# Entropy Wavelet-Based Method to Increase Efficiency in Highway Bridge Damage Identification
## 4 Numerical Results and Analysis
In this way, the average SEI for the CWT coefficients (entropy-CWT diagram) considering all of the measurement points (25%, 50%, and 75% of L) of the current scenario and the same wide scale range (1 to 1000) and scale increment used for the diagram in Figure 9 is obtained. Thus, in Figure 11, the respective results of the average SEI can be observed, where three zones are clearly noticed. Figure 10: CWT coefficients for the scale range from 1 to 1000 with increments of 50, corresponding to the scenario with damage at 25% of L and measuring at 25% of L.  
Figure 9: CWT coefficients for each scale value from 1 to 1000, corresponding to the scenario with damage at 25% of L and measuring at 25% of L.  
SEI zone 1: This has a scale range from 1 to 52, and the maximum SEI value can be located. In this zone, the effects of the natural frequencies of the system are present (if the WEAM were used, they would correspond to the reddish spots, indicating high CWT coefficients, as presented in the lower part of the colored 3D CWT diagrams, see Figure 12). Therefore, the high levels of chaos or entropy in this zone are due to the effect of natural frequencies, and not the effects of damage. Therefore, this zone is not useful and should not be considered to define the scale range in the EWM. Figure 11: Entropy–CWT diagram, plotting the average SEI for the CWT coefficients considering all measurement points (25%, 50%, and 75% of L) for the case of a bridge with a deck damaged at 0.25 L and a scale range from 1 to 1000. Figure 12: CWT diagrams from original acceleration signals for the case of a bridge with a damaged deck at 0.25 L applying the WEAM. Three different measuring positions (from top to bottom: (**a**) 0.25 L, (**b**) 0.50 L, and (**c**) 0.75 L [1]. SEI zone 2: This has a scale range from 53 to 98, where the average SEI value falls from its maximum value to its minimum value. This zone indicates that the effects of natural frequencies begin to decrease until they are completely eliminated, and, on the other hand, the effects of damage have not yet appeared yet. Consequently, the chaos decreases sharply, and the minimum entropy value is due to the edge effects of the signals. This area should also not be considered to establish the scale range to detect damage. SEI zone 3: This has a scale range from 99 to 1000, where the average SEI value gradually increases from its minimum value to a very high value close to its maximum. This zone indicates that the effect of the natural frequencies no longer exists, and the entropy increment is mainly due to the manifestation of the effects of the damage presence. Therefore, the chaos generated in this area is useful to define the scale range to identify damage. It is important to mention that the presence of any kind of defect in any type of bridge will be clearly detectable just via the use of the SEI zone 3, since it is only in this zone that the effect of the natural frequencies, which is the most hostile and makes damage detection impossible, does not exist anymore, and the damage manifestation will be clearer. However, the CWT scale values of this zone, as well as the other two zones, will change according to the analyzed structure, and must then be determined, since the natural frequencies are an intrinsic parameter of any structure, with specific values according to its properties/geometry. Thus, the entropy-CWT diagram presented in Figure 11 was obtained in less than 21 s (less than 7 s for each measurement point), and this is the diagram that allows for the elimination of the most critical stage of the WEAM (stage 5: generation colored 3D CWT diagrams to establish the useful scale range, which implies about 3 min and 8 s of computing time, as in Figure 12 [1], but with a scale range from 1 to 1000) and the subsequent stages that are related to this critical stage, and which consume around 95% of the WEAM post-processing time, as mentioned above.