The dynamics of SSH are simulated by a 1.5-layer quasi-geostrophic (QG) model. This model simulates the dynamics of the first baroclinic mode, known to capture most of the SSH variability. In this model, the conserved potential vorticity q is diagnosed from SSH: 1q=∇2ψ-1LD2ψ, where LD is the Rossby radius of deformation, ∇2=∂2∂x2+∂2∂y2, and ψ is the streamfunction, such as in 2ψ=gfSSH, with g being the gravity constant and f being the Coriolis frequency.

The conservation of potential vorticity is written as follows: 3∂q∂t+ug⋅∇q=0, where ug is the geostrophic velocity vector diagnosed from SSH: 4ug=k×∇ψ=gf0k×∇SSH, where f0 is the mean value of the Coriolis frequency over the domain (f plane), k denotes the vertical direction, and ∇=(∂∂x,∂∂y).

In reality, an altimeter only provides accurate observations of the time-fluctuating part of SSH, called sea level anomaly (SLA). The time-averaged SSH, called mean dynamical topography (MDT), is computed with the combination of in situ data and other satellite observations . 5SSH=MDT+SLA

To formulate the QG dynamics with SLA, we decompose the geostrophic flow and the potential vorticity using the Reynolds decomposition: 6aug=ug‾+ug′,6bq=q‾+q′, where ug‾ and q‾ stand for the mean components (SLA independent), and ug′ and q′ stand the time-fluctuating components (diagnosed from SLA): 7aug′=gf0k×∇SLA,7bq′=gf0∇2SLA-gf0LD2SLA.

The prognostic equation for the potential vorticity fluctuation is then as follows: 8∂q′∂t+ug‾⋅∇q′+ug′⋅∇q‾+ug′⋅∇q′=0.

The SLA observations (denoted as SLAobs) are assimilated into the QG model with the back-and-forth nudging (BFN, ) technique. This technique is based on the nudging strategy, which consists of gently pulling the model trajectory towards the observations. Mathematically, an extra term proportional to the difference between the model SLA and the observations is added in Eq. (): 9∂q′∂t+ug‾⋅∇q′+ug′⋅∇q‾+ug′⋅∇q′=K(SLAobs-SLA), where K is the tunable nudging coefficient. Its value determines the balance between the weights given to the observations and the QG dynamics. As explained in , K varies in time and space to allow a smooth nudging of the model towards the observations. Mathematically, the nudging coefficient at time t and at the model grid point x is computed by the following equation: 10K(x,t)=K0∑i=1Nobse-||x-xi||D2e-t-tiτ2, where K0 is the nominal value of the nudging coefficient, Nobs is the number of observations, and (xi,ti) are the space–time coordinates of the ith observation. D and τ are the spatial and temporal scales at which the model is nudged towards the observations, impacting directly the scales of the reconstructed structures.

The BFN algorithm calls iteratively the forward nudging, defined as a forward-in-time propagation of Eq. () with K>0, and the backward nudging, defined as a backward-in-time propagation of Eq. () with K<0, within a fixed temporal window T. The temporal window T has to be chosen considering the observation sampling, the decorrelation time of the QG model and the computational complexity. At the beginning of each temporal loop, the SLA variable is initialized with the value estimated from the previous loop. In a few iterations (less than 20), the BFN converges towards a trajectory that fits both the observations and the model dynamics. For more details on the BFN-QG technique, the reader is referred to , and .

We assess the BFN-QG performances in a part of the Agulhas Current (25–45∘ S, 10–40∘ E) from 1 January 2010 to 31 December 2019. The Agulhas Current is the major western boundary current of the Southern Hemisphere, transporting large volumes of warm and saline water from the Indian Ocean to the Atlantic Ocean, greatly impacting climate and vessel trajectories .

As input data of the BFN-QG, we use the along-track L3 filtered SLA products from Jason-3, Sentinel-3A, Sentinel-3B, HaiYang-2, CryoSat-2 and SARAL/AltiKa. These SLAs have been distributed by the CMEMS (http://marine.copernicus.eu/, last access: October 2023) after the reprocessing of 25 years of altimetric data . For our analysis, we use the spatially filtered data, whose cutoff has been set to 65 km, corresponding to altimeters' effective resolutions .

