Search Results (12)

We discuss a model of seismic activity that is based on the concept of energy in a cluster of sources of seismic activity. We show that specific cases of the studied model lead to the Gutenberg–Richter relationship and the Omori law. These laws are valid for earthquakes that happen in a single cluster of sources of seismic activity. Further, we discuss the distribution of earthquakes for several clusters containing sources of seismic activity. This distribution contains, as a specific case, a version of the negative binomial distribution. We show that at least a part of the roll-off effect connected to the parameter b of the Gutenberg– Richter law occurs because one records earthquakes that happen in more than one cluster of sources of seismic activity. Full article

Crude oil prices crashed and dropped into negative territory at the onset of the COVID-19 pandemic. This extreme event triggered a series of great-magnitude aftershocks. We seek to investigate the cascading dynamics and the characteristics of the series immediately following the oil market crash. Utilizing a robust method named the Omori law, we quantify the correlations of these events. This research presents empirical regularity concerning the number of times that the absolute value of the percentage change in the oil index exceeds a given threshold value. During the COVID-19 crisis, the West Texas Intermediate (WTI) oil prices exhibit greater volatility compared to the Brent oil prices, with higher relaxation values at all threshold levels. This indicates that larger aftershocks decay more rapidly, and the period of turbulence for the WTI is shorter than that of Brent and the stock market indices. We also demonstrate that the power law’s exponent value increases with the threshold value’s magnitude. By proposing this alternative method of modeling extreme events, we add to the current body of literature, and the findings demonstrate its practical use for decision-making authorities—particularly financial traders who model high-volatility products like derivatives. Full article

We investigated the magnitude–frequency relationship and decay pattern of an aftershock sequence using data from the 2008 Wenchuan earthquake. We analyzed the spatial variations in aftershock activity parameters b and p. The calculated b-value of the aftershock sequence is 0.89 ± 0.02, which is relatively small, probably owing to the absence of small earthquakes in the aftershock catalog. The p-value, indicating the decay rate of aftershock activity, is 1.05 ± 0.02, which is normal. The decay pattern of the Wenchuan aftershock sequence agrees well with the modified Omori law. The b-value of the aftershock sequence mainly spatially varies between 0.6 and 1.2, and the p-value varies between 0.6 and 1.8. Although the physical significance of the spatial variations in b- and p-values has not been clearly defined, in this study, the physical significance of the b-value is mainly related to changes in stress, P-wave velocity, and the density of media in the earthquake area, and that in the p-value is associated with the fault slip amount during the mainshock; the b- and p-values show a strong linear correlation. After the mainshock, stress decreased and increased in areas with large and small b-values, respectively; the regions with large and small b-values were associated with low and high P-wave velocities, respectively. The subsurface media experienced relatively high and low apparent velocities in areas with small and large b-values, respectively. The amount of fault slip was small and large in regions with small and large p-values, respectively, exhibiting a linear correlation between the fault slip amount and p-value. The results indicate that the spatial variations in the b- and p-values were related to the physical properties of the media in the earthquake area and focal earthquake mechanism. Full article

The Omori-Utsu law shows the temporal power-law-like decrease of the frequency of earthquake aftershocks and, interestingly, is found in a variety of complex systems/phenomena exhibiting catastrophes. Now, it may be interpreted as a characteristic response of such systems to large events. Here, hierarchical dynamics with the fast and slow degrees of freedom is studied on the basis of the Fokker-Planck theory for the load-state distribution to formulate the law as a relaxation process, in which diffusion coefficient in the space of the load state is treated as a fluctuating slow variable. The evolution equation reduced from the full Fokker-Planck equation and its Green’s function are analyzed for the subdynamics governing the load state as the fast degree of freedom. It is shown that the subsystem has the temporal translational invariance in the logarithmic time, not in the conventional time, and consequently the aging phenomenon appears. Full article

We develop the notion of Random Domino Automaton, a simple probabilistic cellular automaton model for earthquake statistics, in order to provide a mechanistic basis for the interrelation of Gutenberg–Richter law and Omori law with the waiting time distribution for earthquakes. In this work, we provide a general algebraic solution to the inverse problem for the model and apply the proposed procedure to seismic data recorded in the Legnica-Głogów Copper District in Poland, which demonstrate the adequacy of the method. The solution of the inverse problem enables adjustment of the model to localization-dependent seismic properties manifested by deviations from Gutenberg–Richter law. Full article

