# Prediction of Temperature Distribution for Previous Cement Concrete Pavement with Asphalt Overlay
## 4 Model Verification with Measured Field Data
### The Results of Model Validation  
The input data including solar radiation intensity, thickness of each layer except the subgrade, layer thermal properties and the measured air temperature were applied to develop a MATLAB computer program. Then, this program was used to calculate the temperature distribution of the time dependence. The temperature of the pavement structure from top to down shows gradually a tendency toward stabilization. By analyzing the measured temperature, the initial temperature \(c\) in Equation (10) is assumed to be 36 \({}^{\circ}\)C for August 24 and 20 \({}^{\circ}\)C for 16 December. In order to measure temperature distributions throughout the pavement section, numerous temperature sensors were installed at different depths of the pavement section in this existing cement concrete pavement with an asphalt overlay. The measured and calculated pavement temperatures at \(z\) =0.02 m (at the middle of AC-13 layer), \(z\) = 0.04 m (at the bottom of AC-13 layer), \(z\) = 0.065 m (at the middle of AC-20 layer), \(z\) =0.09 m (at the bottom of AC-20 layer), \(z\) =0.35  
Figure 8: Measured and predicted temperature at \(z=0.02\) m and \(z=0.04\) m on 16 December. Figure 6: Measured and predicted temperature at \(z=0.065\) m and \(z=0.09\) m on 24 August. Figure 7: Measured and predicted temperature at \(z=0.35\) m and \(z=0.55\) m on 24 August. Let \(T_{p}\) and \(T_{m}\) be the predicted and measured pavement temperature, respectively. Table 4 lists the maximum and minimum values of \(T_{p}-T_{m}\), the mean absolute error values, i.e. \(\left|T_{p}-T_{m}\right|_{\text{mean}}\)at six various depth positions during two different time periods (summer and winter). From Table 4, the maximum absolute errors and the maximum mean absolute error values between the predicted and measured temperature at different depth locations are less than 2.89 \({}^{\circ}\)C and 1.58 \({}^{\circ}\)C, respectively. The increase in the pavement depth can lead to lower average relative absolute errors. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Date**} & \multirow{2}{*}{**Error Type**} & \multicolumn{8}{c}{**Depth Location \(z\) (m)**} \\ \cline{3-8}  & & **0.02** & **0.04** & **0.065** & **0.09** & **0.35** & **0.55** \\ \hline \multirow{2}{*}{24 August} & \(\begin{pmatrix}T_{p}-T_{m}\\ T_{p}-T_{m}\\ T_{m}\\ T_{p}-T_{m}\\ \end{pmatrix}_{\text{min}}\) (\({}^{\circ}\)C) & 1.84 & 2.05 & 1.73 & 1.40 & \(-\)0.12 & 0.03 \\  & \(\begin{pmatrix}T_{p}-T_{m}\\ T_{m}\\ \end{pmatrix}_{\text{min}}\) (\({}^{\circ}\)C) & \(-\)2.89 & \(-\)2.45 & \(-\)1.87 & \(-\)2.85 & \(-\)2.54 & \(-\)0.75 \\  & \(\begin{pmatrix}T_{p}-T_{m}\\ T_{p}-T_{m}\\ \end{pmatrix}_{\text{mean}}\) (\({}^{\circ}\)C) & 1.07 & 1.17 & 1.32 & 1.58 & 1.26 & 0.27 \\ \hline \multirow{2}{*}{16 December} & \(\begin{pmatrix}T_{p}-T_{m}\\ T_{p}-T_{m}\\ T_{m}\\ \end{pmatrix}_{\text{min}}\) (\({}^{\circ}\)C) & 2.43 & 1.89 & 1.66 & 0.77 & 0.00 & \(-\)0.20 \\  & \(\begin{pmatrix}T_{p}-T_{m}\\ T_{p}-T_{m}\\ \end{pmatrix}_{\text{min}}\) (\({}^{\circ}\)C) & \(-\)0.48 & \(-\)1.43 & \(-\)1.06 & \(-\)1.10 & \(-\)0.80 & \(-\)0.70 \\  & \(\begin{pmatrix}T_{p}-T_{m}\\ \end{pmatrix}_{\text{mean}}\) (\({}^{\circ}\)C) & 0.71 & 0.67 & 0.53 & 0.41 & 0.38 & 0.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Errors between the predicted and measured temperatures of various depth locations. Figure 10: Measured and predicted temperature at \(z\) = 0.35 m and \(z\) =0.55 m on 16 December. Figure 9: Measured and predicted temperature at \(z\) = 0.065 m and \(z\) = 0.09 m on 16 December. Therefore, the proposed analytical solutions derived by the technique of Laplace transform and its numerical inversion can be applied to conduct the prediction of the temperature distribution in the existing cement concrete pavement with an asphalt overlay with reasonable and high accuracy.