# Entropy Wavelet-Based Method to Increase Efficiency in Highway Bridge Damage Identification
## 3 Damage Identification Methodology and Their Fundamentals
### Entropy-Wavelet Based Method (EWM)  
The EWM, as explained previously, was developed from the need to make the WEAM more efficient for damage identification in highway bridges, in such a way that the necessary stages and the computing time required to apply the WEAM could be significantly reduced [1], but without losing its precision in the damage identification. Figure 8 illustrates the steps of our proposed, the EWM method, and the WEAM method. Thus, the EWM was applied to evaluate the health condition of the RPM bridge is presented in Figure 8, and this consisted of recording the vibration signals at different points along the bridge deck while a vehicle crosses it. Subsequently, the average SEI corresponding to the CWT coefficients of the current case to be analyzed is calculated considering the different measurement points and a wide scale range. Next, starting from the lowest point of the SEI, the useful scale range is defined to identify the damage, and the average curve of the CWT coefficients is obtained considering the selected scale range and the different measurement points. Finally, the curve obtained from the CWT coefficients is compared with the corresponding curve of the healthy case (baseline), and, if there is damage, a sudden increment in the amplitude of the CWT coefficients will be observed in the position of the damage for the current case. This will not happen for the healthy case, since the case without damage will exhibit a curve that will tend to be very flat and of low amplitude. The step-by-step application of the EWM for the detection and localization of damage in bridges is as follows:  
1. Instrument the bridge with vibration sensors distributed proportionally along the deck. 2. Obtain vibration responses for the healthy bridge (baseline) and for the current condition of the bridge while a vehicle or vehicles cross the bridge. 3. Obtain the average SEI for the CWT coefficients of the current case to be analyzed, considering the data from all the measurement points and a very wide scale range, allowing SEI results outside of the effect zone of the structure natural frequencies. 4. Define the most useful CWT scale range for damage identification from the lowest point of the SEI value (end of the effect of the structure natural frequencies) and upwards. 5. Apply a suitable filter to remove as much noise in the signals (Savitzky-Golay filter; order: 2, window length: 19) and eliminate any edge effects by extending the signals on both sides. It should be noted that this step is not entirely necessary, since in the results presented in Section 4, it can be observed that even without filtering  
Figure 8: Schematic diagram of the EWM and WEAM methods. or extending the signals, the identification of damage is possible and very evident. However, it is recommended to filter and extend the signals, so that the detection and location of damage is even clearer. Likewise, the most appropriate MW must be used throughout the application of the EWM, which corresponds with the one previously defined via the WEAM (Mexican hat for the RPB). 6. Obtain the average value of the CWT coefficients for the selected scale range and for the different measurement points. 7. Compare the curve obtained in the previous point with the corresponding curve of the healthy case. If there is damage, the magnitude of the CWT coefficients will increase suddenly at the location of the damage, which will not happen for the healthy case, since the healthy case will demonstrate a curve of low amplitude without significant increments at specific positions. Thus, comparing this new methodology, EWM, with the WEAM presented in [1], the simplification of steps achieved with the EWM is evident (from 11 to 7 stages); however, it should be noted that the steps that were eliminated with the EWM are the ones that consume the most computing time with the WEAM.