## 2 Methods and Data
### Experimental Design  
#### 2.3.1 Comparison of Regression Models  
Three regression models commonly applied to spatial downscaling, LR, random forest (RF), and support vector regression (SVR) models, were applied to estimate the TC in Equation (3). As the effect of different TC estimates on the predictive performance of ATPRK predictions was primarily explored in this work, ATPK was only employed as a residual correction method in all the comparison experiments, based on our previous study [2]. It should be noted that the comparison of the three regression models did not aim to select the optimal regression model but to compare the behaviors of different regression models with respect to input errors.  
LR is the baseline for the TC estimator in ATPRK-based spatial downscaling. RF, as an ensemble learning method combining tree-based predictors, is relatively robust to outliers and effectively avoids overfitting by maximizing diversity through tree ensembles [30]. Furthermore, RF demands relatively a small number of user-specified hyperparameters [16]. In this study, we optimized two hyperparameters: the number of variables for the best splitting and the number of trees to be grown, by minimizing out-of-bag errors. The feature selection procedure was not considered in this study to ensure that each regression model used the same auxiliary variables. SVR, one of the kernel-based learning methods, is known to effectively model nonlinear relationships for noisy data [31]. In this study, we applied \(\varepsilon\)-SVR with the \(\varepsilon\)-insensitive function for TC estimation. The radial basis  
Figure 2: Two datasets used for spatial downscaling experiments: (**a**) Landsat shortwave infrared band image; (**b**) AMSR-2 soil moisture data in July 2017 (unit: %). The black polylines and dots in (**b**) denote the administrative boundary of South Korea and soil moisture observation sites used for accuracy evaluation, respectively.  
function (RBF), widely utilized in remote sensing data processing [31; 32], was selected as the kernel function. The optimal values of three hyperparameters in \(\varepsilon\)-SVR, including \(\varepsilon\), the regularization parameter, and the gamma of the RBF kernel, were determined through a grid search within appropriate ranges identified in preliminary experiments.  
In addition to ATPRK predictions, two other cases were considered spatial downscaling results, as listed in Table 2. Since the TC estimate in Equation (3) has often been used as a spatial downscaling result in previous studies, the direct use of the TC estimate as the downscaling result (C1) was compared with the ATPRK result obtained after a residual correction (C2). Moreover, the impact of the normalization for TC estimated from nonlinear regression on predictive performance was also explored. More specifically, the normalization of TC estimates (C3) was additionally applied when RF and SVR were applied to estimate TCs.  
All spatial downscaling procedures, including data preparation and accuracy evaluation, were implemented using ENVI software version 5.6 (L3 Harris Technologies, Broomfield, CO, USA), the Scikit-learn library [33], and Python/R programming.