Dobricic et al. (2007) found that the vertical EOFs computed from the covariance between temperature, salinity and sea level could produce corrections that are not geostrophically balanced and proved that the enforcement of the geostrophic relationship for the sea level in the error covariance matrix has a significant positive impact on the accuracy of the analysis. Thus, in OceanVar2 the sea surface height increments from EOF projections are overwritten using a sea level operator. Two different sea level operators, with different levels of complexity, are implemented in OceanVar2.

The first is the commonly used Dynamic Height operator Vη=VDH, which is defined as: 4VDH=δηDH=-1ρ0∫-D0-αδT+βδSdz Where D is a uniform reference level, corresponding to a level-of-no-motion, δT and δS are temperature and salinity increments respectively, α and β the expansion and contraction coefficients. The correlation between Sea Level Anomaly (SLA) from altimetry measurements and the dynamic height anomaly computed from in-situ measurements is high for regions deeper than 1000 m (Dhomps et al., 2011), at synoptic, seasonal and interannual time scales.

OceanVar2 allows the application of a more complex linear Vη. This operator is based on the formulation published in DP08 and derives from the steady state results of a linear barotropic model forced by buoyancy anomalies induced by the temperature and salinity increments. While the methodology was previously described in the literature, the barotropic model code itself was fully re-introduced and parallelized to function efficiently within the new OceanVar2 framework, with its general mathematical structure remaining unchanged. Its inclusion as a fully integrated option is a new feature of this version. The barotropic model equations, discretized in time by the semi-implicit scheme (Kwizak and Robert, 1971), are: 5Un+1-Un-12Δt-fVn=-gH∂η∗∂x-∫-H0∫-z0∂δb∂xdz′dz+γ∇2U∗6Vn+1-Vn-12Δt+fUn=-gH∂η∗∂y-∫-H0∫-z0∂δb∂ydz′dz+γ∇2V∗ ηn+1-ηn-12Δt+∂U∗∂x+∂V∗∂y=0 where U and V are vertically integrated velocity components, f is the Coriolis parameter, g is the acceleration due to gravity, H is the bottom depth, η is the surface elevation, δb is the buoyancy anomaly, and γ is the horizontal viscosity coefficient. The superscripts indicate the time step relative to n, and the superscript “*” indicates the weighted average between two timesteps. A more detailed description of the barotropic model and its discretization can be found in DP08. In the present version the barotropic model assumes closed lateral boundaries.

In the previous Eqs. (5) and (6), the buoyancy forcing term is defined as: δb=gδρρ0 and the density perturbation δρ is estimated by the linear equation of state: 7δρ=αδT-βδS Expansion and contraction coefficients (α and β) in Eqs. (4) and (7) can be assumed space independent or spatially variable and estimated by linearizing the equation of state around a user defined background temperature (Tb) and salinity field (Sb): α=∂ρ∂T|T=Tb,S=Sbβ=∂ρ∂S|T=Tb,S=Sb In the latter case the coefficients are read from an external input file.

In OceanVar2 the sea level operators produce the final sea surface height increments, replacing the increment produced by the cross-covariance between temperature, salinity and sea level provided by the EOFs (DP08, Storto et al., 2018).

The choice on the sea level operator has consequences on the velocity operator Vu,v. Vu,v computes the velocity correction assuming geostrophic balance under the Boussinesq and incompressibility approximations: fugz=-1ρ0∂p∂yfvgz=+1ρ0∂p∂x Decomposing the pressure p at any level z as: pz=patm+gρ0η+∫-z0δρρ0dz where patm is the atmospheric pressure, g is the effective gravity, η is the free surface elevation and δρ is the density departure from a reference state ρ0. Neglecting the atmospheric pressure, and rewriting the hydrostatic term as buoyancy term, the geostrophic velocities become: 8fugz=-g∂η∂y+∫-z0∂δb∂ydz9fvgz=+g∂η∂x+∫-z0∂δb∂xdz When adopting the barotropic model as Vη, the sea surface height in Eqs. (8) and (9) is replaced by the increments deriving from the solution of the barotropic model and the lower limit of the baroclinic integral reaches -H(x,y), the ocean floor. On the other hand, when Vη=VDH the horizontal pressure-gradient force must vanish at the level-of-no-motion, D, in Eq. (4). In Eqs. (8) and (9) η is then substituted with δηDH and the velocity increments are computed only up to depth D. Thus, in case of the barotropic model, velocity corrections are provided for the entire water column, while in case of dynamic height, velocity corrections are provided from the surface to the level-of-no-motion.

