## 4 Evaluation
### Osmosis  
As has already been done for the GULFSTREAM domain, we investigate how the different interpolation techniques behave when varying nadir aggregation parameter \(d\) Figure 10a,c for the corresponding aggregations on 4 August 2013 and 5 August 2013. The daily nRMSE as a function of the along-track nadir time window parameter \(d\) (not shown here) leads to the same GULFSTREAM-related optimal values, namely, ANDA behaves best when considering only the data restained to the targetted day \(t_{k}\) and both FP-ConvAE and FP-GENN performs better with \(d=5\). Regarding the GENN configuration, the unsupervised configuration with additional use of DUACS OI as input does not seem to perform well on the OSMOSIS domain, while it was the best option in the GULFSTREAM region. It is especially noticeable in the four 20-day long time series; see Figure 11. However, this result should be qualified because when replicating the same preliminary work to find the best FP-GENN configuration but with no observation errors (see Figure A12 in Appendix B), the unsupervised configuration is again the best solution. Thus, on this less energetic OSMOSIS domain, but with more discernable fine scales, the observational errors seems to have much more consequences than when considering a domain mainly driven by mesoscale energies. To stick with a unique plug and play solution, an interesting idea would be to explicitly implement a multi-scale approach that directly uses a set of \(m\) operators accounting for the \(m\) resolutions of the dynamical process. This should help to better reconstruct the fine scales whatever the dynamical regime of the region and the observational errors. As a consequence, we selected in this section the supervised 2 configuration with additional use of DUACS OI as input. Let us note that this setup could also be used in future operational context, since the GENN inputs are still made of purely observational data: along those lines, this type of configuration is similar to the AnDA setup that needs both observation data and gap-free data to be operated. Figure 9: Global SSH gradient field reconstruction obtained by OI, (post-)AnDA, VE-DINEOF, FP-ConvAE and FP-GENN for a joint assimilation/learning of along-track nadir with wide-swath SWOT data. (**a**) Ground truth (\(\
abla_{\text{SSH}}\)), (**b**) OI, (**c**) post-AnDA, (**d**) VE-DINEOF, (**e**) FP-ConvAE, (**f**) FP-GENN. Figure 11: Daily spatial nRMSE computed for the four 20-day non-continuous validation periods for the six supervised/unsupervised FP-GENN configurations. The spatial coverage of 11 days (\(d=5\)) accumulated along-track nadir (**a**) expanded with wide-swath SWOT data, (**b**) is provided by the red barplot. (**a**) Nadir, (**b**) nadir+swot. Figure 10: One (\(d=0\)) and 11 day (\(d=5\)) accumulated along-track nadir and wide-swath SSH pseudo-observations (meters) on 4 August 2013 (**a,b**) and 5 August 2013 (**c,d**). (**a**) Nadir (\(d=0\)), (**b**) nadir (\(d=0\)) + swot, (**c**) nadir (\(d=5\)), (**d**) nadir (\(d=5\)) + swot. In Figure 12, the daily nRMSE obtained with our set of data-driven interpolators over the validation period, it can be seen that using AnDA with along-track nadir data and wide-swath SWOT observations gets the best scores, which is confirmed in the Taylor diagram (Figure 13a) and also with R/I/AE-scores in Table 4. Still, FP-GENN performs in a very similar way and the single use of nadir data is largely favorable to FP-GENN-MNM + OI. On the spectral analysis in Figure 13b, the signal-to-noise ratios of FP-GENN and AnDA indicate a capability to retrieve spatial scales up to 50-60 km, while the OI clearly only catches again the spatial scales over 100 km. Again, let it be known that when no observational errors are introduced (see Figure A14b in Appendix B), the fully unsupervised configuration of FP-GENN still behaves better. The single use of along-track nadir data clearly downgrades the performance of interpolations, even if the gain remains significant for FP-GENN.