The ROMS 4D-Var Data Assimilation system implemented for this study (version 4.3), builds upon the configuration developed by , with modifications applied to the data assimilation framework, expanding the amount and type of assimilated observations (see Sect. ) and by extending the analysis to a whole year. Specifically, the same numerical grid (Fig. a, 1/36° resolution with 40 terrain-following sigma layers) is used and the following model parameters are adopted for the characterization of the vertical discretization of the grid: Vtransform=2, Vstretching=4, θs=7, θb=2, and for the horizontal eddy viscosity ν=5 m2s-1, and diffusivity κT=1 m2s-1. The k–kl GLS (Generic Length Scale) scheme for turbulence closure parameters is adopted . Since the tidal signal in the area is of the order of centimeters, and considering that we use daily averaged values as boundary conditions, we did not take into account tides in the modelling system.

Initial and boundary conditions are derived from the Mediterranean Sea Physics Reanalysis provided by the Copernicus Marine Environment Monitoring Service (CMEMS), at 1/24° resolution and daily frequency. These are imposed at the southern open boundary. Radiation boundary conditions are applied at all depths to velocity components (u, v), temperature (T) and salinity(S), with a one-day nudging timescale for inflow, three times larger than outflow . A Flather condition is used for barotropic velocities, a Chapman condition for the free surface η, and a zero gradient condition for total kinetic energy (TKE). Atmospheric forcing is provided at hourly frequency from the ERA5 reanalysis dataset , including sea-level pressure, 2 m air temperature and relative humidity, precipitation rate, surface wind components, and downward short- and long-wave radiations. These variables are used to compute air-sea fluxes following the bulk formulation of . Daily river discharge data for the Rhone and Arno rivers were obtained from the Hydro Portal of France Waters (https://www.hydro.eaufrance.fr , last access: November 2023), and the Tuscany Region SIR Dataset (https://www.sir.toscana.it, last access: November 2023), respectively. The nonlinear model without assimilation (freerun, FR) was run from 2019 to 2022, after a three-month spin-up starting in late 2018. The DA experiment spans the period January–December 2022, starting from an initial first-guess provided by the free nonlinear model run on 1 January 2022 at 00:00 LT.

The ROMS state vector comprising all ocean gridpoint values of temperature, salinity, horizontal velocity and free surface height will be denoted by x.

Since we adopt a strong constraint approach, the control vector z includes initial condition, open boundary conditions and surface fluxes but no model error is explicitly considered . The analysis vector za in 4D-Var DA can then be expressed by the following equation: 1za=zb+BGT(GBGT+R)-1d=zb+Kd where K is the so called Kalman gain matrix, zb is the background control vector, B is the background error covariance matrix (incorporating background errors associated with initial and boundary conditions and atmospheric forcing), G is the tangent linear model sampled at observation locations and GT its adjoint. R is the observation error covariance matrix, and d=y-H(xb) is the innovation vector, with y the observation vector, xb the background state vector, and H the observation operator, which maps from the state space to the observation space and includes the nonlinear model in 4D-Var DA. The matrix P=GBGT+R is the stabilized representer matrix describing the total error covariance in observation space .

In practice, the analysis is obtained by minimizing a cost function following the incremental approach of , as implemented in ROMS by . Sequential linearizations of the cost function around a nonlinear solution (an “outer loop”) similar to a Gauss–Newton method, give rise to a series of quadratic problems which are minimized using the Lanczos formulation of the Restricted B-Preconditioned Conjugate Gradient algorithm (RPCG), implemented as a sequence of “inner loops”, as described by . The algorithm factorizes the R-preconditioned stabilized representer matrix P̃=R-1GBGT+I, yielding an approximation of the Kalman gain matrix of the form: 2K≈BGTVmTm-1VmTGBGTR-1 where Vm is the matrix of Lanczos vectors, Tm is a tridiagonal matrix, and m is the number of inner loops.

In this study, we use a single outer loop and 9 inner loops trying to find a compromise between the available computational power, the time required to run the experiments and the reduction of the cost function. The analysis/forecast sequence is characterized by a 3 d long assimilation window followed by a 3 d forecast, resulting in 122 windows over the year 2022. No overlapping between analysis (AN) is performed and each analysis uses the forecast (FC) from the previous window as background; hence, in the following, the terms “forecast” and “background” are used interchangeably. The choice of 2022 is motivated by the broader availability of HFR observations during this year.

