## 3 Case Studies and Analyses
### Case 2: El Centro Earthquake Ground Motion  
The El Centro earthquake ground motion, widely used in civil and earthquake engineering, was obtained from the Imperial Valley earthquake in 1940, recorded at the site of El Centro. The acceleration record is a 2688-point signal with \(\Delta t=0.02\)s. It can be seen from Figure 7a that  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **Average Peak Error (\%)** & **Maximum Peak Error (\%)** & **Root-Mean-Square Error** \\ \hline Trapezoidal integral & 0.32 & 0.33 & 0.12 \\ Matlab Ode45 & 0.05 & 0.05 & 0.01 \\ Low-frequency cutoff & 0.83 & 0.91 & 0.04 \\ Low-frequency attenuation & - & - & - \\ WATEBI & 0.43 & 0.57 & 0.02 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Five methods for the velocity reconstruction of SDOF system.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **Average Peak Error (\%)** & **Maximum Peak Error (\%)** & **Root-Mean-Square Error** \\ \hline Trapezoidal integral & 0.21 & 0.26 & 0.07 \\ Matlab Ode45 & 0.39 & 0.6 & 0.01 \\ Low-frequency cutoff & 0.4 & 0.47 & 0.04 \\ Low-frequency attenuation & 10.85 & 21.43 & 0.12 \\ WATEBI & 0.19 & 0.23 & 0.02 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Five methods for the displacement reconstruction of SDOF system  
Figure 6: Displacement of the SDOF system: **(a)** Full duration; **(b)** Enlarged detail at the tail end.  
the maximum ground motion exceeds 300 Gal (cm/s\({}^{2}\)). Figure 7b shows the amplitude spectrum of acceleration. Obviously, it is wide-banded, and contains numerous frequency components as many other seismic records do and the frequencies of significant components are lower than 10 Hz. Noise may exist in the record due to low-frequency tilt of device, high frequency burr or even direct component, all possibly leading to deviation in the reconstructed velocity and displacement.  
The site-recorded velocity and displacement are used as reference for assessing the accuracy of WATEBI algorithm. Before running WATEBI, the original acceleration history is interpolated with the cubic spline function using \(\Delta t/6\), in order to achieve a finer and smoother curve.  
Although the original ground motion of acceleration is stochastic and transient, it can be seen from Figures 8 and 9 that the reconstructed signals (especially velocity) by WATEBI approach more closely to the site records than the other four methods do. For displacement, the deviation from the real data starts at some 35 s, but the two most useful peak-to-trough cycles (from 12 s to 28 s) for checking structural dynamic responses are well matched. Tables 3 and 4 compares the performance of five methods for numerically derived velocity and displacement. It is obvious that the trapezoidal integral is still efficient in velocity reconstruction, with regard to error control on peaks, but for the overall performance of velocity and displacement, the proposed WATEBI method outruns all the rest four methods. It should be also recognized that the three error evaluating indicators (Equations (10)-(12)) of displacement results are larger than those of the velocity results. After all, displacement reconstruction is based on velocity. Both noise and the tiny trend term in velocity can amplify the deviation in displacement.  
Figure 7: Recorded acceleration of El Centro earthquake: (**a**) Acceleration time history; (**b**) Amplitude spectrum.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **Average Peak Error (\%)** & **Maximum Peak Error (\%)** & **Root-Mean-Square Error** \\ \hline Trapezoidal integral & 6.81 & 12.55 & 0.77 \\ Matlab Ode45 & 7.98 & 13.87 & 1.15 \\ Low-frequency cutoff & 11.04 & 11.09 & 1.33 \\ Low-frequency attenuation & 11.73 & 12.41 & 1.23 \\ WATEBI & 5.93 & 8.71 & 0.74 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Five methods for the displacement reconstruction of El Centro event.  
Figure 8: Velocity of El Centro ground motion: (**a**) Full duration; (**b**) Enlarged detail at largest trough.