Inter-event Time Distribution Research Articles

Self-organization of earthquake swarms

The temporal clustering of swarm activity differs significantly from characteristics of aftershock sequences accompanying mainshocks. This is often assumed to be caused by crustal structure complexities and fluid migration. However, the underlying mechanism is not yet fully understood, especially, the processes and conditions which lead to the apparent differences between the swarm patterns and typical mainshock–aftershock sequences. In previous works, we have shown that the most conspicuous characteristics of tectonic earthquakes can be reproduced by stick-slip block models incorporating visco-elastic interactions. Now, the same model is shown to reproduce an almost periodical occurrence of earthquake swarms in the case of an enlarged postseismic response. The simulated swarms respect not only the Gutenberg-Richter law for the event sizes, they also reproduce several observations regarding their spatio-temporal patterns. In particular, the comparison with the January 1997 and the year 2000 swarm in Vogtland/NW-Bohemia shows a good agreement in the interevent-time distributions and the spatio-temporal spreading of the swarm activity. The simulated seismicity patterns result from self-organization within the swarm due to local stress transfers and viscous coupling. Consequently, the agreement with the Vogtland swarm activity do not allow any decision about the preparatory process of the swarms; in particular, the question whether the swarms are initially triggered by fluid intrusion or tectonic motion cannot be answered. However, the model investigations suggest that the process of self-organization is very important for understanding the activity patterns of earthquake swarms.

Overflow Risk Analysis for Stormwater Quality Control Basins

The operational cycle of a storm water quality control basin can be divided into the waiting period between events and the filling and draining period during an event. In this study, an inherent overflow risk is defined as the probability of having a large event exceed the basin storage capacity. Such a probability is prescribed by the basin storage capacity and the local distribution of rainfall event depths. An operational overflow risk is defined as the probability of having the basin overwhelmed by a subsequent storm event during the draining process. An operational risk is found to be dependent on watershed runoff coefficient, basin drain time, local average rainfall event depth, and average rainfall interevent time. In practice, the selection of a basin drain time is a tradeoff between the removal of suspended solids in storm water and the overflow risk. The concept of “the longer, the better “applies to the sedimentation process, but concern for the overflow risk requires that the basin drain as fast as possible. This paper presents a design method by which the overflow risk associated with a basin storage volume can be evaluated for various drain times. The mathematical models developed to describe the distribution of rainfall interevent time and the runoff capture curve provide good agreement with the long-term continuous rainfall data recorded in seven metropolitan areas in the United States. The risk-based approach developed in this study provides a quantifiable basis for making the decision on the operation of a storm water quality control basin.

Estimating the Expected Total Number of Events in a Process

We consider estimation of the cumulative mean function of a process recurring in time, such as the numbers of arrests or migrations accrued by an individual, as a function of their age. We call this theage profile of a series of events. In some situations we can expect a finite value for the total number of events experienced by an individual, for example, when the distribution of the interevent times is improper, so that the process may cease at a finite time with positive probability. We propose and analyze a new estimator, constructed from Kaplan–Meier (KM) estimators of the interevent time distributions, for such an age profile, and compare it with the Nelson–Aalen (NA) estimator of the cumulative mean function of a process. The KM estimator is proved to be uniformly consistent for the age profile if and only if follow-up in the sample is sufficient in a sense that is manifested in practice by the leveling off of the profiles at their right-side ends, and is asymptotically normally distributed around its true value under mild conditions. Simulation results suggest that it is generally a better estimator than the NA estimator for the total number of events. It also appears to be more stable when applied to real data examples. For accurate estimation, it seems to be important to select a cohort of individuals whose ages at the first event are as similar as possible. The estimators are illustrated on some time-to-arrest data.

Testing a simple stochastic model for the dynamics of waterfowl aggregations.

We consider a simple stochastic model for the dynamics of mixed-species waterfowl aggregations and describe two methods for assessing the fit of this model to field data. The model does not incorporate species-specific behavior. It assumes that all birds act independently and incorrectly predicts an exponential distribution for inter-event times. We reject this model, show that 29% of the birds move in groups of two or more birds, and demonstrate that the distribution of inter-event times between the movements of groups of birds is exponential. We find no difference in movement rates or group sizes between seasons, and no difference between groups arriving into or departing from the observed aggregations. An analysis of group composition suggests that species at low abundance behave differently than species at high abundance: birds with few conspecifics are more likely to move in mixed-species groups than birds with many conspecifics. We suggest that simple stochastic models provide a useful way to explore the dynamics of animal behavior.

