## 4 Evaluation
### Gulf Stream  
We first have to discuss the time window parameter \(d\) related to the aggregation of along-track data over a specific day \(t_{k}\); see Section 2.2. The same value of this parameter may not be optimal for all the interpolators: AnDA exhibits a better performance when considering only along-track nadir data of the day (\(d=0\)), thereby contradicting the previous optimal results of \(d=5\) found \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Name** & **Formula** \\ \hline \multirow{6}{*}{Temporal domain} & nRMSE & nRMSE(\(t_{k}\)) = \(\sqrt{\frac{1}{|\widehat{\mathcal{D}}|}\sum_{\widehat{\mathcal{D}}} (\mathbf{x}_{k}-\hat{\mathbf{x}}_{k})^{2}}\bigg{/}\sigma_{\mathbf{x}}\) \\  & Error variance & \(\sigma_{\mathbf{x}-\hat{\mathbf{x}}}^{2}(t_{k})\) = \(\frac{1}{|\widehat{\mathcal{D}}|}\sum_{\widehat{\mathcal{D}}} \big{[}(\mathbf{x}_{k}-\hat{\mathbf{x}}_{k})-\overline{(\mathbf{x}_{k}- \hat{\mathbf{x}}_{k})}\big{]}^{2}\) \\  & Correlation & COOR(\(t_{k}\)) = \(\frac{\text{Cov}(\mathbf{x}_{k},\hat{\mathbf{x}}_{k})}{\sigma(\mathbf{x}_{k}) \sigma(\hat{\mathbf{x}}_{k})}\) \\  & Reconstruction score & R-score = \( 100\times\left(1-\frac{\sum((\mathbf{x}-\overline{\mathbf{x}})-( \mathbf{\hat{x}}-\overline{\mathbf{x}}))^{2}}{\sum_{\Omega}( \mathbf{x}-\overline{\mathbf{x}})^{2}}\right)\) \\  & Interpolation score & I-score = \( 100\times\left(1-\frac{\sum((\mathbf{x}-\overline{\mathbf{x}})-( \mathbf{\hat{x}}-\overline{\mathbf{x}}))^{2}}{\sum_{\overline{\Omega}}( \mathbf{x}-\overline{\mathbf{x}})^{2}}\right)\) \\  & Auto-encoder score & AE-score = \( 100\times\left(1-\frac{\sum((\mathbf{x}-\overline{\mathbf{x}})-( \psi(\mathbf{x}))-\overline{\psi(\mathbf{x})}))^{2}}{\sum_{\widehat{\mathcal{D }}}(\mathbf{x}-\overline{\mathbf{x}})^{2}}\right)\) \\ \hline \multirow{2}{*}{Spectral domain} & RAPS & RAPS(\(\lambda\)) = DSP(\(\hat{\mathbf{x}}_{k}\))(\(\lambda\)) \\  & Signal-to-Noise Ratio & SNR(\(\lambda\)) = \(\frac{\text{DSP}(\mathbf{x}_{k}-\hat{\mathbf{x}}_{k})(\lambda)}{\text{DSP}( \mathbf{x}_{k})(\lambda)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Temporal and spectral statistics used to assess the performances of the interpolators in the observation system simulation experiment. Figure 6: Daily spatial nRMSE computed for the four 20-day non-continuous validation periodss for OI, (post-)AnDA, VE-DINEOF, FP-ConvAE and FP-GENN. The spatial coverage of one day (\(d=0\)) accumulated along-track nadir and that of wide-swath SWOT data are respectively provided by the red and green barplots. (**a**) Nadir, (**b**) nadir+swot. by [5] over the Mediterranean sea, which may indicate AnDA responds differently to the along-track aggregation strategy depending on the energetic dynamical regime of the region. On the other hand, both FP-ConvAE and FP-GENN interpolators perform better (not shown here) by aggregating nadir data over a 5-day time window. As a consequence, the results presented in what follows will use a value of \(d=0\) for AnDA and VE-DINEOF and \(d=5\) for FP-ConvAE and FP-GENN. Next, to evaluate the behavior of the different interpolators on both along-track nadir samplings and their fusion with wide-swath SWOT datasets and make the comparison possible, we have to preliminarily define whether the NN-based interpolators were used under a supervised learning strategy, i.e., with gap-free SSH anomaly maps used as targets in the training phase, or under an unsupervised setting, with single observations used as targets for the reconstruction criterion. Overall, six possible configurations were tested and listed in Table 2: two supervised versions using either the gap-free maps (supervised 1) or the pseudo-observations (supervised 2) as input and the unsupervised version with both input and target only were of the pseudo-observations. These three configurations are also tested when adding the DUACS OI product as a covariate for input data, because we think figured this may give prior information about how the anomaly field \(d\mathbf{x}\) is distributed.