NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-179-2017Sandpile-based model for capturing magnitude distributions and spatiotemporal clustering and separation in regional earthquakesBatacRene C.rbatac@nip.upd.edu.phhttps://orcid.org/0000-0001-8377-7429Paguirigan Jr.Antonino A.TarunAnjali B.LongjasAnthony G.https://orcid.org/0000-0003-0256-7820National Institute of Physics, University of the Philippines Diliman, 1101 Quezon City, PhilippinesSt. Anthony Falls Laboratory, University of Minnesota, 2 Third Ave. SE, Minneapolis, MN 55414, USARene C. Batac (rbatac@nip.upd.edu.ph)27April20172421791874May20162June201610January201711January2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/179/2017/npg-24-179-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/179/2017/npg-24-179-2017.pdf
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
-1.6, consistent with the expectations of the Gutenberg–Richter (GR) law,
for a wide range of the targeted triggering probability values. Additionally,
for targeted triggering probabilities within the range 0.004–0.007, we
observe spatiotemporal distributions that show bimodal behavior, which is not
observed previously for the original sandpile. For this critical range of
values for the probability, model statistics show remarkable comparison with
long-period empirical data from earthquakes from different seismogenic
regions. The proposed model has key advantages, the foremost of which is the fact
that it simultaneously captures the energy, space, and time statistics of
earthquakes by just introducing a single parameter, while introducing minimal
parameters in the simple rules of the sandpile. We believe that the
critical targeting probability parameterizes the memory that is inherently
present in earthquake-generating regions.
Introduction
The
sandpile model, introduced as a representative system for illustrating
self-organized criticality (SOC; ), has opened up new
avenues for the use of discrete cellular automata (CA) models in capturing
the salient features of many systems in
nature .
Seismicity, which is rife with power-law statistical
distributions , is an interesting test case for
such approaches. Despite the complexity of the processes in the earth's crust
that limit our ability for accurate, short-term prediction of events, it is
worth noting that many statistical features of seismicity, as obtained from
substantially complete earthquake records, can be recovered using simple CA
models.
One of the earliest attempts for sandpile-based modeling of earthquake
distributions is by , who used a two-dimensional
sandpile to show the power-law Gutenberg–Richter (GR) distributions of
earthquake energies . Subsequent authors also
noted that the simple sandpile produces power-law distributions of earthquake
waiting times upon introducing a threshold
magnitude . Additional parameters have been
introduced in the model to account for other features of seismicity.
introduced aftershock triggering to the sandpile
model to recover the aftershock frequencies and the hypocenter distributions,
which also follow power-law decays. To represent a scale-invariant
distribution of earthquake faults,
incorporated a power-law distribution of box sizes in the CA model and
recovered not only the GR distribution but the occurrence of foreshocks. On
the other hand, investigated the effect of a
heterogeneous strength distribution and found that the power-law exponent of
the magnitude distribution is dependent on the degree of the heterogeneity.
Inspired by the sandpile design, used a CA implementation
of the earlier Burridge–Knopoff model that
incorporates dissipative terms and inhomogeneous energy redistribution rules
to capture key elements of seismicity, along with foreshocks and
aftershocks . In another work,
has shown that the GR law can be recovered in a
forest-fire model, with the fires interpreted as the earthquake occurrences.
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.
Model specifications
The model utilizes a two-dimensional space discretized into a grid of L×L cells arranged in a square lattice. The cells contain
continuous-valued information states σ representing the local measure
of susceptibility to rupture. At time t=0, the states are initialized to
have values within [0,σmax), where, in this case, we set
σmax=1.0 as the relative measure of the rupture threshold.
The dynamical evolution of the grid is guided by rules patterned after the
Zhang sandpile that uses continuous-valued states . We
choose an asynchronous update rule, such that every discrete time step the
grid is triggered by adding a constant value ν to a single location (x,y), σ(x,y,t+1)→σ(x,y,t)+ν, with all the
other sites unperturbed. The asynchronicity may represent the nonuniformity
of crustal motion that drives the accumulation of elastic potential energy at
faults. Moreover, the model introduces a targeted triggering probability p
that the most susceptible site, i.e., the site with the highest σ value
in the grid, will receive the driving term ν. Triggering is therefore
applied to the most susceptible site with probability p and to a randomly
chosen site with probability 1-p. The value of p represents a memory
term, and parameterizes the tendency of fracture to occur at more susceptible
locations along an earthquake-generating zone.
