## 3 Results
### Thermal Response Transfer Process  
If air temperature is continuously transferred to the ground, temperature changes will occur at shallow depths in the soil. According to [PERSON] et al. [21], temperature changes in a certain area will induce increases in the size of the heat-transferred zone and decrease that of the zone lacking heat transfer. Assuming that the heat transferred by air temperature can be expressed as the cumulative temperature transfer, the thermal response to heat transferred to a soil can be characterized by the cumulative air temperature. Therefore, cumulative air temperature (\(CAT_{tr,m}\)), which is the thermal response measured for \(m\) days using Equation (1), can be defined as follows:  
\[CAT_{tr,m}=\sum\limits_{t=1}^{m}\left(AT_{tr,t}-T_{ref}\right) \tag{2}\]  
where \(AT_{tr,t}\) is the daily mean air temperature and \(T_{ref}\) is the reference temperature (0 \({}^{\circ}\)C in this study). Normalized cumulative air temperature, \(CAT^{*}{}_{tr,m}\), is used to express the cumulative air temperature measured in the atmosphere from the start of the measurement period to \(m\) days, as a uniform thermal response can be expressed as:  
\[CAT^{*}{}_{tr,m}=\frac{CAT_{tr,m}}{CAT_{tr,m_{\max}}}=\frac{\sum\limits_{t=1}^{ m}\left(AT_{tr,t}-T_{ref}\right)}{\sum\limits_{t=1}^{m_{\max}}\left(AT_{tr,t}-T_{ ref}\right)} \tag{3}\]  
where \(m_{\max}\) is the number of days (160 days in this study) of the entire measurement period (i.e., the total number of samples for the daily mean temperature). On the basis of the \(m\)th measurement day, the expected future normalized thermal response, \(RCST^{*}{}_{tr,m}\) (where RCST represents Reverse Cumulative Soil Temperature), for the rest of the measurement period (\(m_{\max}-m\) days) can be defined using the \(CAT^{*}{}_{tr,m}\) for the measurement period of \(m\) days as:  
\[RCAT^{*}{}_{tr,m}=1-CAT^{*}{}_{tr,m} \tag{4}\]  
The relative expression for representing the portion of net heat transfer is denoted as \(RCAT^{*}{}_{tr,m}\) (where RCAT denotes Reverse Cumulative Air Temperature), and indicates the future normalized thermal response. In other words, and \(RCAT^{*}{}_{tr,m}\) of unity (i.e., equal to 1) indicates that the net heat transfer (cumulative air temperature) for the first \(m\) days is zero. When \(RCAT^{*}{}_{tr,m}=0\), this implies the maximum cumulative air temperature and no additional net heat transfer will occur after the \(m_{\max}\) measurement day.  
Thermal response was defined in this study based on the air temperature because air temperatures greatly influence soil temperatures. Air temperature has been measured over a long period of time and in various regions. However, there has been relatively little effort made to measure soil temperatures, as the measurement of soil temperature is more difficult than that of air temperature [33; 34]. It is because and involves considerably higher costs and time investments [17; 26]. Therefore, a model for predicting soil temperatures uses air temperatures may play an important and effective role in assessing the changes in soil conditions related to global climate change and local disturbances.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Depth** & **Location** & **Mean \(w\)** & **Standard deviation of \(w\) (w\({}_{\text{SD}}\))** \\ \hline \multirow{3}{*}{15 cm} & M1 & 10.63 & 4.07 \\  & M2 & 5.33 & 1.91 \\  & M3 & 20.62 & 6.11 \\  & M4 & 31.10 & 5.84 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Means and standard deviations of measured soil water contents (\(w\)).