# Drive-by Bridge Damage Detection Using Continuous Wavelet Transform
## 3 Continuous Wavelet Transform and Development of the Proposed Method
The wavelet coefficient maps of the residual accelerations obtained by subtracting the accelerations recorded at the rear-axle from those recorded at the front-axle for the undamaged and damaged bridges are shown in \[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]1 Comparing Figure 8 with Figures 4 and 6, we can observe that the approach utilized to remove the road roughness effects indeed leads to a wavelet coefficient map that is closer to that obtained from a vehicle crossing a bridge that has a smooth surface rather than that obtained from a bridge with road roughness. Particularly, the WCM of the residual accelerations on an undamaged bridge in Figure 8a demonstrate that the effect of road roughness is minimized at scales greater than 300, which is the region of interest for us. As such, once we plot the WCM of the residual accelerations recorded on a damage bridge (Figure 8b), the damage leads to disturbances in the WCM at the higher scales that are clearly visible on the WCM. Further, taking a horizontal section of the wavelet coefficient maps at a scale of 600 in Figure 9, we can clearly see the perturbations in the wavelet coefficients close to the damage location on the damaged bridge. On the contrary the wavelet coefficients for the undamaged bridge remains relatively stable at zero even in the presence of road roughness. Next, we will explain how we decided to use the scale 600, where the horizontal section depicted in Figures 5, 7 and 9 is taken at. As indicated before, [PERSON] [50] recommends, in case of direct monitoring, the scale where the horizontal section is taken to be significantly higher than the scale corresponding to the bridge frequency to be able to capture the static response. However, in case of vehicle scanning, where the accelerations are recorded on the vehicle rather than the bridge, vehicle frequencies and driving frequencies also need to be considered. The scale where we will take the horizontal section should be far away from the vibration frequencies of the vehicle because the behavior at scales close to these vibration frequencies will be dominated by the vehicle response itself and not the static response of the bridge. On the other hand, the driving frequency is free from the stiffness properties of the bridge and, thus, the behavior of the bridge at this frequency should be expected to be either unaffected or minimally affected by any damage in the bridge. Here, we should note that the driving frequency is generally a very low frequency, especially for lower vehicle speeds for which the drive-by monitoring  
\[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]2 Wavelet Coefficient Map of residual acceleration response for rough profile (**a**) undamaged case (**b**) 50% damage. methods have been shown to be most effective. Thus, the scale where we will take the horizontal section needs to be higher than those corresponding to the vehicle and bridge frequencies to ensure that we are capturing the static response of the bridge and not dynamic response, but also needs to be lower than the driving frequency. In our study, the vehicle frequency is 5.77 Hz, and the first bridge frequency is 4.35 Hz (Table 1). These frequencies correspond to Scale 44 and Scale 60 in the Wavelet Coefficient Map (WCM), respectively, as computed using Equation (6) with \(F_{c}\) = 0.25 Hz for the Mexican Hat wavelet and \(\Delta t\) = 0.001 s. Figure 10 illustrates the WCM of the residual acceleration response of the vehicle as it moves at a constant speed of 2 m/s across the damaged bridge with road roughness.The driving frequency is 0.08 Hz, aligning with scale 3150 in the WCM. Figure 10 schematically represents a region where the horizontal section can be taken between the scales of the driving frequency and the first bridge frequency. Note that the driving frequency is not shown in Figure 10 because it corresponds to such a high scale.