In the event that a cell matches or exceeds a maximum possible value
σmax, the local region is deemed to rupture. No new trigger is
added to the system during such events; instead, the stress from the
collapsing site is transferred to the four nearest neighbors in the grid,
σ(x±1,y±1,t)→σ(x±1,y±1,t)+14σ(x,y,t), leading to the relaxation of the original site,
σ(x,y,t)→0. Such relaxations may produce a cascade of
subsequent stress redistributions and relaxations in the grid when one or
more of the neighbors are driven to the threshold. As in the
previous sandpile models, the number of affected sites in the grid, A, is
tracked to quantify the relative event size. Additionally, we also recorded
the number of unique activations V, the number of times a cell has been
affected by a cascade, as a proxy for the actual energy or seismic moment of
the relaxation event.
Avalanche size and earthquake energy PDFs. For all figures, lines
corresponding to the power-law trend with exponent α=1.6 are
provided as guides. (a) Model results show similar behaviors
despite the large differences in p, signifying the retention of sandpile
characteristics. The obtained power-law distributions are comparable to the
power-law trends in the energy distributions from (b) Japan (JP),
(c) Philippines (PH), and (d) southern California (SC). In panels (b)–(d), the horizontal
axes scales are preserved; shaded regions denote energy values with
substantial completeness which will be used for subsequent
analyses.
Prior calibrations show that ν=10-3 produces power-law event size
distributions comparable to the GR law, and that tmax={1,4,16}×107 iterations, where the first 10 % are neglected for transient
behavior, produces a substantial number of avalanche events for L={256,512,1024} grids, respectively. We investigated the case of different
targeted triggering probabilities, p={0,1×10k,5×10k,1}, where the integer k ranges from -5 to -1, to scan a wide range of possible
system behaviors. For each of the p values, we track all nonzero Ai and
Vi and their avalanche origins and occurrence times (xi,yi,ti),
where i denotes the temporal index of occurrence of an event. The spatial
and temporal separations of successive events, Ri=[(xi-xi-1)2+(yi-yi-1)2]1/2 and Ti=ti-ti-1, are computed, and the
probability density functions (PDFs) of all A, V, R, and T values are
plotted.
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 Ath={5,10,50,100,500,1000,5000} such that all events with A<Ath are removed from the sequence.
Because the A PDF is just expected to be cut off below Ath, we
observed how the statistical distributions of R and T will be affected
upon employing different Ath values.
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 . The JP records are obtained from
the Japan University Network Earthquake Catalog (JUNEC), with approximately
137 000 events from July 1985–December 1998; the PH earthquakes are composed
of 70 000+ events from 1973 to 2012, as obtained from the Preliminary
Determination of Earthquakes (PDE) Catalog; while the SC records are from the
Southern California Earthquake Catalog (SCEC) containing 516 000+ events from
1982 to 2012 (events due to man-made activities are removed). We compared the
behaviors of the model and data statistics using scaling factors derived from
model parameters.
Model results
Figure a shows the avalanche size PDFs for the different values
of the targeted triggering probability p. For the broad range of p values
considered, the distributions are found to be comparable to a power-law
A-α with α=1.6. Continuous-state sandpiles have been known
to have avalanche size scaling exponents greater than 1.0, the exponent of
the discrete Bak–Tang–Wiesenfeld (BTW) sandpile.
conducted large-scale simulations of a similar Zhang sandpile and obtained
exponents slightly higher than 1.2, which can go even higher for large
driving rates ν. In a similar model that incorporated non-conservation,
obtained power-law exponents that approach 1.6 in
the conservative limit for the same order or magnitude of ν that we used.
The higher exponents and the effect of the driving rates are also verified by
an equivalent conservative model and actual sand avalanche experiments by
, and in other asynchronous updating
models .
