In general terms, earthquakes are the result of brittle
failure within the heterogeneous crust of the Earth. However, the rupture
process of a heterogeneous material is a complex physical problem that is
difficult to model deterministically due to numerous parameters and
physical conditions, which are largely unknown. Considering the variability
within the parameterization, it is necessary to analyze earthquakes by means
of different approaches. Computational physics may offer alternative ways to
study brittle rock failure by generating synthetic seismic data based on
physical and statistical models and through the use of only few free parameters.
The fiber bundle model (FBM) is a stochastic discrete model of material
failure, which is able to describe complex rupture processes in heterogeneous
materials. In this article, we present a computer code called
the stochasTic Rupture Earthquake MOdeL, TREMOL. This code is based on
the principle of the FBM to investigate the rupture process of asperities on
the earthquake rupture surface. In order to validate TREMOL, we carried out a
parametric study to identify the best parameter configuration while
minimizing computational efforts. As test cases, we applied the final
configuration to 10 Mexican subduction zone earthquakes in order to compare
the synthetic results by TREMOL with seismological observations. According to
our results, TREMOL is able to model the rupture of an asperity that is
essentially defined by two basic dimensions: (1) the size of the fault plane
and (2) the size of the maximum asperity within the fault plane. Based on
these data and few additional parameters, TREMOL is able to generate numerous
earthquakes as well as a maximum magnitude for different scenarios within a
reasonable error range. The simulated earthquake magnitudes are of the same
order as the real earthquakes. Thus, TREMOL can be used to analyze the
behavior of a single asperity or a group of asperities since TREMOL considers
the maximum magnitude occurring on a fault plane as a function of the size of
the asperity. TREMOL is a simple and flexible model that allows its users
to investigate the role of the initial stress configuration and the
dimensions and material properties of seismic asperities. Although various
assumptions and simplifications are included in the model, we show that
TREMOL can be a powerful tool to deliver promising new insights into
earthquake rupture processes.
Introduction
Rupture models of large earthquakes suggest significant heterogeneity in slip
and moment release over the fault plane
e.g.,. In order to characterize the seismic
source rupture complexity, two main models have been proposed: the asperity
model and the barrier model . Asperities
are defined as regions on the fault rupture plane that have larger slip and
strength in comparison to the average values on the fault plane
. Asperities also have larger stress drop than the
background area . Understanding the physical
features in the fault zone that produce these high-slip regions is still a
challenge.
The most common method for studying seismic asperities is waveform slip
inversion. However, information obtained from this method is highly variable
due to the inherent nature of the inversion process (see review in
). The slip inversion results depend on the type of data
(such as strong ground motion and geodetic and/or seismic data at different
distances) and the inversion technique used. used average
slip to define asperities. In their criterion, asperities include fault
elements for which slip is 1.5 times or more larger than the average slip. By
using this criterion, it is possible to estimate the asperity area from a
finite-fault slip model. Considering the stress drop for a circular crack
model (Δσ) , the stress drop on an asperity
(Δσa) can be estimated as Δσa=(Aeff/Aa)Δσ, where Aeff and
Aa are the rupture effective area and the asperity area,
respectively . The Aeff/Aa
factor (or its reciprocal value) depends on different features with the most
relevant one being the type of earthquake. For example,
found that on average the total area covered by asperities represents
22 % of the total rupture area for inland crustal events.
showed that Aa/Aeff is
approximately equal to 20 % for plate-boundary events. Similarly, for
subduction events, the value of Aa/Aeff is
approximately equal to 25 % .
The previous average values were determined considering values that range
from 0.09 to 0.35. This last condition means, for instance, that the
reciprocal fraction Aa/Aeff can deviate from these
average values as well (for example, 0.09 to 0.35 for the proportions
mentioned above), which leads to great stress contrasts (factors of 2.8 to
11) .
proposed another definition of asperities based on the maximum displacement,
Dmax. They defined “large-slip” and “very-large-slip”
asperities as regions where the slip D lies between 0.33Dmax≤D<0.66Dmax and 0.66Dmax≤D,
respectively. They found that approximately 28 % of the rupture plane is
occupied by large-slip asperities, whereas very-large-slip areas constitute
only 7 % of the fault plane. Furthermore, different authors agree that
the rupture area of the asperity scales with the seismic magnitude
among
others.
The estimation of seismic magnitude is an essential feature for
characterizing the energy of an earthquake. In fact, an accurate magnitude
estimation is indispensable to conduct both deterministic and probabilistic
seismic hazard assessments.
Earthquakes are the most relevant example of self-organized criticality (SOC)
. The concept of SOC can be
visualized by imagining a natural system in a marginally stable state,
wherein
phases of instability may occur that place the system back into a
meta-stable state . A popular model
representing this process was proposed by and is
well-known as the “sand pile model”. Some models have been proposed to
explain the statistical behavior of earthquake patterns based on the SOC
concept: e.g., , ,
, and . The failure properties
of solids have been modeled by simple stochastic discrete models, which are
based on the SOC framework. The fiber bundle model, FBM, is one of those
models that has been used to reproduce many basic properties of the failure
dynamic within solids . Additionally, the
FBM has been successfully applied to studies of brittle failure of rocks
.
The fiber bundle model
The FBM is a mathematical tool to study the rupture process of heterogeneous
materials that was originally introduced by . Over
the years the FBM has been widely used to study failure in a wide range of
heterogeneous materials .
Regardless of the specific FBM type, there are three basic assumptions that
all FBMs have in common
.
A discrete set of cells (or fibers) is defined on a d-dimensional
lattice. In seismology, the bundle can represent a fault system or seismic
source wherein each fiber is a section of the fault plane
or individual faults .
A probability distribution defines the inner properties of each cell
(fiber), such as lifetime or stress distribution.
A load-transfer rule determines how the load is distributed from the
ruptured cell to its neighbor cells. The most common load-transfer rules are
(a) equal load sharing (ELS), in which the distributed load is equally shared
to the other cells within the material or bundle, and (b) local load sharing
(LLS) whereby the transferred load is only shared with the nearest neighbors.
TREMOL is based on the probabilistic formulation of the FBM, with the failure
rate of a set of fibers given by Eq. ():
.
dU(t)dt=-U(t)K(σ(t)),
where U(t) is the number of fibers that remain unbroken at time t. The
hazard rate K(σ(t)) is a function of the fiber stress σ(t).
Experimental results show that the hazard rate of materials under constant
load can be well-described by the Weibull probability distribution function.
This behavior can be summarized in Eq. ()
:
K(σ(t))=ν0σ(t)σ0ρ,
where ν0 is the reference hazard rate, and σ0 the reference
stress. The Weibull exponent, ρ, quantifies the nonlinearity
. If σ0=ν0=1, the expression in
Eq. () can be simplified to K(σ(t))=σ(t)ρ.
From the probabilistic formulation, two equations arise (Eqs.
and ), which are applied in our algorithm to define
the system dynamics. The details of these two equations are described below.
and developed a
relation to compute the expected rupture time (dimensionless) of the fibers
following Eqs. () and (). This expected rupture
time interval is defined as δk (Eq. ) and can be
applied to any load-transfer rule:
δk=1∑i=1Nσiρ(t),
where N is the total number of cells, and σi is the load in the
ith cell. The dimensionless cumulative time, T, is the sum of
δk.
The failure probability, Fi, which is a function of the load σi in each cell, is
Fi=δkσiρ(t).
The dynamic values δk and Fi are updated with each time step due
to rupture processes and the resulting load transfer.
A suitable FBM algorithm to simulate earthquakes should consider a complex
stress field, physical properties of materials, stress transfer between
faults (at short and long distances), and dissipative effects. Using the FBM
we assume that earthquakes can be considered analogous to characteristic
brittle rupture of a heterogeneous material
.
The previous basic concepts about the FBM were considered for the development
of the TREMOL code, with the purpose of modeling the behavior of seismic
asperities. In the next section, we describe the details of this code.
The TREMOL code
Since the main objective of TREMOL is to simulate the
rupture process of seismic asperities based on the principles of the FBM, we
model two materials with different mechanical properties interacting with
each other.
In order to introduce the features of TREMOL we describe three main stages
during the application of TREMOL.
Preprocessing In this stage we have to assign the following input data:
the size of the fault plane,
the size of the maximum asperity within the fault plane, and
other parameters (load-transfer value π, strength value γ, initial load values σ, and load threshold σth).
