Figure 1: Comparison of the RMS of the differences (a) in current modulus and (b) in direction, of the independent drifting buoy velocities and the altimeter geostrophic velocities obtained using MDT solutions first guess of the CNES-CLS22 (unfiltered at the coast with the Lagrangian filter) and the first guess of the CNES-CLS18, in percent :
\(\mathbf{\%}\mathbf{\text{RMSD}}\mathbf{\,=}\frac{\mathbf{\text{RMS}}\left(\mathbf{U}_{\mathbf{\text{MDT}}_{\mathbf{FG22}}}\right)\mathbf{-}\mathbf{\text{RMS}}\left(\mathbf{U}_{\mathbf{\text{MDT}}_{\mathbf{FG18}}}\mathbf{-}\mathbf{U}_{\mathbf{d}}\right)}{\mathbf{\text{RMS}}\left(\mathbf{U}_{\mathbf{\text{MD}}\mathbf{T}_{\mathbf{FG18}}}\mathbf{-}\mathbf{U}_{\mathbf{d}}\right)}\mathbf{\,}\). These statistics are calculated in boxes of 5◦ by 5◦. Only boxes with more than 100 measurement points and more than 10 different drifters are shown. The bluish colors denote improvement while reddish colors stand for degradation.

2.2 Computation of the synthetic mean heights

The aim of estimating synthetic mean heights is to use temperature and salinity profiles to calculate a mean height whose physical content is the same as the MDT. Temperature and salinity profiles are used to calculate density variations in the water column and to estimate a dynamic height (between the profile’s reference depth and the surface) representing the baroclinic component of the dynamic circulation. Here we use reference depths of 200, 400, 900, 1200 and 1900m, with each dynamic height calculated according to one of these reference depths, the deepest accessible according to the maximum depth of the T/S profile. The dynamic heights thus estimated do not have the same physical content as the Altimeter Sea Level Anomalies (SLA), as the SLA is also influenced by baroclinic processes between the reference depth and the bottom, as well as by barotropic processes not measured by change in temperature and salinity (Rio et al. 2014b). To obtain average heights over the reference period, we need to remove the temporal variability (of the SLA) from the instantaneous dynamic heights; we therefore need to remove the temporal variability by subtracting the part of the SLA corresponding to the baroclinic process from the surface to the reference depth. Then add the missing components: the barotropic signal and the baroclinic signal from the reference depth of the profile to the bottom. The method for these two steps is the same as the one fully described in Rio et al. (2011) and also used in Rio et al. (2014a) and in Mulet et al. (2021).
In particular, the missing quantity, the average contribution of deep baroclinic and barotropic processes, is estimated as the difference between the climatology of dynamic heights relative to the given reference depth, the same as the T/S profile and the first estimate of the CNES-CLS MDT22. For each reference depth considered in this study (200, 400, 900, 1200 and 1900m), a climatology is calculated from the same CORA database (section 3) to remain consistent, this time using the maximum number of profiles possible, i.e. for the reference depth 200m, using all profiles deeper than 200m. These climatologies are then filtered at the same spatial scale as the first guess, i.e. 125km.
The optimal analysis to estimate the CNES-CLS22 MDT is performed on the anomalies with respect to the first guess, so we remove the first guess from the synthetic fields before the optimal analysis and then add this first guess to the result to obtain the final MDT. In this remove-restore framework, the estimation of mean synthetic heights is equivalent to considering only the small scales of synthetic dynamic heights relative to reference depth. Figure 2 shows these fine scales of synthetic mean heights (all reference depths averaged by 1/8° by 1/8° boxes). The signal is intense in the large western boundary currents and in the Antarctic Circumpolar Current (ACC), with anomalies of over +/-10 cm. These mean synthetic heights will tend to accelerate these large currents.
Finally, the synthetic mean heights are averaged in 1/8◦ boxes. The associated errors are computed as described in Rio et al. (2011). These errors are high in areas of high oceanic variability: western boundary currents, the ACC; and near the poles, where there are few data or less confidence in processing. Overall, the error is less than 2cm in the rest of the ocean. As in the Rio et al. (2014b) study, the error associated with these mean synthetic heights does not take into account the first-guess error and is therefore certainly underestimated.