Earthquake aftershocks display spatiotemporal correlations arising from their self-organized critical behavior. Dynamic deterministic modeling of aftershock series is challenging to carry out due to both the physical complexity and uncertainties related to the different parameters which govern the system. Nevertheless, numerical simulations with the help of stochastic models such as the fiber bundle model (FBM) allow the use of an analog of the physical model that produces a statistical behavior with many similarities to real series. FBMs are simple discrete element models that can be characterized by using few parameters.
In this work, the aim is to present a new model based on FBM that includes geometrical characteristics of fault systems. In our model, the faults are not described with typical geometric measures such as dip, strike, and slip, but they are incorporated as weak regions in the model domain that could increase the likelihood to generate earthquakes. In order to analyze the sensitivity of the model to input parameters, a parametric study is carried out. Our analysis focuses on aftershock statistics in space, time, and magnitude domains. Moreover, we analyzed the synthetic aftershock sequences properties assuming initial load configurations and suitable conditions to propagate the rupture. As an example case, we have modeled a set of real active faults related to the Northridge, California, earthquake sequence. We compare the simulation results to statistical characteristics from the Northridge sequence determining which range of parameters in our FBM version reproduces the main features observed in real aftershock series.
From the results obtained, we observe that two parameters related to the initial load configuration are determinant in obtaining realistic seismicity characteristics: (1) parameter

Most earthquakes occur when adjacent blocks move along fractures in the Earth's crust, as a consequence of stress build-up arising from the regional strain and the stress change caused by a preceding earthquake or by the tectonic stress accumulation

Fault systems have a statistical self-similar structure over a wide range of scales

Moreover, earthquakes follow power statistical laws for their observed scaling properties such as the Gutenberg–Richter (GR) distribution

SOC systems have been studied as a means to explain seismicity by several authors

This section gives a general description of the fiber bundle model and the statistical relations used in this work.

The basic components necessary to construct an FBM are

Defining a discrete set of cells located in a regular (

Assigning a probability distribution function for the inner properties of each cell. This failure law will define the probability distribution function of the stress (in the static case) or the probability distribution function of the rupture time (in the dynamic version)

Establishing the load-sharing rule. This component is crucial in a FBM since the model shows a fundamental change depending on the manner of the load transfer after a cell fail

FBM was developed in two versions that simulate material rupture by different effects: (1) the static version in which the fiber strength is time independent

According to laboratory studies, the Weibull distribution describes the hazard rate (

The FBM model simulation starts by discretizing a hypothetical surface in a bi-dimensional array (

From Eq. (

The time of occurrence or cumulative time

In our previous work, we developed a FBM version to simulate spatial and magnitude aftershock patterns. Following the general assumptions proposed in

A local load-sharing rule including the eight nearest neighbors, and a threshold load (

In order to quantify the resemblance between synthetic catalogs and real seismic catalogs, we use statistical measures which are relevant for evaluating the SOC behavior. These measures are represented by power laws in magnitude (GR law), time (MO law), and space (e.g., fractal dimension). In Appendix

Empirical relations, interpretation, and main parameters.

In our previous work

In the Appendix, three pseudo-codes are included to describe the model algorithm. Our model is coded in Julia language

The initial conditions are as follows:

In this work, and as a first attempt, we simplified the real 3-D domain by choosing a bi-dimensional surface to represent the epicentral distribution. Moreover, we assume that considering that the seismicity of southern California is shallow and mostly restricted to the planar strike-slip faults, the two-dimensional approach can be used as a simplification.
Therefore, the 2-D Cartesian grid is a rectangular domain

The Weibull index,

No external load is received after the initial load assignation, so that our model describes the relaxation process after a mainshock.
Therefore, we do not discuss or simulate the mainshock or foreshocks. The load increase in a cell is due to internal load transfer processes. In a companion study

The rupture conditions are as follows:

In this work, we choose the values proposed in

The completion of a simulation is as follows:

A FBM simulation is terminated when any cell in the system is unable to exceed the threshold

The discrete planar faults of a particular region are modeled by using an image of the real fault system. This image is mapped in the domain

In

Map (azimuthal view) of the Northridge fault system. It is digitalized to include in our computing domain where each pixel of the map corresponds to a cell (

The map of faults has a real physical size in km

Careful attention has been given to minimum magnitudes which depend on size of the cell

We are left with three freely varying parameters for our study. Based on previous results, we use

The epicentral location of the simulated aftershocks is the position of the first avalanche event (

We define

Table

Model parameters.

