Recorded data and CA data are compared for the three 10-day intervals with large, medium and low EPR(10days) values shown in Fig. 13 (a). In order to compare dynamic parameters directly, the scale for calculating those parameters is unified. In this section, for both recorded and CA data, the EPR is calculated for the first 1000 data points of the data block for which \(\alpha\)-tremor is calculated. Here, one data block for either the recorded data or the CA data is defined as 1024 data points equidistant in chronological order. For the recorded data, the one data block corresponds to 51.2 seconds of a data acquisition time. This section compares the 16875 data blocks of the recorded data (corresponding to the data length of 10 days) with the CA data blocks and evaluate the degree of agreement between them.

Eq. (2), the definition of EPR, shows that EPR is calculated from transition rate matrix \(W_{ij}\) and state vector \(p_i\). In other words, we need to know the EPR and \(W_{ij}\) to determine \(p_i\). On the other hand, referring to the discussion for Fig. 12, \(W_{ij}\) corresponds to a pair of EPR and \(\alpha\)-tremor. So, we replace \(W_{ij}\) with \(\alpha\)-tremor and EPR in evaluating the \(p_i\) matching. Adopting \(\alpha\)-tremor is advantageous since \(\alpha\)-tremor is observable, is directly related to ground vibrations, and provides insights on earthquakes. Consequently, the EPR, \(\alpha\)-tremor, \(p_i\), and additionally the density of states are evaluated in the comparison of the recorded with the CA data.

## CA data characteristics

CA data of periodic \((d, p)\)-CA184 is generated by the method shown in Fig. 14. The initial density of states “\(d\)” is selected to be 0.5 to be coherent with the density of states of the recorded ground vibration data which fluctuates around 0.5 as shown in Fig. 13 (b)-(d). 2000 CA data is generated for each hop probability “\(p\)” of 0.2, 0.3, 0.5, and 0.87, and the characteristics of the generated data are shown in Fig. 15.

Fig. 15 shows frequency of (\(\alpha\)-tremor, EPR) pairs, frequency of \(\alpha\)-tremor, and density of states. In all figures in Fig. 15, the frequency of occurrence of \(\alpha\)-tremor is concentrated around -1, with small values in the positive region, similar to the characteristics of the recorded ground vibration in Fig. 13.

If "\(p\)” is greater than or equal to 0.5, a larger positive \(\alpha\)-tremor is observed (red down arrow in the \(\alpha\)-tremor frequency plot in Fig. 15 (c) and (d)) than if “\(p\)” is less than 0.5. Therefore, the large “\(p\)” which is the near-equilibrium condition corresponds to more noticeable positive \(\alpha\)-tremor distribution. On the other hand, for the recorded data, the noticeable positive \(\alpha\)-tremor distribution is also seen in Fig. 13 (d) for the other near-equilibrium condition, the small EPR(10days). Consequently, the \(\alpha\)-tremor frequency distribution in near-equilibrium systems is similar in both the 10-day scale recorded data and the 2000-data-point scale \((d=0.5,\,p)\)-CA184 data.

## 10-day comparison of recorded data and CA data

Fig. 16 compares periodic \((d=0.5, p<1)\)-CA184 with ground vibration data recorded for 10 days. The comparison is performed on 16875 recorded data blocks, each data block corresponding to 51.2 seconds. An evaluation vector containing 8 components: EPR, \(\alpha\)-tremor, density of states, \(p_1\), \(p_2\), \(p_3\), \(p_4\), and \(p_5\) is calculated for each data block of the CA and the recorded data. Then, the Pearson correlation between the evaluation vector of the recorded data block and the evaluation vector of the CA data block is calculated. For each evaluation vector of the recorded data block, Pearson correlations are calculated for all CA data blocks. The CA data block with maximum absolute value of the Pearson correlation is selected as the “optimal match”, and compared with the recorded data block as shown in Fig. 16. In Fig. 16, the \(p_i\) components are differentiated by colors so that the values of \(p_i\) can roughly be read.

