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.

Rupture models of large earthquakes suggest significant heterogeneity in slip
and moment release over the fault plane

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

Earthquakes are the most relevant example of self-organized criticality (SOC)

The FBM is a mathematical tool to study the rupture process of heterogeneous
materials that was originally introduced by

A discrete set of cells (or fibers) is defined on a

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. (

The failure probability,

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.

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

Processing TREMOL uses the data from the preprocessing stage to carry out the FBM
algorithm, and by applying Eqs. (

Post-processing In this stage, TREMOL summarizes the results that are computed in the
processing stage and computes the equivalent rupture area (km

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

Schematic representation of the considered local load rule. The
broken cell with load,

In TREMOL, a fault plane is modeled as a rectangle

To define each fault plane

The assumption of Eqs. (

In order to differentiate the parameters of the asperity from the rest of the
fault plane

Results of one realization by TREMOL.

Once the initial information for the entire domain

Due to the integrated strength property some extra rules for rupture are
necessary. The requirement for failure is

When a cell within

After every execution TREMOL outputs a catalog
detailing where the position (

Furthermore, we define the inter-event rate

In the post-processing step we additionally computed the rupture duration of
the largest simulated earthquake,

We used Eq. (

Using these considerations, we can assign a physical unit of time (s) to the
largest simulated earthquake,

The flowchart in Fig.

TREMOL preprocessing: input parameters and their definition.

We performed a sensitivity analysis of the three asperity parameters
(

To explore the influence of

The input data of this experiment are summarized in Table

Input data in order to carry out cases 1 to 12.

To perform the parametric study of

Schematic configuration for the parametric study of

The separation between the two asperities remains constant. We chose a value of

In order to explore how the system behaves when

Main input data in order to carry out cases 13 to 18.

Example of the strength configuration

The modification of the

Main input data in order to carry out cases 19 to 24.

An example configuration of different asperity sizes,

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,

The number of cells was

Example of the domain configuration

In some cases, the number of asperities computed in

In order to study how the asperity size

We carried out 50 realizations per event (Table

The finite-fault source parameters used in this work.

In this case the number of cells is

Main data used for Ev. 1 to Ev. 10.

As reported in

Main data for the case study Ev. 7 test, Ev. 7a test, and Ev. 7b test.

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

Main data for assessing a future earthquake in the Guerrero seismic gap (GG event).

Figure

If

If

Mean of the maximum rupture area (km

Mean and standard deviation of the maximum magnitude over 50
realizations depending on

Mean and standard deviation of the ratio

Using the mean of

Overall, there are three aspects observable.

If

If

If

For each value of

Statistical results of

We observe in Fig.

Figure

Statistical results of

Based on the observations described in the previous section, we used

Figure

Comparison between the real asperity area

Figure

Statistical results of the maximum magnitude for the events from
Table

Statistical results of the maximum magnitude for the events from
Table

Figure

Proportion of simulated ruptured area occupied by the largest
avalanche,

We also computed an equivalent rupture duration,

However, it is worth noting that

Equivalent rupture duration

In the cases in which several effective rupture areas were proposed by different
studies (see Table

A comparison between the data from Table

Estimation of the characteristics of a future earthquake in the
Guerrero seismic gap.

In Fig.

In the results, there were two dominant tendencies visible:
(1)

The parametric study indicates that the largest rupture

As mentioned in Sect.

The parameter

There is an unstable transition zone of

There is a stable zone of

Magnitude, computational time (s), and number of steps to
“activate” the whole asperity for one execution as a function of

The results of Fig.

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.

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

The algorithm of TREMOL enables the model to store stress history and to
simulate static fatigue due to an included strength parameter

The range of values found in the sensitivity analysis are not unique for the
Mexican examples. In fact, the parameters

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

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

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.

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.

A big advantage of our algorithm is the low number of free parameters
(

The TREMOL code is freely available at the home page
(

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.

The authors declare that they have no conflict of interest.

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
(

This paper was edited by Thomas Poulet and reviewed by two anonymous referees.