## 1 Introduction  
The European Space Agency's Sentinel-2 (S2) satellite constellation comprises two satellites that carry arrays of multispectral sensors that cover 13 bands in the visible, near-infrared, and shortwave-infrared part of the spectrum. Spectral characteristics of Sentinel-2 A and B sensors can be assumed to be roughly identical for practical purposes and cover the same wavelengths and resolutions [1]. Four bands have 10 m spatial resolution, bands B2, B3, and B4 in the visible spectrum and B8 in the near-infrared spectrum. Bands B1, B9, and B10 have 60 m resolution. The remaining bands, B5, B6, B7, B8A, B11, and B12, have 20 m resolution. As Figure 1 illustrates, bands B9, B10, and B12 cover low transmittance wavelengths which makes them harder to process. B10, the cirrus band, is a water absorption band and is usually ignored outside of cloud cover detection applications, as it does not contain surface information. Some multispectral satellites, for instance, Landsat-7, capture a high-resolution panchromatic band alongside lower resolution bands of varying spectra which can be used for pansharpening, i.e., transferring detail from the high resolution panchromatic band to the lower resolution bands. S2 does not have high-resolution panchromatic sensors, so pansharpening methods can not be directly applied to S2 data. It is necessary to use S2's other high-resolution bands when generating super-resolution versions of low-resolution bands, which complicates the process somewhat. Popular pansharpening methods have still been applied to S2 data by selecting or generating high resolution bands, for example, in [2]. Methods have also been developed specifically for super-resolution of S2 images, and considerable prior work is available on the subject. Model based methods explicitly define a model of image degradation to be reversed based on prior knowledge of the image acquisition system, while machine learning based methods learn it directly from the data. Numerous methods are based on a variational framework which is based on projections onto a subspace justified by a linear sensor model: Super-resolution for multi-spectral multiresolution estimation (SupReME) [4], Sentinel-2 Sharpening (S2 Sharp) [5], Multiresolution Sharpening Approach (MuSA) [6], and Sentinel-2 Super-resolution via Scene-adapted Self-similarity (SSSS) [7]. Area-To-Point Regression Kriging (ATPRK) [8] and Sen2 res [9] use somewhat different approaches. ATPRK combines regression and area-to-point kriging to predict higher resolution representations of the low-resolution bands. The point spread function (PSF) of the sensor, size of pixels, and spatial correlation are used for predictions based on the high-resolution bands [8]. The Sen2 res method explicitly encodes geometric details from available high-resolution bands as pixel properties independent from their reflectance. It attempts to preserve the spectral content of each low-resolution band independently from the geometry and combine the two to solve the super-resolution problem [9]. A number of deep learning based methods have been proposed for S2 super-resolution. These include Deep Sentinel-2 (DSen2) [10] which was trained at reduced resolution on a random selection of 60 large scenes (\(\sim\)110 \(\times\) 110 km\({}^{2}\)). A Deep Residual Network that fuses the fine and coarse spatial resolution bands is suggested in [11]. Sentinel-2 Symmetric Skip Connection convolutional neural network (S2 SSC) [12] extends on this but uses an unsupervised single image learning (zero-shot) approach in which the training and testing are performed on the same image. The only training input is the image for which a super-resolved version is desired. The Remote Sensing Very Deep Super-Resolution [13] (RS-VDSR) network retrains previously developed deep convolutional networks from [14; 15] to work with S2 data with promising results.