The background error covariance matrix is modeled following . Climatological variances of model state variables, computed from the free run over 2019–2022, are used as proxies for background error variances. To avoid too large standard deviation values for temperature and heat flux, we removed from the original time series the moving average at quarterly timescale, and then the moving average at daily timescale for each cell of the domain. The background error covariance matrix is factorized as B=ΣCΣ where Σ is a diagonal matrix of standard deviations, and C a univariate correlation matrix with a specified decorrelation length, modeled with a diffusion operator. Horizontal and vertical decorrelation length scales were set to 15 km and 10 m, respectively, for all state variables associated with initial and boundary conditions, and to 70 km for surface fluxes. Although B is assumed univariate, flow-dependent cross-covariances emerge during the assimilation window via GBGT in Eq. (). For practical reasons, the observation error covariance matrix R is considered diagonal. Error values for each observation type are discussed in Sect. .

Several observational datasets were assimilated into the model, including surface currents from the Tyrrenian-Ligurian HFR network, in situ velocities, temperature and salinity measurements from various platforms, and satelite-derived SST and SLA.

HFR total surface currents u and v from the European HFR Node (https://www.hfrnode.eu, last access: February 2024) cover parts of the Ligurian Sea (Fig. a) being JERICO (Joint panEuropean Research Infrastructure for Coastal Observatories, https://www.jerico-ri.eu, last access: February 2024) facilities. These data are provided at 2.0 km resolution and hourly frequency for 2022. Before assimilation, a 3 h low-pass filter was applied at each location to the u and v time series to remove high-frequency variability. We did not perform a specific procedure to remove the tidal signal since we assumed the tidally induced velocities to be negligible. Only observations flagged as “good” with a geometric dilution of precision (GDOP) larger than 2 were retained . Furthermore, at each time step, outliers-defined as velocities above the 99.5 %-ile-were excluded. To reduce the HFR data volume and redundancy (Fig. d), the u and v velocities were averaged into 6 km bins and sub-sampled every 3 h. Visual inspection confirmed that the spatially averaged fields retained key patterns. Despite this thinning, HFR data still comprised nearly half of all assimilated observations (Fig. d). The HFR observation error standard deviation was set to 0.1 ms-1 .

In situ velocities were also obtained from the Corsica Channel Mooring (CoCM, Fig. a) , which is also a JERICO facility and includes an upward-facing acoustic doppler current profiler (ADCP) deployed at 450 m depth. The instrument recorded data at 16 m vertical intervals (from 379 m upward) every 2 h. Observations with a multibeam-derived “indicative error” (GLGOPV) outside the [-0.1,0.1] ms-1 range-mostly from depths shallower than 50 m – were excluded due to unrealistic signal shifts. A 6 h moving average was applied to the time series before assimilation, and the observation error standard deviation was set to 0.05 ms-1.

In situ temperature and salinity data were obtained from CMEMS and the European Multidisciplinary Seafloor and water column Observatory (EMSO-ERIC https://emso.eu, last access: June 2024). CMEMS data (product INSITU_GLO_PHY_TS_DISCRETE_MY_013_001) include time series and profile datasets. They are part of the Global Ocean CORA In situ Observations , collected mainly by Argo Floats, XBT, CTD, XCTD and moorings, and we refer to them as TCORA and SCORA to indicate temperature and salinity data, respectively, without making the same distinction unless otherwise noted. Only data with all quality flags equal to “good” were used. EMSO data are temperature (TEMSO) and salinity (SEMSO) time series from four different mooring platforms: Lion (LI) in the Gulf of Lion (https://www.seanoe.org/data/00333/44411, last access: June 2024, ), Albatross (LO) near Toulon (https://www.seanoe.org/data/00720/83244, last access: June 2024, ), Dyfamed (DY) in the western Ligurian Sea (https://www.seanoe.org/data/00326/43749, last access: June 2024, ), and W1M3A (W1) in the central Ligurian Sea (http://www.w1m3a.cnr.it, last access: June 2024). Locations are reported in Fig. a. Observation error standard deviations were set to 0.2 °C for temperature and 0.075 for salinity.

Satellite-derived SST was obtained from the CMEMS Mediterranean Sea High Resolution L3S Sea Surface Temperature product (SST_MED_PHY_L3S_MY_010_042) with a spatial resolution of 1/20° . The assigned observation error standard deviation was 0.4 °C.

Along track SLA data were sourced from the CMEMS SEALEVEL_EUR_PHY_L3_MY_008_061 product, which includes data from multiple altimeter missions at 7 km resolution. The assimilated quantity was the absolute dynamic topography (ADT), computed as a sum of the SLA and the mean dynamic topography (MDT). A bias correction was applied by subtracting the temporal mean of ADT and adding the MDT from the 2019–2022 ROMS free run, interpolated along the satellite tracks. To better constrain the free surface, each SLA observation was repeated one hour before and after its timestamp, following the assumption of slowly varying sea level signals as and .

