## 3 Field Testing
### Discussion of the Correlation between DLTE and **RPSD**  
To further investigate the correlation between DLTE and **RPSD**, the maximum of **RPSD**'s density distribution (denoted by **RPSD\({}_{\text{m}}\)**) was extracted from each subfigure of Figure 10 and compared with the corresponding DLTE from Table 1. Figure 11 shows their correlation in the two tests. Note that DLTE squared (DLTE\({}^{2}\)) is utilized for comparison in Figure 11 because **PSD**'s amplitude is the frequency spectrum's amplitude squared. The green area is the 95% confidence interval, which means that there is a 95% probability that the true best-fit line for the populations lies within the confidence interval. From Figure 11, it is found that:  
Figure 10: Estimated density distribution of **RPSD** at different joints.  
(1) In the first test, the DLTE\({}^{2}\) is between 0.65-0.95 and mainly concentrated around 0.90. The correlation between \(\mathbf{RPSD_{m}}\) and DLTE\({}^{2}\) is about 0.844.  
(2) In the second test, some joints' condition deteriorate significantly so the range of DLTE\({}^{2}\) become wider (0.40-0.95). But the correlation is still 0.875.  
Similarly, Figure 12 illustrates the correlation between \(\mathbf{RPSD_{m}}\) and DLTE\({}^{2}\) in different types of joints. Figure 12a shows the results of dummy joints. Since the samples of rabbet joints and dowel bars are both much fewer than dummy joints, their results are illustrated together in Figure 12b. It is found that:  
(1) Owing to the sensitivity to temperature, the DLTE\({}^{2}\) of dummy joints is distributed between 0.40-0.95 and has more deviations than other types of joints. The correlation is 0.868.  
(2) The DLTE\({}^{2}\) of rabbet joint and dowel bars are mostly concentrated in 0.75-0.90 as the temperature has a slight influence on their DLTE\({}^{2}\). The correlation is around 0.880.  
Figure 11: Correlation between \(\mathbf{RPSD_{m}}\) and DLTE\({}^{2}\) (**a**) In the first test; (**b**) In the second test.  
Figure 12: Correlation between \(\mathbf{RPSD_{m}}\) and DLTE\({}^{2}\) (**a**) In dummy joints; (**b**) In rabbet joints and dowel bars.  
From Figures 11 and 12, the findings indicate that \(\mathbf{RPSD_{m}}\) has a significant correlation with DLTE\({}^{2}\). The correlation is close in different types of joints as well as two different tests that have a four-month gap. It proves the robustness of the vibration-based method.  
Besides, to investigate the accuracy of the \(\mathbf{RPSD_{m}}\), Figure 13a illustrates the correlation between \(\mathbf{RPSD_{m}}\) and DLTE\({}^{2}\) according to the all results of the two tests. Also, the difference between \(\mathbf{RPSD_{m}}\) and DLTE\({}^{2}\) (\(\Delta\)RL)is calculated as  
\[\Delta RL=\sqrt{\mathbf{RPSD_{m}}}-DLTE \tag{27}\]  
and its cumulative distribution is shown in Figure 13b.  
From Figure 13, It is found that: (1) the overall correlation between \(\mathbf{RPSD}\) and LTE is 0.880. It is very close to the results in Figures 11 and 12. (2) the 5% and 95% percentile of \(\Delta\)RL is about \(-\)0.08 and 0.06, respectively, which means 90% of \(\Delta\)RL is between \(-\)0.08-0.06. The main part of the difference is due to DLTE. According to reference [7; 8; 9], the measured deviation of DLTE could usually reach 0.042. It agrees with the measurement results shown in Table 1. The deviation reaches 0.050 from Table 1. Therefore, the \(\Delta\)RL should be mostly below \(\pm\)0.03 regardless of the DLTE-part error. It also agrees with the test results in reference [12], from which the difference is about \(\pm\)0.02.