Inter-event Time Distribution Research Articles

Magnitude and Interevent Time Statistics of Strombolian Activity of Villarrica Volcano and Inference Regarding the Flow Regime

AbstractThe different flow regimes of two‐phase gas‐in‐liquid flow—as it occurs in the shallow conduit of basaltic volcanoes—are characterized by distinct frequency‐size distributions of the liquid and gaseous slugs. Assuming that the ascent of gaseous bubbles is indicated by seismic events, we explore the possibility of inferring the flow regime from the frequency distributions of magnitudes and interevent times. Our data set consists of 20,000 volcanic seismic events recorded in early March 2012 at Villarrica Volcano (Chile), which are commonly attributed to Strombolian activity. One crucial factor is the completeness of the catalog in terms of detectable amplitudes, which we assess using a stochastic simulation of the network output based on statistical properties of the ambient seismicity. Magnitudes, at which we consider the catalog complete, show an exponential occurrence. Yet, the simulation approach indicates that low magnitude events occur indeed more sparsely than expected for an exponential distribution, and that the magnitude distribution does not obey the Gutenberg‐Richter law. Interevent times are log‐normally distributed, which implies a preferred recurrence interval. The distributions are consistent with a slug flow regime. They correlate weakly with the preceding magnitude, suggesting that slugs coalesce while ascending.

A Laboratory Perspective on the Gutenberg‐Richter and Characteristic Earthquake Models

AbstractProbabilistic seismic hazard analysis (PSHA) is the standard method used for designing earthquake‐resistant infrastructure. In recent years, several unexpected and destructive earthquakes have sparked criticism of the PSHA methodology. The seismological part of the problem is the true frequency‐magnitude distribution of regional seismicity. Two major models exist, the Gutenberg‐Richter (G‐R) and the Characteristic Earthquake (CE) model, but it is difficult to choose between them. That is because the instrumental, historical, and paleoseimological data available are limited in many regions of interest. Here, we demonstrate how a friction experiment on aggregates of glass beads can produce both regular (CE equivalent) and irregular (G‐R equivalent) stick‐slip. Using a new rotary shear apparatus we produced and analyzed large catalogs of acoustic emission (AE) events related to stick‐slip. The distributions of AE sizes, interevent times, and interevent distances were found to be sensitive to particle size and the applied normal stress, and, to a lesser degree, the stiffness of the loading apparatus. More importantly, the system spontaneously switched behavior for short periods of time. In the context of PSHA, if faults are able to switch behavior as our experimental system does, then justifying the choice of either the CE or the G‐R model is impossible based on existing observations.

Seismicity Simulation and a Forecast Model Based on Monte Carlo Simulation Techniques and Performance of Distribution Models

In seismic hazard studies, simulation techniques are exploited relatively less compared to other statistical methods. The main purposes of their use were merely scientific: either to reduce uncertainty in seismic hazard or to test the consistency of a forecast. Its limited exploitation in the field does not mean that the method could not be exploited further. Given sufficient data and with the right models, a full-scale simulation is possible, which in fact becomes a forecast with future timing, location and magnitude of future events are simulated.Within this context, the main purpose of this study is to layout a method to create a full scale simulation with all the parameters namely the time, location and magnitude of an earthquake is simulated. For that purpose, in addition to the main parameters defining past seismic patterns, namely the temporal, spatial, magnitude distributions, the combined spatio-magnitude distribution is also modeled. As another improvement compared to the current procedures, occurrence date is assigned to the simulated event according to the inter-event time distribution, which enhances the simulation procedure. The eventual temporal distribution and the associated earthquake rates are checked with the original earthquake rate for its consistency. In the end, a simulation procedure, which can also be used as a forecast model is developed to improve the existing procedures further.

Scaling properties of the Mw7.0 Samos (Greece), 2020 aftershock sequence

On October 30, 2020, a strong and shallow earthquake (Mw = 7.0) hit Samos, an island on the eastern edge of the Aegean Sea (Greece). The epicenter was located on the north offshore of the Greek island of Samos. The goal of our work is to provide a first analysis of the scaling properties observed in the aftershock sequence as reported until December 31, 2020, as numerous seismic clusters activated. Our analysis is focused on the main of the clusters observed in the East area of the activated fault zone and strongly related with the mainshock’s fault. The aftershock sequence follows the Omori law with a value of p ≈ 1.01 for the main cluster which is remarkably close to a logarithmic evolution. The analysis of interevent times distribution, based on non-extensive statistical physics indicates a system in an anomalous equilibrium with a crossover from anomalous (q > 1) to normal (q = 1) statistical mechanics, as great interevent times approached. A discussion of the crossover observed, is given in terms of superstatistics. In addition, the obtained value q ≈ 1.67 suggests a system with one degree of freedom. Furthermore, a scaling of the migration of aftershock zone as a function of the logarithm of time is discussed in terms of rate strengthening rheology that govern the evolution of afterslip process.