Super-observations were created by averaging data within the same model grid cell (horizontally and vertically) and time window (hourly), as described by . A background quality control procedure was applied following , rejecting observations when di2>β2(σb2+σo2), where di is the ith component of the innovation, σb2 is the background error at observation location, σo2 is the observation error variance and β is a coefficient that depends on the type of observation. Here β was assigned a value 5 for all observation types meaning that observations with innovations that exceed five times the square root of the expected total error variance are rejected.

The spatial density of assimilated observations is reported in Fig. c, while the distribution with depth and typology is reported in Fig. b and d, respectively. The amount of observations per assimilation window throughout the year is reported in Fig. e. In total, approximately 4.5×106 observations were assimilated during 2022, with a mean of 37.12×103 observations per assimilation cycle, a maximum of 49.15×103, a minimum of 16.29×103 and a standard deviation equal to 5.82×103. Further temporal and depth-dependent distributions by data source are presented in Figs.  and .

The properties of the adjoint model allow quantification of the impact that each assimilated observation has on a specific scalar metric I(z), following the approach described by . Specifically, the increment in the metric ΔI(z)=I(za)-I(zb) due to DA can be approximated as: 3ΔI(z)≈dTKTMbT(t)∂I/∂z where Mb(t) is the tangent linear model computed around the background solution xb. In this study, we consider the time-averaged volume transport integrated over a transect, expressed as: I(z)=1Nt∑n=1NthnTv⟂,n=Tr‾ where Nt is the total number of time steps, v⟂,n is the velocity component orthogonal to the transect, and hn is a vector comprising the contributions to cross-sectional area at each time step n. After replacing K by the approximation in Eq. (), the impact ΔI(z) on transport becomes: 4ΔI(x)≈1NdTR-1GBGTVmTm-1VmTGB∑n=1NtMbT(tn)hn.

The contribution of a single observation yi is readily computed from the dot-product of the ith component of the innovation d and the ith component of the vector g=KTMbT(t)∂I/∂z. Therefore, the averaged increment for an assimilation window is ΔI(z)=∑i=1Nobsdi⋅gi=ΔTr‾. Moreover, the total observation impact can be decomposed according to the control vector to the 4D-Var increments as ΔI(z)=dT⋅(gx+gf+gb)=∑i=1Nobsdi⋅(gx,i+gf,i+gb,i) where gx, gf and gb represent the contributions to g from the initial conditions (IC), atmospheric forcing (AF) and open boundary conditions (BC), respectively .

In the present work, we focus on the volume transport across the Corsica Channel, evaluated along the transect between (9.4250°E, 43.0204°N) and (9.8369°E, 43.0204°N) reported in Fig. a. We quantify the contribution of the different observation platforms to the resulting transport increments and their respective influence on each component of the 4D-Var control vector.

A first indication of the effectiveness of the assimilation procedure in fitting the model to observations is provided by the behavior of the cost function J . Figure a shows the initial and final values of J, along with the ratio of the final to initial observation-related cost component, Jo=12[y-H(x)]TR-1[y-H(x)]. Final values of J are consistently lower than the initial ones, with the ratio Jo,fin/Jo,ini ranging between 0.2–0.6.

Insight into potential overfitting to the observations is provided by analyzing the eigenvalues of the R-preconditioned stabilized representer matrix P̃ . The majority of the increment δz is typically associated with the smallest eigenvalues, which may introduce spurious noise and degrade the solution. Therefore, a useful rule-of-thumb to prevent overfitting is to retain, for each assimilation cycle, only the contributions from eigenvalues λ satisfying λ/λmax>0.01 . Since the eigenvalues of P̃ match those of the tridiagonal matrix Tm, they can be directly computed as a diagnostic during each 4D-Var cycle. Figure b shows that the number of inner loops used does not lead to significant overfitting, with at most only the final eigenvalue falling below the 0.01λmax threshold.

