Epidemic-type Aftershock Sequence Model Research Articles

Bayesian earthquake forecasting approach based on the epidemic type aftershock sequence model

The epidemic type aftershock sequence (ETAS) model is used as a baseline model both for earthquake clustering and earthquake prediction. In most forecast experiments, the ETAS parameters are estimated based on a short and local catalog, therefore the model parameter optimization carried out by means of a maximum likelihood estimation may be not as robust as expected. We use Bayesian forecast techniques to solve this problem, where non-informative flat prior distributions of the parameters is adopted to perform forecast experiments on 3 mainshocks occurred in Southern California. A Metropolis–Hastings algorithm is employed to sample the model parameters and earthquake events. We also show, through forecast experiments, how the Bayesian inference allows to obtain a probabilistic forecast, differently from one obtained via MLE.Graphical

SuiETAS: Developing and Testing ETAS-Based Earthquake Forecasting Models for Switzerland

ABSTRACT We present the development and testing of multiple epidemic-type aftershock sequence (ETAS)-based earthquake forecasting models for Switzerland, aiming to identify suitable candidate models for operational earthquake forecasting (OEF) at the Swiss Seismological Service. We consider seven model variants: four variants use parameters obtained through fitting the ETAS model to the Swiss earthquake catalog, and three use generic parameters that were fit to Californian seismicity or global seismicity from regions tectonically similar to Switzerland. The model variants use different pieces of information from the current state-of-the-art time-independent earthquake rate forecast underlying the Swiss seismic hazard model (SUIhaz2015), and one is calibrated on a larger local data set that includes smaller earthquakes by allowing a time-dependent estimate of the completeness magnitude. We test all variants using pseudoprospective short-term (7-day) forecasting experiments and retrospective long-term (30-year) consistency tests. Our results suggest that all ETAS-based models outperform the time-independent SUIhaz2015 forecast in the short term, but two of the model variants overestimate event numbers in the long term. ETAS parameters are found not to be universally transferrable across tectonic regimes, and region-specific calibration is found to add value over generic parameters. Finally, we rank all model variants based on six criteria, including the results of the pseudoprospective and retrospective tests, as well as other criteria such as model run time or consistency with the existing long-term model, using a multicriteria decision analysis approach. Based on this ranking, we propose the ETAS model calibrated on the Swiss catalog, and with the spatial background seismicity distribution of SUIhaz2015 as the ideal candidate for the first Swiss OEF system. All procedures and choices involved in the development and testing of the Swiss ETAS model follow recently established expert recommendations and can act as a reference in the establishment of time-variant earthquake forecasting models for other regions.

Comparison of statistical low-frequency earthquake activity models

Slow earthquakes are slow fault slip events. Quantifying and monitoring slow earthquake activity characteristics are important, because they may change before large earthquakes occur. Statistical seismicity models are useful for quantifying seismicity characteristics. However, no standard statistical model exists for slow earthquake activity. This study used a high-quality catalog of low-frequency earthquakes (LFEs), a type of slow earthquake, in the Nankai subduction zone from April 2004 to August 2015 and conducted the first comparison of existing statistical LFE activity models to determine which model better describes LFE activity. Based on this comparison, this study proposes a new hybrid model that incorporates existing model features. The new model considers the LFE activity history in a manner similar to the epidemic-type aftershock sequence (ETAS) model and represents the LFE aftershock rate (subsequent LFE occurrence rate) with a small number of model parameters, as in the Omori–Utsu aftershock law for regular earthquakes. The results show that the proposed model outperforms other existing models. However, the new model cannot reproduce a feature of LFE activity: the sudden cessation of intense LFE bursts. This is because the new model superimposes multiple aftershock activities and predicts extremely high seismicity rates during and after the LFE bursts. I suggest that reproducing and successfully predicting the sudden cessation of intense LFE bursts is critical for the further improvement of statistical LFE activity models. In addition, the empirical equations formulated in this study for the LFE aftershock rates may be useful for future statistical and physical modeling of LFE activity.Graphical

A 3D Spherical ETAS Model: A Case Study in the Alaska-Aleutian Region

Abstract This study proposes a hypocentral version of the spherical epidemic-type aftershock sequence (ETAS) model by incorporating hypocentral depth information into the spherical ETAS model and applies it to earthquake data from the Alaska-Aleutian area. We validate the model using the stochastic reconstruction techniques and illustrate the rationale behind selecting the spatial probability density function. Subsequently, we estimate the 3D spatial variations of the background and offspring seismicity in the study region, revealing that the majority of clusters are located in shallow regions. We also identify an apparent temporal change in the background seismicity caused by artificial effects. Compared with the 3D-ETAS and spherical version of the epidemic-type aftershock sequence (SETAS) models, the new 3D-SETAS model provides a better description of 3D seismicity in high-latitude regions globally.

