## 4 Data and Experimental Setting
### Experimental Setting  
The proposed neural-network-based Kalman scheme involves the following parameter setting. The proposed patch-based and NN-based Kalman filter is applied to SST anomaly fields w.r.t. optimally-interpolated SST fields (see below for the parameterization of the optimal interpolation). These optimally-interpolated fields provide a relevant reconstruction of horizontal scales up to \(\approx\)100 km.  
We exploit patch-level representations with non-overlapping \(20\times 20\) patches. This patch size was particularly tuned for the resolution of fine scale structures for this particular dataset [22]. For each patch \(\mathcal{P}_{s}\), we learn an EOF basis from the training data. We keep the first 50 EOF components, which amount on average to 95% of the total variance. For the patch-level NN model \(\mathcal{F}^{\mathcal{P}_{s}}\), we use a bilinear residual neural network architecture with 60 linear neurons, 100 bilinear neurons and 10 fully-connected layers with a _Relu_ activation. Among other parametrizations [39]. This architecture prove to outperform several other data driven models in the forecasting of patch based SST dynamics. The reconstruction model \(\mathcal{F}_{r}\) is a convolutional neural network with 3 convolutional layers. The first two layers comprise 64 filters of size \(3\times 3\) with a _Relu_ activation and the last layer is a linear convolutional layer with one filter. This parameters were tuned to give the best forecasting performances at a low computational cost.  
Regarding the diagonal covariance model \(\mathcal{F}_{D}^{\mathcal{P}_{s}}\), we consider an MLP with 4 layers, 3 hidden layers with 200 neurones and _Relu_ activations and an output layer with a _softplus_ activation. This parametrization was tunned to give the best trad-off between assimilation results and numerical complexity sins more complicated models lead to the same results illustrated in Section 5. With a view to evaluating the EOF-based covariance parameterization, we consider both PB-NNKF-EOF and PB-NNKF schemes.  
We perform a quantitative analysis of the interpolation performance of the proposed scheme with respect to an optimal interpolation, and the EOF based interpolation method VE-DINEOF [11] which are two of the most popular techniques in spatio-temporal fields interpolation. Furthermore, in order to provide a comparison to an other data-driven data assimilation technique, we also tested the interpolation technique based on analog forecasting. Overall, the considered parameter setting is as follows:  
* Optimal interpolation (OI): We use a Gaussian kernel with a spatial correlation length of 100 km and a temporal resolution length of 3 days. These parameters were empirically tuned for the considered dataset using a cross-validation experiment.
* Analog data assimilation (Local Analog Forecasting(LAF)-EnKF, Global Analog Forecasting(GAF)-EnKF): We apply both the global and local analog data assimilation schemes, referred to as GAF-EnKF, LAF-EnKF [19; 22]. Similarly to the proposed scheme, we consider \(20\times 20\) patches and 50-dimensional EOF decomposition with an overlapping of 10 pixels. We let the reader refer to [19; 22] for a detailed description of this data-driven approach, which relies on nearest-neighbor regression techniques.
* EOF based reconstruction (PB-VE-DINEOF): We also compare our approach to the state-of-the-art interpolation scheme based on the projection of our observations with missing data on an EOF basis [11]. The SST field is here decomposed as described in the analog data assimilation application into a collection of \(20\times 20\) patches with a 10 pixels overlapping. Each patch is then reconstructed using the VE-DINEOF method.