Data mining method in seismology by applying cellular automaton equivalence of ground vibration fluctuations recorded near the epicenter of the 2011 Mw 9 East Japan earthquake

Abstract

We propose a data mining method to extract physical phenomena from the ground vibration velocity fluctuation (GVF) which is a daily observed weak signal and has no specific waveform. Our aim is to detect any traces of the return of ground motion to its normal state after earthquakes, and the evolution of ground motion towards earthquakes in the GVF. We focus on the GVF recorded near the epicenter of the magnitude 9 Great East Japan Earthquake (GEJE). Although the GVF can be discussed in the framework of the master equation which represents the thermodynamics of a stochastically fluctuating non-equilibrium system, the relation between the thermodynamic parameters and seismology-related phenomena is hardly known. So, we introduce cellular automata (CA) to analogically interpret the physical phenomena of interest as thermodynamic parameters, and show the thermodynamic equivalence between GVF and CA. These CA-derived thermodynamic parameters are then expected to deliver physical phenomena similar to those of the target when evaluated on the GVF data. In the demonstrated example of data mining, the stress relaxation signal immediately after GEJE is discovered from the recorded GVF data by deriving the thermodynamic parameters representing the stress relaxation of CA and evaluating these thermodynamic parameters for the recorded GVF data. Furthermore, the elastic rigidity and viscosity of the underground structures are estimated from the stress relaxation signal. The novelty of this study lies in drawing attention to the GVF, and combining standard techniques of CA with GVF through the use of the master equation.

Appendix

Reasons for choosing GVF over ground vibration itself

We have been discussing the thermodynamic properties of GVF without directly involving the earthquake itself. Our focus is primarily on the A-noise, which is defined in the relatively high-frequency range of the Fourier spectrum of GVF. This approach is chosen based on our anticipation of a small difference between the spectrum of the ground motion and the spectrum of GVF (the fluctuation of the ground motion) in the high-frequency range. This is because the fluctuation signal primarily represents the high-frequency component of the original signal.

Figure 20 provides evidence supporting our initial expectations. In Fig. 20 (a), we compare the Fourier amplitude spectrum of the recorded ground vibration velocity (green line) and its GVF (black line). The red and orange lines represent the 10-point Simple Moving Averages of the green and black lines, respectively. Figure 20 (b) and (c) display the GVF data and recorded ground vibration velocity data, respectively. The ground vibration velocity data was recorded in the UD direction at KSN from 5048321 to 5049344 (sec × 20), with the start of the elapsed time set at 2012-03-01 00:00:00.

In the frequency range of 2.97 Hz to 9.8 Hz in Fig. 20 (a), which is where the A-noise is defined, the spectrum of the ground vibration velocity data closely matches the spectrum of GVF, exhibiting a Pearson correlation coefficient (PCC) of 0.8. This indicates that studying the A-noise and the related thermodynamics of ground vibration velocity is equivalent to studying the A-noise and the related thermodynamics of GVF. We have chosen to focus on GVF since fluctuations are closely associated with thermodynamics, as demonstrated by the master equation.

Rationale behind selecting CA184

We extensively explore the equivalence between GVF and CA and utilize CA184 as an appropriate rule for studying the thermodynamic properties of GVF. This section provides a rationale for this approach.

The recorded ground vibration and GVF are wave quantities measured at a fixed point and are likely related to the wave (Eq. 7).

$$\begin{aligned} \frac{d^2v}{dt^2}-cc^2\frac{d^2v}{dx^2}=0 \; , \;\; v=v(x,t) \; , \;\; cc=constant \end{aligned}$$

or equivalently

$$\begin{aligned} \left( \frac{d}{dt}+cc\frac{d}{dx} \right) \left( \frac{d}{dt}-cc\frac{d}{dx} \right) v=0 \\ \end{aligned}$$

or

$$\begin{aligned} \frac{dv}{dt}=-cc\frac{dv}{dx}\; , \;\; \frac{dv}{dt}=cc\frac{dv}{dx} \end{aligned}$$

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On the other hand, Nishinari and Takahashi (1999) showed that CA184 is mathematically derived by transforming the Burgers (Eq. 8) (Nishinari and Takahashi 1999). If the second term on the right side of Eq. 8 is neglected, the Burgers equation exhibits the same form as the wave equation (Eq. 7). Therefore, it is likely that GVF can be expressed using the Burgers equation and, consequently, CA184. As a result, there is a possibility of finding equivalence between GVF and CA184.

$$\begin{aligned} \frac{dv}{dt}=2v\frac{dv}{dx}+\frac{d^2v}{dx^2} \; , \;\; v=v(x,t) \\* \nonumber \end{aligned}$$

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