Model performance in reproducing observations is shown in Figs.  and  for the FR, FC and AN runs. Figure  shows the time evolution of the root mean square error (RMSE), computed over individual assimilation windows, for spatially distributed variables: surface velocities from HFR, uHFR and vHFR, CORA in situ temperature and salinity TCORA and SCORA, and satellite-derived SST and SLA. RMSE is calculated as 1Nobsw∑i=1Nobswyi-yim, where Nobsw is the number of observations for a specific observed variable during an assimilation window, yi is the ith observation value and yim=H(x) the model counterpart in observation space. The number of observations per window is also shown as a gray bar on the secondary y axis. The assimilation improves both FC and AN runs with respect to the FR across all variables (Fig. g). For surface currents measured by the HFR, a difference between the three runs, with the FC falling between FR and AN is clearly observable (Fig. a and b). The weighted average RMSE reduction compared to FR is between 10.1 % (u) and 17.7 % (v) for the forecast (ΔRMSE*‾FC) and between 33.7 % (u) and 40.8 % (v) for the analysis (ΔRMSE*‾AN) (Table ). CORA in-situ observations of T and S (Fig. c and d) exhibit periods where FC and AN are similar (especially for salinity Fig. d) and with a quite large variability in the number of observations, which decreases in the second half of the year. The RMSE reduction for FC and AN in temperature, ΔRMSE*‾FC and ΔRMSE*‾AN for TCORA are equal to 25.6 % and 46.7 %, respectively. A similar result holds for SCORA (Table ). SST and TCORA show a notable RMSE increase for the FR, and less so for the FC, starting in mid-May, while RMSEAN tends to remain more stable (Fig. c and e). Reduction in error ranges from 21.2 % to 52.5 % for SST and from 25.6 % to 46.7 % for TCORA, for FC and AN, respectively (Table ). For the SLA the improvement due to the DA procedure is evident throughout the year for the AN, whereas for the FC it is more pronounced from the second half of June (Fig. f). Indeed, the error reduction is 8.3 % for the forecast and 33.6 % for the analysis (Table ).

For observations collected by fixed platforms (those from the JERICO CoCM and the EMSO platforms), the vertical distribution of RMSE is shown in Fig.  (panels a to d). Also in this case, the data assimilation procedure leads to a general improvement in the model skills for both the AN and FC fields across all observed variables (Fig. e). For the current velocities observed by the CoCM, the RMSE reduction is more pronounced for the meridional (v) component than for the zonal (u) component, with the v velocity reaching a maximum ΔRMSE*‾AN reduction of 62.4 % (Table  and Fig. a and b). In situ temperature and salinity observations from the EMSO platforms show an uneven vertical distribution (Fig. c and d), and a variable number of observations across platforms (Fig. f). The RMSE reduction for both FC and AN is more evident above 450 m depth, while differences between the three simulations become less significant below this depth, down to 2250 m (Fig. c and d). The ΔRMSE*‾ is larger for the SEMSO than for the TEMSO (Table ).

The improvement obtained through DA was also evaluated using the Pearson's correlation coefficient, r=∑i=1Nobsw[yi-avg(y)][yim-avg(ym)]std(y)⋅std(ym), and the global mean difference (BIAS). The change in global BIAS (ΔBIAS‾) and in mean correlation (Δr‾) between the different model runs are reported in Table , further confirming the overall positive impact of data assimilation on model performance.

The correlation between model output and observations consistently increases following the DA procedure, as indicated by positive values of Δr‾ (Table ). In contrast, the global BIAS shows a slight increase in some cases relative to the FR. However, BIAS values remain generally smaller than the standard deviation of the observed variables, regardless of wether model performance improves or slightly deteriorates. For instance, the increase in ΔBIAS‾ for uCoCM ranges between 0.003–0.004 ms-1, compare to the standard deviation of 0.037 ms-1 in the observations. Similarly, for SLA, ΔBIAS‾ varies between 0.001–0.015 m, while the standard deviation of the observed SLA is 0.053 m.

Figure  compares the observed and modeled northward velocity component of the CoCM time series at three different depths to evaluate DA performance in reproducing transport through the Corsica Channel. Figure a–c shows better agreement in AN than FR. Winter velocity peaks are generally captured, albeit the FR shows surface overestimation and bottom underestimation in late 2022. Annual mean profiles of northward velocity across the channel (Fig. d and e) confirm these trends: FR overestimates velocity, while AN aligns well with observations. Negative velocity areas are reduced in AN. Differences between AN and FC are minor and are not shown.

Time series of total northward transport through the CC for the three runs (FR, FC and AN) are shown in Fig. a. The annual averages are 0.49, 0.31, and 0.28 Sv, respectively.

Both DA-informed simulations (FC and AN) show a net reduction in northward transport compared to the free run, although all three simulations capture expected seasonal variability, characterized by higher transport in winter and lower values in summer. The most notable differences occur in summer season and early fall: between June and October, average transport is 0.29 Sv in the FR and only 0.03 Sv in the AN. For the remaining part of the year, averages are 0.63 and 0.45 Sv, respectively. Nonetheless, these results are not necessarily generalizable to other years, as DA could potentially increase transport in different conditions.

Figure b compares the increment in time-averaged transport estimated using the tangent linear approximation (ΔTr‾TL, Eq. ) with that computed directly from the difference between the non-linear runs AN and FC (ΔTr‾NL). The good agreement between the two time series confirms that the tangent linear approximation holds well within the three-day assimilation window. Increments in average transport during an assimilation window range between -0.45–0.39 Sv, with a mean of -0.03 Sv and a standard deviation of 0.16 Sv.