In this article we present a space–time epidemic-type aftershock sequence (ETAS) model for the area of Hungary, motivated by the goal of its application in insurance risk models. High-quality recent instrumental data from the period 1996–2021 are used for model parameterization, including data from the recent nearby Zagreb and Petrinja event sequences. In the earthquake-triggering equations of our ETAS model, we replace the commonly used modified Omori law with the more recently proposed stretched exponential time response form, and a Gaussian space response function is applied with a variance add-on for epicenter error. After this model was tested against the observations, an appropriate overall fit for magnitudes $M\ge 3.0$ was found, which is sufficient for insurance applications, although the tests also show deviations at the $M=2.5$ threshold. Since the data used for parameterization are dominated by Croatian earthquake sequences, we also downscale the model to regional zones via parameter adjustments. In the downscaling older historical data are incorporated for a better representation of the key events within Hungary itself. Comparison of long-term large event numbers in simulated catalogues versus historical data shows that the model fit by zone is improved by the downscaling. Full article

This paper proposes phenomenological equations that describe various aspects of aftershock evolution: elementary master equation, logistic equation, stochastic equation, and nonlinear diffusion equation. The elementary master equation is a first-order differential equation with a quadratic term. It is completely equivalent to Omori’s law. The equation allows us to introduce the idea of proper time of earthquake source “cooling down” after the main shock. Using the elementary master equation, one can pose and solve an inverse problem, the purpose of which is to measure the deactivation coefficient of an earthquake source. It has been found for the first time that the deactivation coefficient decreases with increasing magnitude of the main shock. The logistic equation is used to construct a phase portrait of a dynamical system simulating the evolution of aftershocks. The stochastic equation can be used to model fluctuation phenomena, and the nonlinear diffusion equation provides a framework for understanding the spatiotemporal distribution of aftershocks. Earthquake triads, which are a natural trinity of foreshocks, main shock, and aftershocks, are considered. Examples of the classical triad, the mirror triad, the symmetrical triad, as well as the Grande Terremoto Solitario, which can be considered as an anomalous symmetrical triad, are given. Prospects for further development of the phenomenology of earthquakes are outlined. Full article

Strong crustal earthquakes in Greece are typically followed by aftershocks, the properties of which are important factors in seismic hazard assessment. In order to examine the properties of earthquake sequences, we prepared an earthquake catalog comprising aftershock sequences with mainshocks of M_w ≥ 5.5 from 1995 to 2021. Regional aftershock parameters were estimated to highlight variations in aftershock decay and productivity among regions with similar seismotectonic characteristics. A statistically based method of estimating aftershock duration and a metric of relative aftershock productivity to examine the variations among the different cases were employed. From the detailed analysis of the selected seismic sequences, we attempt to unravel the physical mechanisms behind deviations in aftershock duration and productivity and resolve the relative contribution of background seismicity, the Omori–Utsu law parameters and the mainshock faulting properties. From our analysis, the duration of aftershock sequences depends upon the rupture process of the mainshock, independently of its magnitude. The same applies to aftershock productivity, however, other tectonic setting (e.g., seismic coupling) or source-related (e.g., focal depth, stress drop) parameters also contribute. The estimated regional parameters of the aftershock rate models could be utilized as initial ones to forecast the aftershock occurrence rates at the early stage following a mainshock. Full article

The 27 September 2021 damaging mainshock (Mw6.0) close to Arkalochori village is the strongest earthquake that was recorded during the instrumental period of seismicity in Central Crete (Greece). The mainshock was preceded by a significant number of foreshocks that lasted nearly four months. Maximum ground subsidence of about 18 cm was estimated from InSAR processing. The aftershock sequence is located in an almost NE-SW direction and divided into two main clusters, the southern and the northern ones. The foreshock activity, the deformation area, and the strongest aftershocks are located within the southern cluster. Based on body-wave travel times, a 3-D velocity model was developed, while using combined space and ground-based geodetic techniques, the co-seismic ground deformation is presented. Moreover, we examined the co-seismic static stress changes with respect to the aftershocks’ spatial distribution during the major events of the foreshocks, the Mw = 6.0 main event as well as the largest aftershock. Both the foreshock and the aftershock sequences obey the scaling law for the frequency-magnitude distribution as derived from the framework of non-extensive statistical physics (NESP). The aftershock production rate decays according to the modified Omori scaling law, exhibiting various Omori regimes due to the generation of secondary aftershock sequences. The analysis of the inter-event time distribution, based on NESP, further indicates asymptotic power-law scaling and long-range correlations among the events. The spatiotemporal evolution of the aftershock sequence indicates triggering by co-seismic stress transfer, while its slow migration towards the outer edges of the area of the aftershocks, related to the logarithm of time, further indicates a possible afterslip. Full article