A Multivariate Non-Parametric Hazard Model for Earthquake Occurrences in Turkey

This paper provides an introduction to multivariate non-parametric hazard model for the occurrence of earthquakes since the hazard function defines the statistical distribution of inter-event times. The method is ap plied to the Turkish seismicity since a significant portion of Turkey is subject to frequent earthquakes and presents several advantages compared to other more traditional approaches. Destructive earthquakes from 1903 to 2009 between the longitudes of (39-42)N◦ and the latitudes of (26-45)E◦ are used. The paper demonstrates how seismicity and tectonics/physics parameters that can potentially influence the spatio-temporal variability of earthquakes and presents several advantages compared to more traditional approaches.

Interevent-time distribution and aftershock frequency in non-stationary induced seismicity

The initial footprint of an earthquake can be extended considerably by triggering of clustered aftershocks. Such earthquake–earthquake interactions have been studied extensively for data-rich, stationary natural seismicity. Induced seismicity, however, is intrinsically inhomogeneous in time and space and may have a limited catalog of events; this may hamper the distinction between human-induced background events and triggered aftershocks. Here we introduce a novel Gamma Accelerated-Failure-Time model for efficiently analyzing interevent-time distributions in such cases. It addresses the spatiotemporal variation and quantifies, per event, the probability of each event to have been triggered. Distentangling the obscuring aftershocks from the background events is a crucial step to better understand the causal relationship between operational parameters and non-stationary induced seismicity. Applied to the Groningen gas field in the North of the Netherlands, our model elucidates geological and operational drivers of seismicity and has been used to test for aftershock triggering. We find that the hazard rate in Groningen is indeed enhanced after each event and conclude that aftershock triggering cannot be ignored. In particular we find that the non-stationary interevent-time distribution is well described by our Gamma model. This model suggests that 27.0(± 8.5)% of the recorded events in the Groningen field can be attributed to triggering.

Dynamics of cascades on burstiness-controlled temporal networks

Burstiness, the tendency of interaction events to be heterogeneously distributed in time, is critical to information diffusion in physical and social systems. However, an analytical framework capturing the effect of burstiness on generic dynamics is lacking. Here we develop a master equation formalism to study cascades on temporal networks with burstiness modelled by renewal processes. Supported by numerical and data-driven simulations, we describe the interplay between heterogeneous temporal interactions and models of threshold-driven and epidemic spreading. We find that increasing interevent time variance can both accelerate and decelerate spreading for threshold models, but can only decelerate epidemic spreading. When accounting for the skewness of different interevent time distributions, spreading times collapse onto a universal curve. Our framework uncovers a deep yet subtle connection between generic diffusion mechanisms and underlying temporal network structures that impacts a broad class of networked phenomena, from spin interactions to epidemic contagion and language dynamics.

Non-Markovian majority-vote model.

Non-Markovian dynamics pervades human activity and social networks and it induces memory effects and burstiness in a wide range of processes including interevent time distributions, duration of interactions in temporal networks, and human mobility. Here, we propose a non-Markovian majority-vote model (NMMV) that introduces non-Markovian effects in the standard (Markovian) majority-vote model (SMV). The SMV model is one of the simplest two-state stochastic models for studying opinion dynamics, and displays a continuous order-disorder phase transition at a critical noise. In the NMMV model we assume that the probability that an agent changes state is not only dependent on the majority state of his neighbors but it also depends on his age, i.e., how long the agent has been in his current state. The NMMV model has two regimes: the aging regime implies that the probability that an agent changes state is decreasing with his age, while in the antiaging regime the probability that an agent changes state is increasing with his age. Interestingly, we find that the critical noise at which we observe the order-disorder phase transition is a nonmonotonic function of the rate β of the aging (antiaging) process. In particular the critical noise in the aging regime displays a maximum as a function of β while in the antiaging regime displays a minimum. This implies that the aging/antiaging dynamics can retard/anticipate the transition and that there is an optimal rate β for maximally perturbing the value of the critical noise. The analytical results obtained in the framework of the heterogeneous mean-field approach are validated by extensive numerical simulations on a large variety of network topologies.

