# Drive-by Bridge Damage Detection Using Continuous Wavelet Transform
## 4 Parametric Study
### Damage Location and Level  
First, we will take a look at the efficacy of the proposed framework to detect and locate different levels of damage at different locations. For this, we simulated three different levels of damage that correspond to 20%, 50%, and 80% reduction in the flexural stiffness of the bridge at three different segments. These segments start at 7 m, 12 m, and 16 m away from the left support of the bridge and are 1 m long. For each case, we first conducted a numerical analysis for a vehicle speed of 2 m/s and a sampling frequency of 1000 Hz. The road roughness profile in Figure 2 was considered in the analyses. We then carried out a continuous wavelet transform on the residual accelerations, which are computed by subtracting the accelerations recorded at the back-axle from those recorded at the front axle, to create the wavelet coefficient map for each case. Finally, we took a horizontal section of the wavelet coefficient maps at scale 600, which meets the criteria summarized above. Figure 11 presents the variation of the wavelet coefficients with location on the bridge for different damage locations and levels including the undamaged case. The plots are provided between \(x=5\) m and \(x=20\) m, i.e., 5 m away from each support due to the aforementioned edge effects.  
In Figure 11, the damage location is marked by red-colored vertical dashed lines, and, for each damage location, the wavelet coefficients were plotted for the different damage levels. The wavelet coefficients for the undamaged case remain relatively stable at zero. On the other hand, induced damage creates a clear perturbation in the wavelet coefficient around the damage location, while they revert to zero once the vehicle travels away from the damage. As such, presence of damage can be successfully detected for each damage location and each damage level depicted in Figure 11. Further, the disturbances in the wavelet coefficients follow the unique pattern explained in Figure 11. Accordingly, the wavelet coefficients reach a negative peak before the damage location, start to increase crossing the horizontal axis close to the middle of the damaged segment on their path to a positive peak, which occurs after the damaged section. Hence, the location of the damage can successfully estimated for each damage location by the proposed method.  
Further, the perturbations in the wavelet coefficient maps increase with an increase in the damage level paving the way for potentially quantifying the damage. However, this amplification in the perturbations with damage level, makes it difficult to visually identify the effect of 20% damage on the wavelet coefficients when we plot all three damage levels in one plot. Therefore, we re-plotted the wavelet coefficients for the undamaged case and the 20% damage between 12-13 m and 16-17 m cases in Figure 12. In this figure, we can clearly observe that, the wavelet coefficients for 20% damage follows their counterpartsfor higher damage levels in Figure 11. Thus, we can confidently state that, the proposed framework can detect and locate damage for all damage levels and locations investigated.