The observation error ϵ0 is defined as the difference between the observation vector y and the observation counterpart in the true state xt (Cohn, 1997) y=h∗xt+ϵ0→ϵ0=y-yt-h∗xt-yt=ϵm+ϵr, where yt is the true (unknown) observation value, h∗ is an observation operator, ϵm labels the instrumental or measurement error (distance of the actual value from the true state) and ϵr gathers the different components of the representativeness errors, due to inaccuracies in h∗ and the sampling error of the observations with respect to the true signal. Under the assumption of unbiased error 〈ϵ0〉=0 and that ϵm and ϵr are uncorrelated, the error covariance matrix R can be constructed as the sum of two terms that can be estimated independently: R=〈ϵ0ϵ0〉≈〈ϵmϵm〉+〈ϵrϵr〉=Rm+Rr. If the errors associated to different observations are uncorrelated, the two matrices simplify greatly in diagonal ones. This hypothesis is valid for most of the current global/regional observational datasets and it is generally correct when observations are sampled relatively far in time (say few hours to avoid cross-correlation term in R^m) or are sparse with respect to the model grid resolution (to not include off-diagonal elements in R^r). Observation errors are a function of observation type in OceanVar2.

OceanVar2 contains various procedures for the quality control and preprocessing of observations. A background quality check is included to reject observations that are too far from the model estimate. This quality check uses a threshold on the squared misfit. While OceanVar2 supports approximating the misfit distribution with a Huber norm PDF (Storto, 2016) to mitigate the impact of outliers, this feature was not required for the analysis presented in this work.

To ensure that the assumption of spatially and temporally uncorrelated observation error is satisfied, data thinning and superobbing procedures are employed. Horizontal thinning rejects observations that are too close in space, and in cases where multiple data from the same instrument fall into the same model grid cell, only the observation closest to the analysis time is retained. Along the vertical dimension, a superobbing procedure takes the average of all data falling within the same model layer. Additionally, two important rejection criteria can be activated: coastal rejection prevents the assimilation of altimetric and in-situ coastal observations to avoid inconsistencies between observed and modelled coastal processes, and a bathymetry-based rejection prevents the assimilation of data in shallow areas.

A fundamental aspect to consider when assimilating SLA is the possible presence of tides in the modelled solution and in the observed data. Discrepancies between modelled and observed tides can, as a first approximation, be attributed to inaccuracies in the bathymetry of the model, the bottom and/or the coastal frictional dynamics. If the difference between observed and modelled estimates is due to tides, then this part of the misfit is primarily composed of external gravity waves. In the present OceanVar2 formulation, this high-frequency signal would be incorrectly projected into baroclinic increments by the covariance matrix of the background error (Eq. 3). Since tidal errors are not dynamically linked to the slow baroclinic ocean state, using the standard B matrix leads to spurious analysis increments. It is therefore essential to filter out the tidal signal from both the observed and modelled SLA. This work offers a solution to the assimilation of satellite altimetry in a model with tides, showing that a filtering procedure can be accurate enough and that no additional adjustment is required in the analysis.