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 p
indicates that the targeted triggering probability has minimal effect on the
avalanching mechanism of the grid, such that the system preserves the SOC
characteristics of the original sandpile. In contrast, the Olami–Feder–Christensen
(OFC) model, one of the foremost discrete models of seismicity, tends to lose the universality of the exponents upon the
introduction of non-conservation .
Interevent distance statistics of model, with
rescaling for comparison with actual earthquake separation distance data.
(a) For an L=256 grid, higher p results in the preponderance of short-R
values. The trends of the model closely mimic those of the data for (b) JP,
(c) PH, and (d) SC, where calibration was done by comparing the modes of the
model p=0 and shuffled sequences of the empirical data. Larger grids in
panels (b)–(d) result in the capability to replicate the shorter R
regimes.
In Fig. a, we observe that the original sandpile p=0
produces unimodal statistics, whose tails decay towards the largest possible
distance 2L in the finite grid. The simple sandpile, therefore, is
not capable of replicating the observed earthquake separation distance
distributions, which are found to exhibit bimodality due to the difference in
the characteristic times of the correlated aftershock sequences and the
independent
mainshocks .
This inspired the introduction of p, which is a random occurrence in time
but is inherently affecting the spatial distribution of events in the grid.
We do note here that the parameter p is just the probability to target the
most susceptible site in the lattice, unlike previous implementations that
actually preselect the next targeting location within the vicinity of the
previous avalanche . Indeed, without the
imposition of such a spatial bias, the replication of the short-R regimes
is not guaranteed. Interestingly, however, the plots in Fig. a
show increased probability of occurrence of the short-R distances upon
introducing nonzero p. From this, we can deduce that the most susceptible
sites in the lattice are most likely to be found within the vicinity of a
previous large avalanche, a fact that was not exploited by earlier similar
models. In fact, in the biased case p=1, we recovered unimodal
statistics, as shown in Fig. a, albeit at a shorter
characteristic distance; for the L=256 grid, the average location of the
most susceptible site from the previous avalanche origin was found to be
around 21 cell lengths. Midway between these two extremes (p=0 for the
original and p=1 for the completely biased sandpile), we can find a
suitable value of p where reasonable comparison with empirical data can be
obtained.
Interevent time statistics of model, and rescaling
for comparison with actual earthquake waiting time data. (a) For an L=256
grid, higher p results in the shift of the distribution to shorter T
values. To obtain substantial power-law regimes, we used the results for the
L=1024 grid to replicate the waiting time statistics of (b) JP, (c) PH, and
(d) SC. By preserving the fraction of events left upon imposing thresholding,
we obtained Ath values of (b)5×103 for JP, (c)5×105 for PH, and (d)5×103 for SC. The shortest waiting times in
the data are scaled to be a unit of iteration. The finite total iteration
times resulted in model distributions that are not able to capture the very
long tails of those of the empirical data, especially for (d) SC, which has
the longest period among the catalogs considered.
The interevent time distributions are shown in Fig. a for the
L=256 and tmax=107 iterations. We observe the expected shift of
the tail cutoff towards shorter T values as p is increased; triggering
the highly susceptible sites will more likely result in a new avalanche
event, thereby shortening the average waiting time. The resulting
distributions are for the case wherein all the events are included in the
sequence; we expect a lengthening of the tails of the distributions when we
neglect other events below the threshold Ath.
DiscussionEnergy distributions and the Gutenberg–Richter law
The GR law, which is usually presented in terms of the magnitude m and as
a complementary cumulative distribution function (CCDF) log10m=a-bm, can be shown to be equivalent to an energy E CCDF that behaves as
E-2/3 from the definition of m and by assuming b=1, which is the
case for most complete records . By noting that the CCDF
is effectively an integral of the PDF, the earthquake energy PDF will then
behave as E-5/3. In Fig. b–d, similar power-law trends
have been obtained for the JP, PH, and SC records, which have different
levels of catalog completeness, as indicated by the extent of the power-law
regimes. To minimize the problems associated with the inherent incompleteness
of smaller-energy events , we impose a
threshold magnitude mth for succeeding analyses such that earthquake
events with magnitudes lower than mth are dropped from consideration.
The range of such magnitudes considered, which are well within the power-law
regimes of the plots, is shaded in Fig. b–d: mth∈[2.5,3.5] for JP and SC and mth∈[4.5,4.8] for PH.
