## 4 Validation and Discussion
Through regression analysis, high \(R^{2}\) values (0.769-0.809) between the measured and predicted values were obtained. This result is more also more promising than that of another model previously proposed by [PERSON] et al. [34] and [PERSON] et al. [33]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Location** & \(R^{2}\) & **RMSE** & **MAE** \\ \hline P1 & 0.809 & 0.800 & 0.535 \\ P2 & 0.769 & 2.233 & 1.750 \\ P3 & 0.804 & 1.132 & 0.801 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Correlation coefficients (\(R^{2}\)), root mean square error (RMSE), and mean absolute error (MAE) for validation sites P1–P3. Figure 12: Comparison between measured and predicted temperatures at a depth of 15 cm using additional (validation) data for different locations: (**a**) P1, (**b**) P2, and (**c**) P3. In Figures 11 and 12, P1 and P3 exhibited smaller ranges of variation in their maximum temperatures and soil temperatures than P2, as the soil temperatures were less affected by the dynamic changes in the air temperature. This is because the soil temperatures of P1 and P3 were dominantly affected by the soil water content and properties (i.e., thermal conductivity or diffusivity), and thus the uncertainty of soil temperatures [19] was slightly reduced. Meanwhile, predictions of the soil temperatures at site P2 at a depth of 15 cm exhibited the highest uncertainty, the highest degree of scattering, and the lowest \(R^{2}\) value. Especially for P2, the predicted temperatures deviated from the measured values in late spring and early summer. During this period, the proposed model underestimated soil temperatures when compared to the measured values (Figure 11b). Differences between measured and predicted soil temperatures have also been reported by other researchers (e.g., [PERSON] and [PERSON] [50,51]; [PERSON] et al. [52]). In general, when the soil water content and air temperature increase, soil temperatures increase due to changes in the soil water content, which plays a larger role than temperature alone. The increase in soil water content increases the thermal conductivity of the soil, thereby directly affecting soil temperatures. In contrast, air temperatures indirectly affect soil temperatures as they change the energy available for heating the soil into soil water content [52]. Such dynamics may have caused the differences observed between the parameters measured at M1-M4 and those measured at P2 [33], thereby resulting in underestimated soil temperatures in the late spring and early summer. As shown in Figure 11a,c, the initial portions of soil temperatures at a depth of 15 cm for locations P1 and P3 were closed to 0. In regression analyses, data concentrated near a certain value would produce a bias result or a better correlation between measurements and predictions. For the examination of the effect of using concentrated data on a certain value (here, temperatures near 0 \({}^{\circ}\)C) on model prediction, the first 40 data points from the start of the temperature measurements at locations P1 and P3 were excluded. The same procedure proposed in this study was implemented. The optimal parameters determined from regression analysis are summarized in Table 7. Comparison between the prediction and measurement of soil temperatures at a depth of 15 cm with increasing time is represented in Figure 13. At the beginning of the prediction, the model underestimates the soil temperatures; however, the underestimation trends diminished with increasing time. Figure 14 shows point-to-point comparison between predicted and measured soil temperatures at a depth of 15 cm for locations P1 and P3. From the comparison between \(R^{2}\) values [0.809 in Figure 12a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Location** & \(\mathbf{w_{SD}}\) & \(\boldsymbol{\alpha}\) & \(\boldsymbol{\beta}\) & \(\boldsymbol{CSTRCAT_{15,mmax}}\) \\ \hline P1 & 3.7527 & 4.1847 & 2.5451 & 189.7 \\ P3 & 3.3165 & 4.1698 & 2.5529 & 188.6 \\ \hline \hline \end{tabular}
\end{table}