The Copernicus along-track sea level anomalies are provided together with an estimate of the tidal signal along the tracks so tides can be filtered easily from the observations. To remove the tidal signal from the model background field, the tidal amplitude and phase for the eight components included in the Mediterranean Sea model (M2, S2, K1, O1, N2, P1, Q1 and K2) have been derived from a simulation output by harmonic analysis of the hourly sea level field. Following Cao et al. (2015), six months of hourly data were used for the harmonic analysis. The harmonic analysis was performed using the Pawlowicz et al. (2002) algorithm, based on the Foreman method (Foreman, 1977, 1978) at each model grid point. Knowing the tidal constants, it is possible to estimate the model tidal sea level at the exact time and location of the altimetry data and remove this component from the model outputs.

During the model simulation, misfits between model estimates and observations are computed and before entering OceanVar2, the misfits are updated, removing the tidal signal from both observations and model results. In Fig. 2 an example of an SLA satellite track is provided with model estimates and the satellite observations, the position of the track is shown in Fig. 1. In the upper panel of Fig. 2, the full signal from the model simulation and the observations is drawn as a function of latitude along the track. The middle panel of Fig. 2 shows the signals without tides, in addition to a debiasing procedure described by Dobricic et al. (2012). Dobricic et al. (2012) show that this method is the best for considering differences between the large-scale steric signal and the mean dynamic topography between observations and model. The average difference along the track is removed if the track is continuous, or for individual segments if the track is discontinuous due to the presence of land. Finally, in the bottom panel of Fig. 2 the two tidal components for the observational and modelled SLA are shown indicating the large-scale signal of tides in the open ocean.

In addition to a free-run (non-assimilative model simulation), we present results from two sets of six 1-year assimilative experiments each, comparing them against each other and against observations. All the experiments assimilate ARGO floats and XBT temperature and salinity data in the whole domain including the Atlantic part, while SLA data are assimilated only within the Mediterranean Basin (see Fig. 1). In every experiment the vertical component of the background error covariance matrix is modelled using 25 tri-variate EOFs (temperature, salinity and SLA) computed following Dobricic et al. (2006) for every model grid point. The EOFs are computed from a 30-year timeseries of the Mediterranean Sea reanalysis (Escudier et al., 2021). The horizontal correlation radius was set to a constant value of 27 km, determined through sensitivity experiments. The diffusive filter was iterated five times to model the horizontal covariance. To account for coastal effects, the correlation radius was linearly decreased starting from about 30 km offshore to the minimum grid resolution near the coast. Additionally, a Neumann boundary condition was applied at the coast, setting the normal derivative of the field to zero. Observations are rejected if they are less than 15 km from the coast and if the misfits are larger than fixed thresholds: 5 °C for temperature, 2 psu for salinity; and 30 cm for SLA. The observational error covariance matrix is assumed diagonal. The results presented in Fig. 2 demonstrate that our model accurately reproduces along-track SLA tidal gradients, with the difference between the modelled and observed tidal signal being nearly constant. We effectively remove this along-track bias by subtracting the mean residual along each satellite track. This critical step ensures that the resulting misfit is very similar to the one computed without tides. Based on these considerations, we decided not to modify the SLA observational error, maintaining consistency with previous work (Escudier et al., 2021). Consequently, all the SLA data have an associated error of 3 cm regardless of the satellite and the geographic distribution. The observational errors for in-situ observations were tuned via the Desroziers' method (Desroziers et al., 2005) and vary monthly. We first prescribed the observation errors used in the previous system. The assimilation system was then run repeatedly, using the innovations and residuals to apply the Desroziers' formula to obtain an estimate for the observation error R. This process was iterated until the errors converged (e.g., Escudier et al., 2021). The resulting vertical error profiles are as follows: temperature and salinity observational errors peak at the surface with values of 0.45 °C and 0.14 psu, from 75 to 325 m depth they decrease linearly to values of 0.2 °C and 0.05 psu, starting from 750 m they have constant values of 0.1 °C and 0.02 psu respectively.

In addition to these sensitivity experiments, we also performed a set of idealized tests with synthetic observations to verify the internal consistency of the assimilation system and assess the behaviour of misfits and residuals under controlled conditions. A detailed description of these verification experiments is provided in the Supplement.

