## 5 Results and Discussion
### Global Forecasting and Interpolation Performances  
**Forecasting performances of the proposed data-driven dynamical priors:** We further evaluate the performance of the proposed schemes over the considered case-study region. Table 2 reports the RMSE of the proposed NN-based representation compared with local and global forecasting operator [19]. The proposed patch-level NN-based model outperforms the benchmarked approaches by about 5-15% in terms of forecasting RMSE. This is due to the structure of the proposed model that involves a postprocessing operator learnt from data that combines the predicted patches in order to \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Asmiliation Method** & \multicolumn{3}{c}{**Considered Patch RMSE (\%C)**} \\  & **Patch 1** & **Patch 2** & **Patch 3** & **Patch 4** \\ \hline LAF EnKF & 0.50 & 0.25 & 0.22 & 0.39 \\ Bi-NN-EnKF & 0.55 & 0.23 & 0.22 & 0.30 \\ Bi-NN-NNKF-EOF & **0.46** & **0.20** & **0.19** & **0.27** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Patch-level interpolation experiment: RMSE of the reconstructed anomaly fields for the LAF EnKF (local analog forecasting based ensemble Kalman filter), Bi-NN-EnKF (Bilinear residual neural net model (\(\mathcal{F}^{\mathcal{P}_{s}}\)) used in an ensemble Kalman filter), Bi-NN-NNKF (Proposed NNKF based on a bilinear residual neural net dynamical mean model). Figure 2: Selected patches on the high resolution component of the SST data. (The SST map corresponds to 19 July 2015). minimize the global forecasting error. The analog forecasting models in the other hand, even through the use of more patches with overlapping, process the output patches through EOF projections to get ride of the variability due to the patches interaction. However, This smoothing results in losing some high resolution information which results in diminishing the forecasting results with respect to our proposed model. **Interpolation performances of the proposed model with respect to OI and DINEOF:** We report the mean interpolation performance in Table 3 and the interpolation error time series in Figure 3. The proposed NN-based scheme (PB-NNKF-EOF) leads to very significant improvements with respect to the optimal interpolation and PB-VE-DINEOF schemes in terms of RMSE and correlation coefficients for both the SST and its gradient with a relative improvement of the RMSE above 50% for missing data areas for the SST and its gradient (resp. 40%). This important gain clearly emphasizes retirement of fine scale structures unresolved using OI and DINEOF techniques. From a methodological point of view, this gain was clearly expected. OI and DINEOF schemes rely purely on data to interpolate the SST field. Therefore when provided with observations with a high missing data rate, these techniques are only able to retrive horizontal scales up to \(\approx\)100 km. In the opposite, our proposed framework combines both the observations and the data driven model outputs to reconstruct our SST field which results in better representation of fine scale structures. **Interpolation performances of the proposed model with respect to GAN-EnKF, LAN-EnKF:** A clear gain is also exhibited w.r.t. analog data assimilation schemes with a relative gain greater than 20% in terms of RMSE for both the SST and its gradient. The same conclusion holds in terms of correlation coefficients close to 90% or above for all parameters for PB-NNKF-EOF scheme, all the other ones depicting correlation coefficients below 85% for SST gradient fields. These results reflect the forecasting performances illustrated in Table 2 and the patch based interpolation performances in Table 1. Indeed, the PB-NNKF-EOF scheme outperforms both the analog forecasting operators in terms of one step ahead predictions which suggest better assimilation in a global scale especially for missing data areas. Although the considered NN-based representation exploits non-overlapping patches, we still come up with significant improvements w.r.t AnDA schemes which involve a 50% overlapping rate between patches.