In keeping with the earlier sandpile-based approaches where the avalanche
size A is used for comparison with earthquake
energies , we present in
Fig. the PDFs of A with those of E from the seismogenic
regions considered. It is worth emphasizing that similar power-law trends
result from the introduction of the parameter p, regardless of how large
its relative value is. We note, however, that aside from the avalanche size
A, there are other parameters that can be used to track the extent of the
avalanche event. One such measure is the number of activations V, wherein
the sites repeatedly affected by the avalanching process get to be counted
multiple times. Previous works have shown that V and A in discrete models
may in fact have actual associations with the seismic moment and fracture
area, respectively, and may exhibit nontrivial scaling
relations . We present in Fig. a
the distributions obtained upon tracking V.
Model statistics for V and scaling with A.
(a) The PDF of V (shown here with a power-law V-β where β=1.45
as a guide) also shows minimal changes upon introducing p. (b) The
scaling behavior of V with A is supralinear, with behaviors ranging from
V∝A1.5 for p=0 to V∝A1.3 in the regime of large
A values for p=1.
Empirical earthquake interevent distance
distributions (hollow symbols), along with the corresponding shuffled
sequences (filled symbols) for (a) JP, (b) PH, and (SC). The broken lines
indicate the R* values where the original and the shuffled sequences
begin to show similar trends.
The V PDFs also show a behavior similar to those of their corresponding
A: there are minimal changes upon scanning for different p values. The
distributions also follow power-law behaviors V-β with β
around 1.4 to 1.5 (the case of β=1.45 is plotted as a guide in Fig. a). The parameter V is a better representation
of the energy E in earthquakes, and the obtained scaling exponent β
is still deemed to be close to the earthquake energy scaling exponents. The
fact that the model can replicate the energy statistics is a vital first
requirement for any discrete model of earthquakes. Additionally, the
preservation of the power-law exponent for almost any value of p indicates
that the model does not deviate significantly from the original sandpile
behavior, and may exhibit (self-organized) criticality.
To understand the scaling relations between V and A, we plot the V
(activated cells) versus A (affected cells) in Fig. b and note
that the scaling relations, which are higher than linear, change for higher
p. The case of p=0 (randomly triggered sandpile) results in a V(A)∝A1.5 scaling. On the other hand, for p=1 (sandpile with
targeted triggering), the behavior appears to shift towards V(A)∝A1.3 for very large A values. This lower scaling exponent of the
activation for large avalanche sizes is expected for targeted triggering;
because the most susceptible site is always targeted, there is minimal
accumulation of near-critical sites near the location of the avalanche
origin, which results in a lower number of reactivations of affected sites near
an avalanche event.
Spatial separation of earthquake events
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
Tin and Tout for (a)–(c) earthquake data and (d)–(f) corresponding
rescaled model results, plotted with the relative frequency plot of all T values.
Nearby (far away) events have a higher (lower) chance of having short waiting
times and a lower (higher) chance of having long waiting times, as can be seen
from the modes of the conditional frequency distributions. The insets of
panels (d)–(f) show that the Tin PDFs of model and rescaled data have
significant overlap, signifying the similarities in their correlated
origins.
Interestingly, the addition of the simple targeted triggering probability p
has enabled us to recover statistical distributions that are comparable to
those observed in regional earthquake records up to a scaling factor. It
should be noted that without any form of spatial clustering, the
characteristic separation distance is limited by the finite system size.
Rescaling is therefore conducted by comparing the characteristic sizes
(modes) of the memoryless cases of the model (p=0) and the data (shuffled
sequence). The interevent distance distributions of the shuffled sequences
are shown as the black symbols in Fig. , while the
corresponding model p=0 distribution is shown in Fig. a,
with both clearly showing unimodal statistics.
Upon getting the rescaling factor, we scan through the possible p values to
obtain p values that will result in comparable R distributions between
model and data. We observe that the model parameters that will correspond to
the empirical distributions upon such a simple rescaling range from p∗≈ 0.004 to 0.007. Figure b–d show the interevent
distances between successive earthquakes in the different regional records
considered, superimposed with the rescaled statistics of the model.