In all the experiments, the state vector contains the following model state variables: x=T,S,u,v,ηT where T is the three-dimensional temperature field, S the three-dimensional salinity field, u and v are the total horizontal velocity components and η the two-dimensional sea surface height.

Two sets of experiments are performed to compare the operational stability of mass-field-only T,S,η correction against the maximal impact of full-state variable T,S,u,v,η correction. Each of the two sets consists of six individual experiments (Exp-1 through Exp-6) with different OceanVar2 sea level operators and choices of free parameters. In Exp-1, which is used as a reference experiment, the barotropic model is used as the sea level operator with constant (in space and time) α and β in Eq. (7) and SLA data are rejected when falling in areas shallower than 100 m. In the second experiment (Exp-2), consistently with Adani et al. (2011), we rejected SLA data falling in areas shallower than 150 m. Experiment 3 is similar to Exp-2, but we test the sensitivity to variable expansion and contraction coefficients in Eq. (7). The coefficients are computed by linearizing the equation of state around a monthly mean climatology. In all these three experiments we integrated the barotropic model for 3 d with a time-step of 3600 s and then used the average of the last day as an approximation of the steady state solution. The integration of the barotropic model is fully implicit and the turbulent viscosity is equal to 650 m² s⁻¹.

Experiments 4, 5 and 6 use the dynamic height as the sea level operator. The difference between them is the choice of the level-of-no-motion. In Exp-4 we used a level-of-no-motion equal to 150 m, thus Exp-2 and Exp-4 differ only for the sea level operator. This shallow depth was specifically chosen to test the upper limit of observation inclusion, ensuring that the assimilation scheme uses the maximum possible Sea Level Anomaly (SLA) data coverage, even in areas shallower than the generally accepted level-of-no-motion. In Exp-5 the level-of-no-motion is 350 m in agreement with the Mean Dynamic Topography (Rio et al., 2014) used. Finally, in Exp-6 the level-of-no-motion is set at 1000 m which is the traditional choice for the operational setting of the Mediterranean Sea forecasting system (Coppini et al. 2023). In all these experiments, the depth of the level-of-no-motion naturally coincides with the minimum depth of SLA observations inclusion in the data assimilation scheme.

The configuration of the six individual experiments (Exp-1 to Exp-6) is identical across both sets. The two sets of experiments are designed to test the critical operational trade-off between numerical stability and maximal observational impact. In the first set of experiments, we prioritize stability by applying the analysis increments only to the assimilated variables (T, S and η), even though the state vector contains the velocity field. This forces the model dynamics to adjust the velocities indirectly. In the second set of experiments, which tests the maximal assimilation scope, also velocity corrections are applied directly in the analysis step. Throughout the remainder of the manuscript, experiments marked with an asterisk refer to those with velocity corrections. The first sets of experiments are summarized in Table 1. The second set of experiments has the same naming convention as the first set, with the sole difference being the application of the velocity correction in the analysis definition.

To fully assess the performances of the experiments listed in Table 1, the mean squared error (e.g., Murphy, 1988) is decomposed and the single components are analysed: 10MB=m‾-o‾,11SDE=σm-σo,12CC=1σo1σm1N∑i=1Nmi-m‾oi-o‾, where MB is the mean bias error, SDE is the standard deviation error and CC is the cross correlation between the modelled and observed fields. The ith modelled and observed variable is denoted by mi and oi, respectively; m‾ and o‾ are the respective averages (horizontal and temporal); while σm and σo are the respective standard deviations. In addition, the unbiased root mean squared error (uRMSE) is computed: 13uRMSE=1N∑i=1Nmi-m‾-oi-o‾2. It is important to note that the model results and observations used here are the same as those used to calculate misfits within the assimilation cycle. However, while not all misfits are utilized in the assimilation process, all available observations are included in the error statistics. This ensures that all experiments are evaluated based on the same reference dataset of observations. Furthermore, to evaluate model performance even in very shallow regions, the observational dataset used in the misfits, and thus in the calculation of the error statistics, includes sea level anomaly (SLA) data covering regions up to 10 m depth. This allows for the assessment of model skill in very shallow regions where data are not assimilated in any of the presented experiments.