Processing TREMOL uses the data from the preprocessing stage to carry out the FBM
algorithm, and by applying Eqs. () and
()
the rupture process is computed in the fault plane studied. The asperity
size of each earthquake is used by TREMOL to also compute the magnitude of
each synthetic earthquake.
Post-processing In this stage, TREMOL summarizes the results that are computed in the
processing stage and computes the equivalent rupture area (km2). In
general, TREMOL output generates a synthetic catalog of earthquakes,
which consists of the following:
total number of earthquakes that can occur in the fault plane studied,
size of the asperity of each earthquake, and
magnitude of each earthquake.
In the next sections we describe with more detail each one of the three main
stages during the application of TREMOL. An overview of the entire simulation
process is shown in Fig. .
TREMOL flowchart. At the beginning (preprocess) the algorithm
initiates a domain Ω with Nx×Ny cells
in which every cell is either part of an asperity or of the background or fault
plane. Afterwards (first time step, k=1) a uniform distribution allocates a
random stress load and rupture probability to all cells. In addition,
asperity cells obtain a random strength value from a uniform distribution.
Next (time step k≥2) the failure process starts following the FBM
algorithm. After every failure the stress of the broken cell is redistributed
via the LLS rule and the number of time steps (k) increases by 1 until the
target number of time steps is reached. If the final number of time steps has
been reached the simulation stops. At the end, all information about the
entire failure process is saved in a database or a synthetic catalog that can
be used for statistical analysis. Further details about the algorithm are
given in Sect. .
Schematic representation of the considered local load rule. The
broken cell with load, σF, distributes the largest load
fraction, σO (Eq. ), to its four
orthogonal neighbor cells. The remaining load, σD
(Eq. ), is transferred to its four diagonal neighbor
cells. Afterwards, the load of the broken cell drops to zero, σ=0.
Asperity cells cannot receive any new load.
(a) Spatial distribution of the random initial loads
σ(i,j). RAsp represents a rectangular fault plane of
Nx=Ny=100 cells. The color bar indicates the load
and the threshold load of σth=1. (b) Spatial
distribution of the strength γ(i,j). Two main regions can be
distinguished in this figure: (1) the asperity region defined as the inner
rectangle and (2) a background area or fault plane. While the asperity
contains strength values in the range of 3 to 5, the rest of the fault plane
has a strength value of 1.
Preprocessing: input data and initial conditions
In TREMOL, a fault plane is modeled as a rectangle (Ω), which is
divided into Nx×Ny cells. Each cell is defined
by its position (i, j), where i∈[1,…,Nx] and
j∈[1,…,Ny]. In the fault plane Ω earthquakes can
occur with different magnitudes. Additionally, it is possible to assign to
each fault plane an asperity region (RAsp).
To define each fault plane (Ω) and its respective asperity region
(RAsp) it is necessary to assign specific properties to their
cells. Particularly, it is necessary to define three properties (or values)
for each cell of Ω and RAsp: a load σ(i,j), a
strength value γ(i,j), and a load-transfer value π(i,j).
The loadσ(i,j). At the beginning of each realization, TREMOL randomly assigns
a value of the load σ(i,j) to each cell of Ω using a
uniform distribution function (0<σ(i,j)<1). This assumption simulates
a heterogeneous stress field. Moreover, a load threshold σth=1
is necessary to create a limit at which a cell must fail .
At the end of this step any cell within Ω must have a load value between 0 and 1.
The strength valueγ(i,j). This parameter represents an analogy to the
concept of hardness or strength. In our model, the algorithm will find it
difficult to break a cell if this cell has a value γ>1 since the
strength threshold before failure is set as γth=1 (see a
detailed explanation in Sect. ). As a result, a
strength γ>1 may simulate a hard material that needs to be weakened
before it can fail. This process can be regarded as similar to material
fatigue or creep failure. The strength value for all cells in
RAsp, namely γAsp, is chosen in a discrete
interval of UD=[γRef-1,γRef+1], where UD is an integer
uniformly distributed and γRef is an assigned reference
value.
The load-transfer valueπ(i,j). This parameter represents the percentage
of load that can be distributed from a ruptured cell to its neighbors. In
this study, the load in the ruptured cell is called σF(i,j).
TREMOL uses a local load sharing (LLS) rule considering the eight nearest
neighbors. According to previous studies, such as
, TREMOL redistributes the majority of the load
to the four orthogonal neighbors. The load that is transferred to these
orthogonal neighbors is called σO, and it is defined
according to Eq. ():σO(i,j)=0.98σF(i,j)πF(i,j)4,where πF is the load-transfer value of the failed cell.
Additionally, a small proportion of the load is transferred to the four
diagonal neighbors. The value of this load is called
σD(i,j), and it is defined according to
Eq. ():σD(i,j)=0.02σF(i,j)πF(i,j)4.
The assumption of Eqs. () and ()
is in agreement with what is expected for the maximum shear stress directions
with respect to the main stress orientation that gives rise to both
synthetic and antithetic faulting e.g.,.
Figure is a schematic representation of the load distribution
process from the failed cell, σF(i,j) (in red), to its
nearest neighbors.
In order to differentiate the parameters of the asperity from the rest of the
fault plane Ω, we define πAsp(i,j) and
γAsp(i,j) that refer only to the cells within
RAsp. For the rest of the fault plane Ω, we are using the
same parameters defined previously: π(i,j) and γ(i,j).
Figure a shows an example of the randomly distributed initial load
throughout the fault plane. Figure b displays an example of
differences between the strength of the asperity and the rest of the fault
plane.
Results of one realization by TREMOL. (a) The spatial
distribution of avalanches. Patches of the same color indicate one
temporal-consecutive Avalanche cluster (synthetic earthquake).
(b) Logarithmic representation of the inter-event rate with time.
The red dots represent the inter-event rate when the asperity rupture occurs.
The blue dots indicate foreshocks.
Main computational processes
Once the initial information for the entire domain Ω is defined, the
core algorithm of TREMOL will realize a transfer, accumulation, and rupture
process. While the cells interact with each other, there are two basic
failure processes depending on the load of the cell in comparison with the
threshold load .
Normal event. If all cells within the system have a load
σ(i,j)<σth, a normal event is generated,
and the cell that will fail is randomly chosen considering the individual
failure probability of each cell, F(i,j) (Eq. ).
Avalanche event. If one or more cells have a load value σ(i,j)≥σth, an avalanche event is generated, and the
cell that fails is the one with the greatest σ(i,j) value.
Due to the integrated strength property some extra rules for rupture are
necessary. The requirement for failure is γ(i,j)=1. On the other hand,
if a cell with γ(i,j)>1 is chosen, its strength is reduced by one
unit. This strength condition enables us to simulate a material weakening
process during the load-transfer process. Additionally, this condition offers
the possibility to produce large load accumulations locally, which are more
likely to generate larger ruptures.
When a cell within RAsp breaks it becomes inactive until the end
of the simulation, which means it cannot receive any further load. The large
load concentration within the asperity usually produces a very short time
interval (Eq. ), with the result that there is physically not
enough time available to reload the stress on an asperity right after its
rupture. On the contrary, a cell outside of the asperity region remains
active after its failure but its load drops to zero. The simulation ends when
all the cells within the asperity have become inactive.
Output data and post-processing
After every execution TREMOL outputs a catalog
detailing where the position (x,y) of the failed cell, the rupture time
(Eq. ), the avalanche event or
normal event identification, the mean load, and many other values
are saved for each time step. We cluster avalanche events
considering the time and space criterion. We assume ai-1=(xi-1,yi-1) and ai=(xi,yi) are two consecutive
avalanche events generated in chronological order. If their
Euclidean distance is Δri≤rth (where rth=22), then ai and ai-1 will belong to the same cluster. This
clustering algorithm is applied to all generated avalanche events.
Lastly, we extract a new catalog that shows the size of each cluster, the
position of the first element of each cluster related to the nucleation
point, and the time when it was initiated. This database is our simulated
seismic catalog. Note that the cluster size is given in nondimensional
units. However, we use an equivalence between Ω and an effective area
Aeff in order to obtain a physical rupture area. Finally, each
cell can represent an area in square kilometers. This step is necessary in order to
compute an equivalent magnitude, which is comparable with real earthquake
magnitudes. For this purpose, we use three magnitude–area relations. In
particular, we use Eqs. (), (),
and () obtained by for
Mexican subduction earthquakes:
7log10Aa=-4.393+0.991Mw,8log10Aa=-5.518+1.137Mw,9log10Aa=-6.013+1.146Mw,
where Aa is the asperity area (km2).