Aftershock forecasts based on incomplete earthquake catalogues: ETASI model application to the 2023 SE Türkiye earthquake sequence

SUMMARY The epidemic-type aftershock sequence (ETAS) model is the state-of-the-art approach for modelling short-term earthquake clustering and is preferable for short-term aftershock forecasting. However, due to the large variability of different earthquake sequences, the model parameters must be adjusted to the local seismicity for accurate forecasting. Such an adjustment based on the first aftershocks is hampered by the incompleteness of earthquake catalogues after a mainshock, which can be explained by a blind period of the seismic networks after each earthquake, during which smaller events with lower magnitudes cannot be detected. Assuming a constant blind time, direct relationships based only on this additional parameter can be established between the actual seismicity rate and magnitude distributions and those that can be detected. The ETAS-incomplete (ETASI) model uses these relationships to estimate the true ETAS parameters and the catalogue incompleteness jointly. In this study, we apply the ETASI model to the SE Türkiye earthquake sequence, consisting of a doublet of M 7.7 and M 7.6 earthquakes that occurred within less than half a day of each other on 6 February 2023. We show that the ETASI model can explain the catalogue incompleteness and fits the observed earthquake numbers and magnitudes well. A pseudo-prospective forecasting experiment shows that the daily number of detectable m ≥ 2 can be well predicted based on minimal and incomplete information from early aftershocks. However, the maximum magnitude (Mmax ) of the next day’s aftershocks would have been overestimated due to the highly variable b value within the sequence. Instead, using the regional b value estimated for 2000–2022 would have well predicted the observed Mmax values.

Bootstrapping for parameter uncertainty in the space–time epidemic-type aftershock sequence model

SUMMARY This paper proposes the use of the parametric bootstrap as a simple way of obtaining reliable confidence intervals for the parameters in the space–time epidemic-type aftershock sequence (ETAS) model. Using an earthquake data catalogue spanning almost 20 yr of seismic activity in the Pacific Ocean off the shores of Vancouver Island (British Columbia, Canada), the authors show through simulation that the confidence intervals based on asymptotic maximum-likelihood theory can sometimes be misleading. In contrast, confidence intervals based on empirical quantiles from bootstrap samples have reliable nominal coverage, provided that edge effects are properly taken into account, both in the bootstrap procedure and in simulation studies. An R package called ETASbootstrap, developed by the authors, facilitates the use of this resampling procedure.

Spatio-temporal characterization of earthquake sequence parameters and forecasting of strong aftershocks in Xinjiang based on the ETAS model.

In this paper, the Integrated Nested Laplace Algorithm (INLA) is applied to the Epidemic Type Aftershock Sequence (ETAS) model, and the parameters of the ETAS model are obtained for the earthquake sequences active in different regions of Xinjiang. By analyzing the characteristics of the model parameters over time, the changes in each earthquake sequence are studied in more detail. The estimated values of the ETAS model parameters are used as inputs to forecast strong aftershocks in the next period. We find that there are significant differences in the aftershock triggering capacity and aftershock attenuation capacity of earthquake sequences in different seismic regions of Xinjiang. With different cutoff dates set, we observe the characteristics of the earthquake sequence parameters changing with time after the mainshock occurs, and the model parameters of the Ms7.3 earthquake sequence in Hotan region change significantly with time within 15 days after the earthquake. Compared with the MCMC algorithm, the ETAS model fitted with the INLA algorithm can forecast the number of earthquakes in the early period after the occurrence of strong aftershocks more effectively and can forecast the sudden occurrence time of earthquakes more accurately.