Fig. 16 (b) shows the comparison regarding the 10-day time interval with the medium EPR(10days) value of 0.054. The CA data consists of 10000 data blocks generated for \((d, p)\) of (0.5, 0.2), (0.5, 0.22), (0.5, 0.3), (0.5, 0.5), and (0.5, 0.87). For every \((d, p)\), 2000 data blocks are generated. EPR, \(\alpha\)-tremor, density of states, \(p_1\), \(p_2\), \(p_3\), \(p_4\), and \(p_5\) simultaneously match between the recorded and CA data: 99.9% of the 16875 pairs of CA and recorded data blocks show a Pearson correlation greater than 0.9.

Fig. 16 (a) shows the comparison regarding the 10-day time interval with the high EPR(10days) value of 0.06. To handle the high value in EPR, 4000 data blocks are generated for each of \(p=\) 0.2, 0.22, 0.3, 0.5 and 0.87, and 2000 data blocks are additionally generated for each of \(p=\) 0.15 and 0.17. Total 24000 data blocks are included in the CA data. 99.7% of the 16875 pairs of CA and recorded data blocks show a Pearson correlation greater than 0.9.

Fig. 16 (c) shows the comparison regarding the 10-day time interval with the low EPR(10days) value of 0.032. To handle the low value in EPR, 4000 data blocks are generated for each of \(p=\) 0.2, 0.22, 0.3, 0.5 and 0.87, and 2000 data blocks are additionally generated for \(p=\) 0.7. Total 22000 data blocks are included in the CA data. 99.9% of the 16875 pairs of CA and recorded data blocks show a Pearson correlation greater than 0.9.

In the all cases of the 10-day comparison between the recorded and the CA data, EPR, \(\alpha\)-tremor, density of states, \(p_1\), \(p_2\), \(p_3\), \(p_4\), and \(p_5\) are simultaneously in good agreement. For a given recorded data block, we can find a CA data block that closely matches the recorded data block. It is implied that the recorded data block is a subset of the periodic \((d=0.5,p<1)\)-CA184 data block. It is also implied that by appropriately selecting the pop probability “\(p\)”, the ground vibration states can be reproduced with the periodic \((d=0.5,p<1)\)-CA184, and the ground vibration in the GEJE process is equivalent to the periodic \((d=0.5,p<1)\)-CA184.

It is worth noting that the match between the CA and the recorded data for the small EPR(10days) in Fig. 16 (c) is achieved by introducing the additional CA data generated for \((d,p)\)=(0.5, 0.7). This implies that the small EPR(10days) system is closer to the equilibrium system of \(p\)=1 than the medium or large EPR(10days) system. This is consistent with the discussion for Fig. 13 that the smaller the EPR(10days), the closer to equilibrium. Therefore, both the CA and the recorded data provide similar thermodynamic information in a 10-day scale.

It should also be noted that if the evaluation vector of the recorded data block and the CA data block match, good matches are obtained for detailed dynamic properties such as the Fourier amplitude spectrum, the contour plot of \(W_{ij}\) and the \(t-t\) diagram. With respect to the case of large EPR(10days), Fig. 17 (a1) and (a2) compare the detailed characteristics of the best matched CA data block and the corresponding recorded data block, respectively. In both cases, there is no significant component near 10 Hz in the Fourier spectrum, there is a large yellow distribution in the lower left of the transition rate matrix map, and there are small number of clusters of blue cells in each row of the \(t-t\) diagram.

With respect to the case of low EPR(10days), Fig. 17 (b1) and (b2) compare the detailed characteristics of the best matched CA data block and the corresponding recorded data block, respectively. Both show large frequency components near 10 Hz in the Fourier spectrums, a large yellow distribution in the upper right of the transition rate matrix map, and a sparse distribution of blue cells in the \(t-t\) diagram.

The above comparisons show that there is coherences between the best matched CA and the corresponding recorded data with respect to the Fourier spectrum, the transition rate matrix, and the \(t-t\) diagrams of the binary sequence.