Equation () was obtained from asperities defined by
the average displacement criterion .
Equations () and () were computed
from asperities defined by the maximum displacement criterion for a large
asperity and a very large asperity, respectively .
Furthermore, we define the inter-event rate Δνk as analogous to
the rupture velocity:
Δνk=ΔrkΔδk,
where Δrk is the inter-event Euclidean distance between the
k event located at (xk,yk) and k-1 event in (xk-1,yk-1).
Δrk=(xk-xi-k)2+(yk-yi-k)2
The inter-event time Δδk is computed following
Δδk=δk-δk-1,
where δk is given by Eq. (). Figure a shows
an example of the final spatial distribution of rupture clusters for a
particular example. Each cluster is indicated by the same color and
represents a simulated earthquake. Figure b shows the related
inter-event rate. The inter-event rate largely increases when the asperity
rupture occurs.
In the post-processing step we additionally computed the rupture duration of
the largest simulated earthquake, DAval, using the rupture
velocity and the effective fault dimensions obtained from finite-fault
models (Table ).
We used Eq. () to compute DAval:
DAval=LMaxVr+16WMax×LMax7π3/2β,
where
β≈Vr0.72.
Using these considerations, we can assign a physical unit of time (s) to the
largest simulated earthquake, Asyn=LMax×WMax.
The flowchart in Fig. and the pseudo-codes 1, 2, and 3
summarize the algorithm of TREMOL. A summary of all required parameters to
execute the TREMOL code are shown in Table .
TREMOL preprocessing: input parameters and their definition.
ParameterDefinitionNcellnumber of cells in Ωπasppercentage of transferred load to neighbor cells in the asperity domainγaspstrength at each (i,j) cell in the asperity domainSa-Aspratio of asperity area computed by TREMOLSaratio of asperity area computed by finite-fault modelAeffeffective area (km2) computed by finite-fault modelAaasperity area (km2) computed by finite-fault model
Sensitivity analysisMethods: parametric study
We performed a sensitivity analysis of the three asperity parameters
(γasp, Sa-Asp, and πasp) in
order to identify the best combination that produces the best approximation
to real data, such as the maximum rupture area, Asyn, and its
related magnitude Msyn. In order to investigate the influence
of every single parameter, we statistically determined how the results vary
with different parameter configurations.
To explore the influence of πasp, we analyzed 12 values
(0.67≤πasp≤1.0, with increments of 0.3). The
minimum πasp=0.67 assigns the same value to an asperity
cell and to a background cell. On the other hand, πasp=1.0
means that the load in a failed asperity cell is fully transmitted to the
neighbors (ideal case with no dissipative effects). Note that
πasp=1 does not represent real physical conditions since
dissipative effects are ignored completely. On the other hand, if
πasp=0.67 (case 1) the asperity cells would transfer as
much load as the cells in the background. The objective is to generate a load
concentration within the asperity that corresponds to the largest magnitude.
If the asperity cells transfer as much load as the background cells, no such
load concentration can be obtained. As a result, we can expect that the mean
Asyn for πasp=0.67 (case 1) is the lowest
value in comparison to all other cases.
The input data of this experiment are summarized in Table .
We assigned a strength to the asperity (RAsp)
γasp=4±1 and a value of γBkg=1
to the rest of the fault plane. These values are chosen after experimental
trials, which have shown that the difference is large enough to simulate a
significant strength difference with low computational effort. To define the
effective area and the asperity size, we chose the values computed for the
earthquake of 20 March 2012, Mw=7.4, in
: Aeff=2944.2km2 and
Sa=0.26. We defined the size of Ω consisting of Ncell=10000 cells in
total. We carried out 50 simulations per
πasp configuration. In addition, we modified the random seed
to have different initial load configurations, σ(i,j), to ensure
that the results over πasp are independent of the initial
load conditions σ(i,j).
To perform the parametric study of γasp, we configured two
asperities embedded in Ω. In this experiment, the total size is
Ω=200×100 cells. Afterwards, we located each asperity in the
center of the two sub-domains Ω′ of 100×100 cells.
Figure shows a schematic representation of the
domains Ω and Ω′ used in this experiment.
Schematic configuration for the parametric study of
γasp and Sa-Asp. The size of the domain
Ω is Ω=200×100 cells. Each asperity is located within the
center of the two sub-domains Ω′ of 100×100 cells. The
strength parameter γasp and degree of heterogeneity for
each asperity can be varied according to the material properties.
The separation between the two asperities remains constant. We chose a value of
πasp=0.90 to produce a large contrast between the asperity
and the rest of the fault plane (π=0.67)
. In order to analyze the influence of
γasp (and Sa-Asp), the asperity on the right-hand side (Asp. 2) has varying strength values, while the strength of the left
asperity (Asp. 1) remains constant. Finally, the maximum ruptured area and
magnitude generated in each Ω′ are computed.
In order to explore how the system behaves when γasp
changes, we analyzed six different values of γasp=[2±1,5±1,7±1,9±1,11±1,14±1] (cases 13 to 18). The input
data used in this test are summarized in Table . We defined
the same asperity size for both: Sa1=Sa2=0.22.
In Fig. , we show an example of the spatial configuration of
this analysis. The background strength is considered to be
γbkg=1= constant, and the color bar indicates the
γ(i,j) values.
Main input data in order to carry out cases 13 to 18.
Example of the strength configuration γ(i,j) for the
sensitivity analysis of γasp. Two asperities with the same
size Sa-Asp=0.22 are defined and embedded in Ω
following the schema in Fig. . The conservation
parameters are πasp=0.90 and πasp=0.67. The
color bar indicates different γ(i,j) values. The left asperity
(Asp. 1) contains constant properties, while the right asperity (Asp. 2) has
variable strength values.
The modification of the Sa-Asp parameter was based on the same
configuration as described in the previous section. We analyzed six different
values of the asperity size Sa (cases 19 to 24). In
Fig. we show an example of the asperity configuration in which the
left asperity (Asp. 1) has a constant size Sa2, while the size
of the right one (Asp. 2) increases. In this experiment, we considered
γasp=5±1 and πasp=0.90. The main
data related to these six cases are summarized in Table .
Main input data in order to carry out cases 19 to 24.
An example configuration of different asperity sizes,
Sa-Asp. The color bar indicates the strength γ(i,j)
values used during the test.
Model validation – methods
We evaluated the capability of the model to reproduce the characteristics of
10 Mexican subduction earthquakes (eight shallow thrust subduction events, ST,
and two intra-slab subduction events, IN). The input data of the effective
area,
Aeff, and the asperity ratio size, Sa, are given
from waveform slip inversions and seismic source studies (Aeff=Leff×Weff and Sa=Aa/Aeff) shown in the database of Mexican
earthquake source parameters by . This database
includes results from two different methodologies: spectral analysis and
finite-fault models. From the latter, the database provides estimations of
effective fault dimensions, rupture velocity, source duration, number of
asperities, stress, and radiated seismic energy on the asperities and
background areas. Slip solutions were obtained with teleseismic data for
events with 6.4<Mw<8.2.
The number of cells was Ncell=10000 for a domain Ω
of 100×100 cells. We modeled the size of Ω proportionally to
the size of Leff and Weff for each scenario
according to the following equations, Eqs. () and
():
15Nx=NcellLeffWeff,16Ny=WeffLeffNx,
where Nx and Ny are the number of cells in the x axis and
y axis, respectively. As an example, Fig.
presents the size and aspect ratio of Ev. 3 and Ev. 5 (Table ).
Example of the domain configuration Ω, considering
Leff and Weff. (a) Example configuration
of event 3 and (b) example configuration of event 5. The required
data can be found in Table .
In some cases, the number of asperities computed in
is greater than 1. However, as a first
approximation we simplified the problem by modeling only one asperity per
earthquake.
In order to study how the asperity size Sa affects the maximum
ruptured area, we randomly modified the size as
Sa-Asp=Sa+(α⋅(Sa/2)),
where 0<α<1 is a random value. We introduce this assumption
because we want to avoid a preconceived final size. In future trials it may
be useful to consider the inner uncertainties of finite-fault models. The
asperity aspect ratio follows the same proportion as the effective area,
ΩxΩy=Nx(Sa)Ny(Sa)
(Fig. ).