The contribution of each observation type to changes in the total northward transport is shown in Fig. c, as detailed in Sect. . Observations from the CoCM have the largest overall impact on the transport increment, followed by HFR, SST, in situ T and S, and SLA. The relative impact of observation types also varies seasonally: CoCM data dominate during winter and autumn, while HFR observations have a greater influence during spring and summer. No clear seasonal pattern is observed for the remaining data sources.

Figure d quantifies the contribution of different components of the 4D-Var control vector (initial condition, boundary conditions, surface fluxes of heat, freshwater and momentum) to the CC transport increment ΔTr‾. The southern boundary corrections dominate throughout the year, indicating that transport is influenced by large-scale basin dynamics beyond the model's current spatial extent. Initial conditions have a limited role, except at the beginning and end of the year, while surface forcing becomes more influential from June onward.

An indicator for assessing the effectiveness of the DA procedure is a scatter plot of the contribution of each observation di⋅gi (i.e. its impact on ΔI) vs. the corresponding innovation element di, as illustrated in Fig. . These scatter plots can also be interpreted as contingency diagrams, with the percentage of points in each quadrant indicated.

Overall, the points in Fig.  are approximately evenly distributed between positive and negative innovations for all observation types, except for in situ salinity (Fig. f). This can be viewed as an indication that a clear and evident bias is not present in these observed components of the state vector, but for the in situ salinity, for which a non symmetric distribution for positive and negative innovation values is present (44.4 % negative; 55.7 % positive). The “butterfly” shape of each scatter plot indicates that observations with small innovations tend to produce only minor changes in CC transport. This behaviour is expected, as observations that require only small corrections on the background state should not significantly alter transport estimate. Conversely, observations associated with larger innovations, exhibit a wider range of impacts on CC transport.

By and large, for HFR, SST, SLA, CoCM, and in situ salinity, impacts are approximately symmetrically distributed around di⋅gi=0, meaning that these data sources contribute both positively and negatively to CC transport adjustments. In contrast, in situ temperature observations show a skewed distribution, with approximately 61 % of the points leading to a reduction in transport (di⋅gi<0). Figure  also reveals a pronounced asymmetry in the distribution of points relative to di=0 for SLA and CoCM observations. Specifically, negative SLA innovations (di<0) typically lead to an increase in CC transport (di⋅gi>0), and vice versa. The opposite pattern is observed for CoCM data. While such a behavior is clear for the CoCM v-velocity component, further analysis is needed to fully understand the underlying mechanism driving the SLA signal in the DA system.

An additional metric for evaluating observation impacts is the root mean square (RMS) of transport increments, calculated for both the aggregate effect of each observation type and for the average contribution of a single observation . The Global RMS for each observation typology is calculated as 1Nw∑i=1NwΔTr‾i2, where Nw is the total number of assimilation windows and ΔTr‾i is the average transport increment associated with the ith assimilation window. The average RMS impact per individual observation type (Datum) is given by 1Nobs∑i=1Nobs(di⋅gi)2, where Nobs is the total number of observations of a given type. Results for both metrics are reported in Table .

In situ velocity observations from the CoCM are the most influential in constraining the CC transport, especially when considering the average impact per observation. Aggregated over time, HFR data emerge as the second most effective data source. However, each HFR observation is approximately 3–5 times less impactful than in situ T and S measurements, and about 7 times less impactful than CoCM data. This can be explained by the fact that in situ data directly affect the whole water column, whereas surface data can only influence it indirectly, through the adjoint. While SST observations have greater overall impact than SLA on aggregate, the impact of individual observations of each of the two types is similar (Table ) in agreement with the different amount of observations Nobs.

Recalling that the yearly averaged transport values that we obtained at the CC for FR, FC and AN are equal to 0.49, 0.31, and 0.28 Sv, respectively, these values are lower than previous estimates based on observations, ranging from 0.54 to 0.71 Sv , and numerical models, such as 0.49 Sv in and 0.5 Sv in , the latter based on a multi-decadal time series. However, reanalysis by and obtained an average transport around 0.33–0.34 and 0.35±0.365 Sv, respectively, considering the whole transect including the CC and the area from the Island of Capraia to the Italian coast.

Part of the discrepancy with observational estimates may stem from the assumption of negligible longitudinal variability in the northward velocity across the channel that is not fully supported by our model results (Figs. e and a and e) and was also observed by previous numerical studies . Interestingly, the two reanalyses of and obtained an annual average transport comparable to our result by considering the whole Tyrrhenian transect. These values lower than those from previous literature can be ascribed to a possible negative annual averaged transport for the area between the Island of Capraia and the Italian Coast, or to a more appropriate representation of the flow in the area.

