# Entropy Wavelet-Based Method to Increase Efficiency in Highway Bridge Damage Identification
## 4 Numerical Results and Analysis
This is because, by expanding the scale range, the damage effect is \"strengthened\", and the edge effects of the signal are \"dimmed\", so, as expected, the edge effect that is most inhibited is the closest one to the damage position; that is, the one on the left side of the diagram. Both the diagrams in Figures 15 and 16 are generated very quickly (4 s required for the one in Figure 15 and 10 s for the one in Figure 16). Therefore, the time difference between them is very small, but, with the diagram presented in Figure 16 with a wider scale range, more noticeable evidence of damage and the inhibition of signal edge effects is gained without the need of conditioning the signal (signal extension at both ends). Nevertheless, the diagram shown in Figure 15 is also very useful for identifying damage. Thus, it is recommended to define the scale range using SEI zone 3 of the entropy-CWT diagram, locating the lowest entropy value first, and then establishing the minimum value of the scale range after the lowest entropy value in order to allow the damage to begin to manifest. The maximum value of the scale range should be defined when the SEI value of the entropy-CWT diagram practically no longer increases and remains constant. For the case currently analyzed, the scale range, from 181 to 1000, is excellent, as demonstrated in Figures 11, 15 and 16. On the other hand, in order to demonstrate the reliability, efficiency, quickness, and precision of the EWM, the average CWT coefficients for the current case of damage at 25% of L is obtained for the scale range from 181 to 1000, but now considering the corresponding average of the CWT coefficients for the three measurement points (25%, 50%, and 75% of L)  
Figure 16: Average CWT coefficients for the scale range from 181 to 1000, corresponding to the scenario with damage at 25% of L and measuring at 25% of L.  
instead of just one measurement point at 25% of L. In this way, it is possible to verify that the EWM, like the WEAM, does not require a measurement point at the damage position, and the average of a few measurement points proportionally distributed is sufficient to identify damage. Figure 17 shows the diagram with the corresponding curve to the case mentioned in the previous paragraph, and, additionally, the respective curve of the healthy case considering the three measurement points is also included. In this figure, it can be clearly observed that the curve of the healthy case looks very flat, without any magnitude increments of CWT coefficients in specific positions; however, the curve of the damaged case, similar to the one presented in Figure 16, exhibits an evident sudden increase in the damage area, locating this defect with 99.44% precision. Finally, in order to have a diagram that makes the presence of the damage even more noticeable, the same diagram presented in the previous figure (Figure 17) is obtained, now eliminating the edge effects, for which, before obtaining the average CWT coefficients for the chosen scale range and considering the three measurement points, the original signals (time vs. acceleration) provided by the numerical simulations with the FEM model are extended at both ends. That is, the first and last cycles of the signals are repeated on the left and on the right sides, respectively; smoothing, in this way, the beginnings and ends of the signals in order to avoid the generation of discontinuities, which produce high CWT coefficients. The process of eliminating edge effects via signal extensions is the same as the one used for the WEAM, which is very simple, quick, and can be consulted in [1]. Therefore, the corresponding diagram without edge effects is shown in Figure 18. Comparing this latest diagram without edge effects with the corresponding diagram with edge effects (Figure 17), the benefit of eliminating edge effects is very evident, since the curve of the healthy case becomes even flatter because it does not contain effects that increase the CWT coefficients; there are only residual edge effects, which are almost negligible.