We carried out 50 realizations per event (Table ), changing the
size Sa-Asp in each one (Eq. ).
The finite-fault source parameters used in this work.
Weff and Leff are the effective fault dimensions
(width and length, respectively, according to ).
Areal is the asperity area, Aeff is the effective
rupture area (Weff×Leff). Duration is the
rupture duration computed from the slip inversion, Na is the
number of asperities, and Vr is the rupture velocity. Ratio is the
aspect ratio of the fault area. The type of the event is labeled ST for
shallow thrust and IN for intra-slab events.
Ev.DateMwLeffWeffRatioSa=DurationVrTypeNaReferenceID(km)(km)Areal/Aeff(s)(km S-1)107/06/19827.034.4717.811.940.23–3.2ST1219/09/19858.1158.62115.041.380.31–2.6ST2330/04/19866.838.3137.161.030.26222.5ST1414/09/19957.468.8046.611.480.23322.5ST1509/10/19958.0169.6559.252.860.27922.8ST2618/04/20026.72313.881.660.24302.2ST2720/03/20127.454.9453.591.030.26302.7ST17a7.451.4255.470.930.21–1.8ST1USGS7b7.440.0344.600.890.21–2.0ST1Wei (2012)811/04/20126.521.9521.841.040.23152.8ST1908/09/20178.2125.9571.131.770.34–2.0IN3USGS1019/09/20177.134.4736.120.950.32–2.2IN1USGSModeling the rupture area and magnitude of 10 subduction earthquakes – methods
In this case the number of cells is Ncell=10000
(100×100). We carried out 50 executions per event and in each
execution we randomly changed the size Sa-Asp following
Eqs. (), (), and (). The input data of
the 10 modeled earthquakes in Table are summarized in
Table .
Main data used for Ev. 1 to Ev. 10.
DataValueNumber of asperities1Ncell10 000πasp0.90γasp5±1πbkg0.67γbkg1σth1Sasee Table Sa-AspEq. ()Aeffsee Table Case study (Oaxaca, <inline-formula><mml:math id="M299" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">7.4</mml:mn></mml:mrow></mml:math></inline-formula>, 20/03/2012): using different effective areas <inline-formula><mml:math id="M300" display="inline"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi mathvariant="normal">eff</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> for the same event – methods
As reported in for some events, there are
several solutions that allow us to analyze the variability in the estimated
source parameters (see parameters of events 7, 7a, and 7b in
Table ). In this study, we applied TREMOL to study how the
ruptured area and the assessed magnitude change when we use different input
data to model the same earthquake. The data related to these three events are
summarized in Table .
Main data for the case study Ev. 7 test, Ev. 7a test, and Ev. 7b test.
DataValueNumber of asperities1Ncell10 000 (cells)πasp0.90γasp5±1πbkg0.67γbkg1σth1Sa-AspEq. ()Sasee Table (Ev. 7, 7a, and 7b)Aeffsee Table (Ev. 7, 7a, and 7b)Assessing a future earthquake in the Guerrero seismic gap: rupture area and magnitude – methods
We apply our method for the estimation of possible future earthquakes, in
particular to compute the expected magnitude, since TREMOL may offer new
insights for future hazard assessments. We carried out a statistical test to
assess the size of an earthquake that may occur in the Guerrero seismic gap
(GG) region.
As input parameters, we used the area found by :
Leff=230km×Weff=80 km. We
defined the asperity size ratio Sa as proposed by
for regular subduction zone events (SB) based on
average slip, Sa=0.25. ,
, and proposed a probable
maximum magnitude for this region of Mw≈8.1-8.4.
Therefore, using the effective rupture area (Leff,
Weff, and Sa), we executed the algorithm as in
previous sections. The input data related to this analysis are summarized in
Table . Likewise, we want to estimated the duration
Daval of the event. To compute this value, we used a mean of
the Vr from Table .
Main data for assessing a future earthquake in the Guerrero seismic gap (GG event).
DataValueNumber of asperities1Ncell10 000 (cells)πasp0.90γasp5±1πbkg0.67γbkg1σth1Sa-AspEq. ()Sa0.25Aeff18 400 (km2)ResultsResults: parametric studyPercentage of transferred load, <inline-formula><mml:math id="M331" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">π</mml:mi><mml:mi mathvariant="normal">asp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>
Figure shows the mean (black dots) of the
maximum ruptured area Asyn, including the upper and lower limits
of the standard deviations (blue squares), after the execution of all 12 cases
(Table ) with 50 realizations. The value of
Asyn is related to the largest produced cluster in Ω.
There are two dominant tendencies identifiable.
If πasp<0.76, the mean of the maximum ruptured area increases
continuously more than 1 order of magnitude from 15 to ≈500 km2, i.e., an increase of 3333 %. The standard deviation of
Asyn for πasp=0.7 is ≈35 km2
(100 % error).
If πasp≥0.76 (cases 4 to 12), the Asyn
values remain essentially constant (≈500 km2). Likewise, the
upper and lower limit vary around the same order. The standard deviation for
this interval is ≈100 km2 (20 % error).
Mean of the maximum rupture area (km2), Asyn, for
different values of πasp depicted as black circles. The
minimum and maximum limits of the rupture area are represented by blue
squares.
Mean and standard deviation of the maximum magnitude over 50
realizations depending on πasp.
Mean and standard deviation of the ratio Ssyn=Asyn/Aeff over 50 realizations for different values
of πasp. The red line indicates the asperity ratio
Sa computed for event 7 (Table ).
Using the mean of Asyn obtained in each case, we computed the
corresponding magnitude. The results are given as the mean and standard
deviation of the maximum magnitude in Fig. for all 12 cases
(see Table ). Due to the fact that ruptured area and
magnitude are correlated (see Eqs. ,
, and ), the pattern in
Fig. is very similar to the one in
Fig. .
Overall, there are three aspects observable.
If πasp≥0.76 (cases 4 to 12), the mean magnitudes show a steady value (≈7.2).
If 0.70≤πasp<0.76, a transition with an increasing trend with the largest standard deviation is visible.
If πasp=0.67 (case 1), the mean of the maximum magnitude is the lowest.
In this experiment, the initial value of Sa=0.26 remains
constant; i.e., the asperity size does not increase randomly (red line in
Fig. ). After executing all configurations,
we computed the ratio of Sa-Asp=Asyn/Aeff, relating to the largest ruptured area. We
show the mean and standard deviation of this ratio Sa-Asp in
Fig. . We observed that the ratio of
Sa-Asp is always ≈0.10 lower than Sa.
For each value of γasp (Table ), we
performed 50 executions while changing the initial strength parameter of the
asperity γasp (Fig. b). Likewise, we computed
the maximum magnitude obtained for each Ω′.
Figure indicates the mean and standard deviation of
the computed maximum magnitude with a dependence on γasp. The
upper subplot (blue markers) shows the results for the left (constant)
asperity (Asp. 1). The lower subplot (red markers) shows the results for the
right (variable) asperity (Asp. 2).
Statistical results of γasp for a configuration
similar to Fig. . Markers represent the mean value, while the
error bars indicate the standard deviation for all 50 executions considering
different initial strength configurations. The red markers correspond to the
results of the left asperity (Asp. 1) and blue markers to the results of the
right asperity (Asp. 2) (Fig. ). The strength of the left
asperity is kept constant, whereas the strength of the right asperity is
variable.
We observe in Fig. that the mean magnitude remains
essentially independent for γasp>5±1. Additionally,
the error bars slightly decrease, while γasp increases.
Another observation is that when γasp=2±1 the
average of the maximum magnitude is the lowest in both asperities. Moreover,
there is a transition zone for 2±1≤γasp≤5±1.
We observed that γasp>5±1 has a limited influence on
the results of the maximum magnitude. The maximum magnitude of
γasp=14±1 is approximately 0.3 magnitudes larger than the one
of γasp=5±1.
Figure shows the mean magnitude and standard deviation
as a function of asperity size. The first asperity with the fixed size
indicates a relatively constant magnitude of approximately 7.4. Conversely, the
second asperity with variable size produces only a slight increase in
magnitude. The magnitude of Sa-Asp=0.52 is approximately 0.5
magnitudes larger than the one of Sa-Asp=0.22.