The mean state of the ocean surface needed to advect the QG potential vorticity anomaly q′ through Eq. () is extracted from the CNES-CLS18 mean dynamic products . In this product, the topography (MDT) and velocity (MDV for mean dynamical velocity vector) are estimated with a multivariate objective analysis of a combination of altimeter and space gravity data and in situ measurements. A central step of the analysis lies in the filtering of the ageostrophic component of the in situ velocity measurements.

In Eqs. () and (), both ug‾ and q‾ have to be prescribed. For reasons not investigated during this work, the MDV product is not divergence free. Because ug‾ must be divergence free by construction (after Eq. ), we prescribe it with the divergence-free part of the MDV, called MDVrot (the subscript indicates that MDVrot contains the rotational part of the MDV). The field MDVrot is computed with the geostrophic Eq. () from a mean dynamic topography called MDTrot, obtained after solving the following elliptic equation: 11Δ(MDTrot)=gf∇×MDV. The rotational operator on the right rules out the divergent part of the MDV. For consistency, q‾ is diagnosed from Eq. () and MDTrot. Figure  indicates the significance of this procedure: the original MDT and MDV differ from their divergence-free counterparts by the orders of magnitude of the fields.

At the end of the mapping processing, we add the full MDV to the estimated velocity anomalies.

We compare the performances of the BFN-QG system and the DUACS DT2018 system . We use the global daily product provided by CMEMS on a 0.25∘ latitude × 0.25∘ longitude grid. The BFN-QG is run on a 0.1∘ latitude × 0.1∘ longitude grid, and the output maps are saved every 3 h. The parameters of the BFN-QG (defined in the previous sections) have been prescribed after a sensitivity experiment and are listed in Table .

The comparison focuses on SLA (hereafter called SLA mapping; see Sect. ) and geostrophic currents (hereafter called velocity mapping; see Sect. ). For assessing the SLA-mapping capability, we exclude SARAL/AltiKa from the altimetric observation network to use it as independent data, and we focus only on the year 2019. For assessing the velocity-mapping capability, all the available altimetric observation networks are used, and the validation is performed with independent drifter and SST data over the entire time period (10 years).

In this section, the velocity-mapping performances are assessed using independent drifter data at hourly resolution, selected from . The ageostrophic component of the observed velocities has not been removed in these reference data as we assume that it should affect the performance of the DUACS and BFN-QG methods in the same way. Snapshots of the norm of geostrophic velocities from the BFN-QG and DUACS systems are shown in Fig. . At first sight, the currents estimated by the BFN-QG exhibit finer-scale structures (filaments and small vortices) than the ones derived from DUACS. Figure  is further discussed later.

A performance diagnostic of an Eulerian nature is performed by comparing the estimated currents with the velocities measured by the drifters at each drifter location (in space and time). We use the same methodology as for the SLA-mapping performances: the mapped velocities (meridional and zonal components) are interpolated on the drifters' locations, and the errors with the drifters' velocities are aggregated in 1∘ latitude × 1∘ longitude boxes. The RMSE and the gain / loss ratio R (Eq. ) are then computed in each box. The results are reported in Fig. . The geographical distribution of the gain / loss ratio R shows a net improvement in the estimation of both zonal and meridional currents. This improvement is more pronounced for the meridional component, which is often harder to estimate from altimetry compared to the zonal component due to the nearly meridional orientation of the altimetry tracks. Like the SLA mapping, more improvements (as before, this is a relative comparison in %) occur in high-variability regions, as shown by the intensification of blue pixels in the inner domain delimited by the green contour in Fig. . Besides, the relative performances of the BFN-QG in low-variability regions are better for the velocity mapping than for the SLA mapping. This is probably due to the fact that the low-variability dynamics occur at large scales, which have very little impact on geostrophic currents.