Given that the CC is always positive for all the experiments, the misfit statistics for the experiments listed in Table 1 are also analysed in terms of relative improvement with respect to the free-run or Exp-1 according to the following metrics definition: 14SuRMSEExp#=uRMSERef-uRMSEExp#uRMSERef×100,15SCCExp#=CCExp#-CCRefCCRef×100, where # indicates the different experiments listed in Table 1 and “Ref” denotes the free-run or Exp-1. SCC is only calculated using Exp-1 as the reference. The exclusion of the free-run as a reference is necessary because the very small CC values would lead to excessively large percentage values that hinder meaningful comparison between the different experiments. In the manuscript the relative performance statistics are presented only for the SLA data given that a similar comparison for the temperature and salinity did not provide additional insights.

The performance of Exp-1 is compared with free-run. In Fig. 4 the statistics for the SLA are shown. Every point in the time-series represents 5 d window statistics. That is, the overbars and the standard deviations in Eqs. (10), (11), (12) and (13) are computed over a 5 d time window.

The free-run has an error of about 5 cm, slightly growing during the second half of the year. The model with assimilation underwent a 10–20 d adaptation period, after which the uRMSE of the misfit stabilizes around 3 cm. A consistent improvement is noted in the CC. No seasonal cycle is observed in the CC of Exp-1, whereas the free-run exhibits a distinct summer minimum in the correlations. The SDE in the free-run is on average negative and it is characterized by 5 d oscillations. In Exp-1, the SDE stabilized around values of 0.25 cm, indicating an overestimation of the observed variability.

The SLA yearly averaged statistics were clustered according to ocean bathymetry and are plotted in Fig. 5. In areas with bathymetry between 150 and 2500 m, the free-run exhibited an almost constant uRMSE. However, the error increases in shallower and deeper regions, reaching the maximum in areas deeper than 3500 m. The uRMSE of the Exp-1 was more stable and approximately half of the corresponding free-run statistics. Regardless of the region considered, Exp-1 has better statistics than the free-run, indicating the effectiveness of the assimilation procedure. For the CC differences between free-run and Exp-1 results are also evident. In the free-run CC decreases with depth, while for the Exp-1 we observe an opposite tendency. In terms of SDE, the largest improvements with respect to the free-run are in shallow areas. The free-run underestimates the variance in areas with bathymetry shallower than 1000 m, this is particularly evident in areas shallower than 150 m. The model with data assimilation tends to overestimate the observed variability particularly in shallow areas.

In Fig. 6 the vertical profiles of uRMSE, MB, CC and SDE for salinity (upper panels) and temperature (bottom panels) for the free-run and the Exp-1 are shown. Statistics are computed against all available ARGO and XBT profiles. The vertical structure of the salinity uRMSE is similar between the free-run and the Exp-1. The salinity errors are characterized by a near surface maximum which is reduced in the assimilative run. The MB in the free-run has a subsurface minimum at 100 m depth, while Exp-1 misfits have almost homogenous values through all the water column. The CC resembles the vertical distribution of the uRMSE, with values approaching the unity in both the experiments below 500 m depth. Finally the salinity SDE confirms the large improvement arising from the assimilation procedure. In the near surface layer, the free and the assimilative run have opposite behaviours, with the free-run overestimating the observed variability while the assimilative run underestimating it. Below 100 m depth the SDE salinity values of Exp-1 are noticeably reduced with respect to the free-run.

The temperature uRMSE and CC are characterized by a strong, summer intensified (not shown), subsurface maximum/minimum due to the model difficulties in reproducing the correct stratification. A second uRMSE local maximum (CC minimum) is present around 300 m probably related to the misrepresentation of the Levantine Intermediate Water (LIW) advection in the different Mediterranean regions. A third temperature error relative maximum is present between 1000 and 1500 m depth. The assimilation corrects all the errors by approximately 30 %–50 % down to 500 m, less below this depth.

