## 4 Validation and Discussion  
Figure 10 shows the air temperatures measured at location \(x\) (Figure 2) and the soil water content measured at a depth of 15 cm at locations P1-P3. The additionally obtained wSD = 6.7723 for P1, wSD = 8.9907 for P2, and wSD = 6.4885 for P3. In accordance with the procedure presented in Section 3.6, the parameters of \(\alpha\), \(\beta\), and \(CSTRCAT_{15,\ mmx}\) were obtained using the wSD values obtained from locations P1-P3 (Table 5). The developed model was implemented using the values shown in Table 5, and the measured soil temperatures were compared with the predicted \(ST_{i,m}\)-\(T_{nf}\) data (Figure 11). Figure 10: Additionally, observed air temperature and soil water contents at sites P1–P3 used for model validation  
As shown in Figure 11, dynamic changes in soil temperatures at a depth of 15 cm over time were effectively predicted. In particular, although the temperatures measured for validation at a depth of 15 cm had different tendencies than those measured for the main investigation at the same depth (Figure 3), the proposed model exhibited excellent predictive performance for all of the three locations. For location P1, the measured mean and maximum soil temperatures were 2.002 \({}^{\circ}\)C and 4.687 \({}^{\circ}\)C, while the predicted values were 1.8768 \({}^{\circ}\)C and 4.7486 \({}^{\circ}\)C, respectively. For location P2, the measured mean and maximum soil temperatures were 5.954 \({}^{\circ}\)C and 11.409 \({}^{\circ}\)C, while the predicted values were 5.1139 \({}^{\circ}\)C and 13.0350 \({}^{\circ}\)C. Finally, for location P3, the measured mean and maximum soil temperatures were 2.747 \({}^{\circ}\)C and 6.165 \({}^{\circ}\)C, while the predicted values were 2.5548 \({}^{\circ}\)C and 6.7359 \({}^{\circ}\)C.  
The difference between the measured and predicted mean soil temperatures was 0.125 \({}^{\circ}\)C for P1 and 0.190 \({}^{\circ}\)C for P3, while the difference between the measured and predicted maximum soil temperatures was 0.061 \({}^{\circ}\)C for P1 and 0.571 \({}^{\circ}\)C for P3. These differences in predicted soil temperatures appear to have been caused by the incompleteness of the model. However, the results indicate that it is possible to effectively predict the soil temperatures of all points for depths up to 15 cm at an adjacent location using three simple fitting parameters if air temperature is measured at a location and the wsd of the adjacent location can be measured. To measure the suitability of the predicted and measured temperature data, the root mean square error (RMSE), mean absolute error (MAE), and \(R^{2}\) were calculated for the corresponding data. The RMSE is the performance indicator of a regression equation in which the RMSE value increases \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Location** & **wSD** & **\(\alpha\)** & **\(\beta\)** & _CSTRCAT15,mmax_ \\ \hline P1 & 6.7723 & 4.2562 & 2.5078 & 195.0 \\ P2 & 8.9907 & 4.2909 & 2.4901 & 197.5 \\ P3 & 6.4885 & 4.2509 & 2.5105 & 194.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Values of parameters for model validation. Figure 11: Measured and predicted temperatures for additional (model validation) data of a depth of 15 cm with increasing time for different locations: (**a**) P1, (**b**) P2, and (**c**) P3. as the error increases. Mean average error is an indicator of whether there are many or few areas that appear to be outliers [48]. Table 6 shows the \(R^{2}\), MAE, and RMSE values of the measured and predicted soil temperatures. As shown, the RMSE value for the data of the proposed model and the measured data ranged from 0.800-2.233, and the MAE ranged from 0.535-1.750. Location P1 exhibited the highest accuracy, as the MAE and RMSE values were 0.535 and 0.743, respectively. The results of P1-P3 exhibited better RMSE and MAE results than those of a previously proposed model [49]. Figure 12 presents a comparison of the measured and predicted soil temperature data with one-to-one data (blue dotted lines) for each location.