# Dynamic Stability Assessment of High-Speed Railway Bridges Using Numerical Model Updating
## 3 Estimation of the Dynamic Properties of the High-Speed Railway Bridges  
The cross-correlation function was applied in order to estimate the dynamic properties of the high-speed railway bridges using the response measured from different points in the ambient vibration test [42]. Although the ambient vibration condition is similar whenever a high-speed train passes, the response measured from the bridge is not the same because of the varying load acting on the bridge. Therefore, applying the cross-correlation function in the measurement point roaming method is a useful signal processing method. Cross-correlation function is used for each measurement group in order to determine the same frequency component using the responses measured at different positions of the structure and different times. Cross-correlation signal involves distortion in the time and frequency domains in relation to the original signal. The main characteristics of a signal can be effectively found since such distortion occurs in a direction that further emphasizes parts with high correlations among the components of the two signals. The cross-correlation function can be expressed as Equation (1),  
\[R_{xy}(\tau)=\frac{1}{N}\sum_{k=1}^{N}x(k)y(k+\tau) \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Constraints** & **Imgi 2 nd Bridge** & **Maeaji Bridge** & **Hwalchun Bridge** \\ \hline \(\times\), z & 1, 3 & 1, 3, 5 & 1, 3, 5, 7 \\ \(\times\), y, z & 2, 4 & 2, 4, 6 & 2, 4, 6, 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Boundary conditions of the target bridges. Figure 3: Measurement positions and groups on the target bridges: (**a**) Imgi 2 nd Bridge; (**b**) Maeaji Bridge; (**c**) Hwalchun Bridge. where \(x(k)\) and \(y(k)\) are the \(k\)-th measured values for the measured signals \(x\) and \(y\). \(N\) is the number of data to be used in the cross-correlation function, and \(\tau\) is the time newly defined in the cross-correlation function. The most common method for estimating the natural frequency of a structure is peak-picking analysis, which involves applying the Fourier transform to the measured response. The natural frequency was estimated in this study by using the response to which the cross-correlation function was applied. In order to exclude the mass effect caused by high-speed trains, natural frequency was estimated by using the response to which the cross-correlation function was applied in the ambient vibration section. The natural frequency to be used for numerical model updating was determined by calculating the average value of the estimated natural frequency values. Figure 4 shows the results of applying the Fourier transform to the response obtained by applying the cross-correlation function to estimate the natural frequency in the Imgi 2 nd Bridge. Extended Kalman filter was used in this study in order to estimate the damping ratio of each high-speed railway bridge [43]. Methods based on the half-power bandwidth and logarithmic decrement, which are commonly used to estimate the damping ratio, cannot be properly applied when the natural frequency of the response measured from a structure changes. However, the estimation of the damping ratio using the extended Kalman filter shows stable damping ratio analysis results compared to the methods based on the half-power bandwidth and logarithmic decrement because the natural frequency of response is estimated along with the damping ratio. The mode shape of each high-speed railway bridge was estimated by applying proper orthogonal decomposition to the response obtained via the cross-correlation function [44]. Proper orthogonal decomposition, a method of finding a coordinate system to express temporally and spatially ir-regular fluctuation fields, is used to estimate the mode shape of each mode through mode decomposition. If the linear sum of a specific mode in the response of a structure is expressed as Equation (2), the response of the entire structure, \(X\), can be expressed as Equation (3).