The second performance diagnostic is Lagrangian: simulated drifter trajectories are compared with the real ones. For one drifter at one time, we compute the distances between the real drifter location and the locations predicted with the evaluated velocity fields. The predictions are initialized with the real drifter locations at earlier times ranging from 0 to 20 d, every 3 h. As an example, Fig.  displays the results for one drifter and one time. The BFN-QG-derived geostrophic currents improve the prediction of short-term Lagrangian transport compared to the DUACS-derived geostrophic currents: the blue dots, representing Lagrangian predictions with lead time up to 5 d, are much closer to the real location of the drifter for the BFN-QG system than for DUACS. But the red dots, representing Lagrangian predictions higher than 10 d, are as far as the ones predicted by DUACS. To investigate the lead time dependency of the relative performances of BFN-QG and DUACS in this Lagrangian diagnostic, the gain / loss ratios are plotted as a function of lead time in Fig. . The BFN-QG improves the Lagrangian prediction by more than 7 % for 1–3 d lead times compared to DUACS, with an enhanced improvement for drifters located in high-variability regions. The gain / loss ratio becomes positive (which means better performances for DUACS compared to the BFN-QG) for lead times higher than 10 d. The standard deviation of the Lagrangian errors (not shown) is higher for the BFN-QG than for DUACS, and this is accentuated for long lead times. This is qualitatively visible in Fig.  where the distances between the expected locations and the real location of the drifter increase almost linearly with the lead times for DUACS, while they are much more scattered for the BFN-QG (especially for lead times higher than 10 d). This can be explained by the higher spatial resolution of the BFN-QG fields.

This section compares the positions of fronts and eddies diagnosed from our reconstructions with those diagnosed from high-resolution SST observations. To do so, we use the Fronts Derived from Remote Sensing SST Observations by SEVIRI (Spinning Enhanced Visible and InfraRed Imager) over the Agulhas Region dataset created within the ESA World Ocean Circulation (WOC) project (10.12770/6c776c43-425b-4d29-9934-0822696f15d8) as ground truth. For each point (Pi=lonilati) of the detected frontal structures, we compute the flow crossing the fronts using either BFN-QG or DUACS geostrophic currents: 15Flow[Pi]=v[Pi]⋅δi‖vPi‖‖δi‖=cos⁡(v[Pi],δi)^, where v[Pi] is the velocity vector at point Pi, δi=lati+1-lati-1-(loni+1-loni-1)cos⁡(lati) is the normal vector of the front at point Pi, and (v[Pi],δi)^ is the angle between v[Pi] and δi. The values of the flow range from 0 to 1. Assuming that SST fronts and currents are aligned, the lower the flow, the more consistent the current estimation is with the SST. For this analysis to be valid, the currents have to be in geostrophic balance, and the advection of frontal structures must be negligible.

The Agulhas Current area is an excellent natural laboratory for this kind of analysis, with a strong geostrophic current, strong SST gradients and relatively weak advection of the frontal structures. However, the presence of Natal pulses in the nearshore side of the Agulhas Current core and complicated dynamics occurring in the retroflection area may advect significantly the frontal structures and thus limit this analysis. An illustration of the comparison of fronts and velocity is shown in Fig. . The direction of the streamlines derived from the BFN-QG (left) and DUACS (right) can be compared. One can visually see that the sharp turn of the Agulhas Current is better represented in the BFN-QG than in DUACS.

Figure shows statistics of the crossing flow computed from the geostrophic currents derived both from the BFN-QG and DUACS techniques within the 10-year study period. As for the previous diagnostics, the statistics are aggregated in 1∘ latitude × 1∘ longitude boxes. The left panel of Fig.  indicates that all pixels of the region are covered by several thousands of SST front occurrences, hence providing reliable statistics. The number of occurrences depends on the probability of detecting frontal structures and the cloud cover.

The statistics show mixed performances of the BFN-QG compared to DUACS, with stronger geographical patterns than with the previous diagnostics (see Sects.  and ). In particular, the meanders of the Agulhas Current are well captured by the BFN-QG, and the improvement in the main current is significant. On the other hand, in some regions, DUACS significantly outperforms the BFN-QG. We note that these regions are mostly characterized by weak crossing currents, as shown in the middle panel of Fig. . One example is the Agulhas Bank, i.e., the coastal region south of Africa characterized by very shallow waters. In this region, the weak performances of the BFN-QG are probably due to the non-representation of the bathymetry in the QG model (whose variations strongly affect the value of the Rossby radius of deformation LD, which modulates the potential vorticity field, through Eq. ). This shows that the method needs to be improved to better perform in coastal areas. Finally, Fig.  also depicts weak performances of the BFN-QG (compared to DUACS) in the southwestern part of the study domain, which is in contradiction with the other diagnostics. This can be due to non-reliable statistics because of the weaker density of observations and/or too-strong advection of the fronts by the currents that limits the validity of the analysis.