# Drive-by Bridge Damage Detection Using Continuous Wavelet Transform
## 3 Continuous Wavelet Transform and Development of the Proposed Method
We selected scale 600 as it is close to the middle of the window but any scale within this window can be a viable option as they correspond to relatively low frequencies, and thus, capturing static response. In each application of the proposed framework, we recommend that the user creates a similar figure and determines the scale where the horizontal section will be taken accordingly. \[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]3 Wavelet Coefficient Map between using vehicle’s residual acceleration response at 2 m/s for 50% damage. \[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]4 Wavelet coefficients at scale 600 for 50% damage. Finally, we will focus on localizing the damage. For this, we will use Figure 9, which depicts the wavelet coefficients at a scale of 600 extracted from the WCM of the residual accelerations recorded on the undamaged and damaged bridges; see \[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]5 As expected, in the absence of damage, the wavelet coefficients remain stable around zero while damage causes a disturbance in these coefficients. Further, in our numerical analysis, we observed that these disturbances follow a unique pattern as will be demonstrated for other cases in the upcoming sections. There are two, nearly identical, peaks in terms of absolute value: one in the negative direction and one in the positive direction as indicated in \[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]6 The negative peak occurs just before the vehicle reaches the damaged location followed by an increase in the wavelet coefficients leading to the second peak (the positive peak) which is observed after the damage location. While the wavelet coefficients are increasing from the negative peak to the positive one, they cross the horizontal axis (wavelet coefficients = 0) within the damaged zone, generally very close to the middle of the damage location. As such, using this unique pattern, we can estimate the location of the damage. Here, we should note that, the negative and positive peaks used to locate the damage is observed only when the WCM is computed for the residual vehicle response. On the other hand, we cannot observe the same behavior when the WCM is computed for the accelerations recorded only on the rear-axle (or front-axle) of the vehicle traveling on a smooth bridge (Figure 5), because using residual response is not necessary in this case due to the lack of road roughness. We should also note the smooth road profile case only presents an academic case as all bridges exhibit a certain level of road roughness. Thus, tracing the negative and positive peaks of the WCM of residual vehicle response as shown in Figure 9 will lead to successful damage location in real-life applications. To explain why the disturbances in the wavelet coefficients (Figure 9) start before the damaged zone and extend beyond it, we need to take a closer look at the static displacements under the moving load depicted in \[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]7 It is clear from this figure that the changes in the static displacements due to damage also extend to a range that is much longer than the damage location. Since the scale we focus on is carefully selected to reveal the pseudo-static behavior of the bridge, the disturbances in the wavelet coefficients occur approximately at the region of the bridge where the damage significantly impacts the static displacements. At the end of this section, we should also mention a recognized phenomenon in continuous wavelet transform that leads to high wavelet coefficients at the two ends of the bridge. These relatively high coefficients are attributed to boundary effects due the finite length of the input signal [50; 51]. These effects are generally limited to the first and final 20% of the bridge length.