Statistical results of Sa-Asp for a configuration
similar to Fig. . The markers indicate mean and standard
deviation for 50 realizations. Red markers correspond to the results of the
right and variable asperity and blue markers to the results of the left and
stable asperity.
Results: model validationModeling 10 Mexican subduction zone earthquakes
Based on the observations described in the previous section, we used
γasp=5±1 and πasp=0.90 in order
to validate the model. We chose γasp=5±1 because it
represents the strength interval of 5±1≤γasp≤14±1 with less computational cost. We chose πasp=0.90
because it represents the relatively constant magnitude for the parameter
range 0.76≤πasp≤0.90. In addition, πasp=0.90 enables us to obtain the best approximation to the ratios of
Sa-Asp. Both parameter choices ensure an appropriate
reproduction of the asperity rupture area, the maximum magnitude, and least
computational payload.
Figure depicts a comparison between the (real) asperity
area Areal (Table ) and the area of the largest
simulated earthquake, Asyn. We plot the mean (blue dots), the
minimum (green triangles), and the maximum (red triangles) of all 50
realizations for each real earthquake event. Black squares represent the real
asperity size. The results in Fig. point out that
Asyn is almost identical to Areal from
Table for the majority of earthquakes. Only three events show
significant differences between synthetic and realistic maximum rupture area.
Even in these cases, however, Areal is located within the upper
and lower limit of Asyn.
Comparison between the real asperity area Areal and
the synthetic values (mean and standard deviation) of the largest ruptured
event Asyn. Black squares depict the real asperity area from
Table , whereas blue circles indicate the mean area of 50
executions. Red and green triangles represent the maximum and minimum
Asyn.
Figure shows the statistical results of the synthetic maximum
magnitude, Msyn, determined for all 10 events. The real
magnitudes from Table are given as red markers. Black circles
indicate the mean of Msyn of 50 realizations using
Eq. (), whereas blue and green markers indicate the
magnitude following the Eqs. () and
(), respectively. The error bars represent the standard
deviation. We observed that the statistical parameters computed with TREMOL
fit the magnitudes shown in Table . However, the computed
magnitudes depend on the scale relation employed
(Eqs. , , and
). Figure includes the
mean of the three scale relations. Overall, the mean magnitude
M‾syn and the expected magnitude Mw show
similar values. Given that the difference between the mean and the expected
value (Table ) is lower than ΔMw<0.5 for the
10 events, we can confirm that the results of assessing the magnitude by means
of TREMOL using a randomly modified asperity size, Sa-Asp
(Eq. ), are reasonable.
Statistical results of the maximum magnitude for the events from
Table . Red squares depict the real estimated magnitudes from
Table , while black circles, blue triangles, and green triangles
represent the synthetic mean magnitude of 50 executions following
Eqs. (), (), and
(), respectively. The error bars are the standard
deviation of the scale relations.
Statistical results of the maximum magnitude for the events from
Table . Red squares represent the magnitudes from
Table , whereas black circles represent the mean magnitude
(M‾syn) value of all 50 executions following
Eqs. (), (), and
(). The error bars stand for the standard deviation of
the mean for the three scale relations.
Figure shows the real ratio size Sa from
Table (black squares) in comparison to the mean of the largest
simulated earthquake, Ssyn (blue squares). The standard
deviation is represented as error bars. The results indicate that in most of
the cases the computed Ssyn range fits the expected
Sa well. Note that for events 3, 7, and 8 the mean values are
lower than the reported Sa, while Ssyn is
overestimated for events 2, 5, and 9. For events 1, 4, 6, and 10 the estimated
value of Ssyn coincides with the expected one. However, the
error bars encompass the expected values in all cases (Fig. ).
Moreover, if we compare Fig. with
Fig. we observe that the employed strategy
of randomly increasing asperity size (using Eq. ) generates
rupture areas similar to the ones proposed by .
Proportion of simulated ruptured area occupied by the largest
avalanche, Ssyn, in comparison with the real ratio size
Sa from Table . The real ratio size Sa
from Table is represented by black squares and the mean of the
largest simulated earthquake, Ssyn, by blue squares. The
standard deviation is represented as error bars.
We also computed an equivalent rupture duration, DAval, using
the equation proposed by to calculate the rise time
(Eqs. and ). determined
the rupture velocity Vr (Ev. 3–8), which is a useful parameter
in order to validate our results. Figure shows the results of
this analysis. In red we plot the values Vr calculated by
and in blue the DAval based on
Eq. () with Vr provided by Table . The
equivalent DAval using Eqs. () and
() is printed in black. In cases in which we have the reference
values, Vr, computed by , we observe
that the reference values are always larger than the modeled
DAval values.
However, it is worth noting that Vr is the mean rupture time that
considers the rupture of the whole effective area (Aeff). For
the simulated rupture duration, DAval, we only consider the
rupture length of the largest rupture cluster Asyn. As a
result, smaller values than those proposed in are
expected. Nevertheless, the rupture duration shows a clear dependency on the
magnitude.
Equivalent rupture duration DAval (s) calculated via
the rupture velocity by using the size of the largest rupture cluster. Red
squares represent the reference values proposed by
, while blue squares and black circles depict the
synthetic rupture duration computed by means of Vr based on
Eqs. () and (), respectively.
Case study (Oaxaca, <inline-formula><mml:math id="M432" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">7.4</mml:mn></mml:mrow></mml:math></inline-formula>, 20/03/2012)
In the cases in which several effective rupture areas were proposed by different
studies (see Table ), it is possible to assess which set of
parameters is better in order to simulate an event by means of TREMOL. We
tested TREMOL by using three different combinations of Leff,
Weff, and Sa according to results for Ev. 7 in
Table . A comparison of these three combinations is visualized
in Fig. : panel (a) shows the comparison of the ruptured
areas, Areal and Asyn; panel (b) shows the mean and
standard deviation of the maximum magnitude, Msyn, in
comparison to the reference magnitude; and panel (c) shows the ratio Ssyn
of the simulated events compared to Sa, the real scenarios.
Although the three combinations express similar results, the closest
approximation between real and synthetic data is generated based on (Ev. 7).
A comparison between the data from Table and the
results by TREMOL for events 7, 7a, and 7b. (a) Maximum ruptured
area, Asyn; (b) mean maximum magnitude,
Msyn; (c) ratio of maximum event size,
Ssyn.
Assessing a future earthquake in the Guerrero seismic gap: rupture area and magnitude
Estimation of the characteristics of a future earthquake in the
Guerrero seismic gap. (a) Estimated rupture area,
(b) rupture duration, (c) average of the mean magnitude
considering three scale relations in Eqs. (),
(), and ().
In Fig. a, we compare the mean of maximum ruptured area,
Asyn, including error bars with the reference area,
Aa. The rupture area computed in TREMOL shows a possible range
from 4000 to 7000 km2. This interval is based on a considered size of
Sa=0.25. In the subplot of Fig. b, we
estimated the duration Daval of the rupture event. The results in
Fig. b indicate that the duration Daval is
similar to that of the other events of magnitude Mw≈8.
The duration may range from 80 to 110 s, while a rupture duration between 90
and 100 s is most likely. Figure c shows the mean of the
estimated magnitude using Eqs. (),
(), and (). TREMOL outputs a possible
range of 8.1≤Mw≤8.5, which matches the proposed value
by , , and
of Mw≈8.1-8.4.
In the results, there were two dominant tendencies visible:
(1) πasp<0.76 and (2) 0.76≤πasp. If
πasp<0.76 the mean of the maximum ruptured area increased
continuously more than 1 order of magnitude from 15 to ≈500 km2, i.e., an increase of 3333 %. Therefore, the range of
πasp is both crucial and sensitive. A parameter increase of
only 15 % affects the size of the biggest earthquake within the system
by 3333 %. Considering the large standard deviation of ≈35 km2 (100 % error) a parameter configuration based on
πasp<0.76 would be unsuitable for further simulations due to
the unstable properties obtained for that range. The second tendency,
however, offers the possibility to determine a stable conservation parameter
that can be freely chosen in the range of
0.76≤πasp≤1.0. The stable state of maximum rupture area
is caused by a self-organized critical avalanche size of Acrit≈500km2 based on a grid of 100×100 cells with
Aeff=2944.2 km2 and Sa=0.26. As soon as
Acrit is achieved by the system, the largest avalanche will
stop increasing in size, whereas other avalanches within the system will be
favored to grow. On the other hand, this means that TREMOL breaks the
asperity in patches rather than completely during one unique rupture event
(see Figs. and ). This last
condition is reasonable considering that the algorithm of FBM used in TREMOL
favors clustering the rupture of cells. Therefore, it is reasonable that some
cells remain outside of a unique rupture group because they do not satisfy
the failure conditions. As a consequence, we think that it is necessary to
define an initial area greater than the expected area of the asperity where
the asperity rupture can occur. This result also justifies the proposed
Eq. (), wherein the size of the asperity increases randomly up
to 50 % larger than the value proposed by .
