## 4 A Soil Temperature Model Based on Thermal Response  
### Thermal Response Transfer Process  
During the heat transfer from air to soil or that from the soil surface to lower-depth soil due to radiation, lower-depth soils experience temperature variation followed by a change of the thermal response of soil due to heat transfer. During the process (i.e., at the initial moment) of heat transfer, theoretically or conceptually, the soil medium can be classified into two zones: the heat-transferred zone and the heat-untransferred zone. When continuous heat transfer occurs in the soil medium, the heat-transferred zone increases compared to the heat-untransferred zone and eventually, the heat-untransferred zone will diminish. When heat transfer changes in a timely manner, the portions of the heat-transferred zone increase compared to that of the heat-untransferred zone, and their heat  
Figure 5: Relationships between soil temperature at a depth of 5 cm and those observed at depths of 10, 15, 20, and 30 cm in the (**a**) site A and (**b**) site B.  
changes will have integrated into a thermal response. These thermal responses of soil medium can be alternatively and effectively represented by the cumulative soil temperature. Specifically, soils without temperature change can be defined as having thermal response of zero. This suggests that the transferred zone can be evaluated based on the identification of temperature change, and the degree of thermal response can be assessed from the cumulative soil temperature measured from the reference temperature (i.e., the temperature at the start of measurement or the initial temperature).  
From Equation (1), the cumulative soil temperature (\(CST_{ud,m}\)) for the uppermost layer (at a depth of 5 cm) for duration of \(m\) days can be defined as:  
\[CST_{ud,m}=\sum\limits_{t=1}^{m}\left(T_{ud,t}-T_{ref}\right) \tag{2}\]  
where \(T_{ud,t}\) is the average daily soil temperature at the depth of 5 cm and \(T_{ref}\) is assumed as 0 \({}^{\circ}\)C in this study.  
The normalized cumulative thermal response (\(CST_{ud,m}\)) at the uppermost depth (5 cm) from the start of temperature measurement to time \(m\), in days, can be described as:  
\[CST_{ud,m}^{*}=\frac{CST_{ud,m}}{CST_{ud,m_{\max}}}=\frac{\sum\limits_{t=1}^{m }\left(T_{ud,t}-T_{ref}\right)}{\sum\limits_{t=1}^{m_{\max}}\left(T_{ud,t}-T_{ ref}\right)} \tag{3}\]  
where \(m_{\max}\) is the entire duration of temperature measurement duration (or the entire number of samples of average daily temperature) in days. Then, with respect to the measurement duration of \(m\) days, the expected thermal response (\(RCST_{ud,m}\)) at the uppermost level for the rest of the measurement period (i.e., in this study, this is 160 subtracted by \(m\) days) can be defined as:  
\[RCST_{ud,m}^{*}=1-CST_{ud,m}^{*} \tag{4}\]  
The \(RCST_{ud,m}^{*}\) indicates the portion of future normalized thermal response, which is deduced using the total potential-normalized thermal response (which is \(1-CST_{ud,m}^{*}\)) and the current normalized cumulative thermal response (\(CST_{ud,m}^{*}\)).  
Practically, \(RCST_{ud,m}^{*}\) can be considered, as the remaining normalized thermal response soils are expected to experience. In other words, if \(RCST_{ud,m}^{*}\) is 1, no heat transfer occurs. On the other hand, if \(RCST_{ud,m}^{*}\) is 0, heat is completely transferred to the material and a certain temperature is maintained.