Inter-event Time Distribution Research Articles

Empirical analysis of individual popularity and activity on an online music service system

Quantitative understanding of human behaviors supplies basic comprehension of the dynamics of many socio-economic systems. Based on the log data of an online music service system, we investigate the statistical characteristics of individual activity and popularity, and find that the distributions of both of them follow a stretched exponential form which interpolates between exponential and power law distribution. We also study the human dynamics on the online system and find that the distribution of interevent time between two consecutive listenings of music shows the fat tail feature. Besides, with the reduction of user activity the fat tail becomes more and more irregular, indicating different behavior patterns for users with diverse activities. The research results may shed some light on the in-depth understanding of collective behaviors in socio-economic systems.

Solution of the nonlinear theory and tests of earthquake recurrence times

We develop an efficient numerical scheme to solve accurately the set of nonlinear integral equations derived previously in [A. Saichev and D. Sornette, J. Geophys. Res. 112, B04313 (2007)], which describes the distribution of interevent times in the framework of a general model of earthquake clustering with long memory. Detailed comparisons between the linear and nonlinear versions of the theory and direct synthetic catalogs show that the nonlinear theory provides an excellent fit to the synthetic catalogs, while there are significant biases resulting from the use of the linear approximation. We then address the suggestions proposed by some authors to use the empirical distribution of interevent times to obtain a better determination of the so-called clustering parameter. Our theory and tests against synthetic and empirical catalogs find a rather dramatic lack of power for the distribution of interevent times to distinguish between quite different sets of parameters, casting doubt on the usefulness of this statistic for the specific purpose of identifying the clustering parameter.

Seismic Cycles and Earthquake Predictability on East Pacific Rise Transform Faults

The concept of a seismic cycle, where the stress on a fault repeatedly builds up over a long period of time and then is rapidly released in a large earthquake, influences studies of both the basic physics of faulting and applied research aimed at estimating earthquake hazards. This hypothesis suggests that large earthquakes might be quasi periodic and that the probability of a particular portion of a fault rupturing twice in quick succession should be low. However, this basic hypothesis has been difficult to verify owing to the long repeat times of the largest earthquakes on most faults. East Pacific Rise (EPR) transform faults are an advantageous location to evaluate the seismic cycle hypothesis owing to their fast slip rates and the moderate size (∼ M w 6) of their largest earthquakes. Using surface-wave based determinations of the relative separations between earthquake centroids, I document 16 pairs of M w≥5.5 events that had overlapping ruptures. The distribution of interevent times for these pairs is tightly clustered around 5 yr (with a coefficient of variation ∼0.2) indicating that quasi periodicity may be prevalent for the largest events on these faults. Moreover, I find no pairs of overlapping M w 5.5–6.2 earthquakes separated by less than 50 cm of elapsed plate motion, indicating that the two basic features of the seismic cycle hypothesis are evident in the timing of large EPR transform mainshocks. I have also confirmed earlier results demonstrating a high degree of short-term predictability of EPR mainshocks by combining teleseismic and hydroacoustic earthquake catalogs. Thus, there appears to be a high degree of both short and long-term predictability on EPR transforms.

Complexity in Sequences of Solar Flares and Earthquakes

In this paper the statistical properties of solar flares and earthquakes are compared by analyzing the energy distributions, the time series of energies and interevent times, and, above all, the distributions of interevent times per se. It is shown that the two phenomena have different statistics of scaling, and even the same phenomenon, when observed in different periods or at different locations, is characterized by different statistics that cannot be uniformly rescaled onto a single, universal curve. The results indicate apparent complexity of impulsive energy release processes, which neither follow a common behaviour nor could be attributed to a universal physical mechanism.

Distribution of tsunami interevent times

The distribution of tsunami interevent times is analyzed using global and site‐specific (Hilo, Hawaii) tsunami catalogs. An empirical probability density distribution is determined by binning the observed interevent times during a period in which the observation rate is approximately constant. The empirical distributions for both catalogs exhibit non‐Poissonian behavior in which there is an abundance of short interevent times compared to an exponential distribution. Two types of statistical distributions are used to model this clustering behavior: (1) long‐term clustering described by a universal scaling law, and (2) Omori law decay of aftershocks and triggered sources. The empirical and theoretical distributions all imply an increased hazard rate after a tsunami, followed by a gradual decrease with time approaching a constant hazard rate. Examination of tsunami sources suggests that many of the short interevent times are caused by triggered earthquakes, though the triggered events are not necessarily on the same fault.