The rescaled model statistics for p=0.007 show good agreement with
interevent distances from the three seismogenic regions. As expected, larger
grid sizes will result in a better discrimination of shorter R; i.e., one
pixel unit will correspond to shorter actual distance units. In our case, for
the largest grid size used (L=1024), we find that the scaling factors
obtained by matching the modes result in the following correspondence with a
unit cell length: 1.3 for JP, 1.2 for PH, and 0.5 km for SC. The
distributions are found to be similar regardless of the threshold magnitude
Ath considered due to the finite system size; even upon removing the
weakest events, the avalanche origins are confined within the grid, resulting
in the same probability density distribution of R.
Temporal separation of earthquake events
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 .
The proposed model, therefore, must also show these features to be able to
compare reasonably well with the temporal distributions of seismicity. In the
following, we compare the results of the model having p*=0.007 and
grid dimension L=1024, which has been shown to have comparable R
statistics with empirical data.
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 T regimes first;
theoretically, the longest T will be recovered if the model is allowed to
run for very long iteration times. Additionally, in rescaling the time, one
should take into account the fact that the earthquake record is thresholded
by mth, effectively lengthening the average time between the occurrence
of two events. Ideally, if all the events, no matter how weak, can be
detected and recorded, we would not have long tails in the waiting time
distribution of earthquakes. This is also observed in sandpile-based models;
previous approaches have shown that the waiting time distribution will be
Poisson distributed when all the events are considered, but will begin to
show apparent power-law characteristics upon
thresholding .
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 mth=2.5, we used Ath=5×103; on the other
hand, for PH, with relative completeness beyond mth=4.5, Ath=5×105 is used. The values of Ath are obtained by maintaining the
fraction of events left after neglecting the weaker events. Still, because of
the limited number of regional data sets considered, which does not allow for
further testing their correspondence, we emphasize that the values of the
Ath obtained do not necessarily translate into an exact equivalence
with the threshold magnitude mth for the data.
Upon removing the events with A<Ath, we obtained the modes of both the
data and the model for visual comparison. This resulted in slight differences
in the rescaling factors for the different data sets. One iteration of the
model corresponds to 0.006 s for JP, 0.004 s for PH, and 0.002 s for SC.
Fig. b–d above show the rescaled model distributions
alongside the those of the empirical data, showing qualitative similarities
in their trends.
Apart from recovering the qualitative trends in the T PDFs, we conducted
additional analyses to check if the model results also show spatiotemporal
clustering and separation behaviors. In Fig. , we mark the
location of R*, the characteristic separation distance where the
empirical distributions and those of the shuffled sequences begin to show
comparable trends. The R* values of the region considered, which are
similar to the results of , are deemed to be a good
marker for separating “nearby” and “far away” events. We note that a
similar procedure done using the rescaled model statistics results in
comparable R* values. Using R*=164 km for JP, R*=125 km
for PH, and R*=79 km for SC, we separated the corresponding waiting
times T into the sets Tin={T|R≤R*} and Tout={T|R>R*}. Figure shows the relative frequency plots of
T, superimposed with those of Tin and Tout, for the empirical
data and the rescaled model values.
As shown in Fig. a–c, for all the seismogenic regions
considered, the distributions of Tin and Tout differ significantly
from that of the total T. The relative frequency plots of T in all cases
can be shown to be a crossover between Tin and Tout that have
different modes. As expected, the Tout distributions do not coincide due
to the different periods involved in the catalogs considered. On the other hand, the Tin
distributions all have modes at short T values,
suggesting a strong dependence among the interevent properties in space and
time . This conditional distribution therefore
quantifies the spatiotemporal clustering observed in earthquakes,
particularly among aftershock sequences that result from the correlated
mechanisms: “nearby” events are also more likely to be separated by shorter
waiting times.
In Fig. d–f, we observe that despite the shorter
iteration times being considered, the model was able to show the separation
of the Tin and Tout distributions, a feature that is also found in
empirical data and in other earthquake
models . Moreover, it is particularly interesting to
note that the rescaled Tin statistics of model and corresponding
Tin from the earthquake data show comparable trends, especially for
shorter waiting times, as shown in the insets. The Tin statistics
have been shown to correspond with the statistics of aftershocks, as shown in
studies of fresh aftershock statistics from empirical
data . This suggests that the correlated mechanisms in
actual earthquake systems that produce the Tin distributions are also
present in the model.