The transport across the whole transect linking the Tyrrhenian and Ligurian Seas, and the interaction between the two sub-transects, also in light of basin-scale mass budgets, deserve further studies. Furthermore, additional analysis, particularly over a longer time period, will help determine whether the year 2022 reflects inter-annual variability , or is the indication of a longer-term trend.

Comparing the non-assimilative (FR) and the assimilative (FC and AN) runs it is important to stress that the latter are the cumulative result of corrections applied to the background by DA over each assimilation window. Observing Fig. , it is clear that DA systematically reduces the northward velocity in the channel to better match the observations, leading to a corresponding reduction in total transport (Fig. a). However, Fig. b, shows that northward transport reduction is dominant up to mid-August 2022, after which positive transport increments become more frequent than negative ones. This evolving pattern is further dissected in Fig. c: from January to early April, CoCM data primarily drive the transport corrections; between April and early August, HFR data become more influential, despite not being directly located at the CC transect, where the transport is computed. Figure a and c show the average velocity increment δv during the periods dominated by the JERICO facilities, CoCM (from 1 January to 9 April), and HFR data (from 22 April to 11 August), respectively. CoCM observations tightly constrain the flow near their location but with the side effect of accelerating the northward flow in the surface western part of the transect. In fact, the δv values, averaged over the whole year, are characterized by an overall reduction for the most part of the CC transect and a slight increase at the upper western flank (not shown). This may be the result of insufficient corrections to the boundary conditions, redirecting the volume transport to unconstrained portions of the domain (Figs. a and e). Interestingly, it can help to explain why the total transport (Fig. a) is less affected than the velocity alone (Fig. a–c) when the FR and the AN runs are compared. Figure d and e show that while the northward flow weakens across most of the transect in AN with respect to FR, particularly on the eastern side (that observed by the mooring), a partial compensation occurs through a weaker (in absolute value) southward flow on the western flank and the already mentioned slightly northward acceleration at the upper western flank.

Figure b, d, and f show the average spatial increment in transport per unit width, defined as ΔF(x,y)avg=ΔFu,avg2+ΔFv,avg2, where ΔFu,avg=1Tf-Ti∫TiTf∫hδudzdt and ΔFv,avg=1Tf-Ti∫TiTf∫hδvdzdt. Here, h is water depth, and Ti, Tf define the time interval, and the total transport can be recovered multiplying ΔFu,avg and ΔFv,avg by the cell widths Δy and Δx, respectively (we prefer to consider this quantity rather than depth averaged velocity since the latter is independent of the volume of water moved). CoCM related transport increments show a pronounced southward pattern (Fig. a and b), while HFR-driven increments manifest as surface- and bottom-intensified reductions in velocity, with weaker overall transport changes (Fig. c and d). Furthermore, ΔF(x,y)avg in Fig. b is in general stronger than that of Fig. d, despite the two show a similar pattern.

Another key period is from 18 November to 23 December, when CoCM-driven positive increments are counterbalanced by HFR-driven reductions (Fig. c). The corresponding average increment in velocity at the Corsica Channel cross section δv and the spatial distribution of ΔF(x,y)avg for the eastern part of the modeled domain are reported in Fig. e and f, respectively. Figure e reveals surface deceleration from HFR and subsurface acceleration from CoCM, with an additional northward acceleration provided by SST (Fig. c), which is reasonably concentrated mainly at the surface. The corresponding average transport increment shows an east-west contrast, with acceleration near Corsica and deceleration across the Tuscany Archipelago (Fig. f). This acceleration at the eastern flank of Corsica, and deceleration at the Tuscany coast, are present at the southern boundary, where an anticyclonic structure in the average transport increment is formed. We argue it is a modulation of the Bonifacio Gyre whose cyclonic structure, if weakened, tends to favor the transport in the CC for this particular current pattern. This happens in correspondence of a period (18 November–23 December) where both IC and AF (other than BC) play a role in constraining the transport in the CC (Fig. d) and it is indeed the atmospheric forcing that acts to modulate the intensity of the Bonifacio Gyre .

This behavior exemplifies a key feature of 4D-Var: the adjoint model MbT uses the derivative ∂I/∂z of the CC transport to inform the system what modifications are required at the control vector to influence I during the 3 d assimilation window. For a more detailed explanation of the observation impact mechanism see . Additional analyses might expand the computational domain to include an additional portion of the domain which potentially influences the transport through the CC. However, it is not straightforward to identify the extension which maximizes the impact of initial condition and atmospheric forcing at the expense of boundary condition. Furthermore, extending the domain to contain the whole Western Mediterranean basin would excessively increase computational costs.