Temperature MB is largely improved by assimilation. The free-run tends to overestimate the observed temperature variability, and the SDE has a marked vertical structure. In general, the assimilation seems capable of correcting most of the model errors except in the upper thermocline/mixed layer depth. Analysis of the corresponding time-series (not shown) indicates a clear summer maximum in all the error statistics in proximity of the mixed layer depth. This behaviour is shared between all the different experiments, but it is clearly reduced in the assimilative runs. However, the persistence of this error maximum suggests a limitation in the current formulation of the Background Error Covariance. Specifically, the static, climatological nature of the EOFs used to model the vertical error component struggles to fully capture the rapidly evolving stratification and strong vertical gradients characteristic of the summer mixed layer. Future development will prioritize replacing these EOFs with more dynamic and stratification-aware operators to address this deficiency.

Figure 7 illustrates the performance of the six different data assimilation experiments (without velocity corrections) in terms of correlation coefficient (CC) and improvements in root-mean-square error (SuRMSE) relative to the model free-run. The top panels display the time evolution of these improvements, highlighting both short-term fluctuations and overall trends. The bottom panels present the time-averaged SuRMSE and CC clustered by bathymetric depth ranges, revealing how the effectiveness of each experiment varies with depth. All assimilative experiments outperformed the model free-run. Experiments using the dynamic height as the sea level operator, with levels of no motion set at 150 m (Exp-4) or 1000 m (Exp-6), generally performed worse than the other experiments. For Exp-6, the differences in both uRMSE and CC were particularly noticeable in regions shallower than 1000 m, where SLA data were not assimilated. However, even in these regions, Exp-6 substantially outperformed the model free-run, suggesting that corrections applied in deeper areas effectively propagated into shallower regions. In Exp-4, the deterioration in results compared to other experiments was primarily confined to deeper regions. In areas shallower than 150 m Exp-1 outperforms the other experiments, however the performances of the experiments are similar indicating that the coastal areas are strongly constrained by the open ocean dynamics. A clear dependence of the CC on model bathymetry was evident in all experiments with the most significant improvements (with respect to the free-run) observed in areas deeper than 3500 m, where the model free-run exhibited the smallest CC. In shallow regions, Exp-6 generally provided the smallest improvement in CC compared to the other experiments. The results demonstrate that certain experiments achieve substantial improvements in deep-ocean regions, while others show more consistent performance across all depths. These results highlight the challenges associated with choosing an appropriate level-of-no-motion in data assimilation of SLA. The requirement to define a spatially constant level-of-no-motion, which is inherent to the dynamic height operator, demonstrates the critical limitations imposed by its theoretical assumptions, and our experimental results explicitly highlight these deficiencies. This constraint imposes a critical methodological compromise. For instance, maintaining a conservative, deep level-of-no-motion (e.g., 1000 m in Exp-6) ensures physical consistency but mandates the exclusion of a large fraction of the available SLA observations. Conversely, adopting a shallower level-of-no-motion (e.g., 150 m in Exp-4) maximizes observation coverage, yet the resulting analysis is compromised due to the violation of the theoretical assumptions in deeper water, ultimately leading to degraded accuracy. The choice of the level-of-no-motion can significantly impact the accuracy of model analysis, especially in complex regions with varying bathymetry and ocean dynamics, thereby confirming the methodological advantage of the barotropic model operator which successfully assimilates all available SLA data without imposing a constant level-of-no-motion constraint.

Figure 8 presents the relative performance of five data assimilation experiments (Exp-2 through Exp-6) compared to a baseline assimilative experiment (Exp-1), now used as the reference. The top panels illustrate the time series of the relative change in root-mean-square error (SuRMSE) and correlation coefficient (SCC). The bottom panels depict the time-averaged SuRMSE and SCC changes, categorized by bathymetric depth ranges. In contrast to the previous figure, where improvements were relative to a model free-run, this figure demonstrates the relative performance of each experiment against the initial data assimilation run. Negative values indicate a decrease in performance (higher uRMSE or lower CC) compared to Exp-1, while positive values indicate improvement. This comparison highlights the incremental benefits or drawbacks of different experimental configurations in relation to a specific data assimilation configuration.

