# Drive-by Bridge Damage Detection Using Continuous Wavelet Transform
## 3 Continuous Wavelet Transform and Development of the Proposed Method
These disturbances can be better observed when we take a horizontal section at a scale that is far away from driving, bridge and vehicle frequencies, which is plotted at a scale of 600; see Figure 5. In this figure, the location on the bridge in the abscissa corresponds the location of the vehicle on the bridge at a given instant and computed as \(x=v*t\), where \(v\) is the vehicle speed and \(t\) is time. Figure 5 clearly shows that, in the absence of road roughness, the induced damage leads to a significant disturbance in the wavelet coefficients close to the damage location while the wavelet coefficients for the undamaged bridge remain stable at zero. When we repeat the exercise summarized above on a bridge with the road roughness profile depicted in Figure 2, we can observe that the wavelet coefficient maps for the damaged and undamaged case are significantly affected by the road roughness; see Figure 6. As a result, the wavelet coefficient maps, especially in the higher scales, become more chaotic and the effect of damage on the WCM becomes negligible compared to the effect of road roughness. Accordingly, when we take a horizontal section from WCMs for the undamaged and damaged bridges at the same scale (Figure 7), the wavelet coefficients fail to provide any useful information about the state of the bridge yet alone the location of the damage. Figure 4: Wavelet Coefficient Map of the rear-axle accelerations for smooth profile (**a**) Undamaged case (**b**) 50% damage. The wavelet coefficient maps and the wavelet coefficients at a scale of 600 shown in Figures 4-7 suggest that these tools can be effective in detecting and locating damage if only we can suppress the adverse effects of road roughness. Since the negative impacts of road roughness on the drive-by monitoring methods have been known for a long time, researchers have invested significant time and energy to negate them [58; 59; 60; 61]. One of the most successful methods to minimize the negative effects of road roughness was shown to drive the vehicle twice on the same bridge and subtract the accelerations recorded in the same bridge. Figure 5: Wavelet coefficients at scale 600 of the rear-axle accelerations for smooth profile and 50% damage. Figure 6: Wavelet Coefficient Map of the rear-axle accelerations for rough profile (**a**) Undamaged case (**b**) 50% damage. \[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]0 Wavelet coefficients at scale 600 of the rear-axle accelerations for rough profile and 50% damage. the second run from the first run. However, this method assumes that the vehicle drives over the exact same roughness profile, a challenging task to achieve in practice. A more practical application of this approach is to use two sensors on the same vehicle, one at the front axle and one at the back axle, and subtract the acceleration response recorded at the back axle from that recorded at the front axle. Several studies [15; 26; 62] demonstrate, both theoretically and numerically, that the residual acceleration thus calculated is free from the adverse effects of road roughness and provides much improved results for tasks like frequency and mode shape estimation using drive-by monitoring methods. Compared to its counterpart where the vehicle is driven over the bridge twice, this method is much more easier to apply in practice, because the front and back axles of the car should be expected to travel over the same roughness profile unless the direction of the vehicle changes sharply. In this study, we leverage this approach to minimize the effects of road roughness.