The forecasting of the evolution of natural hazards is an important and critical problem in natural sciences and engineering. Earthquake forecasting is one such example and is a difficult task due to the complexity of the occurrence of earthquakes. Since earthquake forecasting is typically based on the seismic history of a given region, the analysis of the past seismicity plays a critical role in modern statistical seismology. In this respect, the recent three significant mainshocks that occurred in Alaska (the 2002, Mw 7.9 Denali; the 2018, Mw 7.9 Kodiak; and the 2018, Mw 7.1 Anchorage earthquakes) presented an opportunity to analyze these sequences in detail. This included the modelling of the frequency-magnitude statistics of the corresponding aftershock sequences. In addition, the aftershock occurrence rates were modelled using the Omori–Utsu (OU) law and the Epidemic Type Aftershock Sequence (ETAS) model. For each sequence, the calculation of the probability to have the largest expected aftershock during a given forecasting time interval was performed using both the extreme value theory and the Bayesian predictive framework. For the Bayesian approach, the Markov Chain Monte Carlo (MCMC) sampling of the posterior distribution was performed to generate the chains of the model parameters. These MCMC chains were used to simulate the models forward in time to compute the predictive distributions. The calculation of the probabilities to have the largest expected aftershock to be above a certain magnitude after a mainshock using the Bayesian predictive framework fully takes into account the uncertainties of the model parameters. Moreover, in order to investigate the credibility of the obtained forecasts, several statistical tests were conducted to compare the performance of the earthquake rate models based on the OU formula and the ETAS model. The results indicate that the Bayesian approach combined with the ETAS model produced more robust results than the standard approach based on the extreme value distribution and the OU law. Full article

The acoustic emission, AE, from avalanches of local cracks and microstructural changes of sandstone under confined compression have been reported. These avalanches soften the underlying minerals and play a key role as indicators for the prediction of geo-engineering disasters, such as mining collapses, rock outbursts caused by high ground stress, and man-made quakes by fracking. Compressed sandstone is a model material for the investigation of avalanches. The avalanche energies, amplitudes, and waiting times show the probability distributions that allow us to distinguish between three compression stages; namely, (I) pre-failure, (II) correlated failure, and (III) post-failure. The energy of stage I and stage II is power-law distributed and scale invariant, while post-failure experiments show power laws with high exponential damping (friction). The scaling behavior is close to the predictions of a mean-field (MF) model (stage II) and a force-integrated mean-field model (stage I). Confinement shifts the value of the energy exponent closer to the MF prediction. Omori’s law and waiting time distributions are independent of stress during the compression; their scaling exponents are very similar to those found in seismological studies. Full article

After the Tohoku earthquake in 2011, we observed that aftershocks tended to occur in a wide region after such a large earthquake. These aftershocks resulted in secondary damage or delayed rescue and recovery activities. In addition, it has been reported that there are regions where the intensity of the vibrations owing to the aftershocks was much stronger than those associated with the main shock. Therefore, it is necessary to consider the seismic risk associated with aftershocks. We used the data regarding aftershocks that was obtained from the Tohoku earthquake and various other historically large earthquakes. We investigated the spatial and temporal distribution of the aftershocks using the Gutenberg–Richter law and the modified Omori law. Subsequently, we previously proposed a probabilistic aftershock occurrence model that is expected to be useful to develop plans for recovery activities after future large earthquakes. In this study, the probabilistic aftershock hazard analysis is used to create aftershock hazard maps. We propose a hazard map focusing on the probability of aftershocks on the scale of the main shock for use with a recovery activity plan. Following the lessons learned from the 2011 Tohoku earthquake, we focus on the simultaneous occurrence of tsunamis and aftershocks just after a great subduction earthquake. The probabilistic aftershock hazard analysis is used to derive load combination equations of the load and resistance factor design. This design is intended to simultaneously consider tsunamis and aftershocks for tsunami-resistant designs of tsunami evacuation buildings. Full article