## 1 Introduction  
Satellite images of multiple spectral bands acquired from various satellites, such as MODIS, VIIRS, ASTER, Worldview-3, and Sentinel 2, have different spatial resolutions. The resolution varies by a factor of 2-6 based on the instrumentation. This work focuses on the Sentinel 2 satellites, but the proposed methods for super-resolution, WDESen2 and its variants, are not limited to Sentinel 2 data. The constellation consists of two satellites that capture images of the Earth's surface, where each point on the surface is observed for a period of 5 days. Each image contains information from 13 spectral bands, some of which partially overlap, ranging from ultraviolet/visible to infrared. Band B10 is used for atmospheric corrections and was therefore omitted in the data used for this study. The bands are available in several different resolutions: bands B2, B3, B4, and B8 have a GSD of 10 and will be referred to as the _A bands_; B5, B6, B7, B8a, B10, and B11 have a GSD of 20 m and will be referred to as the _B bands_; while bands B1 and B9 have a GSD of 60 m and will be referred to as the _C bands_. The data can be used in various applications, such as hydrology and water resource management [1; 2; 3]. Even though there are reasons for the different spatial resolutions, such as storage and transmission bandwidth restrictions, often it is desirable to have all bands available at the highest spatial resolution. The problem of bringing all bands to a GSD of 10 m can be calledsuper-resolution. In the work of [PERSON] et al. [4], the authors demonstrated that a CNN-based super-resolution method obtained higher accuracy for all super-resolved bands compared to state-of-the-art algorithms. State-of-the-art algorithms for the super-resolution of multi-spectral satellite data can be divided into three types: pan-sharpening per band, as reviewed in [5; 6]; inverting an explicit imaging model, as demonstrated in [7; 8], with a reduced-rank method applied for Sentinel 2 data in [9]; and machine learning approaches, such as [4]. The approaches in the second category are often based on the reformulation of super-resolution as a convex optimization problem by re-projecting each image in a subspace determined by the corresponding algorithm, such as in [9]. These methods rely on linear sensing model assumptions and are computationally expensive, as the super-resolution of each image requires a separate optimization. While being computationally more efficient, the CNN-based approach of [4] also better preserves spectral characteristics, achieves a favorable computational speed when run on GPUs, and is globally applicable for Sentinel 2 data without retraining. Thus, our work focuses on the DSen2 method from [4], which combines the information of the bands with lower and higher resolutions for super-resolution. The method relies on the assumption of the _self-similarity_ of the upsampling process. This means that upsampling from 20 m to 10 m GSD is assumed to have the same mapping as upsampling from 40 m to 20 m GSD. Based on this assumption, in this work we address the features in B bands that cannot be super-resolved by exploiting the correlation between the bands with higher and lower resolutions. This is achieved in two ways: firstly, by studying extensions of the DSen2 method that predict uncertainty, such as WDESen2, and secondly, by studying the kernel of the inverse problem inherent to the proposed super-resolution method. Specifically, predictive uncertainty is evaluated with deep ensembles, as proposed in [10], and based on the success of weighted versions of deep ensembles for classification problems, as in [11; 12], we propose a novel method of weighted deep ensembles for super-resolution. Uncertainty quantification (UQ) methods in DL usually distinguish between epistemic, or model, uncertainty and aleatoric, or data, uncertainty [13]. However, there is an ongoing debate as to whether these two types of uncertainty can be disentangled [14].