## 2 Methods
### Tuning Parameter Selection  
Since ground truth data is not available for real-world applications of S2 super-resolution, tuning parameter selection is performed using [PERSON]'s protocol [24]. The process is outlined in the flow graph in Figure 2. Two reduced resolution images are generated, with all bands at both 2\(\times\) and 6\(\times\) original pixel size for each image, respectively. Parameters are selected with Hyperopt and used for super-resolution of the full scale data to get the final image. For small images, padding the data may be necessary so that all methods can process the reduced results during the tuning parameter search. Padding is performed using reflective boundary conditions. Appropriate filters are applied to their respective bands to simulate the PSF before downsampling by a factor of 2 and 6, respectively, resulting in 2\(\times\) and 6\(\times\) pixel scales. S2 sensor data from [49] is used to estimate the variance of the PSF of each S2 band based on the measured modulation transfer function (MTF) values for each band. The standard deviation is calculated as \(\sigma=s(\frac{-2\ln(\text{MTF})}{\pi^{2}})^{1/2}\), where \(s\) is the scale difference. Tuning parameters are selected with Bayesian optimization (BO) using the TPE algorithm of the Hyperopt Python package [25; 26; 27]. The parameters that result in the highest average SRE are found using 150 iterations of optimization. During each iteration, the method being tuned is run twice with the parameters suggested by the TPE algorithm, super-resolving the appropriate bands of both reduced images back to the resolution of the original data, \(\mathbf{\mathrm{Y}}_{20m}^{40m}\rightarrow\mathbf{\mathrm{Y}}_{20m}^{{}^{\prime}}\) and \(\mathbf{\mathrm{Y}}_{60m}^{360m}\rightarrow\mathbf{\mathrm{Y}}_{60m}^{{}^{\prime}}\). SRE is calculated between the original data and \(\mathbf{\mathrm{Y}}_{20m}^{{}^{\prime}}\) and \(\mathbf{\mathrm{Y}}_{60m}^{{}^{\prime}}\) and the average SRE is used as a loss for the optimization process. The tuning parameters that give the highest SRE, selected in this manner, at a reduced scale, can then be applied to super-resolve the original data to 10 m resolution. This parameter tuning method has previously been shown to improve the results obtained using S2 Sharp [28]. BO attempts to build a probabilistic surrogate model from previous evaluations that allows it to predict which parameters might be ideal. It works best for search spaces with moderate dimensions [50]. For a specific function and a given set of parameters, BO treats the objective function as a black box and attempts to find the parameters that minimize (or maximize) the output. The aim is to find a good set of parameters as quickly as possible [25]. Tunable parameters and the objective or loss function can be defined to suit dissimilar tasks. For super-resolution using a quantitative image quality metric is appropriate. Tuning the parameters of a method for individual images can, in some cases, improve results dramatically. Different tuning parameters may also be needed for different image subjects. Remote sensing images can differ greatly depending on the subject, e.g., whether the area being captured is urban or rural, mountainous or oceanic, tropical or arctic, etc. The TPE, or Tree-structured Parzen Estimator algorithm, is a hyperparameter optimization algorithm suggested in [26]. Standard BO using Gaussian Processes models the conditional probability distribution \(p(y|x)\), the probability of output \(y\) given input  
Figure 2: Flow graph showing the parameter tuning process. Scale reduced data is super-resolved back to its original resolution and evaluted against the unscaled data for parameter selection. The parameters are then used to super-resolve the unscaled data to 10 m resolution. \(x\), directly. TPE models \(p(x|y)\), the probabilty of input \(x\) given output \(y\) and \(p(y)\), the probabilty distribution of \(y\).