Analysing the Interevent Time Distribution to Identify Seismicity Phases: A Bayesian Nonparametric Approach to the Multiple-Changepoint Problem

SUMMARY In the study of earthquakes, several aspects of the underlying physical process, such as the time non-stationarity of the process, are not yet well understood, because we lack clear indications about its evolution in time. Taking as our point of departure the theory that the seismic process evolves in phases with different activity patterns, we have attempted to identify these phases through the variations in the interevent time probability distribution within the framework of the multiple-changepoint problem. In a nonparametric Bayesian setting, the distribution under examination has been considered a random realization from a mixture of Dirichlet processes, the parameter of which is proportional to a generalized gamma distribution. In this way we could avoid making precise assumptions about the functional form of the distribution. The number and location in time of the phases are unknown and are estimated at the same time as the interevent time distributions. We have analysed the sequence of main shocks that occurred in Irpinia, a particularly active area in southern Italy: the method consistently identifies changepoints at times when strong stress releases were recorded. The estimation problem can be solved by stochastic simulation methods based on Markov chains, the implementation of which is improved, in this case, by the good analytical properties of the Dirichlet process.

Multifractal Approach to Time Clustering of Earthquakes. Application to Mt. Vesuvio Seismicity

The clustering structure of the Vesuvian earthquakes occurring is investigated by means of statistical tools: the inter-event time distribution, the running mean and the multifractal analysis. The first cannot clearly distinguish between a Poissonian process and a clustered one due to the difficulties of clearly distinguishing between an exponential distribution and a power law one. The running mean test reveals the clustering of the earthquakes, but looses information about the structure of the distribution at global scales. The multifractal approach can enlighten the clustering at small scales, while the global behaviour remains Poissonian. Subsequently the clustering of the events is interpreted in terms of diffusive processes of the stress in the earth crust.

Synthetic seismicity models of multiple interacting faults

A synthetic seismicity model for multiple, interacting faults, of any strike and dip in a three‐dimensional elastic half‐space has been developed. The ultimate purpose is to determine how hazard is modified by elastic interactions of events in complex fault systems and to examine the synthetic seismicity for unusual behavior before large events that might be observable in the real world. Each fault is subdivided into some number of equal‐sized patches, and a coefficient of static friction, random within a specified range, is assigned to each. The patch static strength is then the product of friction and differential normal stress. Stress (both shear and normal) accumulates due to some predefined tectonic driving force until one or more patches fail and slip instantaneously. The slip can be in any direction within the fault plane and its magnitude is determined by a specified fractional stress drop. This initial slip induces shear stress and strength changes on all other patches and faults and perhaps results in other patches failing. Patches can slip more than once; once having failed, their strength is reduced, resulting in an overshoot that approximates the effect of dynamic stress enhancement. An event is over when all patches are stable. Although present computational constraints do not allow models with more than about 1500 patches, or a large number of faults, experiments with simple models of two parallel strike‐slip faults (25 km length, 15 km depth, 694 m × 714 m patch size), driven at different rates and separated by 10 or 15 km, give some indicative results. A range of friction between 0.1 and 0.2, with a stress drop of 10% (about 3 MPa or 30 bars), produces a b value (500,000 events) close to 1. The models generate a distinct class of large “characteristic” events that rupture the full fault plane. The distribution of interevent times for the background seismicity is very similar to that for a Poisson process, while the characteristic events are quasiperiodic. Nearly half the larger events directly induce some activity on the other fault. The effect of a characteristic event on the occurrence of another characteristic event on the opposite fault is small and depends on the separation: at 10 km distance other events are retarded while at 15 km distance they are advanced. At 10‐km separation the probability of a pair of characteristic events within 1 year is reduced to 0.3% from 1.7% for the case with no interaction. Consideration of interactions between slip on the faults and the driving mechanism, as well as higher stress drops and more overshoot, would increase the magnitude of the interaction effects.

A real-time simple analyser for point processes

So far, the main limit of simple statistical analysers for point processes has been their inability to perform in real time. In this paper we present a device based on the combined use of a first in, first out memory, a commercial digital signal processor, the evaluation board supplied with it by the manufacturer plus a standard personal computer. By processing a sequence of pulses while they arrive, the instrument outputs (and continually updates) a function that may be selected in advance among those commonly used in the theory of point processes: inter-event time, waiting time, correlation function or counting distributions. This real-time device is able to compute the correlation function in the case of pulse flows with intensities as high as 105 pulses per second; for other statistical functions the acceptable rate may be even higher, of the order of 3*105 pulses per second. The instrument has been tested both on deterministic signals and on laser scattering from rough surfaces; it is currently employed in zero-crossing spectral investigations, whose preliminary results are concisely reported.

