted to the levels of the real ADCPs with data every 8 m (from -12 to -148 m). Since the surface layer was set at -0.5 m, all the used numerical simulation fields were linearly interpolated to this vertical configuration (i.e. -0.5, -12, -20, -28, …  -148 m).

The skills of the data-reconstruction methods were assessed by means of the root-mean-square difference (RMSD) between the reconstructed fields (x) and the reference fields (y). The RMSDs were computed at each point of the 3D grid for each study period and for U and V. Thus, for one grid point and N time steps, 7RMSD=∑t=1Nxt-yt2N, where xt and yt are the reconstructed and reference fields at each time step, respectively.

The relative RMSD, relative to the root-mean-square (rms) current (hereinafter RRMSD), was also considered, since the strength and variability of the current are different at different locations of the study area and therefore influence the magnitude of the RMSDs. Therefore, the considered relative value is 8RRMSD=RMSDrms, where 9rms=∑t=1Nyt2N. Since RMSD and RRMSDs were computed for each study period and for each velocity component, hereinafter we use RMSD-U and RRMSD-U as RMSD and RRMSD computed for U and RMSD-V and RRMSD-V as RMSD and RRMSD computed for V. When the RRMSD is equal to 1 at one point for a study period, it means that the RMSD equals the rms of the studied period at that point.

The results, in terms of RMSDs and RRMSDs, are summarized in Table 3. It is observed that the RMSDs and the RRMSDs are affected by the spatial and temporal variability of the slope current regime. The mean RMSDs are, in general, higher in winter than in summer due to more intense currents in that period. However, the rms values are also higher and in relative terms the reconstructions show, overall, better results in winter (lower mean RRMSDs). This dependence of the results on the current regime can be also observed if we compare the reduced and the entire grid cases. For the reduced grid case, that covers an area of intense zonal slope currents, highest mean RMSDs and lowest mean RRMSDs are generally obtained for U. Since V is much weaker for this grid, it provides the lowest mean RMSDs. Nevertheless, the expected increase in the mean RRMSDs is not so clear compared to the entire grid case due to lower rms values.

Regarding the comparison between data-reconstruction methods, for the entire grid case, the mean RRMSD-U values are remarkably higher for the DCT-PLS. Conversely, for the reduced grid case, the results for the RRMSD-U for the DCT-PLS are better. The mean RRMSD-V values do not show any specific trend. This shows that the DCT-PLS performs better in well-sampled areas, whereas the ROOI performs well also out of these areas.

All these results, in addition to more specific analyses, are shown below in terms of RRMSDs by means of maps (Figs. 6–9) and horizontal mean value profiles along the water column (Figs. 10–11). The results of the RMSDs are shown in Sect. S4. For the ROOI RRMSD maps, the results with the spatial covariances from GLORYS-LR are the ones presented in Figs. 6–7, because those are the ones that most challenge the method. In fact, for the ROOI with GLORYS-HR, the RRMSDs are even lower (see Sect. S2), with the main conclusions being very similar.

For the ROOI, the RRMSD spatial distribution is more uniform in summer (Fig. 6) than in winter (Fig. 7) due to the more variable summer current regime. The Spanish slope area shows the lowest RRMSD-Us due to the strong signal of the along-slope current, with lower values in winter than in summer. This suggests that the reconstructed fields are more accurate in well-sampled areas and that U is well resolved in the numerical simulations used for the definition of the spatial covariances. For the RRMSD-V, the French slope and part of its platform show the lowest values in winter, indicating that the slope current is well reconstructed for that period. Since the density of the observations is much higher at the surface, it is expected that the method performs better in the upper layers; in fact, it is observed that the RRMSDs increase with depth. This increase is sharper in summer than in winter, probably due to higher vertical shear in the currents due to the stratification conditions. It is shown that for the ROOI with GLORYS-LR, the RRMSDs are, in general, below 1.25; that is, the RMSD is below 1.25 times the rms value at each point, except for some concrete areas.

Regarding the DCT-PLS, RRMSD maps (Figs. 8–9) show the lowest values near to the surface and the mooring locations, showing that this method works better in well-sampled areas. The RRMSDs are lower in winter (Fig. 9) than in summer (Fig. 8). For the RRMSD-U, the Spanish slope area shows the lowest values for both periods, whereas low RRMSD-Vs are observed over the French slope in winter, showing that this method is also able to reconstruct the slope current. Overall, RRMSDs increase with depth; nevertheless, in summer the RRMSD-Vs are higher for -52 m (Fig. 8d) than for -100 m (Fig. 8f). This could be related to a stronger vertical shear related to the seasonal thermocline, which in this period is located between -30 and -50 m. For the DCT-PLS, the RRMSDs are not as smooth as for the ROOI, with RMSDs near (off) the observation areas lower (higher) than half (twice) the rms value at each point.