If we focus on the physical processes affecting the performance of the DA system we can try to explain, for example, why the error reduction is more pronounced for the v than the u velocity component observed by the CoCM (Fig. a, b, and e and Table ). Since the latter has a lower observed average magnitude (0.008 ms-1) and standard deviation (0.022 ms-1) with respect to the former (0.05 and 0.09 ms-1, respectively), the DA has more chances to correct v than u, also because we made the assumption that the background error covariance is proportional to the standard deviation of the state variables. Furthermore, being the flow orographically constrained, the eastward velocity (u) cannot reach large values and possible small variations are more hardly caught by the model.

Looking at Fig. c and e, a regime shift is observable in the RMSE for TCORA and SST, which is larger for both FR and FC with respect to AN, starting approximately from mid-May. We argue it can be attributed to the marine heat wave (MHW) that affected the Western Mediterranean sea from mid-May to the end of summer , despite the regime shift in RMSE is present until the end of the year for both TCORA and SST. The effect of the MHW, although to a lesser extent, can be traced on the two RMSE peaks for SCORA in May and August, probably due to strong evaporation processes (Fig. d).

The difference in RMSE between AN and FR for TEMSO and SEMSO, is remarkable for surface and intermediate water, while it reduces or disappears for deeper water (Fig. c and d). It is likely explainable considering that AW, mAW and EIW are more sensitive than WMDW or TDW to seasonal changes that can be captured by DA.

Regarding the circulation, Fig.  presents the time- and depth-averaged currents from FR and AN over two vertical layers: surface to 200 m (AW or mAV), and 250 to 450 m (EIW) .

DA slightly intensifies the NC, primarily by strengthening the WCC, which is barely noticeable in the FR, while the ECC flow is reduced (a and b). This pattern can be seen as a shift from a Western Mediterranean gyre, which principally involves the Tyrrhenian and Provencal basins, toward a North-Western Mediterranean gyre, which partially substitutes the Tyrrhenian inflow with waters coming directly from the Algerian basin. The ECC behavior aligns with previous studies showing its high variability , whereas the pronounced WCC signal was also found by and . In the HFR domain, DA deflects the flow more westward, reducing the curvature observed in FR. The weak cyclonic structures between 4–8°E in FR are more pronounced in AN. Similar differences are seen in the EIW layer (Fig. c and d).

We also looked for a peculiar pattern of the summer circulation in the area, that is the presence of the Ligurian Anticyclone, for its potential role in modulating the ECC . The determination of monthly averaged surface velocity fields allowed us to exclude the presence of the LA in the summer season, for both FR and AN. However, in October, a cyclonic feature north of Corsica was present in the AN and not in the FR run (not shown). Significant current reversal episodes were not observed in the corresponding period, but the presence of the LA might have concurred to maintain the transport lower for the analysis than for the freerun (Fig. a).

Observation impact can allow us to investigate in which ways DA modifies the model state with respect to a particular metric (transport across a section in our case) by optimally balancing between observations and background solution.

Table  reports the RMS transport increments considering each typology of observation (Global) and the single observation per typology (Datum). Apart from CoCM and SLA data, it seems that the impact of a single observation is inversely proportional to Nobs. It could be explained by considering that the larger the amount of observations, the higher the probability that some of them are redundant and do not provide additional information content. On the contrary, CoCM/SLA observations are characterized by large/small Global and Datum RMS, given a comparable Nobs (Table ), showing that the former are crucial to constrain ECC, whereas the latter have a small effect for the present modeling setup.

Developing more the analysis, we can even estimate the contribution of each observation to the transport increment through ΔTr‾TL=dT⋅g. The presence of a seasonality in ΔTr‾ when we consider observation typology (Fig. c), that is roughly cold months controlled by CoCM while warm months until late autumn by HFR, could be explained by the corresponding seasonality in the CC transport. Small northward velocity values in warm months tend to produce small innovations d and, consequently, possible small dT⋅g. To better control the dynamic at the channel, DA relies on another source of observations that is HFR. In cold months, the opposite occurs and the role of CoCM observations is predominant.

Moreover, the variability of the elements of g reveals the location of observations which have the greatest influence on the model. Figure  shows the spatial distribution of the root mean squared values of the elements of g (RMSg) calculated over each computational cell for each observation type.

RMSg for HFR peaks near the eastern part of the coverage region and close to the coast (Fig. a), aligning with the CC's longitudinal range (9.45–9.8°E).

The degree of sensitivity of the transport to SST and in situ T data is particularly high in the Gulf of Lion (GoL). Traces of the upwelling system induced by north-westerly winds are detectable for both SST and in situ T (Fig. c and e), whereas those associated to south-easterly winds at the western coast of the GoL are mainly present for in situ T RMSg (Fig. e). Similarly, it appears that salinity variations in the region of freshwater influence (ROFI) of the Rhône river can contribute to the evolution of the ECC (Fig. f). High sensitivities in the area may be partly explained by the background-error covariance matrix B that we assumed to be proportional to the standard deviation of the freerun. For temperature, it displays large values at the surface in the eastern part of the GoL area and lower, but significant values, at the western part, not only at the surface (not shown). For salinity, large standard deviation values are restricted to the ROFI of the Rhône river (not shown). However, the physical mechanisms through which upwelling/downwelling episodes or salinity variations in the GoL can influence CC transport, deserve and require further investigation.