Effects of amphetamine on differential reinforcement of low rates of responding

A multiple differential reinforcement of low rate 8 s discrete-trial, differential reinforcement of low rate 8 s free-operant procedure was used to investigate the effects of D-amphetamine on interval timing in pigeons. On the discrete trial differential reinforcement of low rate early responses terminated the trial; on the free-operant differential reinforcement of low rate early responses reset the reinforcement timer but had no other effects. Frequency distributions of log interevent times for responses following reinforcement on both the free-operant and discrete-trial components showed a single-peaked distribution. The interevent time distribution for responses preceded by unreinforced responses on the free-operant differential reinforcement of low rate showed two peaks. The pigeons were administered two doses of D-amphetamine for 20 successive sessions. The means of interevent time distributions did not change under drug for either condition but widths of the distributions increased in both conditions and for both types of interevent times in the free-operant differential reinforcement of low rate. In the free-operant condition the mean interevent time for responses preceded by prior responses declined significantly with drug owing to a reduction in the proportion of responses in the shorter interevent time distribution. In effect, response bouts were shortened by drug administration. The rate-dependent hypothesis well described the effect of amphetamine on responses in the free-operant procedure. These results are discussed in the context of amphetamine's effects on the temporal structure of behavior.

Dynamical Scaling in Branching Models for Seismicity

We propose a branching process based on a dynamical scaling hypothesis relating time and mass. In the context of earthquake occurrence, we show that experimental power laws in size and time distribution naturally originate solely from this scaling hypothesis. We present a numerical protocol able to generate a synthetic catalog with an arbitrary large number of events. The numerical data reproduce the hierarchical organization in time and magnitude of experimental interevent time distribution.

Analysis of Temporal Structures of Seismic Events on Different Scale Levels

A statistical model for describing the energy scaling of the distribution of inter-event times is described. By considering the diverse region seismicity (natural and induced) on different scale (energy/magnitude) levels the self-similarity of the distribution has been determined. A comparison between the distribution of inter-event times on different scale levels and the most popular distributions of reliability theory has been carried out. The distribution of inter-event times for different scale levels is well approximated by the Weibull distribution. The Weibull distribution, with parameters which obey the scaling model and the Gutenberg-Richter law, has been tested.

“Universal” Distribution of Interearthquake Times Explained

We propose a simple theory for the "universal" scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks, and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of interevent times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforce the view that triggering earthquakes by other earthquakes is a key physical mechanism to understand seismicity.

Pure Applied Geophysics 162 (2005), 1135–1150. Interevent Time Distribution in Seismicity: A Theoretical Approach by G. Molchan

Pure Applied Geophysics 162 (2005), 1135–1150. Interevent Time Distribution in Seismicity: A Theoretical Approach by G. Molchan

Physics of the Omori law: Inferences from interevent time distributions and pore pressure diffusion modeling

Empirical laws and statistics of earthquakes are valuable as a basis for a better understanding of the earthquake cycle. In this paper we focus on the postseismic phase and the physics of aftershock sequences. Using interevent time distributions for a catalogue of Icelandic seismicity, we infer that the parameter C 2 in the Omori law, often considered to represent incomplete detection of aftershocks, is at least in part related to the physics of the earthquake process. We investigate the role of postseismic pore pressure diffusion after two Icelandic earthquakes on the rate of aftershocks and what we can infer about the physical meaning of C 2 from the diffusion process. Using the Mohr–Coulomb failure criterion we obtain a rate of triggered points in our diffusion model that agrees with the modified Omori law, with a value of C 2 that is consistent with data. Our pore pressure diffusion model suggests that C 2 is related to the process of reducing high pore pressure gradients existing across a fault zone at short times after a main shock.

Impact of memory on human dynamics

Our experience of web access slowing down is a consequence of the aggregated web access pattern of web users. This is just one example among several human-oriented services which are strongly affected by human activity patterns. Recent empirical evidence is indicating that human activity patterns are characterized by power-law distributions of inter-event times, where large fluctuations rather than regularity is the common case. I show that this temporal heterogeneity can be explained by two mechanisms: (i) humans have some perception of their past activity rate and (ii) based on that they react by accelerating or reducing their activity rate. Using these two mechanisms I explain the inter-event time statistics of Darwin's and Einstein's correspondence and the email activity within an university environment. Moreover, they are typical examples of the accelerating and reducing class, respectively. These results are relevant to the system design of human-oriented services.