Future studies may be useful to better determine the influence of
Acrit.
The parametric study indicates that the largest rupture πasp
is produced as long as it is within the range 0.76<πasp<1.0. So even though as πasp increases, large rupture
clusters are generated because a large amount of load is transferred to the
neighboring cells, thereby producing critical local load concentrations in the
system, the particular lower bound is critical. In our simulation short-range interactions convert to long-range processes through the avalanche
mechanism in TREMOL v0.1. The explicit interaction range is given by the
parameter π and the local load sharing rule, since this produces a load
concentration in neighboring cells, promoting ruptures in a local manner
(short range). However, the long range is also captured in a more implicit way.
As mentioned in Sect. , the algorithm searches for a
cell to fail that fulfills two different criteria based on the stress and
the strength values of the cells. This property results in long-range
interactions since the randomness of the initial stress distribution allows
cells at large distances to be activated after a sufficient amount of
subsequent steps.
The parameter γasp quantifies the “hardness” of the
asperity in comparison to the background material. Its value is given as
γasp=γref±1. The value
γref=2 indicates that the strength in the asperity is
twice as big as in the background area. The explored range of
γref (2, 5, 7, 9, 11, 14) is based on an experimental trial.
Figure presents the mean of the maximum magnitude of
an event as a function of γasp with two tendencies being
visible.
There is an unstable transition zone of 2±1≤γasp≤5±1
where the maximum rupture has a strong variation. Therefore, a strength value
within this range should be avoided.
There is a stable zone of 5±1<γasp<14±1, where
γasp can be freely chosen. However, due to computational
costs it is recommended to use the lowest value of γasp=5±1, since the number of necessary time steps to activate the whole
asperity increases strongly with the applied asperity strength (see
Algorithm 1).
Moreover, as γasp increases the simulation requires a
larger number of iterations to break a cell in the asperity, thus implying a
larger computational cost. Our selection (γref=5) ensures a
“stable” maximum magnitude in the lowest computational time.
Figure visualizes the magnitude, the number of steps
required to activate the whole asperity, and the computational time in
seconds for one execution as a function of γref. In this
sense, we considered a value of γref=5 to be adequate from
a computational point of view and also to ensure relatively constant
values of maximum magnitude.
Magnitude, computational time (s), and number of steps to
“activate” the whole asperity for one execution as a function of
γref.
The results of Fig. indicate that asperity size has a
significant influence on the maximum magnitude. We emphasize the importance
of these results because they show that the parameter Sa-Asp is
critical to control the generated magnitude. At the same time, these results
provide the appropriate range of values that TREMOL requires to do a
reasonable assessment of the maximum rupture area and magnitude of an
earthquake.
Discussion: model validation
The model validation by means of 10 different subduction earthquakes showed
that TREMOL is capable of reproducing rupture area and magnitude
appropriately – by means of only few input data – in comparison to the
results from inversion studies. The computed rupture duration by TREMOL
differs from the reference values. The reason may be that the calculation of
the rupture duration is based on the largest (critical) rupture area that is
not equal to the available asperity area (see
Figs. and ).
Nevertheless, the rupture duration shows a clear dependency on the magnitude.
Since TREMOL only requires few input data, it is a powerful tool to simulate
future earthquakes, such as those that might take place in the Guerrero gap
region. The determination of the magnitude of an earthquake based on the
asperity area depends on the scale relation used. We considered it more
appropriate that the relation used be related to the tectonic regime to be
modeled. For example, other possible relations to be applied for a subduction
earthquake regimen could be and
. However, if the user wants to include another
empirical relation it is possible to add it in the script
TREMOL_singlets/postprocessing/calcuMagniSpaceTime Singlets.jl,
such as in the example presented by . It is worth mentioning that the
relation proposed by was analyzed as part of our
tests. However, we found that the magnitude values are lower than the ones
reported by Eqs. (), (), and
(). discussed similarities and
differences with in detail. Nevertheless, the rupture
area is not model sensitive (Fig. ), so in order to
compare real data and simulations it is more appropriate to use the rupture area.
After validating the capability of the model, constraining the input
parameters, and analyzing the results, we consider the conceptual basis
of TREMOL to be expandable to model other tectonic regimes. For example, the
FBM may be applied to study the rupture process in active fault systems and
its effect on aftershock production. Likewise, a three-dimensional version
can be developed to simulate mainshock rupture and its aftershocks as
reported in , who tested a first prototype of a 3-D version.
TREMOL: advantages and disadvantages
The algorithm of TREMOL enables the model to store stress history and to
simulate static fatigue due to an included strength parameter γ. The
vast majority of asperity parameters have already been examined in previous
inversion studies and are usually accessible from online databases.
The range of values found in the sensitivity analysis are not unique for the
Mexican examples. In fact, the parameters π, γ, and σ are
generic for any simulation of similar types of earthquakes. The only
information that needs to be defined beforehand for a unique earthquake is
the effective area size and the asperity area, which may come from
finite-fault models.
Dynamic deterministic modeling of aftershock series is still a challenge due
to both the physical complexity and uncertainties related to the current
state of the system. In seismicity process simulations the lack of knowledge
of some important features, such as the initial stress distribution or the
strength and material heterogeneities, generates a wide spectrum of
uncertainties. One way to address this issue may be to consider a simple
distribution such as a uniform distribution. We think that the validity of
this assumption is given by the comparison of the simulated results with real
data. It is possible that other distributions might also give similar
results. However, the intention of TREMOL v0.1 is to propose a model that
can be used to assign values to the unknown properties mentioned before,
including different distributions. Therefore, we encourage users to try other
distributions and investigate their effects.
The FBM, on the other hand, produces similar statistical and fractal
characteristics as real earthquake series, and its parameters can be regarded
as analogous to physical variables. Likewise, the FBM is able to simulate
failure through static fatigue, creep failure, or delayed rupture
.
One disadvantage of TREMOL is that its output is highly dependent on the
input, which is based on information from kinematic models and therefore
contains inherent uncertainties from inversion studies (see
Table ). TREMOL may be able to compensate for some errors, but how
far this possibility can be exploited needs to be investigated in the future.
Further steps in the advance of the model have just started, which includes the analysis of
a machine-learning approach that will exploit all the
possibilities of this technique.
There are still issues that will likely be addressed in future tests, as outlined below.
For our validations, we used earthquakes for which a suitable amount of
information is available. How can the technique be applied to other events
for which little information is available through, for instance, far-field
recordings of seismicity?
For our validation study, we used a simplified geometry of the real complex
asperity geometries. However, other irregular asperity geometries may be
introduced in future works.
The FBM is a pure statistical model and therefore gives only hints about
underlying physical processes. So far, it does not take into consideration
physical effects such as pore fluid pressure, soil amplification, stress
relaxation of the upper mantle, reactivation of existing faults, volcanic
activity, and many more. One strength of the FBM is that an endless number of
information layers can be included into the model that would allow us to
include physical properties and topography as well.
As it currently stands, TREMOL is not able to simulate complete seismic
cycles. Rate-and-state friction models such as by and
have the ability to reload stress. TREMOL is still in an
early stage of development and thus lacks a reloading feature.
Additional setbacks of TREMOL are that (1) the number of time steps needs to
be adjusted manually for every grid resolution and case scenario, and (2) it
is based on a sequential algorithm. In order to save the stress history
within every cell of the system, a consecutive algorithm is necessary that
changes the state of the system with every time step. This limits the
integration of a parallel domain, but a parallel distributed memory is a good
approach to solve the problem of large domains. As a result high-performance
computing facilities are required when very large grid sizes are used
.