Secondary hot-spots of RMSg for in situ T and S near the LI station and around the eastern coast of Corsica, especially for T, emphasize that distant observations can significantly affect CC transport (Fig. e and f).

The large RMSg values associated with SST observations (Fig. c) at the eastern coast of the Ligurian sea roughly correspond to the pattern observed for the RMSg associated with HFR data (Fig. a). We believe the contribution of those SST observations to ΔTr‾ is linked to that of HFR as their interaction is already observed in a similar 4D-Var application as reported by .

The impact of SLA observations shows no clear spatial pattern (Fig. d), although certain satellite tracks exhibit a larger RMSg values, suggesting that during specific periods the barotropic control on CC transport is significant. Indeed, if we plot the gi values as a function of time (not shown), we observe larger values at the year's start and end, suggesting that when the Tr‾ is higher, corrections of the barotropic modes are larger.

Figure b shows a Hovmöeller diagram of the absolute values of the elements of g associated with the v-velocity component of observations from the CoCM. The sensitivity of the CC transport to these v-velocity observations appears relatively uniform with depth and displays a clear seasonal signal, consistent with the temporal variability seen in Fig. c, where the contribution of CoCM data diminishes during late spring and summer. Although the vertical distribution of sensitivity is generally homogeneous, it tends to be more pronounced at the bottom. This may result from the fact that the g values are weighted by the P matrix (through the Kalman gain, see Eq. ), thus depending on R, B, and the spatial distribution of the other observations. As a result, the sensitivity of the Tr‾ to CoCM data near the surface may be partially influenced by HFR, SST, SLA observations whereas at depth this effect is not present.

Focusing on the gi values helps to identify observation locations that are potentially influential for ΔTr‾ and, by extension, for understanding the physical mechanisms driving transport in the CC. In contrast, analyzing the product di⋅gi (i.e. individual increments) may mask this insight, as a large sensitivity gi could be masked by a small innovation di, or vice versa. A more targeted approach to quantify the influence of different regions or variables on the CC transport would be an adjoint sensitivity study . However, such an approach does not offer information about how observational data significantly shape the model solution.

For most assimilation windows the primary way DA constrains the CC transport is the modification of the boundary conditions (Fig. d). By selecting those with the largest positive and negative transport increments largely attributable to BC modifications (90th and 10th percentiles of the ΔTr‾BC distribution, respectively), we calculated the northward velocity increment at the southern boundary δv (Fig. a and b), the free surface height increment δη (Fig. c and d), and net heat flux increment δQnet (Fig. e and f), between analysis and background, averaged over the two sets of assimilation windows. A positive ΔTr‾BC is indeed associated with a northward-inducing δη (Fig. c) that is consistent with a clear northward barotropic acceleration on the Tyrrhenian side and a less intense southward barotropic acceleration, distributed over a much wider cross sectional area, at the Ligurian-Provençal basin side (Fig. a). For the negative ΔTr‾BC such a mechanism is more marked for both δv and δη. The barotropic nature of these increments is confirmed by the almost uniform increment velocity distribution along depth (Fig. a and b). Interestingly, the δQnet is characterized by a pronounced surface heating/cooling in the Ligurian-Provençal basin when the CC transport increases/decreases (Fig. e and f). Since the ROMS model adopts the Boussinesq approximation , such a mechanism cannot be related to steric adjustments, but rather to a modification to the baroclinic pressure field to maintain the consistency between the barotropic correction at the boundaries and observations in the interior. Additional analysis is required to make explanatory hypotheses at the base of the observed mechanism.

proposed that CC dynamics are governed by atmospheric conditions over the western part of the Ligurian-Provençal basin, where winter heat loss enhances the steric gradient between the Tyrrhenian and Ligurian-Provençal basins, driving Tyrrhenian waters into the Ligurian Sea and strengthening the ECC. Our results cannot directly mimic the processes described by , however, focusing now on the largest positive and negative transport increments attributable solely to the correction to atmospheric forcing, ΔTr‾AF (95th and 5th percentiles of the ΔTr‾AF distribution, respectively), we observe that δη (Fig. a and b) and δQnet (Fig. c and d) align with the mechanism proposed by : an increase (decrease) in transport is triggered by a differential surface cooling (warming) with associated eastward (westward) positive gradient in the free surface increment. Even in this case, additional analyses are required to delve into the mechanism responsible for the agreement with literature results.