Time-Dependent Probabilistic Seismic Hazard Analysis for Seismic Sequences Based on Hybrid Renewal Process Models

ABSTRACT Probabilistic seismic hazard analysis (PSHA) is a methodology with a long history and has been widely implemented. However, in the conventional PSHA and sequence-based probabilistic seismic hazard analysis (SPSHA) approaches, the occurrence of mainshocks is modeled as the homogeneous Poisson process, which is unsuitable for large earthquakes. To account for the stationary occurrence of small-to-moderate (STM) mainshocks and the nonstationary behavior of large mainshocks, we propose a time-dependent sequence-based probabilistic seismic hazard analysis (TD-SPSHA) approach by combining the time-dependent mainshock probabilistic seismic hazard analysis (TD-PSHA) and aftershock probabilistic seismic hazard analysis, consisting of four components: (1) STM mainshocks, (2) aftershocks associated with STM mainshocks, (3) large mainshocks, and (4) aftershocks associated with large mainshocks. The approach incorporates an exponential-magnitude, exponential-time model for STM mainshocks, and a renewal-time, characteristic-magnitude model for large mainshocks to assess the time-dependent hazard for mainshocks. Then nonhomogeneous Poisson process is used to model the occurrence of associated aftershocks, in which the aftershock sequences can be modeled using the Reasenberg and Jones (RJ) model or the epidemic-type aftershock sequence (ETAS) model. To demonstrate the proposed TD-SPSHA approach, a representative site of the San Andreas fault is selected as a benchmark case, for which five time-dependent recurrence models, including normal, lognormal, gamma, Weibull, and Brownian passage time (BPT) distributions, are chosen to determine the occurrence of large mainshocks. Then sensitivity tests are presented to show the effects on TD-SPSHA, including (1) time-dependent recurrence models, (2) mainshock magnitude, (3) rupture distance, (4) aftershock duration, (5) escaped time since the last event, and (6) future time interval. Furthermore, the bimodal hybrid renewal model is utilized by TD-SPSHA for another case site. The comparison results illustrate that the sequence hazard analysis approach ignoring time-varying properties of large earthquakes for long periods and the effects of associated aftershocks will result in a significantly underestimated hazard. The TD-SPSHA-based hazard curves using the ETAS model are larger than those of the RJ model. The proposed TD-SPSHA approach may be of significant interest to the field of earthquake engineering, particularly in the context of structural design or seismic risk analysis for the long term.

Seasonally hydrological load and background seismicity in the intraplate collision zone, the Longmen Shan, China

SUMMARY Hydrological modulation of background seismicity provides a way to understand the seismogenic process and interaction mechanism between the hydrosphere and lithosphere. To elucidate the hydrological modulation of background seismicity in the Longmen Shan, the monthly spatio-temporal distribution of terrestrial water storage (TWS) variations is inverted from the seasonal term of continuous GNSS (Global Navigation Satellite System) data and compared with monthly precipitation. In the Longmen Shan, monthly TWS variations are in phase with precipitation variations in the northern area but lag precipitation variation by 2 months in the southern area. The annual amplitude of the inverted TWS is about 0.2 m, which is consistent with the amplitude of precipitation. To investigate the stress perturbations in the crustal interior induced by the hydrological load, monthly Coulomb failure stress (CFS) variation caused by TWS and precipitation load is calculated on a simplified fault model of the Longmen Shan, respectively. The maximum CFS variations of TWS and precipitation are about 2.4 kPa. To obtain the background seismicity, different earthquake declustering methods are employed on a two-decade earthquake catalogue (M ≥ 2.5). The declustered results are subjected to null hypothesis tests, including assessing whether the earthquake distribution follows a homogeneous Poisson distribution in time or exhibits a temporally homogeneous, spatially heterogeneous Poisson distribution in space and time. The only declustered catalogue derived by the epidemic type aftershock sequence model pass all the tests. Furthermore, the monthly CFS variation and its rate caused by TWS and precipitation are compared with the monthly background seismic events. Notably, a correlation is observed between CFS variation rate due to precipitation and background seismicity. This correlation suggests the presence of long-term hydrological modulation of seasonal background seismicity in the Longmen Shan.

Viscoelastic Slider Blocks as a Model for a Seismogenic Fault.

