## 4 Pavement Temperature-Evaluation Model  
Because pavement surface temperature is strongly influenced by weather conditions and actual pavement conditions such as shadow [16], the atmospheric temperature was employed as a basis for establishing the temperature-evaluation model in this study. Temperature data of the test site from 8 February 2018 to 8 February 2019 were collected to perform the regression analysis and develop the model. Figure 6 presents the over-time variation of atmospheric temperature and temperature at various road depths on 9 February 2018 as an example, indicating that the 1-day temperature variation has the form of a sine function. Such functions have frequently been employed as the form of a temperature-evaluation model for estimating the temperature at various road depths [16; 17]. Accordingly, this study also used a sine function as the basis of the temperature-evaluation model. Moreover, as illustrated in Figure 6, the highest atmospheric temperature occurred around 1 pm, whereas the highest pavement temperatures at depths of 3.5 and 24.5 cm were at 2 pm and 5 pm, respectively. Comparison of the atmospheric temperature and pavement temperature at depths of 3.5 and 24.5 cm as examples (Figure 6) revealed a delay of temperature variation, and the delay time was longer at deeper depths. This temperature delay was caused by the thermal conduction effect. Conducting the heat received on the road surface to various depths requires time, and hence, a temperature transmission delay phenomenon was observed. According to the literature on the temperature-estimated model [16; 17], this research attempted to consider the effect of temperature transmission-delay as a parameter \(b_{4}\) inside the sine function as shown in Equation (1). In Equation (1), \(b_{1}\)-\(b_{5}\) are parameters in the temperature model and \(T_{Z}\) is the evaluated temperature (\({}^{\circ}\)C) at depth Z. The term \(T_{atmo}\) is the atmospheric temperature (\({}^{\circ}\)C), while \(t\) is the time of the day for which the temperature evaluation is being conducted (e.g., for 1:30 pm, \(t\) = 13.5). The model parameters such  
Figure 5: Falling weight deflectometer (FWD) PRIMAX 1500.  
as \(b_{1}\)-\(b_{5}\) in Equation (1) for different depths were obtained by performing regression analysis on the temperature measurements at various depths using the least squares method. \[T_{Z}=b_{1}\times T_{atmo}-b_{2}\times\sin\biggl{(}b_{3}\times\frac{t}{24}+b_{4 }\biggr{)}+b_{5} \tag{1}\]  
The correlation between model-estimated and measured temperature for various depths was summarized in Table 2. The results of the correlation coefficient showed that the correlation coefficient decreased with increasing depth and the correlation coefficient dropped to 0.53 at a depth of 24.5 cm. Furthermore, the slope and intercept in Table 2 were the linear regression function between measured and estimated temperature. The slope and intercept of the regression function closer to 1 and 0, respectively, indicates that the estimated temperatures were more correlated with measured temperatures. However, the results showed that the slope deviated from 1 with increasing depth, while the intercepts diverged away from 0. These results indicated that the temperature model (Equation (1)) cannot accurately estimate the temperature at a deep depth. Hence, this research performed the correlation analysis between the road temperature at various depths and atmospheric temperature considering the temperature transmission-delay. Table 3 summarizes the correlation analysis results of temperatures measured at various depths and the atmospheric temperature. The results show that if the temperature transmission-delay effect was excluded (e.g., 0 h), the correlation between the measured temperature at various depths and atmospheric temperature was significantly decreased with increasing depth.