Two-Dimensional Probabilistic Cell Automaton Model for Broadband Slow Earthquakes

Abstract

Slow tectonic events, such as deep tectonic tremor, low-frequency earthquakes, very low frequency earthquakes, and slow slip events, are regarded as “broadband slow earthquakes”. Various observational properties of these events have previously been explained using a one-dimensional (1D) stochastic model that mimics Brownian motion in the time domain. Here we present a simple two-dimensional (2D) slow earthquake model using a probabilistic cell automaton (PCA), where each cell possesses one of two states (slip or hold) at a given time step. The status of each cell is updated as a Markov process dependent on its own status and that of neighboring cells. The seismic moment rate of the event in the 2D PCA model is assumed to be proportional to the number of slipping cells, which is approximated using a stochastic differential equation equivalent to the previous 1D Brownian model approach. This 2D PCA model is, therefore, able to explain the various observational properties of slow earthquakes. Moreover, it includes an additional degree of freedom to separate the effects of loading and dissipation, which enables us to explain the variations in episodic slow slip activity. The size of the slow earthquake is limited by either the system size or the characteristic size, which is controlled by the degree of dissipation. Such a simple representation of slow earthquakes is useful for characterizing the broad spectrum of slow earthquake behavior in different environments.

Appendix 1

Derivation of (2) from (1).

For an Ito process $X$ defined by

$${\text{d}}X = u{\text{d}}t + v{\text{d}}B,$$

with random processes $u$ and $v$, $Y = g(t,X)$ satisfies the Ito’s lemma (e.g., Øksendal 1998),

$${\text{d}}Y = \frac{\partial g}{\partial t}\left( {t,X} \right){\text{d}}t + \frac{\partial g}{\partial x}\left( {t,X} \right){\text{d}}X + \frac{1}{2}\frac{{\partial^{2} g}}{{\partial x^{2} }}\left( {t,X} \right)\left( {{\text{d}}X} \right)^{2} .$$

(A1)

In case

$${\text{d}}r = - \alpha r{\text{d}}t + \sigma {\text{d}}B,$$

(1)

and $S = C_{0} r^{2}$, we can write

$$\frac{\partial S}{\partial t} = 0, \frac{\partial S}{\partial r} = 2C_{0} r, \frac{{\partial^{2} S}}{{\partial r^{2} }} = 2C_{0} .$$

(A2)

Since $\left( {{\text{d}}r} \right)^{2} = \sigma^{2} {\text{d}}t$, substituting (A2) into (A1), we obtain

$${\text{d}}S = \left( {C_{0} \sigma^{2} - 2\alpha S} \right){\text{d}}t + 2\sigma \sqrt {C_{0} S} {\text{d}}B.$$

(2)