In this work, a model is proposed to examine the role of viscoelasticity in the generation of simulated earthquake-like events. This model serves to investigate how nonlinear processes in the Earth's crust affect the triggering and decay patterns of earthquake sequences. These synthetic earthquake events are numerically simulated using a slider-block model containing viscoelastic standard linear solid (SLS) elements to reproduce the dynamics of an earthquake fault. The simulated system exhibits elements of self-organized criticality, and results in the generation of avalanches that behave similarly to naturally occurring seismic events. The model behavior is analyzed using the Epidemic-Type Aftershock Sequence (ETAS) model, which suitably represents the observed triggering and decay patterns; however, parameter estimates deviate from those resulting from natural aftershock sequences. Simulated aftershock sequences from this model are characterized by slightly larger p-values, indicating a faster-than-normal decay of aftershock rates within the system. The ETAS fit, along with realistic simulated frequency-size distributions, supports the inclusion of viscoelastic rheology to model the seismogenic fault dynamics.

SimplETAS: A Benchmark Earthquake Forecasting Model Suitable for Operational Purposes and Seismic Hazard Analysis

Abstract The epidemic-type aftershock sequence (ETAS) model is the most effective mathematical description of the short-term space–time earthquake clustering. However, the use of such a model is sometimes hampered by the difficulty in estimating the high number of its unknown correlated parameters. Moreover, the most recent ETAS formulations introduce the space–time variability of some parameters that makes their estimation even more arduous. Here, we investigate the model in an opposite perspective, looking for the simplest ETAS parameterization that can satisfactorily describes the earthquake clustering in crustal tectonic regions; we named this model simplETAS. We show that simplETAS calibrated with the Italian seismicity of the last decades adequately describes the space–time occurrence of the out-of-sample largest earthquakes in the instrumental and historical catalog, confirming the validity of the assumptions made to build the model. Owing to its simplicity, simplETAS is easily applicable in most regions, and it has some important properties that are worth being remarked. First, simplETAS can be used as a benchmark model to assess the relative predictive skill of more complex earthquake forecasts. Second, it may be used for operational earthquake forecasting purposes in regions with limited earthquake catalogs. Third, it provides a straightforward, flexible, and effective approach to generate synthetic earthquake catalogs of variable length to be implemented in seismic hazard and risk analysis, overcoming all the declustering-related problems and the controversial Poisson assumption.

Using Deep Learning for Flexible and Scalable Earthquake Forecasting

AbstractSeismology is witnessing explosive growth in the diversity and scale of earthquake catalogs. A key motivation for this community effort is that more data should translate into better earthquake forecasts. Such improvements are yet to be seen. Here, we introduce the Recurrent Earthquake foreCAST (RECAST), a deep‐learning model based on recent developments in neural temporal point processes. The model enables access to a greater volume and diversity of earthquake observations, overcoming the theoretical and computational limitations of traditional approaches. We benchmark against a temporal Epidemic Type Aftershock Sequence model. Tests on synthetic data suggest that with a modest‐sized data set, RECAST accurately models earthquake‐like point processes directly from cataloged data. Tests on earthquake catalogs in Southern California indicate improved fit and forecast accuracy compared to our benchmark when the training set is sufficiently long (>104 events). The basic components in RECAST add flexibility and scalability for earthquake forecasting without sacrificing performance.

EM algorithm for the estimation of the RETAS model*

The Renewal Epidemic-Type Aftershock Sequence (RETAS) model is a recently proposed point process model that can fit event sequences such as earthquakes better than pre-existing models. Evaluating the log-likelihood function and directly maximizing it has been shown to be a viable approach to obtain the maximum likelihood estimator (MLE) of the RETAS model. However, the direct likelihood maximization suffers from numerical issues such as premature termination of parameter searching and sensitivity to the initial value. In this work, we propose to use the Expectation-Maximization (EM) algorithm as a numerically more stable alternative to obtain the MLE of the RETAS model. We propose two choices of the latent variables, leading to two variants of the EM algorithm. As well as deriving the conditional distribution of the latent variables given the observed data required in the E-step of each EM-cycle, we propose an approximation approach to speed up the E-step. The resulting approximate EM algorithms can obtain the MLE much faster without compromising on the accuracy of the solution. These newly developed EM algorithms are shown to perform well in simulation studies and are applied to an Italian earthquake catalog.