Earthquake activity related to seismic cycles in a model for a heterogeneous strike-slip fault

We investigate the evolution of seismicity within large earthquake cycles in a model of a discrete strike-slip fault in elastic solid. The model dynamics is governed by realistic boundary conditions consisting of constant velocity motion of regions around the fault, static/kinetic friction and dislocation creep along the fault, and 3D elastic stress transfer. The fault consists of brittle parts which fail during earthquakes and undergo small creep deformation between events, and aseismic creep cells which are characterized by high ongoing creep motion. This mixture of brittle and creep cells is found to generate realistic aftershock sequences which follow the modified Omori law and scale with the mainshock size. Furthermore, we find that the distribution of interevent times of the simulated earthquakes is in good agreement with observations. The temporal occurrence, however, is magnitude-dependent; in particular, the small events are clustered in time, whereas the largest earthquakes occur quasiperiodically. Averaging the seismicity before several large earthquakes, we observe an increase of activity and a broadening scaling range of magnitudes when the time of the next mainshock is approached. These results are characteristics of a critical point behavior. The presence of critical point dynamics is further supported by the evolution of the stress field in the model, which is compatible with the observation of accelerating moment release in natural fault systems.

Lindmanet al.Reply:

In seismology, an important goal is to attain a better understanding of the earthquake process. In this study of the physics of aftershock generation, I couple statistical analysis with modelling of physical processes in the postseismic period. I present a theoretical formulation for the distribution of interevent times for aftershock sequences obeying the empirically well established Omori law. As opposed to claims by other authors, this work demonstrates that the duration of the time interval between two successive earthquakes cannot be used to identify whether or not they belong to the same aftershock sequence or occur as a result of the same underlying process. This implies that a proper understanding of earthquake interevent time distributions is necessary before conclusions regarding the physics of the earthquake process are drawn.In a discussion of self-organised criticality (SOC) in relation to empirical laws in seismology, I find that Omori's law for aftershocks cannot be used as evidence for the theory of SOC. Instead, I consider that the occurrence of aftershocks in accordance with Omori's law is a result of a physical process that can be modelled and understood.I analyse characteristic features in the spatiotemporal distribution of aftershocks in the south Iceland seismic zone, following the two M6.5 June 2000 earthquakes and a M4.5 earthquake in September, 1999. These features include an initially constant aftershock rate, whose duration is larger following a larger main shock, and a subsequent power law decay that is interrupted by distinct and temporary deviations in terms of rate increases and decreases. Based on pore pressure diffusion modelling, I interpret these features in terms of main shock initiated diffusion processes. I conclude that thorough data analysis and physics-based modelling are essential components in attempts to improve our understanding of processes governing the occurrence of earthquakes.

A simulation model for the study of the within-animal infection dynamics of E. coli O157

Escherichia coli 0157 can cause serious illness, even death, in humans. There is some consensus that the main reservoirs of this harmful bacterium are the rumens and intestines of cattle. Hence, a stochastic model of the bovine gut was developed to investigate the in vivo population dynamics of E. coli O157. Because bacterial numbers can reach minimal levels, a stochastic system was considered, with a birth–death process being used to represent bacterial growth and decay dynamics throughout the gastrointestinal tract. Reinfection through ingestion of bacteria present in the environment was allowed to occur and the required clustered distribution of inter-event times was implemented through the use of a random hazard doubly stochastic Poisson process. Due to the inclusion of multiple compartments, a feedback mechanism and an interest in the non-equilibrium dynamics of the process, it was not possible to obtain an analytical representation of the process and therefore, a simulation study was used to obtain results. The within-animal model can be used to explore the efficacy of control measures which act at an individual animal level.

Estimating Background Activity Based on Interevent-Time Distribution

The statistics of time delays between successive earthquakes has re- cently been claimed to be universal and to show the existence of clustering beyond the duration of aftershock bursts. We demonstrate that these claims are unjustified. Stochastic simulations with Poissonian background activity and triggered Omori- type aftershock sequences are shown to reproduce the interevent-time distributions observed on different spatial and magnitude scales in California. Thus the empirical distribution can be explained without any additional long-term clustering. Further- more, we find that the shape of the interevent-time distribution, which can be ap- proximated by the gamma distribution, is determined by the percentage of main- shocks in the catalog. This percentage can be calculated by the mean and variance of the interevent times and varies between 5% and 90% for different regions in California. Our investigation of stochastic simulations indicates that the interevent- time distribution provides a nonparametric reconstruction of the mainshock magnitude-frequency distribution that is superior to standard declustering algorithm.