In order to validate and compare our synthetic seismic catalogs with real seismicity, we modeled as an example case the fault system geometry and the seismic properties of the Northridge aftershock sequence (Fig.

Statistical parameters of the real catalog of Northridge aftershocks using different threshold magnitudes (

Map that includes the seismicity of magnitude larger than 2.0 during 1981–2006. The yellow star indicates the Northridge epicenter (

In Table

Fitting of the

Fitting of the Hurst exponents (Sect.

We divide the results and their analysis in three domains:

space: fractal capacity dimension,

magnitude:

time: inter-event times

For each parameter, we list its observed properties to facilitate the reading. We are analyzing simultaneously three parameters (the size of the domain

The first analysis is related to the fractal capacity dimension,

As

For

Fractal capacity dimension,

The

The smallest array of

For values of

the number of events included in the statistical fit and

the size of the earthquake-simulated avalanches,

The

Frequency–magnitude distribution computed in one synthetic series as example. Each marker indicates different

The results of the mean and maximum magnitudes are depicted in Fig.

From Fig.

Fig.

Figure a and b show the results of the Hurst exponent for inter-event distance

The re-scaled range analysis of the

The analysis of

Hurst exponent values

The MO empirical law requires careful analysis in our FBM implementation. First of all, we must take into account that we are using dimensionless time (see Eq.

Modified Omori relation (Eq.

MO fitting for the leading aftershock (LA) series computed for the same synthetic series of Fig.

Modified Omori parameters computed in the

The MO parametric results provide information of the temporal behavior in the simulated series. We observe that

The load-transfer value

Epicentral spatial distribution for two examples:

Gutenberg–Richter fit of simulated events for the same series used in Fig.

Lastly, we estimated the error between the real and synthetic statistical values using a measure similar to the Euclidean distance

Minimum Euclidean distance

Euclidean parametric distance

The main goal of this study is to integrate prior knowledge of the spatial geometry of faults in the implementation of the FBM algorithm, improving the model previously proposed in

The results are sensitive to the size of the domain. An exhaustive parametric analysis using machine learning techniques to classify the synthetic series as function of the input parameters (the size

Mean error of three different ML classification algorithms (random forests, supported vector machine, and flexible discriminant analysis) as a function of the domain size (figure from

Considering the example of Northridge, our results suggest that the best combination of parameters to approximate to real cases depends on the minimum magnitude of the real catalogs, as shown in Table

The usefulness of this stochastic model is its capability to generate a large number of scenarios with statistical properties similar to real cases, with low computational cost and a low number of free parameters.

We present a novel model simulation of aftershock sequences that incorporates a 2-D spatial distribution of faults. The representation of faults is carried out by assigning weak cells embedded in a background of “normal” cells. However, this model fulfills statistical properties of aftershock when it is well tuned.
We choose statistical relations which describe the aftershocks' behavior in space, magnitude, and time. By means of a parametric study, we have found the range of values that generates synthetic series capable of reproducing the statistical relations of real aftershock events. In particular, we have used the Northridge fault system geometry projected on the surface and its aftershock sequence as a study case.
We conclude that the initial load configuration (quantified by parameter

The data and the numerical code can be obtained upon request to the author, Marisol Monterrubio-Velasco (marisol.monterrubio@bsc.es, marisolmonterrub@gmail.com).

Fractured systems, including lithospheric faults, are scale invariant in a large scale range being characterized by the power law

The generalized fractal dimension,

The re-scaled range (

With

Then,

The GR (sometimes referred to as the Gutenberg–Richter Ishimoto–Ida) law is considered one of the major manifestations of self-organized criticality in a natural system. It has been observed that earthquake magnitude distributions fit a GR power law

In particular,

In the rest of the present work, we apply the maximum likelihood method (MLE) to estimate

The temporal behavior of aftershocks is commonly described by the MO law

The main algorithm (Algorithm 1; Sect.

MMV developed the numerical code. MMV, FRZ, JCCJ, VMR, and JdlP provided guidance and theoretical advice during the study. All the authors contributed to the analysis and interpretation of the results. All the authors contributed to the writing and editing of the paper.

The authors declare that they have no conflict of interest.

We would like to thank the editor and the reviewers for their insights and suggested improvements of this work. Marisol Monterrubio-Velasco thanks CONACYT for initially supporting this research project. 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. This project has received funding from the European Union's Horizon 2020 Programme under the
ChEESE Project (

The research leading to these results has received funding from the European Union's Horizon 2020 Programme under the ChEESE Project (

This paper was edited by CharLotte Krawczyk and reviewed by Philippe Jousset, CharLotte Krawczyk, and one anonymous referee.