## 4 A Soil Temperature Model Based on Thermal Response
### Determination of the Fitting Parameters of \(A\) and \(Z\) to Evaluate \(CRCST_{i,mmax}\)  
Using the curve fitting procedure (least squares method), the fitting parameters \(A\) and \(Z\) of Equation (7) as functions of depth \(D_{i}\) were determined from regression analysis. The fitting parameters can be represented by using the following equations:  
\[A=3.456(D_{i})^{-0.957}(D_{i})=3.456(D_{i})^{0.043} \tag{11}\]  
\[Z=0.468(D_{i})^{1.469}/(D_{i})=0.468(D_{i})^{0.469} \tag{12}\]  
The fitting parameters of sites A and B at different depths are summarized in Table 1. It should be noted that Equations (11) and (12) were used for up to 30 cm of soil depth. Figure 8a,b illustrate the relationships of \(A\)/\(D_{i}\) and \(Z\)\(D_{i}\) at a depth of \(D_{i}\). The R\({}^{2}\) of the relationships were higher than 0.97.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Depth (cm)** & **Test Site** & **Fitting Parameter \(A\)** & **Fitting Parameter \(Z\)** & \(A\)/\(D_{i}\)** & \(ZD_{i}\) \\ \hline \multirow{2}{*}{10} & A & 3.926 & 1.444 & 0.393 & 14.437 \\  & B & 3.729 & 1.432 & 0.373 & 14.317 \\ \hline \multirow{2}{*}{15} & A & 3.970 & 1.568 & 0.265 & 23.522 \\  & B & 3.836 & 1.638 & 0.256 & 24.569 \\ \hline \multirow{2}{*}{20} & A & 3.942 & 1.705 & 0.197 & 34.099 \\  & B & 3.898 & 1.956 & 0.195 & 39.120 \\ \hline \multirow{2}{*}{30} & A & 3.982 & 2.043 & 0.133 & 61.280 \\  & B & 4.057 & 2.842 & 0.135 & 85.266 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Values of fitting parameters \(A\) and \(Z\).  
Each value of \(CRCST_{i,mmax}\) can be obtained from a curve fitting process (Table 2). Figure 9 illustrates the values of the parameter \(CRCST_{i,mmax}\) for varying soil depths, which is the normalized value of \(CRCST_{i,\,mmax}\) (shown in Table 2) divided by the respective soil depth.  
Figure 9 shows correlations between the maximum cumulative soil temperature (\(CRCST_{i,mmax}/D_{i}\)) normalized by depth and depth (\(D_{i}\)) at a depth of \(i\) cm. Since the R\({}^{2}\) of the \(CRCST_{i,mmax}/D_{i}\) and \(D_{i}\) relationship for different depths was 0.97 or above, the \(CRCST_{i,mmax}\) of each depth can be accurately  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Depth (cm)** & **Test Site** & \(CRCST_{i,mmax}\) & \(CRCST_{i,mmax}/D_{i}\) \\ \hline \multirow{2}{*}{10} & A & 483.3 & 48.3 \\  & B & 606.0 & 60.6 \\ \hline \multirow{2}{*}{15} & A & 399.4 & 26.6 \\  & B & 515.2 & 34.3 \\ \hline \multirow{2}{*}{20} & A & 312.1 & 15.6 \\  & B & 389.4 & 19.5 \\ \hline \multirow{2}{*}{30} & A & 180.3 & 6.0 \\  & B & 207.5 & 6.9 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of fitting parameters \(A\) and \(Z\).  
Figure 8: Correlations between fitting parameters (\(A\) and \(Z\)) and depth (\(D_{i}\)). (a) Relationship between \(A/D_{i}\) and \(D_{i}\); and (b) relationship between \(ZD_{i}\) and \(D_{i}\).  
Figure 9: Correlations between the maximum values of cumulative soil temperature (\(CRCST_{i,mmax}/D_{i}\)) normalized by depth and depth \(D_{i}\).  
evaluated when the depth is given. Then, from Equation (12), the \(CRCST_{i,mmx}\) at a depth of \(i\) cm was calculated as:  
\[CRCST_{i,m_{\max}}=5164.3(D_{i})^{-1.935}(D_{i})=5164.3(D_{i})^{-0.935} \tag{13}\]  
The fitting parameter of \(CRCST_{i,mmx}\) could be obtained without fully considering soil depth or soil conditions. Accordingly, further testing and investigations were needed.