## 3 Results
### Soil Temperature Dynamic Model Using CAT\({}^{*}\)\({}_{tr,m}\)  
As shown in Figure 7, a curve representing the relationship between \(CSTRCAT^{*}\)\({}_{15,m}\) and \(CAT^{*}\)\({}_{tr,m}\) can be approximated using a simple exponential function, as follows:  
\[CSTRCAT^{*}\)\({}_{15,m}=1-EXP(-\alpha\cdot CAT^{*}\)\({}_{tr,m}\)\({}^{\beta}), \tag{7}\]  
where \(\alpha\) and \(\beta\) are fitting parameters. In Equation (7), \(\alpha\) and \(\beta\) represent the scale and overall geometry of the relationship between \(CSTRCAT^{*}\)\({}_{15,m}\) and \(CAT^{*}\)\({}_{tr,m}\), respectively. As \(CSTRCAT^{*}\)\({}_{15,m}\) represents a cumulative value in Equation (7), the actual soil temperature at a certain time point can be obtained by differentiating Equation (7). The mathematical difference between \(CSTRCAT^{*}\)\({}_{15,m}\) and \(CSTRCAT^{*}\)\({}_{15,m}\) can be expressed as follows:  
\[\Delta CSTRCAT^{*}\)\({}_{15,m}=CSTRCAT^{*}\)\({}_{15,m}-CSTRCAT^{*}\)\({}_{15,m-1}. \tag{8}\]  
By multiplying all of the items in Equation (8) by the denominator (\(CSTRCAT_{15,mmax}\)) of Equation (6), the former may be modified into the product of the expected normalized thermal response and the current temperature, as follows:  
\[ \begin{array}{ll}CSTRCAT_{15,m_{\max}}\Delta CSTRCAT^{*}\)\({}_{15,m}&=CSTRCAT_{15,m_{\max}}(CSTRCAT^{*}\)\({}_{15,m}-CSTRCAT^{*}\)\({}_{15,m-1})\\ &=CSTRCAT_{15,m}-CSTRCAT_{15,m-1}\\ &=(ST_{i,m}-T_{ref})(RCAT\)\({}_{tr,m})\end{array}  \tag{9}\]  
Summarizing Equation (9) using Equations (3), (4), (6), and (8) for the current temperature, the soil temperature at a depth of 15 cm on the \(m\)th day can be expressed as:  
\[ \begin{array}{ll}ST_{i,m}&=ST_{i,m}-T_{ref}\\ &=CSTRCAT_{15,m_{\max}}\Big{[}EXP(-\alpha\cdot CAT^{*}\)\({}_{tr,m-1}\)\({}^{\beta})-EXP(-\alpha\cdot CAT^{*}\)\({}_{tr,m}\)\({}^{\beta})\Big{]}(1-CAT^{*}\)\({}_{tr,m})^{-1}\end{array}  \tag{10}\]  
In this instance, \(ST_{i,m}\) can be evaluated based on the values of \(\alpha\), \(\beta\), \(CSTRCAT_{15,mmax}\), and \(CAT^{*}\)\({}_{tr,m}\). Determination of the Fitting Parameters, \(\alpha\) and \(\beta\), for Evaluating \(CSTRCAT_{15,mmax}\)  
The proposed predictive model for soil temperature using air temperature should account for the wide spatial heterogeneities of different soil properties (e.g., different properties and thermal responses at each location). To represent this, a property that represents the ground is required. The soil water content was selected to improve soil temperature modeling. The process of defining fitting parameters using the wSD was included in this study. Equations (11)-(13) were employed to identify fitting parameters for which the predicted values could be calculated by comparing the wSD measured at a depth of 15 cm with \(\alpha\), \(\beta\), and \(CSTRCAT_{15,mmax}\). Table 3 shows the values of \(\alpha\) and \(\beta\) in Equation (7) at a depth of 15 cm according to the wSD and Figure 8 shows the \(\alpha\) and \(\beta\) values based on Table 3. Specifically, Figure 8a,b shows \(\alpha\)wSD and \(\beta\)/wSD. In Figure 8, wSD represents the standard deviation of the soil water content for each location. As the Pearson's correlation coefficients (\(R^{2}\)) of \(\alpha\) and \(\beta\) for the wSD were \(>\) 0.967, \(\alpha\) and \(\beta\) can be easily evaluated if only the wSD is known. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Depth** & **Location** & \(\alpha\) & \(\beta\) & \(\alpha\) **(w\({}_{\text{SD}}\)\({}^{\text{4}}\))** & \(\beta\)/(w\({}_{\text{SD}}\)\({}^{\text{4}}\))** \\ \hline \multirow{4}{*}{15 cm} & M1 & 4.2312 & 2.0416 & 1164.6 & 0.0074 \\  & M2 & 4.0853 & 2.7328 & 54.3 & 0.2057 \\ \cline{1-1}  & M3 & 4.0581 & 1.7266 & 5663.9 & 0.0012 \\ \cline{1-1}  & M4 & 4.4143 & 4.3087 & 5119.7 & 0.0037 \\ \hline \end{tabular}
\end{table} 
Table 3: Values of model fitting parameters for the scale (\(\alpha\)) and overall geometry (\(\beta\)) of the relationship between \(CSTRCAT^{*}\)\({}_{15,m}\) and \(CAT^{*}\)\({}_{tr,m}\) with the standard deviation of \(w\) (wSD).