# Optimization Method for Improving Efficiency of Thermal Field Reconstruction in Concrete Dam
## 3 Optimization Method for Thermal Field Reconstruction
### Interpolation Algorithm Evaluation and Selection
#### 3.1.3 Interpolation Algorithm Selection  
To evaluate the interpolation accuracy, the reconstructed temperatures were compared with the measured values collected from the digital thermometers at the same location (Figure 9).  
Figure 8: Thermal field based on Inverse Distance Weighting interpolation algorithm. (**a**) Day \(1\), (**b**) Day 3, (**c**) Day 5, (**d**) Day 7, (**e**) Day 14, and (**f**) Day 28.  
Figure 9: _Cont._At 7#-1, the maximum error between the reconstructed temperature and the measured temperature was 1.89 \({}^{\circ}\)C. After Day 14, the errors decreased and the reconstruction results converged toward the measurement. At 7#-2, the Natural Neighbor and Kriging interpolation worked well before Day 14. From Day 14 to Day 28, the error of Natural Neighbor gradually increased to 1 \({}^{\circ}\)C on Day 28. The Inverse Distance Weighting result has a significant error when compared with the measurement. At 7#-3, Natural Neighbor has the maximum error of 1.55 \({}^{\circ}\)C. Table 1 presents the errors between the measured temperature values and the reconstructed temperature values for the three interpolation algorithms. The box plots were obtained by calculating the absolute error, as shown in Figure 10. As can be seen, the errors were the greatest by the Inverse Distance Weighting and the smallest by the Kriging, indicating that the Kriging interpolation algorithm works best.  
Further, the statistical indicators, mean absolute error (MAE), and root mean squared error (RMSE) were calculated.  
\[MAE=\frac{1}{3}\frac{3}{j=1}|T_{j,t}-T_{t}|,j\in[1,2,3]. \tag{9}\]  
\[RMSE=\sqrt{\frac{\sum\limits_{j=1}^{3}\left(T_{j,t}-T_{t}\right)^{2}}{3}}. \tag{10}\]  
Figure 9: Temperature comparison at three digital thermometer locations. (**a**) 7#-1, (**b**) 7#-2, and (c) 7#-3.  
where \(T_{j,t}\) is the temperature of the \(j-th\) measurement point of the digital thermometer at time \(t\), and \(T_{t}\) is the reconstructed temperature at the same point. The MAE and RMSE values are listed in Table 2. The average MAE and average RMSE values obtained by Kriging were much less than the other two, indicating that Kriging is the most accurate among the three interpolation algorithms.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **Interpolation** \\ **Algorithm** \\ \end{tabular} & **Position** & \begin{tabular}{c} **Minimum Error** \\ **(\({}^{\circ}\)C)** \\ \end{tabular} & \
\begin{tabular}{c} **Maximum** \\ **Error (\({}^{\circ}\)C)** \\ \end{tabular} &
\begin{tabular}{c} **Average Error** \\ **(\({}^{\circ}\)C)** \\ \end{tabular} \\ \hline \multirow{3}{*}{Kriging} & 7\#-1 & 0.07 & 1.36 & \multirow{3}{*}{0.76} \\  & 7\#-2 & 0.10 & 2.13 & \\  & 7\#-3 & 0.75 & 1.55 & \\ \hline \multirow{3}{*}{Natural Neighbor} & 7\#-1 & 0.17 & 1.84 & \multirow{3}{*}{0.81} \\  & 7\#-2 & 0.00 & 2.88 & \\ \cline{1-1}  & 7\#-3 & 0.50 & 1.40 & \\ \hline \multirow{3}{*}{Inverse Distance Weighting} & 7\#-1 & 0.09 & 2.25 & \multirow{3}{*}{0.96} \\  & 7\#-2 & 0.08 & 2.58 & \\ \cline{1-1}  & 7\#-3 & 0.15 & 1.20 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Errors of three interpolation algorithms.  
Figure 10: Absolute errors of the three interpolation algorithms.