# Eigenstate Transition of Multi-Channel Time Series Data around Earthquakes
## 2 Materials and Methods
### Methodology
We selected three earthquakes as listed in Table 1 to be analysis objects whose waveforms were observed by DONET, and visualized location of the epicenters and the KMDB1 station of DONET observatory in Figure 2. Hereafter, we term each earthquake simply the Mie, Kumamoto, or Fukushima earthquake. As listed in Table 1, the Mie earthquake occurred very near the location of the KMDB1 station and its depth was shallow at 29 km. Thus, we expect that the observed data include information regarding some changes in the crustal state. The others, Kumamoto and Fukushima earthquakes, have similar energy scales to that of the Mie earthquake at moment magnitude 7 and 6.9, but they occurred far from the station. As an example of the reproduced waveform from a SEED file, we show representative waveforms, in which the average values are subtracted from original data, from 20:00 during the day before the Mie earthquake to 0:00 during the day of the earthquake in Figure 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Southeastern Offshore Mie** & **Kumamoto** & **Offshore Fukushima** \\ \hline Day (GMT) & 1 April & 15 April & 21 November \\ Time (GMT) & 02:39 & 16:25 & 20:59 \\ Latitude N [deg.] & 33.19 & 32.78 & 37.35 \\ Longitude E [deg.] & 136.2 & 130.8 & 141.6 \\ Depth [km] & 29 & 12 & 25 \\ Moment Magnitude & 5.8 & 7 & 6.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Earthquakes of 2016 analyzed [20]. Here, BHX, BHY, BHZ, ELX, ELY, and ELZ are the waveforms in the x, y, z direction of the velocity type seismometer, and x, y, z direction of the accelerometer, respectively. Since the change of the other quantities are not at the time scale for expected preceding signals, We adopt only these waveform data for EWS analysis. We provide channel numbers 1, 2, 3, 4, 5, and 6 to BHX, BHY, BHZ, ELX, ELY, and ELZ, respectively. We adopt the channel number L, the time step width M, and the time window N as 6, 200, and 400, respectively. As we treat the sampling frequency as 1 Hz, the unit of M and N is the same with time (seconds). We set the time window N to 400, so we have 400 data points for each time window. This amount of data is sufficient to evaluate the variance, that is enough to discuss EWSs. We note that the data are completely replaced when the time step is different by more than 2. We show the time we analyzed in Table 2, which is 24 h of data before and after the earthquakes. For each earthquake, we took a new normalization constant \(k_{i}\) for channel \(i\). For the programming software, we used GNU Octave (version 5.1.0) [21], and used the internal function \"svd\" for the eigenvalue analysis of the covariance matrix. We made  
Figure 3: Representative waveforms for the Mie earthquake. The origin of the time is 20:00 on March 31 st.  
Figure 2: Location of the epicenters and the KMDB1 station of DONET observatory. The epicenters are shown as filled circle, and the KMDB1 station is shown as a pentagram. normalization factor \(k_{i}\) in Equation (3) from data in the first window of 4 h before the first objective time in Table 2. Namely, we adopt \(k_{i}\) from the first window in 91 st 12:00 data for Mie earthquake, in 106 th 04:00 data for Kumamoto earthquake, in 326 th 08:00 data for Fukushima earthquake, respectively.