"Universal" distribution of interearthquake times explained

Connection between Variations of the Probability Distribution of the Recurrence Time and Phases of the Seismic Activity

The probability distribution of the interevent time between two successive earthquakes has been the subject of numerous studies for its key role in seismic hazard assessment. In recent decades, many distributions have been considered, and there has been a long debate about the possible universality of the shape of this distribution when the interevent times are properly rescaled. In this work, we aim to discover if there is a link between the different phases of a seismic cycle and the variations in the distribution that best fits the interevent times. To do this, we consider the seismic activity related to the Mw 6.1 L'Aquila earthquake that occurred on 6 April 2009 in central Italy by analyzing the sequence of events recorded from April 2005 to July 2009, and then the seismic activity linked to the sequence of the Amatrice-Norcia earthquakes of Mw 6 and 6.5, respectively, and recorded in the period from January 2009 to June 2018. We take into account some of the most studied distributions in the literature: q-exponential, q-generalized gamma, gamma and exponential distributions and, according to the Bayesian paradigm, we compare the value of their posterior marginal likelihood in shifting time windows with a fixed number of data. The results suggest that the distribution providing the best performance changes over time and its variations may be associated with different phases of the seismic crisis.

Modeling Exact Frequency-Energy Distribution for Quakes by a Probabilistic Cellular Automaton

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

Seismic magnitude clustering is prevalent in field and laboratory catalogs

Clustering of earthquake magnitudes is still actively debated, compared to well-established spatial and temporal clustering. Magnitude clustering is not currently implemented in earthquake forecasting but would be important if larger magnitude events are more likely to be followed by similar sized events. Here we show statistically significant magnitude clustering present in many different field and laboratory catalogs at a wide range of spatial scales (mm to 1000 km). It is universal in field catalogs across fault types and tectonic/induced settings, while laboratory results are unaffected by loading protocol or rock types and show temporal stability. The absence of clustering can be imposed by a global tensile stress, although clustering still occurs when isolating to triggered event pairs or spatial patches where shear stress dominates. Magnitude clustering is most prominent at short time and distance scales and modeling indicates >20% repeating magnitudes in some cases, implying it can help to narrow physical mechanisms for seismogenesis.

Earthquake magnitude distribution and aftershocks: A statistical geometry explanation

The emergence of a power-law distribution for the energy released during an earthquake is investigated in several models. Generic features are identified which are based on the self-affine behavior of the stress field prior to an event. This field behaves at large scale as a random trajectory in one dimension of space and a random surface in two dimensions. Using concepts of statistical mechanics and results on the properties of these random objects, several predictions are obtained and verified, in particular the value of the power-law exponent of the earthquake energy distribution (the Gutenberg-Richter law) as well as a mechanism for the existence of aftershocks after a large earthquake (the Omori law).

Hidden jerk in universal creep and aftershocks

Most materials exhibit creep memory under the action of a constant load. The memory behavior is governed by Andrade's creep law, which also has an inherent connection with the Omori-Utsu law of earthquake aftershocks. Both empirical laws lack a deterministic interpretation. Coincidentally, the Andrade law is similar to the time-varying part of the creep compliance of the fractional dashpot in anomalous viscoelastic modeling. Consequently, fractional derivatives are invoked, but since they lack a physical interpretation, the physical parameters of the two laws extracted from curve fit lack confidence. In this Letter, we establish an analogous linear physical mechanism that underlies both laws and relates its parameters with the material's macroscopic properties. Surprisingly, the explanation does not require the property of viscosity. Instead, it necessitates the existence of a rheological property that relates strain with the first order time derivative of stress, which involves jerk. Further, we justify the constant quality factor model of acoustic attenuation in complex media. The obtained results are validated in light of the established observations.

Analyzing the Correlations and the Statistical Distribution of Moderate to Large Earthquakes Interevent Times in Greece

Seismic temporal properties constitute a fundamental component in developing probabilistic models for earthquake occurrence in a specific area. Earthquake occurrence is neither periodic nor completely random but often accrues into bursts in both short- and long-term time scales, and involves a complex summation of triggered and independent events (ΔT). This behavior underlines the impact of the correlations on many potential applications such as the stochastic point process for the earthquake interevent times. In this respect, we intend firstly to determine the appropriate magnitude thresholds, Mthr, indicating the temporal crossover between correlated and statistically independent earthquakes in each 1 of the 10 distinctive sub-areas of the Aegean region. The second goal is the investigation of the statistical distribution that optimally fits the interevent times’ data for earthquakes with M≥Mthr after evaluating the Gamma, Weibull, Lognormal and Exponential distributions performance. Results concerning the correlations analysis evidenced that the temporal crossover of the earthquake interevent time data ranges from Mthr≥ 4.7 up to Mthr≥ 5.1 among the 10 sub-areas. The distribution fitting and comparison reveals that the Gamma distribution outperforms the other three distributions for all the data sets. This finding indicates a burst or clustering behavior in the earthquake interevent times, in which each earthquake occurrence depends upon only the occurrence time of the last one and not from the full seismic history.