Overall, the results of TREMOL are promising. However, the results also point
out the need for further modifications of the algorithm and more intensive
studies. Likewise, many questions are still left to be answered due to the
model's early development stage. In the very near future, however, TREMOL may
be a true alternative to classical approaches in seismology. The simple
integration of layers of information makes TREMOL a simple model that can be
easily modified to simulate the most complex scenarios. At the moment, TREMOL
cannot compete with state-of-the-art and widely accepted rate-and-state
friction-based models, but it is a totally different, complementary, and
promising approach that can provide important insights into earthquake physics
and hazard assessment from a completely different perspective. The
development of TREMOL and similar models should therefore be strongly
encouraged and supported.
Conclusions
In this study, we present an FBM-based
computer code called the stochasTic Rupture Earthquake MOdeL, TREMOL, in
order to investigate the rupture process of seismic asperities. We show that
the model is capable of reproducing the main characteristics observed in real
scenarios by means of few input parameters. We carried out a parametric study
in order to determine the optimal values for the three most important initial
input parameters.
πasp. as long as the fault plane has a conservation parameter
of πbkg=0.67, the conservation parameter of the asperity must
be πasp≥0.76 to ensure a realistic maximum rupture area.
γasp. the best strength interval for the asperity is 5±1<γasp<14±1. However, due to computational
costs it is recommended to use the lowest value of γasp=5±1, since the number of necessary time steps to activate the whole
asperity increases strongly with the applied asperity strength (see
Algorithm 1).
Sa-Asp. the generated magnitude can be controlled by parameter
Sa-Asp. This parameter is dependent on the earthquake of
interest and follows results with data from inversion studies.
We also carried out a validation study employing 10 subduction earthquakes
that occurred in Mexico. TREMOL proved that it is able to reproduce those
scenarios with an appropriate tolerance.
A big advantage of our algorithm is the low number of free parameters
(Leff, Weff, and Sa) to obtain an
appropriate rupture area and magnitude assessment. Our code TREMOL allows
its users to investigate the role of the initial stress configuration and
the material properties over the seismic asperity rupture. Both
characteristics are key factors for understanding earthquake dynamics. The
strengths of our FBM model are the simplicity of implementation, the
flexibility, and the capability to model different rupture scenarios (i.e.,
asperity configurations) with varying mechanical properties within the
asperities and/or background area or fault plane. Although we simplified the
expected complex asperity geometries, irregular asperity geometries may be
introduced in future works. Another advantage is the analysis of earthquake
dynamics from the point of view of deformable materials that break under
critical stress. The results of TREMOL are promising. However, various
assumptions and simplifications require further experiments and modifications
of the algorithm to cover various tectonic settings. Likewise, the machine-learning application by needs to be incorporated
into the model to determine the optimal parameter ranges for different fault
types and tectonic regimes. Although many questions are still left to be
answered due to the model's early development stage, TREMOL proved to be a
powerful tool that can deliver promising new insights into earthquake
triggering processes. Our future work will investigate complex asperity
configurations, earthquake doublets, and stress transfer in three-dimensional
domains.
Code availability
The TREMOL code is freely available at the home page
(10.5281/zenodo.1884981, Monterrubio-Velasco, 2018b), from its GitHub
repository (https://github.com/monterrubio-velasco), or by request to
the author (marisol.monterrubio@bsc.es, marisolmonterrub@gmail.com). In all
cases, the code is supplied in a manner to ease the immediate execution under
Linux platforms. User's manual documentation is provided in the archive as
well.
Author contributions
MMV developed the code TREMOL v0.1.
MMV, QRP, RZ, AAM, and JP provided guidance and theoretical advice during the study.
All the authors contributed to the analysis and interpretation of the results.
DS helped in the improving of the paper's style and the user manual. All the authors contributed to the writing and editing of the paper.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors are grateful to two anonymous reviewers and the editor for their
relevant and constructive comments that have greatly contributed to improving
the paper. M. Monterrubio-Velasco and J. de la Puente thank the
European Union's Horizon 2020 Programme under the ChEESE Project
(https://cheese-coe.eu/, last access: 1 May 2019), grant agreement no. 823844, for partially funding this work.
M. Monterrubio-Velasco and A. Aguilar-Meléndez thank CONACYT for support
of this research project. Quetzalcoatl Rodríguez-Pérez was supported by the
Mexican National Council for Science and Technology (CONACYT) (Catedras
program, project 1126). This project has received funding from the European
Union's Horizon 2020 research and innovation program under Marie Skłodowska-Curie grant agreement no. 777778, MATHROCKS, and from the Spanish
Ministry project TIN2016-80957-P. Initial funding for the project through
grant UNAM-PAPIIT IN108115 is also gratefully acknowledged.
Review statement
This paper was edited by Thomas Poulet and reviewed by two
anonymous referees.
ReferencesAndersen, J. V., Sornette, D., and Leung, K.-T.: Tricritical behavior in
rupture induced by disorder, Phys. Rev. Lett., 78, 2140–2143, 10.1103/PhysRevLett.78.2140, 1997.Aochi, H. and Ide, S.: Conceptual multi-scale dynamic rupture model for the
2011 off the Pacific coast of Tohoku Earthquake, Earth Planet. Space,
63, 761–765, 10.5047/eps.2011.05.008, 2011.
Astiz, L. and Kanamori, H.: An earthquake doublet in Ometepec, Guerrero,
Mexico, Phys. Earth Planet. In., 34, 24–45, 1984.
Astiz, L., Kanamori, H., and Eissler, H.: Source characteristics of
earthquakes in the Michoacan seismic gap in Mexico, B. Seismol. Soc. Am., 77,
1326–1346, 1987.
Bak, P. and Tang, C.: Earthquakes as a self-organized critical phenomenon,
J. Geophys. Res.-Sol. Ea., 94, 15635–15637, 1989.Barriere, B. and Turcotte, D.: Seismicity and self-organized criticality,
Phys. Rev. E, 49, 1151–1160, 10.1103/PhysRevE.49.1151, 1994.
Biswas, S., Ray, P., and Chakrabarti, B. K.: Statistical physics of fracture,
breakdown, and earthquake: effects of disorder and heterogeneity, John Wiley
& Sons, USA, 2015.
Blaser, L., Krüger, F., Ohrnberger, M., and Scherbaum, F.: Scaling
relations of earthquake source parameter estimates with special focus on
subduction environment, B. Seismol. Soc. Am.,
100, 2914–2926, 2010.Caruso, F., Pluchino, A., Latora, V., Vinciguerra, S., and Rapisarda, A.:
Analysis of self-organized criticality in the Olami-Feder-Christensen model
and in real earthquakes, Phys. Rev. E, 75, 055101, 10.1103/PhysRevE.75.055101, 2007.
Chakrabarti, B. K. and Benguigui, L.-G.: Statistical physics of fracture and
breakdown in disordered systems, vol. 55, Oxford University Press, UK, 1997.
Coleman, B. D.: Statistics and time dependence of mechanical breakdown in
fibers, J. Appl. Phys., 29, 968–983, 1958.
Daniels, H. E.: The statistical theory of the strength of bundles of threads,
I, Proc. R. Soc. Lond. A, 183, 405–435, 1945.
Das, S. and Aki, K.: Fault plane with barriers: a versatile earthquake model,
J. Geophys. Res., 82, 5658–5670, 1977.
Das, S. and Kostrov, B.: Fracture of a single asperity on a finite fault: a
model for weak earthquakes?, Earthq. Source Mech., 37, 91–96, 1986.
Eshelby, J. D.: The determination of the elastic field of an ellipsoidal
inclusion, and related problems, Proc. R. Soc. Lond. A, 241, 376–396, 1957.
Geller, R. J.: Scaling relations for earthquake source parameters and
magnitudes, B. Seismol. Soc. Am., 66, 1501–1523,
1976.Gómez, J., Moreno, Y., and Pacheco, A.: Probabilistic approach to
time-dependent load-transfer models of fracture, Phys. Rev. E, 58,
1528–1523, 10.1103/PhysRevE.58.1528,
1998.
Hansen, A., Hemmer, P. C., and Pradhan, S.: The fiber bundle model: modeling
failure in materials, John Wiley & Sons, USA, 2015.
Iwata, T. and Asano, K.: Characterization of the heterogeneous source model
of
intraslab earthquakes toward strong ground motion prediction, Pure
Appl. Geophys., 168, 117–124, 2011.
