## 2 Methods
### Architecture of a Weighted Deep Ensemble for the Super-Resolution of Satellite Data: WDESen2  
The weighted deep ensemble (WDESen2) architecture and its extensions, shown in Figure 1, use the architecture of the DSen2 network [4] as a backbone. By ensembling and weighting the predictions, WDESen2 improves the predictive accuracy compared to DSen2. The selective prediction applied to the predictive uncertainties from WDESen2, which are given as variances according to (6), is referred to as variance-based WDESen2 (VWDESen2). The selective prediction applied on the kernel projection is referred to as kernel-based WDESen2 (KWDESen2). The proposed methods are described below. The ensemble members of WDESen2 are the DSen2 networks that super-resolve the B and C bands to 10 m GSD. For this, two networks are used: a CNN, \(\mathbb{S}_{\times 6}\), which performs the \(6\times\) super-resolution of the C bands from 60 m to 10 m GSD, and another CNN, \(\mathbf{\Psi}_{\theta}\), which performs the \(2\times\) super-resolution of the B bands from 20 m to 10 m GSD. As the \(6\times\) resolution network training procedure is structurally the same as that of the \(2\times\) resolution network, we restrict ourselves to describing the latter. For training a CNN to super-resolve the B bands, there do not exist ground-truth data with a 10 m GSD for the B bands. In order to generate training data, we followed the approach of [4]. The A bands and B bands are downsampled to 20 m and 40 m GSD, respectively. For each band and patch of size \(W/2\times H/2\), the linear downsampling operator, \(DS:\mathbb{R}^{W\times H}\rightarrow\mathbb{R}^{W/2\times H/2}\), is given below in Figure 2.  
Figure 1: Architecture and pipeline of the KWDESen2 and VWDESen2 methods with \(N\) ensemble members.  
Then, the downsampled 40 m GSD and 20 m GSD B and A bands are used as inputs, and the 20 m GSD B bands can be used as targets in the training procedure. Based on the assumption of the self-similarity of super-resolution from 40 m GSD to 20 m GSD compared with that from 20 m GSD to 10 m GSD, the resulting network can then be used for 2\(\times\) super-resolution from 20 m to 10 m GSD. We let \(y_{A}\in\mathbb{R}^{4\times W\times H}\), where \(W\times H\) is the patch-size, and \(y_{B}\in\mathbb{R}^{6\times W/2\times H/2}\), where \(W/2\times H/2\) is the patch-size, denote the observed intensities for the A and B bands, respectively. The Dsen2 method for 2\(\times\) super-resolution is given by  
\[\Psi_{\theta}:\mathbb{R}^{4\times W\times H}\times\mathbb{R}^{6 \times W/2\times H/2} \rightarrow\mathbb{R}^{6\times W\times H},\] \[(y_{A},y_{b}) \to x_{B},\]  
where the B bands, as depicted in Figure 1, are upsampled to the same resolution as the A bands with _bi-linear interpolation_. The associated linear operator, \(M:\mathbb{R}^{H/2\times W/2}\rightarrow\mathbb{R}^{H\times W}\), is applied to each B band.  
By retraining \(N\) times on the same data splits with the \(\ell^{1}\) loss, due to the stochasticity of the training procedure, we obtain \(N\) different CNNs, as shown in Table 1. As visualized in Figure 1, the prediction of WDSen2 is composed of a weighted mean, according to (5), and the corresponding variance (6).  
The predictions of WDESen2 are post-processed using two different methods, kernel- and variance-based partial prediction, which will be described in the following. For 6\(\times\) super-resolution to have the same dimension inputs in the CNN \(\Psi_{\theta}\), the B and C bands are upsampled. However, this implies that there is no ground truth with which to compare the predictions.