Improvements to the modelling of British rainfall using a modified Random Parameter Bartlett-Lewis Rectangular Pulse Model

The Random Parameter Bartlett-Lewis Rectangular Pulse Model (RPBLRPM) was shown in a previous paper to provide generally good results when applied to a 38.5 year record of hourly rainfall from the Elmdon raingauge, Birmingham, UK: second-order properties of the depth and the interevent time distributions and the proportion of dry periods for different time-scales were on the whole well reproduced by the optimized parameters. However, two main problems arose, namely an overestimation of the autocorrelations for daily data in particular, and relatively poor performance in reproducing the extreme values of the depth distribution. The first of these problems is here addressed by superposition of a jitter process on the rectangular pulse, thus providing a more realistic representation of the continuous time process. Applying the jitter to each rainfall cell appears more effective than applying it to the instantaneous rainfall depth and enables a more satisfactory reproduction of the autocorrelations. The use of a gamma variable to replace the exponential distribution of the cell intensity provides a good reproduction of the extreme rainfall depths at different time-scales. Moreover, the values of the parameters providing an optimal reproduction of the extreme values of the depth distribution are in most cases close to those obtained by optimizing on the interevent time features, which provides a method for estimation and ensures that the two objectives do not conflict.

Strategies in strong earthquake prediction

Abstract The prediction problem of a stochastic point process in terms of loss function γ is considered. Losses depend on prediction errors: n , the fraction of failures-to-predict; and τ, the fraction of alarm time. The structure of γ-optimal prediction strategies for some functions γ is described. This allows analysis of the announced prediction of strong earthquakes in the southern part of the San Andreas fault. We consider ( n , τ) as a property of prediction strategy and set problems on the description of ‘prediction capability’ and comparison of prediction algorithms. In addition, the distribution of empirical prediction errors for the renewal process model and the type of attenuation of inter-event time distribution, are considered.

Non‐parametric estimation for renewal processes from event count data

AbstractA procedure is developed for estimating the interevent time distribution of a stationary renewal process that uses only data in the form of counts (the number of events occurring in each of several adjacent disjoint intervals) that are predominantly zeros and ones. Some properties of the procedure are derived, and the procedure's performance is assessed via Monte Carlo simulation.

A new estimation approach based on periodic inspection

The authors present a novel approach for estimating the failure-time distribution using count data, the number of failures per interval, from periodic inspection of a standby redundant system. The procedure is based on a result for stationary renewal processes that related the forward recurrence time to the interevent time distribution. The procedure performed well in a simulation study for five Weibull distributions and is general enough to work for any failure-time distribution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Statistical properties of cosmic-ray showers at ground level determined from photomultiplier-tube background registrations.

The fluctuations of cosmic-ray particles resulting from extensive air showers at ground level are well described by the two-parameter Poisson-driven Yule-Furry and negative-binomial counting probability distributions. The background signal from a single photomultiplier tube has been used to experimentally verify these results with remarkable precision, in spite of the simplicity of the underlying pure-birth stochastic process. Counting distributions from three different photomultiplier-tube detectors operated in the dark are presented, together with the theoretical predictions. Probability distributions of interevent times have also been obtained and these are found to be consistent with the observed clustering properties at the detector output. Our approach is expected to be of importance in quantum optics where cosmic-ray-shower particles can pose a significant limitation on the detection of squeezed light.

Advances in Geophysics

Observations related to tsunami catalogs are reviewed and described in a phenomenological framework. An examination of scaling relationships between earthquake size (as expressed by scalar seismic moment and mean slip) and tsunami size (as expressed by mean and maximum local run-up and maximum far-field amplitude) indicates that scaling is significant at the 95% confidence level, although there is uncertainty in how well earthquake size can predict tsunami size (Rw 0.4–0.6). In examining tsunami Advances in Geophysics, Volume 53 2012, Published by Elsevier Inc. ISSN 0065-2687, Doi: http://dx.doi.org/10.1016/B978-0-12-380938-4.00002-1 35 j event statistics, current methods used to estimate the size distribution of earthquakes and landslides and the inter-event time distribution of earthquakes are first reviewed. These methods are adapted to estimate the size and inter-event distribution of tsunamis at a particular recording station. Using a modified Pareto size distribution, the best-fit power-law exponents of tsunamis recorded at nine Pacific tide-gauge stations exhibit marked variation, in contrast to the approximately constant power-law exponent for inter-plate thrust earthquakes. With regard to the inter-event time distribution, significant temporal clustering of tsunami sources is demonstrated. For tsunami sources occurring in close proximity to other sources in both space and time, a physical triggering mechanism, such as static stress transfer, is a likely cause for the anomalous clustering. Mechanisms of earthquake-to-earthquake and earthquake-to-landslide triggering are reviewed. Finally, a modification of statistical branching models developed for earthquake triggering is introduced to describe triggering among tsunami sources.