Kanamori, H. and Stewart, G. S.: Seismological aspects of the Guatemala
earthquake of February 4, 1976, J. Geophys. Res.-Sol. Ea.,
83, 3427–3434, 1978.Kloster, M., Hansen, A., and Hemmer, P. C.: Burst avalanches in solvable
models
of fibrous materials, Phys. Rev. E, 56, 2615–2625, 10.1103/PhysRevE.56.2615, 1997.
Kun, F., Hidalgo, R. C., Raischel, F., and Herrmann, H. J.: Extension of
fibre
bundle models for creep rupture and interface failure, Int. J.
Frac., 140, 255–265, 2006a.Kun F., Raischel F., Hidalgo R., and Herrmann H.: Extensions of Fibre
Bundle Models. In: Modelling
Critical and Catastrophic Phenomena in Geoscience, edited by: Bhattacharyya P. and Chakrabarti B. K., Lecture Notes in Physics,
vol 705, Springer, Berlin, Heidelberg,
10.1007/3-540-35375-5_3, 2006b.
Lapusta, N. and Rice, J.: Nucleation and early seismic propagation of small
and
large events in a crustal earthquake model, J. Geophys. Res.-Sol. Ea., 108, 1–18, 2003.
Lapusta, N., Rice, J. R., Ben-Zion, Y., and Zheng, G.: Elastodynamic analysis
for slow tectonic loading with spontaneous rupture episodes on faults with
rate-and state-dependent friction, J. Geophys. Res.-Sol.
Ea., 105, 23765–23789, 2000.
Lee, M. and Sornette, D.: Novel mechanism for discrete scale invariance in
sandpile models, Eur. Phys. J. B, 15, 193–197, 2000.
Madariaga, R.: On the relation between seismic moment and stress drop in the
presence of stress and strength heterogeneity, J. Geophys.
Res.-Sol. Ea., 84, 2243–2250, 1979.
Mai, P. M. and Beroza, G. C.: Source scaling properties from
finite-fault-rupture models, B. Seismol. Soc.
Am., 90, 604–615, 2000.
Mai, P. M., Spudich, P., and Boatwright, J.: Hypocenter locations in
finite-source rupture models, B. Seismol. Soc.
Am., 95, 965–980, 2005.
Mendoza, C. and Hartzell S. H.: Slip
distribution of the 19 September 1985 Michoacán,
México, earthquake: near source and teleseismic contraints, Bull. Seismol Soc. Am., 79,
655–669, 1989.
Monterrubio, M., Lana, X., and Martínez, M. D.: Aftershock sequences of
three seismic crises at southern California, USA, simulated by a cellular
automata model based on self-organized criticality, Geosci. J., 19,
81–95, 2015.
Monterrubio-Velasco, M., Zúñiga, F., Márquez-Ramírez, V. H.,
and Figueroa-Soto, A.: Simulation of spatial and temporal properties of
aftershocks by means of the fiber bundle model, J. Seismol., 21,
1623–1639, 2017.Monterrubio-Velasco, M., Carrasco-Jiménez, C., Castillo-Reyes, O.,
Cucchietti, F.,
and la Puente J., D.: A Machine Learning Approach for Parameter Screening in
Earthquake Simulation, in: High Performance Machine Learning Workshop, 24–27 September, Lyon, France, 10.1109/CAHPC.2018.8645865,
2018a.Monterrubio-Velasco, M.: TREMOL_singlets, Zenodo,
10.5281/zenodo.1884981, 2018b.Moral, L., Moreno, Y., Gómez, J., and Pacheco, A.: Time dependence of
breakdown in a global fiber-bundle model with continuous damage, Phys.
Rev. E, 63, 066106, 10.1103/PhysRevE.63.066106, 2001.
Moreno, Y., Correig, A., Gómez, J., and Pacheco, A.: A model for complex
aftershock sequences, J. Geophys. Res.-Sol. Ea., 106,
6609–6619, 2001.
Murotani, S., Miyake, H., and Koketsu, K.: Scaling of characterized slip
models
for plate-boundary earthquakes, Earth Planet. Space, 60, 987–991, 2008.Olami, Z., Feder, H., S., J., and Christensen, K.: Comparison of average stress drop
measures for ruptures with heterogeneous stress change and implications for
earthquake physics, Phys. Rev. Lett., 68, 1244, 10.1103/PhysRevLett.68.1244, 1992.
Peirce, F. T.: Tensile tests for cotton yarns: “the weakest link” theorems
on the strength of long and of composite specimens, J. Textile Inst., 17,
355–368, 1926.
Phoenix, S. L.: Stochastic strength and fatigue of fiber bundles, Int.
J. Frac., 14, 327–344, 1978.
Phoenix, S. L. and Beyerlein, I.: Statistical strength theory for fibrous
composite materials, Comprehens. Composite Mat., 1, 559–639, 2000.
Phoenix, S. L. and Tierney, L.: A statistical model for the time dependent
failure of unidirectional composite materials under local elastic
load-sharing among fibers, Eng. Fract. Mech., 18, 193–215,
1983.
Pradhan, S. and Chakrabarti, B. K.: Failure properties of fiber bundle
models,
Int. J. Modern Phys. B, 17, 5565–5581, 2003.Pradhan, S., Hansen, A., and Chakrabarti, B. K.: Failure processes in elastic
fiber bundles, Rev. Modern Phys., 82, 499–555, 10.1103/RevModPhys.82.499, 2010.
Rodríguez-Pérez, Q. and Ottemöller, L.: Finite-fault scaling
relations in Mexico, Geophys. J. Int., 193, 1570–1588,
2013.
Rodríguez-Pérez, Q. and Zúñiga, F.: Båth's law and its
relation to the tectonic environment: A case study for earthquakes in Mexico,
Tectonophysics, 687, 66–77, 2016.Rodríguez-Pérez, Q., Márquez-Ramírez, V., Zúñiga, F.,
Plata-Martínez, R., and Pérez-Campos, X.: The Mexican Earthquake
Source Parameter Database: A New Resource for Earthquake Physics and Seismic
Hazard Analyses in México, Seismol. Res. Lett., 89, 1846–1862, 10.1785/0220170250, 2018.
Scholz, D. N.: Numerical simulations of stress transfer as a future
alternative
to classical Coulomb stress change? Investigation of the El Mayor Cucapah
event, Master's thesis, University College London, London, 2018.
Singh, S. and Mortera, F.: Source time functions of large Mexican subduction
earthquakes, morphology of the Benioff zone, age of the plate, and their
tectonic implications, J. Geophys. Res., 96, 21487–21502, 1991.
Somerville, P., Irikura, K., Graves, R., Sawada, S., Wald, D., Abrahamson,
N.,
Iwasaki, Y., Kagawa, T., Smith, N., and Kowada, A.: Characterizing crustal
earthquake slip models for the prediction of strong ground motion,
Seismol. Res. Lett., 70, 59–80, 1999.
Somerville, P. G., Sato, T., Ishii, T., Collins, N., Dan, K., and Fujiwara,
H.:
Characterizing heterogeneous slip models for large subduction earthquakes for
strong ground motion prediction, in: Proceedings of the 11th Japan Earthquake
Engineering Symposium, 20–22 November, Tokio, vol. 1, 163–166, Architectural Institute of
Japan, 2002.
Stein, S. and Wysession, M.: An introduction to seismology, eartquakes, and
Earth structure, John Wiley & Sons, USA, 2008.
Strasser, F., Arango, M., and Bommer, J.: Scaling of the source dimensions of
interface and intraslab subduction-zone earthquakes with moment magnitude,
Seismol. Res. Lett., 81, 941–950, 2010.
Turcotte, D. L. and Glasscoe, M. T.: A damage model for the continuum
rheology
of the upper continental crust, Tectonophysics, 383, 71–80, 2004.Vázquez-Prada, M., Gómez, J., Moreno, Y., and Pacheco, A.: Time to
failure of hierarchical load-transfer models of fracture, Phys. Rev. E,
60, 2581–2594, 10.1103/PhysRevE.60.2581, 1999.
Wells, D. L. and Coppersmith, K. J.: New empirical relationships among
magnitude, rupture length, rupture width, rupture area, and surface
displacement, B. Seismol. Soc. Am., 84,
974–1002, 1994.Yewande, O. E., Moreno, Y., Kun, F., Hidalgo, R. C., and Herrmann, H. J.:
Time
evolution of damage under variable ranges of load transfer, Phys. Rev.
E, 68, 026116, 10.1103/PhysRevE.68.026116, 2003.