All the experimental configurations employing the barotropic model perform similarly to each other. In terms of time-series comparison, Exp-4 (with dynamic height as sea level operator and level-of-no-motion equal to 150 m) has performance worse than all the other experiments. Among the experiments using the dynamic height operator, Exp-5 generally has better results both in terms of uRMSE and CC. The analysis per bathymetric class highlights the differences among the experiments. None of the experiments outperform Exp-1 in regions shallower than 150 m both in terms of uRMSE and CC. In deeper areas we see that Exp-4 and Exp-5 produce, in general, worse results, and the worsening is amplified as the depth increases and the level-of-no-motion decreases; 1000 m depth is a clear boundary for the effectiveness of Exp-6. Employing the barotropic model as a sea level operator yields consistent results, with small sensitivity to the minimum depth used in the rejection criterion or to the choice of constant or variable expansion/contraction coefficients. This confirms the difficulty of establishing a constant level-of-no-motion and highlights the benefit of using the barotropic model as a balancing mechanism.

The second set of experiments (Exp-1^* to Exp-6^*) was carried out including velocity corrections in the analysis estimates. This second set represents the maximal scope of multivariate assimilation, providing a benchmark against the mass-field-only correction of the first set. The OceanVar2 configuration used in Exp-4^* generated velocity increments that led to numerical instabilities in the ocean model, preventing this simulation from completing. This critical instability confirms the operational risk associated with introducing direct momentum corrections and highlights the sensitivity of Exp-4. In contrast, the other experiments in this set ran without such issues, underscoring the challenges associated with dynamic height methods. Consequently, our focus is on the stable experiments. Across all the stable configurations, the successful experiments in the second set, which included velocity corrections, demonstrated improved performance compared to their counterparts in the first set, without velocity corrections. The extent of improvement varies depending on the specific experiment and the region analysed.

Figure 9 shows the time series and Fig. 10 presents temporally averaged statistics by bathymetric class for the second set of experiments. In terms of time series (Fig. 9), the error components exhibit the same characteristics as those discussed for Exp-1 (Figs. 4 and 5). The uRMSE exhibits a summer minimum in all experiments. Exp-6^∗ performs significantly worse than the others throughout the year. All experiments using the barotropic model have similar uRMSE values, with Exp-1^∗ generally appearing slightly better than the others. The correlation coefficient (CC) increases throughout the experiment's length. For this statistic as well, Exp-6^∗ is the worst, showing consistently lower values than the other experiments. Even for the SDE, which is generally reduced compared to the previously studied experiments without velocity corrections, the relative performance of the different experiments is confirmed.

Figure 10 presents the temporally averaged statistics clustered according to the bathymetry. All the stable experiments benefited greatly from the inclusion of the velocity corrections. The overall reduction in error confirms the importance of the velocity increments when the system remains stable. Exp-6^∗ confirms its poor performance in areas shallower than 1500 m. However, by also correcting the velocities, its statistics in deep areas are now similar or slightly better than those obtained from experiments using the barotropic model as operator in the background error covariance matrix. Exp-1^∗ is now the best among those analysed for all the bathymetric classes shallower than 500 m.

In terms of correlation coefficient, the results previously obtained by analysing uRMSE seem to be confirmed. For shallow areas (<1000 m), Exp-6^∗ is significantly worse than the others. In all other bathymetric classes, while confirming the previous findings, the differences between the experiments are less pronounced. A different behaviour is observed when analysing the standard deviation of the error. Exp-1^∗ remains the configuration that shows significantly lower error values than the others in almost all bathymetric classes. Notably, Exp-6^* is the one that benefits the most from velocity corrections, demonstrating that for this specific configuration, the direct momentum increments overcome other methodological weaknesses.