## Materials and Methods
### Method of Predicitng Thermal Conductivity of Sandy Soil in the Near-Phase-Transition Zone  
Previous investigations have shown that indicators such as excess temperature (\(T_{e}\)), unfrozen soil thermal conductivity (\(\lambda_{u}\)), and completely frozen soil thermal conductivity (\(\lambda_{f}\)) have a significant influence on the soil's thermal conductivity in the near-phase-transition zone (\(\lambda_{T}\)) [42]. A partial correlation analysis of the above factors was performed, and the results are shown in Table 4. The results show that the thermal conductivity of the sand sample had certain correlations with \(T_{e}\), \(\lambda_{u}\) and \(\lambda_{f}\) both in the drastic and stable phase transition zones (see Section 3.1 for details). The partial correlation coefficients between \(\lambda_{T}\) and \(T_{e}\), \(\lambda_{u}\) and \(\lambda_{f}\) were \(-\)0.772, 0.228 and 0.934 in the stable phase transition zone, and \(-\)0.685, 0.462, 0.912 in the drastic phase transition zone, respectively. Furthermore, the significance coefficients were all less than 0.01, which indicates that the above three factors can be used to predict the soil's thermal conductivity in the near-phase-transition zone. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Temperature Zone** & **Related** & **Correlation** & **\(\lambda_{T}\)** & \(T_{e}\) & \(\lambda_{u}\) & \(\lambda_{f}\) \\ \hline \multirow{8}{*}{Stable phase transition zone} & \(\lambda_{T}\) & \(p\)\({}^{1}\) & 1.000 & \(-\)0.772 & 0.228 & 0.934 \\  & & _Sig._\({}^{1}\) & — & 0.000 & 0.000 & 0.000 \\ \cline{1-1} \cline{2-6}  & \(T_{e}\) & \(P\) & \(-\)0.772 & 1.000 & \(-\)0.232 & \(-\)0.319 \\ \cline{1-1} \cline{2-6}  & \(T_{e}\) & \(Sig._{}\) & 0.000 & — & 0.002 & 0.000 \\ \cline{1-1} \cline{2-6}  & \(\lambda_{u}\) & \(P\) & 0.228 & \(-\)0.232 & 1.000 & 0.876 \\ \cline{1-1} \cline{2-6}  & \(\lambda_{f}\) & \(Sig._{}\) & 0.000 & 0.002 & — & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Partial correlation analysis between \(\lambda_{T}\) and \(T_{e}\), \(\lambda_{u}\) and \(\lambda_{f}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Subsystem** & **Indicator Type** & **Parameter Range** \\ \hline Temperature control system & Range & \(-\)30 \({}^{\circ}\)C\(-\)120 \({}^{\circ}\)C \\ Accuracy & \(\pm\)0.01 \({}^{\circ}\)C \\ Permanent magnet & Magnetic field intensity & 0.3 \(\pm\) 0.05 T \\ Magnet temperature & 25 \({}^{\circ}\)C\(-\)35 \({}^{\circ}\)C \\ Pulse frequency range & 1\(-\)30 MHz \\ Radio frequency unit & Frequency control accuracy & 0.1 Hz \\ Pulse accuracy & 100 ns \\ Data acquisition system & Sampling frequency & 50 MHz \\ Frequency resolution & \(7\times 10^{-7}\) Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the NMR testing system. The multivariate regression (MR) fitting method was used to predict the soil's thermal conductivity in the near-phase-transition zone. The unfrozen water content and thermal conductivity in frozen soil in the near-phase-transition zone changed drastically close to the phase transition temperature and were stable at low temperatures. Thus, the MR fitting formula for the thermal conductivity of frozen soil in the near-phase-transition zone was set as a piecewise function of temperature, namely stable and drastic phase transition zones. In order to select an appropriate multivariate regression function, fitting of the curve between \(\lambda_{T}\) and the above three factors was performed. It was found that the soil thermal conductivity in the stable phase transition zone was linearly related to these three factors, while in the drastic phase transition zone, the thermal conductivity was linearly related to the frozen and unfrozen soil thermal conductivities and exponentially related to the excess temperature. The MR prediction formulas of \(\lambda_{T}\) are as follows:  
\[\left\{