Seismicity clusters in Central Chile: investigating the role of repeating earthquakes and swarms in a subduction region

SUMMARYSeismicity along subduction interfaces is usually dominated by large main-shock–aftershock sequences indicative of a continuum distribution of highly coupled large asperities. In the past decades, however, the increased resolution of seismic catalogues at some subduction zone seems to indicate instead a more complex rheological segmentation of the interface. Large and megathrust earthquake ruptures seem interspersed among regions of low seismic coupling and less stress buildup. In this weaker zone, the strain is primarily released via a combination of moderate-size swarm-like seismicity and aseismic slip. Along the Chilean subduction zone, the densification of the seismic network allowed for the identification of localized seismic clusters, some of them appearing in the form of swarms before megathrust earthquakes. The origin and driving processes of this seismic activity have not yet been identified. In this study, we follow a systematic approach to characterize the seismicity at two persistent clusters in Central Chile, one located offshore Navidad and one inland, at ∼40 km depth beneath Vichuquén, which occurred throughout ∼20 yr. We investigated these clusters, by deriving high-resolution hypocentral locations and moment tensors and performing a detailed analysis of spatio-temporal patterns, magnitude and interevent time distributions of the clustered earthquakes. Both clusters are characterized by weak to moderate seismicity (below Mw 6) and stand out as clear seismicity rate and Benioff strain anomalies. At the Navidad cluster, seismicity occurs in the form of swarms, with a characteristic duration of 2–7 d and location and thrust mechanisms compatible with activity on the slab interface. Conversely, we find at Vichuquén activity dominated by thrust earthquakes occurring as repeaters on the slab interface, with a slip rate of approximately ∼5.0 cm yr−1. We attribute these clusters to local features of the subducting plate: the Navidad swarms are likely driven by repeated high pore pressure transients along a pre-fractured patch of the slab, while the seismicity at the Vichuquén cluster is interpreted as the result of a subducting seamount. Both clusters have been active before and after the Mw 8.8 Maule earthquake and persisted afterwards with the seismicity decay following the Omori law. These interactions are especially evident for the Vichuquén cluster, where the seismicity rate increased considerably after the Maule earthquake and continues to be an area of clearly elevated seismicity rate compared to its surroundings.

Periodicity and Clustering in the Long‐Term Earthquake Record

AbstractElastic rebound theory forms the basis of the standard earthquake cycle model and predicts large earthquakes to recur regularly through cycles of strain accumulation and release. Yet few individual earthquake records are sufficiently long to test the theory. Here we characterize the distribution of earthquake interevent times from a global compilation of 80 long‐term records. We find that large earthquakes recur more regularly than a random Poisson process on individual fault segments. The majority of Earth's well‐studied faults shows weakly periodic and uncorrelated large earthquake recurrence, consistent with the expectations of elastic rebound theory. However, many low activity‐rate (annual occurrence rates < 2 × 10−4) faults show random or clustered earthquake recurrence, which cannot be explained by elastic rebound theory.

Generative models of simultaneously heavy-tailed distributions of interevent times on nodes and edges.

Intervals between discrete events representing human activities, as well as other types of events, often obey heavy-tailed distributions, and their impacts on collective dynamics on networks such as contagion processes have been intensively studied. The literature supports that such heavy-tailed distributions are present for interevent times associated with both individual nodes and individual edges in networks. However, the simultaneous presence of heavy-tailed distributions of interevent times for nodes and edges is a nontrivial phenomenon, and its origin has been elusive. In the present study, we propose a generative model and its variants to explain this phenomenon. We assume that each node independently transits between a high-activity and low-activity state according to a continuous-time two-state Markov process and that, for the main model, events on an edge occur at a high rate if and only if both end nodes of the edge are in the high-activity state. In other words, two nodes interact frequently only when both nodes prefer to interact with others. The model produces distributions of interevent times for both individual nodes and edges that resemble heavy-tailed distributions across some scales. It also produces positive correlation in consecutive interevent times, which is another stylized observation for empirical data of human activity. We expect that our modeling framework provides a useful benchmark for investigating dynamics on temporal networks driven by non-Poissonian event sequences.