Quantifying the Endogeneity in Online Donations

Charitable crowdfunding provides a new channel for people and families suffering from unforeseen events, such as accidents, severe illness, and so on, to seek help from the public. Thus, finding the key determinants which drive the fundraising process of crowdfunding campaigns is of great importance, especially for those suffering. With a unique data set containing 210,907 crowdfunding projects covering a period from October 2015 to June 2020, from a famous charitable crowdfunding platform, specifically Qingsong Chou, we will reveal how many online donations are due to endogeneity, referring to the positive feedback process of attracting more people to donate through broadcasting campaigns in social networks by donors. For this aim, we calibrate three different Hawkes processes to the event data of online donations for each crowdfunding campaign on each day, which allows us to estimate the branching ratio, a measure of endogeneity. It is found that the online fundraising process works in a sub-critical state and nearly 70-90% of the online donations are endogenous. Furthermore, even though the fundraising amount, number of donations, and number of donors decrease rapidly after the crowdfunding project is created, the measure of endogeneity remains stable during the entire lifetime of crowdfunding projects. Our results not only deepen our understanding of online fundraising dynamics but also provide a quantitative framework to disentangle the endogenous and exogenous dynamics in complex systems.

Continuous Time Random Walk with Correlated Waiting Times. The Crucial Role of Inter-Trade Times in Volatility Clustering

In many physical, social, and economic phenomena, we observe changes in a studied quantity only in discrete, irregularly distributed points in time. The stochastic process usually applied to describe this kind of variable is the continuous-time random walk (CTRW). Despite the popularity of these types of stochastic processes and strong empirical motivation, models with a long-term memory within the sequence of time intervals between observations are rare in the physics literature. Here, we fill this gap by introducing a new family of CTRWs. The memory is introduced to the model by assuming that many consecutive time intervals can be the same. Surprisingly, in this process we can observe a slowly decaying nonlinear autocorrelation function without a fat-tailed distribution of time intervals. Our model, applied to high-frequency stock market data, can successfully describe the slope of decay of the nonlinear autocorrelation function of stock market returns. We achieve this result without imposing any dependence between consecutive price changes. This proves the crucial role of inter-event times in the volatility clustering phenomenon observed in all stock markets.

Is Seismicity Operating at a Critical Point?

Seismicity and faulting within the Earth's crust are characterized by many scaling laws that are usually interpreted as qualifying the existence of underlying physical mechanisms associated with some kind of criticality in the sense of phase transitions. Using an augmented epidemic-type aftershock sequence (ETAS) model that accounts for the spatial variability of the background rates μ(x,y), we present a direct quantitative test of criticality. We calibrate the model to the ANSS catalog of the entire globe, the region around California, and the Geonet catalog for the region around New Zealand using an extended expectation-maximization (EM) algorithm including the determination of μ(x,y). We demonstrate that the criticality reported in previous studies is spurious and can be attributed to a systematic upward bias in the calibration of the branching ratio of the ETAS model, when not accounting correctly for spatial variability. We validate the version of the ETAS model that possesses a space varying background rate μ(x,y) by performing pseudoprospective forecasting tests. The noncriticality of seismicity has major implications for the prediction of large events.

Statistical physics approaches to the complex Earth system

Global warming, extreme climate events, earthquakes and their accompanying socioeconomic disasters pose significant risks to humanity. Yet due to the nonlinear feedbacks, multiple interactions and complex structures of the Earth system, the understanding and, in particular, the prediction of such disruptive events represent formidable challenges to both scientific and policy communities. During the past years, the emergence and evolution of Earth system science has attracted much attention and produced new concepts and frameworks. Especially, novel statistical physics and complex networks-based techniques have been developed and implemented to substantially advance our knowledge of the Earth system, including climate extreme events, earthquakes and geological relief features, leading to substantially improved predictive performances. We present here a comprehensive review on the recent scientific progress in the development and application of how combined statistical physics and complex systems science approaches such as critical phenomena, network theory, percolation, tipping points analysis, and entropy can be applied to complex Earth systems. Notably, these integrating tools and approaches provide new insights and perspectives for understanding the dynamics of the Earth systems. The overall aim of this review is to offer readers the knowledge on how statistical physics concepts and theories can be useful in the field of Earth system science.

