## 4 Pavement Temperature-Evaluation Model
According to the correlation analysis results, the atmospheric temperatures had the strongest correlations with the temperature at depths of 3.5 and 5.5 cm when its time delay was 1 h, whereas for depths of 7.5-11.5 cm, the correlation was the strongest with 2 h of time delay. The atmospheric temperature was most strongly correlated with the temperature at depths of 13.5-15.5, 18.5-20.5, and 22.5-24.5 cm when its time delay was 3, 4, and 5 h, respectively. Thus, the atmospheric temperature with a longer delay was considered to employed for evaluating the temperature at deeper depths. Based on the correlation analysis shown in Table 3, this research introduced a transmission-delay atmospheric temperature (\({}^{\circ}\)C) \(T_{atmo-trans}\) considering temperature-transmission effect (e.g., for evaluating the temperature of depth 3.5 cm at 4 pm, \(T_{atmo-trans}\) should be selected as the atmospheric temperature at 3 pm). \(T_{atmo}\) in Equation (1) was then replaced by \(T_{atmo-trans}\) as shown in Equation (2). In Equation (2), \(c_{1}\)-\(c_{5}\) are the temperature model parameters and these parameters can be obtained by regression analysis of the measured temperatures at various depths. \[T_{Z}=c_{1}\times T_{atmo-trans}-c_{2}\times\sin\biggl{(}c_{3}\times\frac{t}{24 }+c_{4}\biggr{)}+c_{5} \tag{2}\]  
The correlation between measured temperature and estimated temperature using Equation (2) were shown in Table 2. The results showed that the correlation coefficient decreased with increasing depth and the correlation coefficient remained high (0.82) at depth of 24.5 cm. Moreover, the slope and intercept of regression equation between model and measured temperatures were close to 1 and 0, respectively. These results indicated that the estimated temperatures using Equation (2) had a high correlation with measured temperatures. Hence, considering the temperature-transmission effect by the term \(T_{atmo-trans}\) can efficiently estimate the temperature at a deep depth. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**The Estimated Temperature of Equation (1)**} & \multicolumn{2}{c}{**The Estimated Temperature of Equation (2)**} \\ \hline Depth & R\({}^{2}\) & Slope & Intercepts & R\({}^{2}\) & Slope & Intercepts \\
3.5 cm & 0.88 & 1.13 & \(-\)4.41 & 0.88 & 0.95 & 1.98 \\
5.5 cm & 0.83 & 1.05 & \(-\)1.72 & 0.90 & 1.00 & 0.26 \\
7.5 cm & 0.80 & 0.95 & 1.13 & 0.89 & 0.99 & 0.47 \\
9.5 cm & 0.74 & 0.88 & 3.01 & 0.89 & 0.99 & 0.18 \\
11.5 cm & 0.69 & 0.82 & 4.50 & 0.88 & 0.99 & 0.25 \\
13.5 cm & 0.62 & 0.62 & 6.18 & 0.88 & 1.00 & 0.04 \\
15.5 cm & 0.61 & 0.76 & 6.23 & 0.86 & 0.99 & 0.07 \\
18.5 cm & 0.56 & 0.73 & 7.20 & 0.86 & 0.98 & 0.66 \\
20.5 cm & 0.54 & 0.74 & 6.71 & 0.82 & 1.00 & 0.24 \\
22.5 cm & 0.53 & 0.75 & 6.35 & 0.82 & 1.00 & 0.09 \\
24.5 cm & 0.53 & 0.77 & 5.36 & 0.82 & 1.02 & \(-\)0.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation between measured and estimated temperature for various depths. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Delay Time**} & \multicolumn{2}{c}{**0 h**} & \multicolumn{2}{c}{**1 h**} & \multicolumn{2}{c}{**2 h**} & \multicolumn{2}{c}{**3 h**} & \multicolumn{2}{c}{**4 h**} & \multicolumn{2}{c}{**5 h**} & \multicolumn{2}{c}{**6 h**} \\ \hline
3.5 cm & 0.857 & **0.872** & 0.80  
According to the analysis results of Equations (1) and (2), the term \(T_{atmo-trans}\) significantly affected the estimation of temperature at deeper depths. This research attempted to consider the temperature-transmission effect only by the term \(T_{atmo-trans}\) as shown in Equation (3). In Equation (3), \(a_{1}\)-\(a_{4}\) are temperature model parameters. The model parameters such as \(a_{1}\)-\(a_{4}\) in Equation (3) for different depths were obtained by performing regression analysis on the temperatures measured at various depths using the least squares method.