Non-Extensive Statistical Analysis of Acoustic Emissions Recorded in Marble and Cement Mortar Specimens Under Mechanical Load Until Fracture.

Non-extensive statistical mechanics (NESM), which is a generalization of the traditional Boltzmann-Gibbs statistics, constitutes a theoretical and analytical tool for investigating the irreversible damage evolution processes and fracture mechanisms occurring when materials are subjected to mechanical loading. In this study, NESM is used for the analysis of the acoustic emission (AE) events recorded when marble and cement mortar specimens were subjected to mechanical loading until fracture. In total, AE data originating from four distinct loading protocols are presented. The cumulative distribution of inter-event times (time interval between two consecutive AE events) and the inter-event distances (three-dimensional Euclidian distance between the centers of successive AE events) were examined under the above concept and it was found that NESM is suitable to detect criticality under the terms of mechanical status of a material. This was conducted by evaluating the fitting results of the q-exponential function and the corresponding q-indices of Tsallis entropy and , along with the parameters and . Results support that for AE data recorded from marble and cement mortar specimens of this work, which is in good agreement with the conjecture previously found in seismological data and AE data recorded from Basalt specimens.

Bayesian Approach for Estimating the Distribution of Magnitudes, Interevent Times and Distances of Earthquake Sequences

A Bayesian approach has been applied to estimate the distribution of magnitudes, interevent distances and times of earthquakes occurred in 2017 in central Italy by using a small amount of random samples drawn from the distribution of the same seismic parameters for the earthquakes occurred in 2014-2016. We applied the method to the whole and aftershock-depleted seismicity by using the exponential and the normal model to fit the distributions of the seismic parameters. Our findings indicate that the exponential model fits the distributions of the seismic parameters much better than the normal model. Furthermore, in the whole seismicity case, the method requires at least 2100 to 2300 random samples to estimate the distributions of the seismic parameters of earthquakes occurred in 2017 with an estimation error less than 0.01; while in the aftershock-depleted case, a minimum number of random samples varying between 360 and 1470 occurred in 2014-2017 is required to estimate the distributions of the seismic parameters of earthquakes occurred in 2017 with an estimation error less than 0.01.

Burst-tree decomposition of time series reveals the structure of temporal correlations

Comprehensive characterization of non-Poissonian, bursty temporal patterns observed in various natural and social processes is crucial for understanding the underlying mechanisms behind such temporal patterns. Among them bursty event sequences have been studied mostly in terms of interevent times (IETs), while the higher-order correlation structure between IETs has gained very little attention due to the lack of a proper characterization method. In this paper we propose a method of representing an event sequence by a burst tree, which is then decomposed into a set of IETs and an ordinal burst tree. The ordinal burst tree exactly captures the structure of temporal correlations that is entirely missing in the analysis of IET distributions. We apply this burst-tree decomposition method to various datasets and analyze the structure of the revealed burst trees. In particular, we observe that event sequences show similar burst-tree structure, such as heavy-tailed burst-size distributions, despite of very different IET distributions. This clearly shows that the IET distributions and the burst-tree structures can be separable. The burst trees allow us to directly characterize the preferential and assortative mixing structure of bursts responsible for the higher-order temporal correlations. We also show how to use the decomposition method for the systematic investigation of such correlations captured by the burst trees in the framework of randomized reference models. Finally, we devise a simple kernel-based model for generating event sequences showing appropriate higher-order temporal correlations. Our method is a tool to make the otherwise overwhelming analysis of higher-order correlations in bursty time series tractable by turning it into the analysis of a tree structure.

Short-Term Seismic Activity in Vrancea. Inter-Event Time Distributions

Short-term seismic activity in Vrancea is analyzed by means of the inter-event magnitude distributions (distributions of the next seismic event following a conditional seismic event). This seismic activity in Vrancea is fitted by Omori-type time-power laws, with possible correlations over a range of 20−25 days, for magnitudes M < 4−5. Such temporal patterns can be employed in seismic hazard and risk estimation, providing the data set is statistically significant.

Modeling temporal networks with bursty activity patterns of nodes and links

The concept of temporal networks provides a framework to understand how the interaction between system components changes over time. In empirical communication data, we often detect non-Poissonian, so-called bursty behavior in the activity of nodes as well as in the interaction between nodes. However, such reconciliation between node burstiness and link burstiness cannot be explained if the interaction processes on different links are independent of each other. This is because the activity of a node is the superposition of the interaction processes on the links incident to the node and the superposition of independent bursty point processes is not bursty in general. Here we introduce a temporal network model based on bursty node activation and show that it leads to heavy-tailed inter-event time distributions for both node dynamics and link dynamics. Our analysis indicates that activation processes intrinsic to nodes give rise to dynamical correlations across links. Our framework offers a way to model competition and correlation between links, which is key to understanding dynamical processes in various systems.