Detecting fluid signals in seismicity data through statistical earthquake modeling

According to the well‐known Coulomb failure criterion the variation of either stress or pore pressure can result in earthquake rupture. Aftershock sequences characterized by the Omori law are often assumed to be the consequence of varying stress, whereas earthquake swarms are thought to be triggered by fluid intrusions. The role of stress triggering can be analyzed by modeling solely three‐dimensional (3‐D) elastic stress changes in the crust, but fluid flows which initiate seismicity cannot be investigated without considering complex seismicity patterns resulting from both pore pressure variations and earthquake‐connected stress field changes. We show that the epidemic‐type aftershock sequence (ETAS) model is an appropriate tool to extract the primary fluid signal from such complex seismicity patterns. We analyze a large earthquake swarm that occurred in 2000 in Vogtland/NW Bohemia, central Europe. By fitting the stochastic ETAS model, we find that stress triggering is dominant in creating the observed seismicity patterns and explains the observed fractal interevent time distribution. External forcing, identified with pore pressure changes due to fluid intrusion, is found to directly trigger only a few percent of the total activity. However, temporal deconvolution indicates that a pronounced fluid signal initiated the swarm. These results are confirmed by our analogous investigation of model simulations in which earthquakes are triggered by fluid intrusion as well as stress transfers on a fault plane embedded in a 3‐D elastic half‐space. The deconvolution procedure based on the ETAS model is able to reveal the underlying pore pressure variations.

Viscoelasticity, Postseismic Slip, Fault Interactions, and the Recurrence of Large Earthquakes

The Brownian Passage Time (bpt) model for earthquake recurrence is modified to include transient deformation due to either viscoelasticity or deep postseismic slip. Both of these processes act to increase the rate of loading on the seismogenic fault for some time after a large event. To approximate these effects, a decaying exponential term is added to the bpt model’s uniform loading term. The resulting interevent time distributions remain approximately lognormal, but the balance between the level of noise (e.g., unknown fault interactions) and the coefficient of variability of the interevent time distribution changes depending on the shape of the loading function. For a given level of noise in the loading process, transient deformation has the effect of increasing the coefficient of variability of earthquake interevent times. Conversely, the level of noise needed to achieve a given level of variability is reduced when transient deformation is included. Using less noise would then increase the effect of known fault interactions modeled as stress or strain steps because they would be larger with respect to the noise. If we only seek to estimate the shape of the interevent time distribution from observed earthquake occurrences, then the use of a transient deformation model will not dramatically change the results of a probability study because a similar shaped distribution can be achieved with either uniform or transient loading functions. However, if the goal is to estimate earthquake probabilities based on our increasing understanding of the seismogenic process, including earthquake interactions, then including transient deformation is important to obtain accurate results. For example, a loading curve based on the 1906 earthquake, paleoseismic observations of prior events, and observations of recent deformation in the San Francisco Bay region produces a 40% greater variability in earthquake recurrence than a uniform loading model with the same noise level.

Earthquake statistics in a Block Slider Model and a fully dynamic Fault Model

Abstract. We examine the event statistics obtained from two differing simplified models for earthquake faults. The first model is a reproduction of the Block-Slider model of Carlson et al. (1991), a model often employed in seismicity studies. The second model is an elastodynamic fault model based upon the Lattice Solid Model (LSM) of Mora and Place (1994). We performed simulations in which the fault length was varied in each model and generated synthetic catalogs of event sizes and times. From these catalogs, we constructed interval event size distributions and inter-event time distributions. The larger, localised events in the Block-Slider model displayed the same scaling behaviour as events in the LSM however the distribution of inter-event times was markedly different. The analysis of both event size and inter-event time statistics is an effective method for comparative studies of differing simplified models for earthquake faults.

Time-clustering behavior of spreading-center seismicity between 15 and 35°N on the Mid-Atlantic Ridge: observations from hydroacoustic monitoring

An earthquake catalog derived from the detection of seismically-generated T-waves is used to study the time-clustering behavior of moderate-size (≳3.0 M) earthquakes between 15 and 35°N along the Mid-Atlantic Ridge (MAR). Within this region, the distribution of inter-event times is consistent with a non-periodic, non-random, clustered process. The highest degrees of clustering are associated temporally with large mainshock-aftershock sequences; however, some swarm-like activity also is evident. Temporal fluctuations characterized by a power spectral density P( f) that decays as 1/ f α are present within the time sequence, with α ranging from 0.12 to 0.55 for different regions of the spreading axis. This behavior is negligible at time scales less than ∼5×10 3 s, and earthquake occurrence becomes less clustered (smaller α) as increasing size thresholds are applied to the catalog. A power-law size-frequency scaling for Mid-Atlantic Ridge earthquakes also can be demonstrated using the distribution of acoustic magnitudes, or source levels. Although fractal seismic behavior has been linked to the structure of the underlying fault population in other environments, power–law fault size distributions have not been observed widely in the mid-ocean ridge setting.