A neural encoder for earthquake rate forecasting

Forecasting the timing of earthquakes is a long-standing challenge. Moreover, it is still debated how to formulate this problem in a useful manner, or to compare the predictive power of different models. Here, we develop a versatile neural encoder of earthquake catalogs, and apply it to the fundamental problem of earthquake rate prediction, in the spatio-temporal point process framework. The epidemic type aftershock sequence model (ETAS) effectively learns a small number of parameters to constrain the assumed functional forms for the space and time correlations of earthquake sequences (e.g., Omori-Utsu law). Here we introduce learned spatial and temporal embeddings for point process earthquake forecasting models that capture complex correlation structures. We demonstrate the generality of this neural representation as compared with ETAS model using train-test data splits and how it enables the incorporation additional geophysical information. In rate prediction tasks, the generalized model shows >4% improvement in information gain per earthquake and the simultaneous learning of anisotropic spatial structures analogous to fault traces. The trained network can be also used to perform short-term prediction tasks, showing similar improvement while providing a 1000-fold reduction in run-time.

Comparison between alarm-based and probability-based earthquake forecasting methods

SUMMARY In a recent work, we applied the every earthquake a precursor according to scale (EEPAS) probabilistic model to the pseudo-prospective forecasting of shallow earthquakes with magnitude $M\ 5.0$ in the Italian region. We compared the forecasting performance of EEPAS with that of the epidemic type aftershock sequences (ETAS) forecasting model, using the most recent consistency tests developed within the collaboratory for the study of earthquake predictability (CSEP). The application of such models for the forecasting of Italian target earthquakes seems to show peculiar characteristics for each of them. In particular, the ETAS model showed higher performance for short-term forecasting, in contrast, the EEPAS model showed higher forecasting performance for the medium/long-term. In this work, we compare the performance of EEPAS and ETAS models with that obtained by a deterministic model based on the occurrence of strong foreshocks (FORE model) using an alarm-based approach. We apply the two rate-based models (ETAS and EEPAS) estimating the best probability threshold above which we issue an alarm. The model parameters and probability thresholds for issuing the alarms are calibrated on a learning data set from 1990 to 2011 during which 27 target earthquakes have occurred within the analysis region. The pseudo-prospective forecasting performance is assessed on a validation data set from 2012 to 2021, which also comprises 27 target earthquakes. Tests to assess the forecasting capability demonstrate that, even if all models outperform a purely random method, which trivially forecast earthquake proportionally to the space–time occupied by alarms, the EEPAS model exhibits lower forecasting performance than ETAS and FORE models. In addition, the relative performance comparison of the three models demonstrates that the forecasting capability of the FORE model appears slightly better than ETAS, but the difference is not statistically significant as it remains within the uncertainty level. However, truly prospective tests are necessary to validate such results, ideally using new testing procedures allowing the analysis of alarm-based models, not yet available within the CSEP.

Aftershocks following the 2011 Tohoku-Oki earthquake driven by both stress transfer and afterslip

Aftershocks are a fundamental characteristic of seismicity, and their generation mechanism is mainly characterized by two physical models, stress transfer from large earthquakes and afterslip-induced stress loading. However, the contribution of each mechanism to aftershock generation remains unclear. Here we investigate the spatiotemporal variations in aftershock activity following the 2011 Tohoku-Oki earthquake by applying the Hierarchical Space–Time Epidemic-Type Aftershock Sequence (HIST-ETAS) model to the decade of recorded seismicity since the mainshock. Using the estimated HIST-ETAS model, we categorize the aftershocks into background earthquakes (which are caused by aseismic phenomena) and triggered earthquakes (which are caused by earthquake-to-earthquake interactions). Most of the earthquakes that occurred updip of the large coseismic slip zone along the Japan Trench are triggered earthquakes, consistent with the lack of afterslip in this area. Conversely, background earthquakes are the predominant earthquake type in the long-term downdip of the large coseismic slip zone, and they positively correlate with the afterslip evolution. Our results suggest the importance of combining these two end-member aftershock generation models to explain aftershock activity and thus provide new insights into the relationship between afterslip and spatiotemporal aftershock distribution. Our classifications may also contribute to the monitoring of afterslip in a given region that hosts a large earthquake, particularly where geodetic observation networks are too sparse to evaluate afterslip evolution.