Dynamics of calling activity to toll-free numbers in China.

The quantitative understanding of human behavior is a key issue in modern science. Recently, inhomogeneous human activities have been described by bursts (consecutive activities separated by long periods of inactivity) and characterized by fat-tailed inter-event time (interval between two activities) distributions. However, the dynamics between number of activities and activity duration are still unclear. In this study, we analyzed 133 million toll-free call records from China to study the dynamics between call frequency and call duration. We confirmed that both call frequency and call duration exhibit circadian cycles and weekly cycles. By analyzing intraday patterns of these two metrics, we found the opposite volatility and clustered distributions. Results of clustering analysis showed that calling activity to toll-free numbers can be clustered into four clusters. In the “Work” cluster, the distribution of call duration was significantly different from that in the other clusters. The corresponding time of “Work” cluster was much shorter than estimates based on common sense. Intraday patterns and clustering results showed that both call frequency and call duration are primarily related to circadian cycles, the nature of human beings, and that work is a secondary factor that affects these variables. Moreover, we found a strong positive correlation between call frequency and call duration, as well as polarization of joint probability. The polarization indicates two extremes in inhomogeneous calling activity to toll-free numbers, i.e., either people are very busy or very idle. The empirical probability of the extreme was approximately four times that of random probability. Our findings may have great usage for studying the dynamics of inhomogeneous human behavior.

Long-tailed distributions of inter-event times as mixtures of exponential distributions.

Inter-event times of various human behaviour are apparently non-Poissonian and obey long-tailed distributions as opposed to exponential distributions, which correspond to Poisson processes. It has been suggested that human individuals may switch between different states, in each of which they are regarded to generate events obeying a Poisson process. If this is the case, inter-event times should approximately obey a mixture of exponential distributions with different parameter values. In the present study, we introduce the minimum description length principle to compare mixtures of exponential distributions with different numbers of components (i.e. constituent exponential distributions). Because these distributions violate the identifiability property, one is mathematically not allowed to apply the Akaike or Bayes information criteria to their maximum-likelihood estimator to carry out model selection. We overcome this theoretical barrier by applying a minimum description principle to joint likelihoods of the data and latent variables. We show that mixtures of exponential distributions with a few components are selected, as opposed to more complex mixtures in various datasets, and that the fitting accuracy is comparable to that of state-of-the-art algorithms to fit power-law distributions to data. Our results lend support to Poissonian explanations of apparently non-Poissonian human behaviour.

On interevent time distributions of avalanche dynamics

Physical systems characterized by stick-slip dynamics often display avalanches. Regardless of the diversity of their microscopic structure, these systems are governed by a power-law distribution of avalanche size and duration. Here we focus on the interevent times between avalanches and show that, unlike their distributions of size and duration, the interevent time distributions are able to distinguish different mechanical states of the system. We use experiments on granular systems and numerical simulations of emulsions to show that systems having the same probability distribution for avalanche size and duration can have different interevent time distributions. Remarkably, these interevent time distributions look similar to those for earthquakes and, if different from an exponential, are indirect evidence of non trivial space-time correlations among avalanches. Our results therefore indicate that interevent time statistics are essential to characterise the dynamics of avalanches.

Mixed distribution model of human communication and its impacts on the spreading process

The heterogeneous inter-event time of human contacts may fundamentally alter spreading dynamics. This generally assumes that the inter-event time distribution can be depicted with power-law–like decays. In empirical human communication data, the shape of the inter-event distribution is more complicated. Particularly, both the head and the tail of the inter-event distribution deviates from the power-law–like decay. In this paper, we examine two communication databases and propose a mixed distribution to depict the inter-event distributions, which agrees better with the empirical data. We then show how the inter-event distributions shape the co-evolved SIR spreading. Especially, the SIR dynamical equations are extended to adapt to the general inter-event distribution via introducing the new infected rate. By a numerical analysis of the newly infected individuals at each time step, we illustrate how the spreading size is determined by the inter-event time distribution and the parameters of the SIR dynamic.