The Olami, Feder and Christensen (OFC) spring-block model has proven to be a powerful tool for analyzing and comparing synthetic and real earthquakes. This work proposes the possible reproduction of Utsu’s law for earthquakes in the OFC model. Based on our previous works, several simulations characterizing real seismic regions were performed. We located the maximum earthquake in these regions and applied Utsu’s formulae to identify a possible aftershock area and made comparisons between synthetic and real earthquakes. The research compares several equations to calculate the aftershock area and proposes a new one with the available data. Subsequently, the team performed new simulations and chose a mainshock to analyze the behavior of the surrounding events, so as to identify whether they could be catalogued as aftershocks and relate them to the aftershock area previously determined using the formula proposed. Additionally, the spatial location of those events was considered in order to classify them as aftershocks. Finally, we plot the epicenters of the mainshock, and the possible aftershocks comprised in the calculated area resembling the original work of Utsu. Having analyzed the results, it is likely to say that Utsu’s law is reproducible using a spring-block model with a self-organized criticality (SOC) model.

The self-organized critical (SOC) spring-block models are accessible and powerful computational tools for the study of seismic subduction. This work aims to highlight some important findings through an integrative approach of several actual seismic properties, reproduced by using the Olami, Feder, and Christensen (OFC) SOC model and some variations of it. A few interesting updates are also included. These results encompass some properties of the power laws present in the model, such as the Gutenberg-Richter (GR) law, the correlation between the parameters a and b of the linear frequency-magnitude relationship, the stepped plots for cumulative seismicity, and the distribution of the recurrence times of large earthquakes. The spring-block model has been related to other relevant properties of seismic phenomena, such as the fractal distribution of fault sizes, and can be combined with the work of Aki, who established an interesting relationship between the fractal dimension and the b-value of the Gutenberg-Richter relationship. Also included is the work incorporating the idea of asperities, which allowed us to incorporate several inhomogeneous models in the spring-block automaton. Finally, the incorporation of a Ruff-Kanamori-type diagram for synthetic seismicity, which is in reasonable accordance with the original Ruff and Kanamori diagram for real seismicity, is discussed.

With the spring-block model proposed by Olami, Feder, and Christensen (OFC), we obtained a time series of synthetic earthquakes with different values of the conservation level ($\beta$), which measures the fraction of the energy that a relaxing block passes to its neighbors. The time series have multifractal characteristics, and we analyzed them with the Chhabra and Jensen method. We calculated the width, symmetry, and curvature parameters for each spectrum. As the value of conservation level increases, the spectra widen, the symmetric parameter increases, and the curvature around the maximum of the spectra decreases. In a long series of synthetic seismicity, we located earthquakes of the greatest magnitude and built overlapping windows before and after them. For the time series in each window, we performed multifractal analysis to obtain multifractal spectra. We also calculated the width, symmetry, and curvature around the maximum of the multifractal spectrum. We followed the evolution of these parameters before and after large earthquakes. We found that the multifractal spectra had greater widths, were less skewed to the left, and were very pointed around the maximum before rather than after large earthquakes. We studied and calculated the same parameters and found the same results in the analysis of the Southern California seismicity catalog. This suggests that there seems to be a process of preparation for a great earthquake and that its dynamics are different from the one that occurs after this mainshock based on the behavior of the parameters mentioned before.

The multifractal spectra analysis of time series has been used for several applications, for example, in tachograms, that are the graph of the time intervals between heartbeats throughout a monitoring, there are differences between the complexity of the tachograms of healthy and young subjects and the tachograms of elderly or subjects with heart disease. Recent research has shown that parameters of the multifractal spectrum of tachograms, such as width (degree of multifractality) and curvature of the spectrum around the maximum, could be used to classify the health status of a subject. On the other hand, tachograms obtained from stress test subjects decrease in complexity compared to those obtained at rest, and this can also be characterized with multifractal analysis. In this work, we created a database of 131 electrocardiographic signal recordings and tachograms of subjects at rest for one hour and in a stress test for 30 minutes, and we performed the analysis of the aforementioned multifractal spectrum parameters. The results of the multifractality analysis confirmed that the complexity in the tachograms of the stress tests decreases significantly with respect to the series at rest, especially in the series of subjects who reported a sedentary lifestyle. Therefore, these parameters could be useful to classify the physical condition of a subject.

By analyzing earthquake catalogs, it has been reported that some changes were noted in earthquakes’ distribution both spatial and temporal before and after a great-magnitude earthquake in the Mexican Pacific Coast. In a previous work (Entropy 2018, 20, 961), we performed a statistical analysis of a Mw = 8.2 earthquake (EQ) that took place on September 7, 2017 in Mexico. We presented the concept of pseudo-velocity as the ratio between the distance and the time of consecutive earthqueakes. We measured the inter-event time and distance between successive events for a period of 19 years in the circular vicinity of the earthqueake of magnitude 8.2. The main result was that the multifractality degree of the pseudo-velocities series exhibits noticeable changes before the Mw = 8.2 earthquake. On the other hand, the Olami, Feder and Christensen (OFC) model has been applied to obtain catalogs of synthetic earthquakes and by analyzing them, many properties of real seismicity have been qualitatively reproduced. In this work, catalogs of synthetic earthquakes with one hundred million events were obtained, for each event its magnitude, the coordinates of its epicenter, the duration time and the time that elapses between each synthetic earthquake are known, in such a way that it is possible to evaluate the pseudo-velocities. In the synthetic seismicity catalog, the largest magnitude events were identified, and exploration circles were defined around each of them. Sliding windows were defined before and after such large magnitude earthquakes and in each of these windows the multifractal spectrum was evaluated and three important parameters were calculated: the width of the spectrum or multifractality degree, the symmetry of the spectrum and the curvature around the maximum. The objective is to analyze the variations of such parameters before and after the great synthetic earthquakes. Unfortunately, the significant changes that were expected as in the real seismicity were not observed. This is probably because according to many authors this model cannot produce aftershocks. We discuss some proposed modifications of the OFC model that could possibly produce aftershocks.

Many complex systems exhibit self-organizing criticality (SOC). In fact, there is a consensus that the Earth’s crust is a SOC system. The Olami, Feder and Christensen (OFC) spring-block model is a non-conservative SOC model that is used successfully to simulate the dynamics of seismic faults. In this model the system of coupled differential equations representing the spring-block model is mapped to a cellular automaton. In this work we include the idea of asperity, which is an important concept in real seismicity, by varying the distribution in the spring-block network. Considering that in real life seismicity faults are composed of different elements, it is necessary to have a model with these characteristics. We were able of reproduce the Gutenberg-Richter behavior (previously obtained in the classic OFC model) in this non-homogenous distribution.