## 3 Case Studies and Analyses
### Case 3: Kaikoura Earthquake in The South Island of New Zealand  
The Kaikoura earthquake that occurred in the South Island of New Zealand on 13 November 2016 was a very strong seismic event with magnitude of 7.8 (M\({}_{\rm w}\)) and a long duration of approximately two minutes. The seismic data used in this paper is a horizontal component that has been corrected after occurrence. The data was recorded with a constant time interval of 0.005 s. Figure 10 shows the time history of acceleration and its amplitude spectrum. Compared with the El Centro record in Figure 7, the energy of Kaikoura record is more concentrated, not only for time history but also for spectrum.  
Owing to the fine time interval, the total number of logging is 14,327, such that interpolation of data is unnecessary. The reconstructions by WATEBI only require seconds of computational time, using an ordinary computer. Figures 11 and 12 illustrate the waveforms of original and derived velocity and displacement. The good performance of WATEBI algorithm is again demonstrated. The low-frequency cutoff method shows the worst degree of reconstruction among all methods, especially in displacement. Considering that more and more advanced technologies have been applied in earthquake monitoring, rich and accurate seismic recordings are available. This in turn helps to improve the accuracy in deriving velocity and displacement in terms of acceleration. Therefore, very good agreement between the derived and recorded data can be expected.  
Figure 9: Displacement of El Centro ground motion: (**a**) Full duration; (**b**) Enlarged detail at largest peak.  
Figure 11: Velocity of Kaikoura ground motion: (**a**) Full duration; (**b**) Enlarged detail at largest trough.  
Figure 10: Recorded acceleration of Kaikoura earthquake: (**a**) Full time history; (**b**) Amplitude spectrum.  
Tables 5 and 6 give the comparison of five methods in deriving the velocity and displacement of Kaikoura earthquake. Again, the three error indicators show that the proposed WATEBI algorithm is most robust. For displacement, both the average peak error and the maximum peak error by WATEBI are even below 1%. The low-frequency attenuation method also provides good accuracy, because the target frequency has been tuned according to the target accuracy \(\alpha_{\text{T}}\) = 0.97. By contrast, the low-frequency cutoff is the worst among all five methods, not only for velocity but also displacement.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **Average Peak Error (\%)** & **Maximum Peak Error (\%)** & **Root-Mean-Square Error** \\ \hline Trapezoidal integral & 0.63 & 0.69 & 0.61 \\ Matlab Ode45 & 2.17 & 2.45 & 0.77 \\ Low-frequency cutoff & 4.35 & 7.80 & 1.58 \\ Low-frequency attenuation & 0.91 & 0.96 & 0.12 \\ WATEBI & 0.64 & 0.73 & 0.35 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Five methods for the displacement reconstruction of Kaikoura event.  
Figure 12: Displacement of Kaikoura ground motion: (**a**) Full duration; (**b**) Enlarged detail at largest trough.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **Average Peak Error (\%)** & **Maximum Peak Error (\%)** & **Root-Mean-Square Error** \\ \hline Trapezoidal integral & 0.33 & 0.65 & 1.68 \\ Matlab Ode45 & 0.32 & 0.51 & 0.10 \\ Low-frequency cutoff & 1.77 & 2.46 & 1.38 \\ Low-frequency attenuation & - & - & - \\ WATEBI & 1.04 & 1.94 & 0.75 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Five methods for the velocity reconstruction of Kaikoura event.