*3.5. Performance Metrics and Evaluation*

The performance of models and the retrieval accuracy of downscaled SM are evaluated in this paper using three indicators: Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and correlation coefficient (*R*). MAE is calculated as the average of the absolute differences between each observation and the mean. This method circumvents the issue of error cancellation, thereby providing a more accurate representation of the actual prediction error magnitude. RMSE is commonly used as a standard to measure the prediction results of machine learning models. The *R* can be used to measure the degree of correlation between two variables. The calculation formulas for the three indicators are as follows:

$$\text{MAE} = \frac{1}{n} \sum\_{i=1}^{n} |X\_i - Y\_i| \tag{8}$$

$$\text{RMSE} = \epsilon \sqrt{\frac{1}{n} \sum\_{i=1}^{n} (X\_i - \mathbf{Y}\_i)^2} \tag{9}$$

$$R(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var[X]Var[Y]}} \tag{10}$$

where *n* represents the amount of data used for modeling, *X* is the reference value of SM, and *Y* is the retrieved value of SM. These three values are crucial for us to evaluate the

prediction accuracy of the model. Among them, *X* is the known true value, and *Y* is the value predicted by our models. To more accurately evaluate the performance of the model, we introduce several key statistical indicators. Among them, *C*ov(*X*,*Y*) represents the covariance of *X* and *Y*, which describes the degree of joint variation of *X* and *Y*. At the same time, *Var*[*X*] is the variance of *X*, and *Var*[*Y*] is the variance of *Y*. These two indicators describe the range of variation of *X* and *Y*, respectively. These three indicators jointly evaluate the performance and prediction accuracy of the model. Covariance describes the correlation between the model's predicted values and the true values, while variance shows the dispersion of the data. The changes in these data directly affect the predictive ability of the model.