Model advantages and insights on empirical modeling
Introducing the parameter p into the sandpile driving is a straightforward
way of incorporating memory into the system. This simple parameter holds a
distinct advantage over other models that introduced additional parameters,
because it spans a wide range of possible statistical distributions in event
size, space, and time, without actually biasing the location of the next
triggering event. Being a single parameter, the correspondence between p
and actual properties of the earthquake-generating system may be difficult,
if not impossible, to ascertain. At best, we may think of p as a combined
effect of many different factors on the ground that lead to the preferential
triggering of a location.
We believe that this parameter, which, for earthquakes, show comparable
statistics for the range p*≈ 0.004–0.007, may be introduced in
other sandpile-based models of other events in nature deemed to show
self-organized (critical) characteristics. It may be possible to quantify the
extent of “memory” of these systems through the value of the parameter p
that best replicates their statistical distributions.
Moreover, a deeper analysis of the other regimes of p may lead to a better
comparison between the model and other similar protocols. For example, for
higher values of p, the model may exhibit extremal dynamics, resulting in
more avalanche events due to the tendency to always trigger the most
susceptible site. On the other hand, for very low values of p, the dynamics
may be comparable to other models that employ uniform loading. Knowing these
limits, and establishing how similar and/or different the model is from other
discrete models may help put the results in a better context.
Conclusions
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 (p*≈ 0.004–0.007), we have
observed that the model statistics show comparable trends with empirical
distributions of earthquake occurrences in energy, space, and time, upon
simple rescaling.
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 ftp://ftp.eri.u-tokyo.ac.jp/pub/data/junec/hypo/. The Philippine earthquake data
are accessed from the Preliminary Determination of
Earthquakes catalog at ftp://hazards.cr.usgs.gov/NEICPDE/olderPDEdata, with earthquakes within 4–20∘ N latitude and
115–130∘ E longitude used for analysis. Finally, the southern California earthquake data are taken from
https://service.scedc.caltech.edu/ftp/catalogs/SCEC_DC/.
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.
Acknowledgements
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
ReferencesBaiesi, M. and Paczuski, M.: Scale-free networks of earthquakes and
aftershocks, Phys. Rev. E, 69, 066106, 10.1103/PhysRevE.69.066106, 2004.Bak, P. and Tang, C.: Earthquakes as a Self-Organized Critical Phenomenon, J.
Geophys. Res., 94, 15635–15637, 10.1029/JB094iB11p15635, 1989.Bak, P., Tang, C., and Wiesenfeld, K.: Self-organized criticality – An
explanation of the 1/f noise, Phys. Rev. Lett., 59, 381–384,
10.1103/PhysRevLett.59.381, 1987.Barriere, B. and Turcotte, D. L.: A scale-invariant cellular automata model
for distributed seismicity, Geophys. Res. Lett., 18, 2011–2014,
10.1029/91GL02415, 1991.Batac, R. C.: Statistical Properties of the Immediate Aftershocks of the 15
October 2013 Magnitude 7.1 Earthquake in Bohol, Philippines, Acta Geophys.,
64, 15–25, 10.1515/acgeo-2015-0054, 2016.Batac, R. C. and Kantz, H.: Observing spatio-temporal clustering and separation using interevent distributions of regional earthquakes,
Nonlin. Processes Geophys., 21, 735–744, 10.5194/npg-21-735-2014, 2014.
Burridge, R. and Knopoff, L.: Model and theoretical seismology, B. Seismol.
Soc. Am., 57, 341–371, 1967.Drossel, B. and Schwabl, F.: Self-organized critical forest-fire model, Phys.
Rev. Lett., 69, 1629–1632, 10.1103/PhysRevLett.69.1629, 1992.
