## Proposed Interpolation Model
### Data Assimilation Procedure  
Given a trained patch-based NN representation as described in the previous section, we derive the associated Kalman-like filtering procedure. As summarized in Algorithm 1, at time step \(t\), given the Gaussian approximation of the posterior likelihood \(P(x_{t-1}|y_{0},\ldots,y_{t-1})\) with mean \(x_{t-1}^{+}\) and covariance \(\Sigma_{t-1}^{+}\), we first compute the forecasted Gaussian approximation at time \(t\) with mean field \(\mathcal{F}(x_{t-1}^{+})\) and patch-based covariance \(\mathcal{F}_{\Sigma}(x_{t-1}^{+},\Sigma_{t-1}^{+})\). The assimilation of the new observation \(y_{t}\) is performed at a patch-level. For each patch \(\mathcal{P}_{s}\), we update the patch-level mean \(x_{\mathcal{P}_{s}t}^{+}\) and covariance \(\Sigma_{\mathcal{P}_{s}t}^{+}\) using Kalman recursion (8) with observation \(y_{\mathcal{P}_{s}t}\). We then combine these patch-level updates to obtain global mean \(x_{t}^{+}\) and covariance \(\Sigma_{t}^{+}\). Whereas we compute global mean \(x_{t}^{+}\) using trained reconstruction network \(\mathcal{F}_{r}\), \(\Sigma_{t}^{+}\) just comes to store the collection of patch-level covariances. This procedure is iterated up to the end of the observation sequence.  
```
1:procedurePB-NNKF(\(\mathcal{F}\),\(\mathcal{F}_{\Sigma}\),\(y\),\(R\))
2:for\(t\)in \([0,...,T]\):
3:\(x_{t}^{-}\leftarrow\mathcal{F}(x_{t-1}^{+})\)
4:\([\Sigma_{\mathcal{P}_{0},t^{\prime}}^{-},...,\Sigma_{\mathcal{P}_{N_{p}},t}^ {-}]\leftarrow\mathcal{F}_{\Sigma}(x_{t-1}^{+},\Sigma_{t-1}^{+})\)
5:\([x_{\mathcal{P}_{0},t}^{-},...,x_{\mathcal{P}_{N_{p}},t}^{-}]\gets Extract Patches(x_{t}^{-})\)
6:\([y_{\mathcal{P}_{0},t},...,y_{\mathcal{P}_{N_{p}},t}]\gets Extract Patches(y_{t})\)
7:for\(s\)in \([1,...,N_{p}]\):
8:\(K_{\mathcal{P}_{s}t}=\Sigma_{\mathcal{P}_{s},t}^{-}H_{\mathcal{P}_{s},t}^{ \dagger}[H_{\mathcal{P}_{s},t}\Sigma_{\mathcal{P}_{s},t}^{-}H_{\mathcal{P}_{ s},t}^{\dagger}+R_{t}]^{-1}\)
9:\(X_{\mathcal{P}_{s},t}^{+}=\Sigma_{\mathcal{P}_{s},t}^{-}+K_{\mathcal{P}_{s},t}[ y_{\mathcal{P}_{s},t}-H_{\mathcal{P}_{s},t}x_{\mathcal{P}_{s},t}^{-}]\)
10:\(\Sigma_{\mathcal{P}_{s},t}^{+}=\Sigma_{\mathcal{P}_{s},t}^{-}-K_{\mathcal{P}_{ s},t}H_{\mathcal{P}_{s},t}\Sigma_{\mathcal{P}_{s},t}^{-}\)
11:\(x_{t}^{+}\gets Reconstruct([x_{\mathcal{P}_{0},t}^{+},...,x_{\mathcal{P} _{N_{s}},t}^{+}])\)
12:\(\Sigma_{t}^{+}\gets Reconstruct([\Sigma_{\mathcal{P}_{0},t}^{+},..., \Sigma_{\mathcal{P}_{N_{s}},t}^{+}])\)
```  
**Algorithm 1** Patch-based NNKF reconstruction  
Compared with the patch-based analog data assimilation [22], it might be noted that we iterate patch-level assimilation steps and global reconstruction steps thanks to the NN-based propagation of the patch-based covariance structure. This procedure potentially allows information propagation from one patch to neighboring ones after each assimilation step. By contrast, in the patch-based analog data assimilation, each patch is processed independently, such that no such information propagation can occur. This is regarded as a key feature to account for the propagation of geophysical structures (e.g., fronts, eddies, filaments,...).  
We refer to the patch-based NNKF reconstruction model using the EOF block-diagonal parameterization of the covariance model \(\mathcal{F}_{\Sigma}\), as model PB-NNKF-EOF. The model using the diagonal parameterization of the covariance model \(\mathcal{F}_{\Sigma}\) in the patch space is referred to as PB-NNKF.