We propose a cellular automata model for earthquake occurrences patterned
after the sandpile model of self-organized criticality (SOC). By
incorporating a single parameter describing the probability to target the
most susceptible site, the model successfully reproduces the statistical
signatures of seismicity. The energy distributions closely follow power-law
probability density functions (PDFs) with a scaling exponent of around

The
sandpile model, introduced as a representative system for illustrating
self-organized criticality (SOC;

One of the earliest attempts for sandpile-based modeling of earthquake
distributions is by

The introduction of additional parameters to subsequent models indicates that the simplest rules of the original sandpile are not able to capture key features of seismicity. In the sandpile model, the stress in the grid is released in a single avalanche event resulting from small-neighborhood cascades; for seismicity, the energy is released in a sequence of correlated events. Additionally, the single triggering at random locations will tend to produce normal distributions of interoccurrence distances and times, which, again, deviate from those observed in records of seismicity. Finally, the conservative sandpile with symmetric nearest-neighbor redistribution rules does not take into account the memory that may be present in actual earthquake-generating zones.

In this work, we adhere to the key features of the sandpile model, and introduce a very simple modification: for a fraction of the iteration times, determined randomly, we direct the triggering into the most susceptible site in the grid. In this case, the avalanches in the grid are deemed to be analogous to the energy release during an earthquake occurrence. Interestingly, this very simple modification in the sandpile rule enabled us to recover, simultaneously, the distributions of event sizes, interevent distances, and interevent times that are comparable to those obtained from substantially complete earthquake records.

The model utilizes a two-dimensional space discretized into a grid of

The dynamical evolution of the grid is guided by rules patterned after the
Zhang sandpile that uses continuous-valued states

In the event that a cell matches or exceeds a maximum possible value

Avalanche size and earthquake energy PDFs. For all figures, lines
corresponding to the power-law trend with exponent

Prior calibrations show that

Records of very low-magnitude earthquakes are oftentimes incomplete because
they are both too weak for detection and their occurrence is orders
of magnitude in frequency as compared with the higher-magnitude ones. In the
model, however, we can resolve all the avalanches, even the smallest ones
that affect only single neighborhoods. To mimic the effect of the
non-retention of the smallest earthquakes, we employed a thresholding
procedure in the analyses by setting

Finally, as a way of comparison and verification, we compare the model
statistics with those obtained from actual earthquake catalogs from Japan
(JP), Philippines (PH), and southern California (SC), as investigated in a
previous work by

Figure

The resulting power-law exponent is deemed to be a result of the accumulation
of stress at various locations; because the triggering is done at only a
single site every time, there is little global connectivity among critical
sites, resulting in a preponderance of smaller, isolated avalanches. The fact
that the distributions are almost similar regardless of the value of

Interevent distance statistics of model, with
rescaling for comparison with actual earthquake separation distance data.

In Fig.

Interevent time statistics of model, and rescaling
for comparison with actual earthquake waiting time data.

The interevent time distributions are shown in Fig.

The GR law, which is usually presented in terms of the magnitude

In keeping with the earlier sandpile-based approaches where the avalanche
size

Model statistics for

Empirical earthquake interevent distance
distributions (hollow symbols), along with the corresponding shuffled
sequences (filled symbols) for

The

To understand the scaling relations between

In the original asynchronous sandpile models, one only recovers unimodal statistics for interevent distances. This is due to the stochastic nature of the triggering: the next location to be perturbed is drawn from an oftentimes uniform distribution; i.e., all sites are likely to be triggered next. Additionally, the nature of internal cascading within the sandpile grid results in the depletion of all the critical sites within the extent of the avalanche area. The same cannot be said of earthquakes: after the release of elastic potential energy at a fault location, the subsequent crustal motion may tend to favor other fractures near the vicinity of the earlier event to release the remaining stored energy.

Conditional relative frequency distributions of

Interestingly, the addition of the simple targeted triggering probability

Upon getting the rescaling factor, we scan through the possible

The rescaled model statistics for

The temporal separation of aftershocks and mainshocks that have different
characteristic waiting times is an intuitive result that is both well known
and widely
studied

In comparing model and empirical temporal interevent statistics, one does not
have the similar advantage of having a finite “space.” The goal of
rescaling in time is to recover the relatively short

For our purpose, we arbitrarily chose the following threshold avalanche sizes
for removing weaker events: for comparison with JP and SC, which are both
taken to have

Upon removing the events with

Apart from recovering the qualitative trends in the

As shown in Fig.

In Fig.

Introducing the parameter

We believe that this parameter, which, for earthquakes, show comparable
statistics for the range

Moreover, a deeper analysis of the other regimes of

In summary, we have presented a simple cellular automata model inspired by
the original sandpile model. The model avoids introducing biased rules and
instead incorporates a probability of targeting the most susceptible site in
the grid, reminiscent of the assumed fracture mechanism of actual earthquake
systems. Within a small range of values (

The work has also uncovered an important property of the sandpile grid: the most susceptible sites lie within the vicinity of a previous large avalanche event. Previous sandpile-based models that synchronously update all lattice sites, or those that asynchronously update at random locations, are not able to exploit this important property, preventing the possibility of directly modeling earthquakes using the sandpile paradigm. The introduction of such a targeting probability without destroying the sandpile properties may hint at self-organized critical mechanisms at work in the grid. The fact that the simple targeted triggering probability simultaneously recovers these important statistical features of earthquakes is a simple yet novel concept that has not been exploited by previously proposed discrete models based on the sandpile.

Deeper analyses and comparisons with other established models of seismicity may help further establish similarities and differences and put the model results in a better context. Additionally, the parameterization of memory in the form of the targeted triggering probability may be extended to other similar models to possibly capture the statistical distributions of other self-organized (critical) events in nature and society.

The Japan University Network Earthquake Catalog (JUNEC) can be accessed
at

Rene C. Batac devised the model and Antonino A. Paguirigan Jr. ran the large-scale simulations. Rene C. Batac and Anjali B. Tarun wrote the paper. Anthony G. Longjas and Anjali B. Tarun provided the empirical data and model comparisons. Anjali B. Tarun and Antonino A. Paguirigan Jr. conducted statistical goodness-of-fit tests.

The authors declare that they have no conflict of interest.

The authors would like to acknowledge financial support from the University of the Philippines Diliman (UPD) Office of the Vice Chancellor for Research and Development (OVCRD) through a PhD Incentive Award with project title “Quantifying the clustering characteristics of complex, self-organizing systems in nature and society.” Antonino A. Paguirigan Jr. acknowledges the Department of Science and Technology (DOST) for his Advanced Science and Technology Human Resources Development Program (ASTHRDP) scholarship.

We extend our gratitude to R. Gloaguen (editor), F. Landes, S. Hergarten, and one anonymous referee for recommendations that significantly improved the content and the presentation of the manuscript. Edited by: R. Gloaguen Reviewed by: S. Hergarten and two anonymous referees