Gutenberg, B. and Richter, C. F.: Seismicity of the Earth and Associated
Phenomena, 2nd Edn., Princeton, N.J.: Princeton University Press, 1954.Hergarten, S. and Neugebauer, H. J.: Foreshocks and Aftershocks in the
Olami-Feder-Christensen Model, Phys. Rev. Lett., 88, 238501,
10.1103/PhysRevLett.88.238501, 2002.Ito, K. and Matsuzaki, M.: Earthquakes as self-organized critical phenomena,
J. Geophys. Res., 95, 6853–6860, 10.1029/JB095iB05p06853, 1990.Jagla, E. A.: Forest-Fire Analogy to Explain the b Value of the
Gutenberg-Richter Law for Earthquakes, Phys. Rev. Lett., 111, 238501,
10.1103/PhysRevLett.111.238501, 2013.
Juanico, D. E., Longjas, A., Batac, R., and Monterola, C.: Avalanche
statistics of granular driven slides in a miniature mound, Geophys. Res.
Lett., 35, L19403, 10.1029/2008GL035567, 2008.Landes, F. P. and Lippiello, E.: Scaling laws in earthquake occurrence:
Disorder, viscosity, and finite size effects in Olami-Feder-Christensen
models, Phys. Rev. E., 53, 051001, 10.1103/PhysRevE.93.051001, 2016.Livina, V. N., Havlin, S., and Bunde, A.: Memory in the Occurrence of
Earthquakes, Phys. Rev. Lett., 95, 208501, 10.1103/PhysRevLett.95.208501,
2005.Lübeck, S.: Large-scale simulations of the Zhang sandpile model, Phys.
Rev. E, 56, 1590–1594, 10.1103/PhysRevE.56.1590, 1997.Malamud, B. D. and Turcotte, D. L.: Cellular automata models applied to
natural hazards, Comput. Sci. Eng., 2, 42–52, 10.1109/5992.841795, 2000.Olami, Z., Feder, H. J. S., and Christensen, K.: Self-organized criticality
in a continuous, nonconservative cellular automaton modeling earthquakes,
Phys. Rev. Lett., 68, 1244–1247, 10.1103/PhysRevLett.68.1244, 1992.Paczuski, M., Boettcher, S., and Baiesi, M.: Interoccurrence Times in the
Bak-Tang-Wiesenfeld Sandpile Model: A Comparison with the Observed Statistics
of Solar Flares, Phys. Rev. Lett., 95, 181102,
10.1103/PhysRevLett.95.181102, 2005.Paguirigan Jr., A. A., Monterola, C., and Batac, R. C.: Loss of criticality in
the avalanche statistics of sandpiles with dissipative sites, Commun. Nonlin.
Sci. Numer. Simulat., 20, 785–793, 10.1016/j.cnsns.2014.06.020, 2015.Piegari, E., Cataudella, V., Di Maio, R., Milano, L., and Nicodemi, N.: A
cellular automaton for the factor of safety field in landslides modeling,
Geophys. Res. Lett., 33, L01403, 10.1029/2005GL024759, 2006.Saichev, A. and Sornette, D.: “Universal” Distribution of Interearthquake
Times Explained, Phys. Rev. Lett., 97, 078501,
10.1103/PhysRevLett.97.078501, 2006.Steacy, S. J., McCloskey, J., and Bean, C. J.: Heterogeneity in a
self-organized critical earthquake model, Geophys. Res. Lett., 23, 383–386,
10.1029/96GL00257, 1996.Touati, S., Naylor, M., and Main, I. G.: Origin and Nonuniversality of the
Earthquake Interevent Time Distribution, Phys. Rev. Lett., 102, 168501,
10.1103/PhysRevLett.102.168501, 2009.Zaliapin, I. and Ben-Zion, Y.: Artefacts of earthquake location errors and
short-term incompleteness on seismicity clusters in southern California,
Geophys. J. Int., 202, 1949–1968, 10.1093/gji/ggv259, 2015.Zaliapin, I., Gabrielov, A., Keilis-Borok, V., and Wong, H.: Clustering
Analysis of Seismicity and Aftershock Identification, Phys. Rev. Lett., 101,
018501, 10.1103/PhysRevLett.101.018501, 2008.Zhang, Y.-C.: Scaling theory of self-organized criticality, Phys. Rev. Lett.,
63, 47073, 10.1103/PhysRevLett.63.470, 1989.