## Materials and Methods
### Method of Predicitng Thermal Conductivity of Sandy Soil in the Near-Phase-Transition Zone
\begin{array}{l}\lambda_{T}=a_{1}+b_{1}T_{e}+c_{1}\lambda_{u}+d_{1} \lambda_{f}\left(\text{{Stable phase transition zone}}\right)\\ \lambda_{T}=a_{2}T_{e}^{b_{2}}+c_{2}\lambda_{u}+d_{2}\lambda_{f}\left(\text{{D drastic phase transition zone}}\right)\end{array} \right., \tag{5}\]  
where \(\lambda_{T}\) is the soil thermal conductivity in the near-phase-transition zone; \(T_{e}\) is the excess temperature, which is calculated according to \(T_{e}=T-T_{f}\); \(\lambda_{u}\) is the unfrozen soil thermal conductivity; \(\lambda_{f}\) is the frozen soil thermal conductivity; and \(a\), \(b\), \(c\) and \(d\) are all fitting coefficients. The RBFNN model was adopted in this work due to its strong local approximation, fast learning convergence speed, low computational burden and extensive generalization ability in order to establish a soil thermal conductivity prediction model in the near-phase-transition zone [43; 44]. The RBFNN model has a three-layer structure, involving an input layer, a hidden layer and an output layer. Considering the relatively small amount of sampling data, the \(K\)-fold cross-validation method was applied to obtain the optimal model. The following Gaussian kernel function was used as the activation function in the RBFNN model:  
\[R(x_{p}-c_{i})=\exp\biggl{(}-\frac{1}{2\sigma^{2}}\left\|x_{p}-c_{i}\right\|^ {2}\biggr{)}, \tag{6}\]  
where \(x_{p}\) is the input unit; \(c_{i}\) is the central vector; and \(\sigma\) is the width parameter of the function. The least squares method was chosen as the loss function formula:  
\[\sigma=\frac{1}{p}\sum_{j}^{m}\left\|d_{j}-y_{j}c_{j}\right\|^{2}, \tag{7}\]  
where \(p\) is the number of input samples; \(d_{j}\) is the expected output value; and \(y_{j}\) is the output unit. Similarly, the factors of \(T_{e}\), \(\lambda_{u}\) and \(\lambda_{f}\) were selected as the predictors of the input layer nodes of the RBFNN model. \(\lambda_{T}\) (for drastic and stable phase transition zones) was set as \begin{table}
\begin{tabular}{c c c c c c} \hline
**Temperature Zone** & **Related** & **Correlation** & \(\lambda_{T}\) & \(T_{e}\) & \(\lambda_{u}\) & \(\lambda_{f}\) \\ \hline \multirow{4}{*}{Drastic phase transition zone} & \(\lambda_{T}\) & \(p\) & \(1.000\) & \(-0.685\) & \(0.462\) & \(0.912\) \\  & \(\text{{Sig.}}\) & — & \(0.000\) & \(0.000\) & \(0.000\) \\ \cline{2-6}  & \(T_{e}\) & \(p\) & \(-0.685\) & \(1.000\) & \(-0.256\) & \(-0.278\) \\  & \(\text{{Sig.}}\) & \(0.000\) & — & \(0.013\) & \(0.004\) \\ \cline{2-6}  & \(\lambda_{u}\) & \(p\) & \(0.462\) & \(-0.256\) & \(1.000\) & \(0.889\) \\  & \(\text{{Sig.}}\) & \(0.000\) & \(0.013\) & — & \(0.000\) \\ \cline{2-6}  & \(\lambda_{f}\) & \(p\) & \(0.912\) & \(-0.278\) & \(0.889\) & \(1.000\) \\  & \(\text{{Sig.}}\) & \(0.000\) & \(0.004\) & \(0.000\) & — \\ \hline \end{tabular} \({}^{1}\)\(p\) represents partial correlation coefficient; _Sig._ represents significance coefficients.  
\end{table} 
Table 4: _Cont._the output layer, and the hidden layer was assigned a two-layer structure, with 13 nodes in the first layer and 10 nodes in the second layer. We compared the prediction accuracy of the RBFNN model with different combinations of node numbers in each hidden layer, and finally determined that the ideal number of nodes was 13 for the first layer and 10 for the second layer. Then, all of the test results were randomly divided into a 7:3 ratio, with 70% of the test data used for training samples and the remaining 30% used to validate the RBFNN predictive model. A schematic diagram of RBFNN model's structure is shown in Figure 5.  
To analyze the accuracy of the MR and RBFNN prediction models, three statistical indexes, namely the mean absolute error (_MAE_), root mean square error (_RMSE_), and determination coefficient (\(R^{2}\)), were used as evaluation indicators.