Thus, for both methods, lower RRMSDs are observed in winter than in summer, along the slope for the along-slope component of the velocity and close to the surface. While the DCT-PLS is more effective at well-sampled areas, the ROOI performs better in the rest of the areas. In general, the best performances are located in the well-sampled areas (Figs. 3–5), showing that the a priori analysis, shown in Sect. 3, can provide an approximate idea about the areas where the reconstructions could, in principle, perform better.

It is observed that the results for the DCT-PLS worsen quickly as we get away from the observation points. Considering the -52 m depth layer, we observe that RRMSD values obtained with the DCT-PLS method increase from 0 to 0.25 at ∼31 km (6.3 km) for the U(V) component in the zonal (meridional) direction.

A further analysis of the spatial mean of the RRMSDs with depth (Figs. 10–11) is performed to evaluate the methods' skills, regardless of the spatial variability shown in previous figures. Note that the same grid points were considered for both data-reconstruction methods and that the ROOI with IBI, GLORYS-LR, and GLORYS-HR are shown in this analysis. The analysis was carried out in the entire grid and the reduced grid (see Fig. 1b) in order to explore the sensitivity of the results to the choice of different areas.

For the entire grid case (Fig. 10), the ROOI with GLORYS-LR provides similar results as the DCT-PLS for V (Fig. 10b and d), whereas it provides much better results for U (Fig. 10a and c). On the other hand, the ROOI with IBI and GLORYS-HR performs better for both velocity components. In addition, as it could be noticed in Table 3 and in Figs. 6–9, the mean RRMSDs show RMSDs around or less than 1 times the rms value at each point, except for U for the DCT-PLS.

In the reduced grid case (Fig. 11), the lowest mean RRMSD-Us are observed for the DCT-PLS, performing significantly better than the ROOI. In general, the mean RMSDs are around or less than 0.75 times the rms value at each point, with values around or less than 0.5 times the rms value for the DCT-PLS. This provides quite a satisfactory reconstruction of the along-slope velocity component in the Spanish slope area. Thus, if the whole water column is considered, the ROOI provides again smaller RRMSDs than the DCT-PLS for the entire grid case, whereas the DCT-PLS provides better results in well-sampled areas.

With regard to the seasonal analysis, lower RRMSDs are observed in winter (Figs. 10–11). The only exception is the RRMSD-U in the entire grid case for the DCT-PLS (Fig. 10a and c) due to the high RRMSDs over the French shelf and slope for that period (see Sect. S3), since this method expands the zonal component to that area of meridional regime.

Considering all the analysed depths and study periods, satisfactory reconstructions are obtained by both methods. These reconstructions provide mean RMSDs for each depth (Figs. 10–11) ranging from 0.55 (0.7) cm s-1 to 10.94 (9.58) cm s-1 for the entire (reduced) grid and mean RRMSDs ranging from 0.07 (0.12) to 3.47 (1.31) with typical values around 1 or less, i.e. with reconstructed field errors around the rms value or less at each point. In general, the RRMSDs are increased with depth and thus RMSDs up to 10.94 cm s-1 are obtained at -150 m.

An analysis with two additional ADCPs was carried out in order evaluate the sensitivity of the data-reconstruction methods to an increased number of observations. The two extra ADCPs were located over the French slope, since this could be a strategic area to monitor the winter slope current downstream of the Capbreton Canyon.

Only the winter period is shown, when the slope current is the strongest and the effects of the new scenario are more noticeable. Also, we show here the results obtained for the -52 m layer, due to its representativeness of the entire water column. The performance of the data-reconstruction methods for this configuration is assessed by subtracting the RRMSD maps of the 2-mooring case to the 4-mooring case. Therefore, the negative (positive) values in Fig. 12 show that the RRMSD is lower (higher) for the 4-mooring configuration, thus showing a better (worse) performance. In general, in this new scenario the performance of both data-reconstruction methods improves, with smoother changes for the ROOI, since it already uses historical information of the covariances in the whole study area.

For U, the addition of two extra ADCP profiles does not affect the Spanish slope area where there are already two moorings that capture the slope current. In the rest of the grid, for the DCT-PLS (Fig. 12b), the performance of the reconstruction is remarkably improved; whereas, for the ROOI (Fig. 12a), although in general the reconstruction is improved, there are some specific areas where the RRMSD-Us are slightly increased. For V, the results improve along the French slope, which are more remarkable for the DCT-PLS (Fig. 12d). However, for this method, the RRMSD-Vs are increased in the areas close to that slope, probably due to the spread of the information from the slope observations to those nearby areas which are not affected by the slope current regime.