## 3 Results  
### Spatial Downscaling of the Landsat Dataset  
The \(R^{2}\) values of three regression models for TC estimation are listed in Table 4. Regardless of the magnitude of errors, RF achieved the highest explanatory power, followed by SVR. LR showed the lowest explanatory power. The addition of errors resulted in a decrease in the explanatory power for all three regression models.  
Table 5 summarizes the accuracy assessment results for different error levels and spatial downscaling results for three regression models. For all cases, accuracy decreased as input errors increased, regardless of regression models and types of spatial downscaling results. For the LR model, the ATPRK result (C2) achieved the best accuracy for error-free input data (E0), indicating that residual correction could improve predictive performance when the input data have no or few errors. However, when the input coarse resolution data were contaminated by severe noise, residual correction led to the worst predictive performance (E3). Notably, the regression-based predictions of LR and SVR (C1_LR and C1_SVR) were less sensitive to the magnitude of input errors. For the RF and SVR models, the TC normalization yielded the best predictive accuracy for error-free input data (E0). As the input errors increased, the TC normalization could not improve accuracy, yielding the worst accuracy for the input data with severe errors (E3). Furthermore, the use of the TC estimate as a downscaling result (C1) could achieve better accuracy than the ATPRK results with residual correction (C2) and with normalization (C3). When comparing the prediction performance of the three regression models, the application of SVR for the TC estimation achieved the best accuracy for all input error levels. Although RF showed the highest explanatory power (Table 4), the higher explanatory power of RF did not always lead to better prediction accuracy. Instead, overfitting to the error-contaminated input data might degrade the predictive performance.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Dataset** & **LR** & **RF** & **SVR** \\ \hline E0 & 60.43 & 90.36 & 76.72 \\ E1 & 54.07 & 86.66 & 68.71 \\ E2 & 41.31 & 79.63 & 62.63 \\ E3 & 21.13 & 68.33 & 47.04 \\ \hline \end{tabular}
\end{table}
Table 4: Coefficient of determination (\(R^{2}\)) values of three regression models for the Landsat dataset (unit: %).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Error Level** & **Case** & **MAE** & **MRAE** & **Error Level** & **Case** & **MAE** & **MRAE** \\ \hline \multirow{8}{*}{E0} & C1\_LR & 0.019 & 13.86\% & & C1\_LR & 0.019 & 13.89\% \\  & C2\_LR & 0.013 & 9.31\% & & C2\_LR & 0.021 & 15.28\% \\  & C1\_RF & 0.015 & 11.33\% & & C1\_RF & 0.017 & 13.23\% \\  & C2\_RF & 0.013 & 9.79\% & & C2\_RF & 0.018 & 13.63\% \\  & C3\_RF & 0.012 & 8.92\% & & C3\_RF & 0.021 & 15.31\% \\  & C1\_SVR & 0.015 & 11.03\% & & **C1\_SVR** & **0.015** & **11.07\%** \\  & C2\_SVR & 0.012 & 8.92\% & & C2\_SVR & 0.020 & 14.74\% \\  & C3\_SVR & **0.012** & **8.55\%** & & C3\_SVR & 0.020 & 14.47\% \\ \hline \end{tabular}
\end{table}
Table 5: Accuracy statistics of all cases for the Landsat dataset (MAE: mean absolute error; MRAE: mean relative absolute error). The best case for each error level is shown in bold. The case name can be referred to in Tables 2 and 3.  
In summary, when the magnitude of input data errors was small, the residual correction in ATPRK was effective, and TC normalization was required for the nonlinear regression model. In contrast, using the TC estimate as a downscaling result was more effective than the ATPRK result when the input data were severely contaminated by errors.