# Entropy Wavelet-Based Method to Increase Efficiency in Highway Bridge Damage Identification
## 4 Numerical Results and Analysis
In Figure 12, the yellow dotted ellipses indicate where energy differences can be observed according to the measurement point and the damage point. Continuing with the development of the bases that gave rise to the EWM, and taking into account the entropy-CWT diagram presented in Figure 11, the usefulness of the SEI zone 3 to detect damage is verified. To do this, the average of the CWT coefficients is obtained considering the measurement position of 25% of L and the scale range from 1 to 100 (Figure 13). This scale range includes the complete SEI zone 1 and SEI zone 2, as well as the beginning of the SEI zone 3, where the SEI is just beginning to increase due to the damage. In this way, looking at Figure 13, as expected, there is no indication of damage. On the other hand, if the scale ranges from 101 to 181 is considered, that is, a scale range that is already completely in SEI zone 3 but still at the beginning of this zone, and  
Figure 13: Average CWT coefficients for the scale range from 1 to 100, corresponding to the scenario with damage at 25% of L and measuring at 25% of L.  
the average of the CWT coefficients is obtained considering the measurement position of 25% of L, then, in Figure 14, it can be seen that the identification of the damage is already possible due to the magnitude increment of CWT coefficients around 25% of L. However, since the damage manifestation is just beginning in this zone, the edge effects of the signals still predominate, which is why the largest magnitude CWT coefficients are found at the extremes of the diagram. Now, if the scale ranges from 181 to 500 is considered, that is, from the last scale value of the previous figure (Figure 14) and up to half of the maximum scale value considered, then the entropy increment in this range is already very significant (see Figure 11), and, therefore, the effects of the damage presence are of great significance, which can be verified by generating the corresponding diagram of the average CWT coefficients (see Figure 15). In Figure 15, it can be noted that, for this scale range of 181 to 500, the sudden increase in the magnitude of the average CWT coefficients around the damage zone is very evident, reaching its maximum value practically at the exact damage position (at 25.13% of L). Additionally, it is important to highlight that the damage effect in this region of SEI zone  
Figure 14: Average CWT coefficients for the scale range from 101 to 181, corresponding to the scenario with damage at 25% of L and measuring at 25% of L.  
Figure 15: Average CWT coefficients for the scale range from 181 to 500, corresponding to the scenario with damage at 25% of L and measuring at 25% of L.  
3 is already very important, meaning that even the maximum value of the average CWT coefficients at the damage zone exceeds the maximum values of the CWT coefficients generated at the ends of the diagram due to the signal edge effects. On the other hand, since it was found that the scale range, from 181 to 500, which is located in SEI zone 3, is very useful for identifying damage, this scale range is extended up to the maximum value considered in Figure 11, taking advantage of the influence of more damage effects. Then, the scale range, from 181 to 1000, is now considered, and the respective average CWT coefficients are obtained (see Figure 16). In Figure 16, it can be observed that the curve of the average CWT coefficients increased its magnitude even more in the damage region because a larger damage influence zone is considered and, therefore, the chaos or entropy of the CWT coefficients is also higher. Likewise, the precision in locating the damage is the same as in Figure 15 (99.48% precision). The other situation that must be highlighted in Figure 16 is the reduction of edge effects, even though the signal has not yet been treated/conditioned for this purpose.