Testing the forecasting skills of aftershock models using a Bayesian framework

The Epidemic Type Aftershock Sequence (ETAS) model and the modified Omori law (MOL) are two aftershock rate models that are used for operational earthquake/aftershock forecasting. Previous studies have investigated the relative performance of the two models for specific case studies. However, a rigorous comparative evaluation of the forecasting performance of the basic aftershock rate models for several different earthquake sequences has not been done before. In this study, forecasts of five prominent aftershock sequences from multiple catalogs are computed using the Bayesian predictive distribution, which fully accounts for the uncertainties in the model parameters. This is done by the Markov Chain Monte Carlo (MCMC) sampling of the model parameters and forward simulation of the ETAS or MOL models to compute the aftershock forecasts. The forecasting results are evaluated using five different statistical tests, including two comparison tests. The forecasting skill tests indicate that the ETAS model tends to perform consistently well on the first three tests. The MOL fails the same tests for certain forecasting time intervals. However, in the comparison tests, it is not definite whether the ETAS model is the better performing model. This work demonstrates the use of forecast testing for different catalogs, which is also applicable to catalogs with a higher magnitude of completeness.

Depth‐Dependent Aftershock Trigger Potential Revealed by 3D‐ETAS Modeling

AbstractCurrent earthquake catalogs provide high‐precision depth values that contain valuable information. Thus, we extend the spatiotemporal Epidemic Type Aftershock Sequence model to 3D by considering hypocentral instead of epicentral distances. To explore the most appropriate parametric form of the spatial kernel, we first examine different triggering functions for the depth difference of the aftershock‐mainshock pairs identified by the Nearest‐Neighbor method, showing that a magnitude‐dependent power‐law kernel fits best the earthquakes data in Southern California. Therefore, we incorporate the corresponding kernel into the 3D‐ETAS model with space‐dependent background activity, where we additionally allow for depth‐dependent aftershock productivity. The application to Southern California shows that the model fits the data well. We also find that the aftershock productivity strongly depends on depth, similar to the seismic moment released by the mainshocks, which may be related to depth‐dependent seismic coupling.

Development of an Epidemic‐Type Aftershock‐Sequence Model Explicitly Incorporating the Seismicity‐Triggering Effects of Slow Slip Events

AbstractSlow slip events (SSEs) at subduction zone plate boundaries sometimes trigger earthquake swarms and megathrust earthquakes. The causal relationship between SSEs and seismicity has been studied worldwide, but the epidemic‐type aftershock‐sequence (ETAS) model, which is a standard statistical model of seismicity, does not explicitly consider the seismicity‐triggering effect of SSEs. Therefore, if an SSE occurs at a plate boundary, probabilistic earthquake forecasts based on the ETAS model fail to predict observed seismicity. Here, we constructed a statistical model named the SSE‐modulated ETAS model by incorporating SSE moment rates estimated from observation data from the global navigation satellite system into the original ETAS model. Our model assumes a linear or power‐law relationship between the SSE moment rates and seismicity rates and estimates its proportionality constant as a new ETAS parameter. We applied this new model to three SSEs and M 2.5 or greater earthquakes in the shallow part of the Hikurangi Trench, New Zealand. The results show that it is better than the original ETAS model, giving a significant reduction in the Akaike information criterion. In addition, we examined the functional forms (e.g., lag time and power exponent) of the equation relating the moment rate of the SSEs to the seismicity rate. We also examine the influence of SSEs on aftershock productivity. Our model can improve short‐term forecasts of seismicity associated with SSEs if the detection and characterization of SSEs can be done in near real time. Our model is also useful for quantifying characteristics of SSE‐induced seismicity.

Incorporating Foreshocks in an Epidemic-like Description of Seismic Occurrence in Italy

The Epidemic Type Aftershock Sequence (ETAS) model is a widely used tool for cluster analysis and forecasting, owing to its ability to accurately predict aftershock occurrences. However, its capacity to explain the increase in seismic activity prior to large earthquakes—known as foreshocks—has been called into question due to inconsistencies between simulated and experimental catalogs. To address this issue, we introduce a generalization of the ETAS model, called the Epidemic Type Aftershock Foreshock Sequence (ETAFS) model. This model has been shown to accurately describe seismicity in Southern California. In this study, we demonstrate that the ETAFS model is also effective in the Italian catalog, providing good agreement with the instrumental Italian catalogue (ISIDE) in terms of not only the number of aftershocks, but also the number of foreshocks—where the ETAS model fails. These findings suggest that foreshocks cannot be solely explained by cascades of triggered events, but can be reasonably considered as precursory phenomena reflecting the nucleation process of the main event.