## 3 Methodology
### Machine Learning-Based Downscaling Models  
Three machine learning algorithms including Cubist, Support Vector Regression (SVR) and Random Forest (RF) were examined in this study. Cubist is a software package that implements a modified regression tree algorithm, which combines instance- and model-based techniques to create rule-based multivariate regression models from training data [52]. It allows overlap between multivariate regression models, thus predictions from multiple rules are averaged to produce the final output. Cubist has been widely used in different applications of remote sensing, such as biomass estimation, impervious surface quantification, drought monitoring, water quality estimation, and land cover estimation [53; 54; 55; 56].  
SVR is a regression version of Support vector machines (SVMs). The basic principle of SVM is that independent variables are mapped to a higher dimensional space where nonlinear correlations are transformed to be linearly separable using kernel functions, and the optimal linear separator is selected to maximize the margin between classes [57]. SVR, as an extension of SVM, uses the same principles as SVM for classification, while it can produce real numbers as output given a set of independent input variables by defining a margin of tolerance (\(\epsilon\)). Both SVM and SVR have been proven robust for classification and the estimation of continuous properties in remote sensing [58; 59; 60; 61; 62; 63; 64]. In this study, LibSVM (version 3.20) tools developed by [PERSON] and [PERSON] [65] were adopted to implement SVR, for which the radius basis function (RBF) kernel was used and the optimal combination of parameters such as cost, gamma and \(\varepsilon\) were selected using a grid search approach.  
RF is an improved algorithm based on a classification and regression trees (CART) method, by combining a large set of CARTs in an ensemble. It has been widely used for various remote sensing applications for both classification and regression [66; 67; 68]. Compared to the CART algorithm, RF uses a bootstrap aggregating technique to improve model performance. Each tree is built from a random subset of training data with a random subset of predictor variables, and the aggregation of output from many trees may smooth the variance between trees and yield more accurate prediction results [69]. Variable importance is obtained by the percentage increase in the mean squared error (MSE), which is calculated using out-of-bag (OOB) data when each variable is permuted. In this study, randomForest add-on package provided in R statistical software was used to implement the method.  
Using the eleven upscaled 1 km Landsat indices as predictor variables and MODIS 1 km ET product as the response variable, prediction models were established based on Cubist, SVR and RF algorithms, respectively. The developed models were then applied to the eleven 30 m Landsat indices in order to generate 30 m ET product. In order to evaluate the model performances, ten-fold cross validation with fixed subsets of samples was conducted for each model, and the average RMSE and relative RMSE (rRMSE) as a percentage of mean MODIS 8-day ET of validation datasets were assessed and compared. The developed machine learning models were then applied to the original 30 m Landsat indicator layers to predict downscaled ET at 30 m resolution. The models were also applied to the upscaled indicators to create Landsat-modeled 1 km ET. The downscaled 30 m ET was then compared with MODIS 1 km ET and Landsat modeled 1 km ET in terms of the ET spatial patterns. The predicted 30 m ET was aggregated to 1 km resolution and compared to MODIS ET. All four products, including MODIS 1 km ET, Landsat-downscaled 30 m ET, Landsat-modeled 1 km ET, and Landsat-aggregated 1 km ET were assessed using _in situ_ ET measurements.