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.

Quantifying uncertainty in a predictive model for popularity dynamics

The Hawkes process has garnered attention in recent years for its suitability to describe the behavior of online information cascades. Here we present a fully tractable approach to analytically describe the distribution of the number of events in a Hawkes process, which, in contrast to purely empirical studies or simulation-based models, enables the effect of process parameters on cascade dynamics to be analyzed. We show that the presented theory also allows predictions regarding the future distribution of events after a given number of events have been observed during a time window. Our results are derived through a differential-equation approach to attain the governing equations of a general branching process. We confirm our theoretical findings through extensive simulations of such processes. This work provides the basis for more complete analyses of the self-exciting processes that govern the spreading of information through many communication platforms, including the potential to predict cascade dynamics within confidence limits.

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.

Possible origin of memory in earthquakes: Real catalogs and an epidemic-type aftershock sequence model

Earthquakes are one of the most devastating natural disasters that plague society. Skilled, reliable earthquake forecasting remains the ultimate goal for seismologists. Using the detrended fluctuation analysis (DFA) and conditional probability (CP) methods, we find that memory exists not only in interoccurrence seismic records but also in released energy as well as in the series of the number of events per unit time. Analysis of a standard epidemic-type aftershock sequences (ETAS) earthquake model indicates that the empirically observed earthquake memory can be reproduced only for a narrow range of the model's parameters. This finding therefore provides tight constraints on the model's parameters and can serve as a testbed for existing earthquake forecasting models. Furthermore, we show that by implementing DFA and CP results, the ETAS model can significantly improve the short-term forecasting rate for the real (Italian) earthquake catalog.

Development and regression of a large fluctuation

We study the evolution leading to (or regressing from) a large fluctuation in a statistical mechanical system. We introduce and study analytically a simple model of many identically and independently distributed microscopic variables n_{m} (m=1,M) evolving by means of a master equation. We show that the process producing a nontypical fluctuation with a value of N=∑_{m=1}^{M}n_{m} well above the average 〈N〉 is slow. Such process is characterized by the power-law growth of the largest possible observable value of N at a given time t. We find similar features also for the reverse process of the regression from a rare state with N≫〈N〉 to a typical one with N≃〈N〉.

Effects of burstiness on the air transportation system

The effects of burstiness in complex networks have received considerable attention. In particular, the effects on temporal distance and delays in the air transportation system are significant owing to their huge impact on our society. Therefore, in this paper, the temporal distance of empirical U.S. flight schedule data is compared with that of regularized data without burstiness to analyze the effects of burstiness. The temporal distance is calculated by a graph analysis method considering flight delays, missed connections, flight cancellations, and congestion. In addition, we propose two temporal distance indexes based on passengers' behavior to quantify the effects. As a result, we find that burstiness reduces both the scheduled and the actual temporal distances for business travelers, while delays caused by missed connections and congestion are increased. We also find that the decrease of the scheduled temporal distance by burstiness is offset by an increase of the delays for leisure passengers. Moreover, we discover that the positive effect of burstiness is lost when flight schedules are overcrowded.

Generalized statistical mechanics approaches to earthquakes and tectonics

Despite the extreme complexity that characterizes the mechanism of the earthquake generation process, simple empirical scaling relations apply to the collective properties of earthquakes and faults in a variety of tectonic environments and scales. The physical characterization of those properties and the scaling relations that describe them attract a wide scientific interest and are incorporated in the probabilistic forecasting of seismicity in local, regional and planetary scales. Considerable progress has been made in the analysis of the statistical mechanics of earthquakes, which, based on the principle of entropy, can provide a physical rationale to the macroscopic properties frequently observed. The scale-invariant properties, the (multi) fractal structures and the long-range interactions that have been found to characterize fault and earthquake populations have recently led to the consideration of non-extensive statistical mechanics (NESM) as a consistent statistical mechanics framework for the description of seismicity. The consistency between NESM and observations has been demonstrated in a series of publications on seismicity, faulting, rock physics and other fields of geosciences. The aim of this review is to present in a concise manner the fundamental macroscopic properties of earthquakes and faulting and how these can be derived by using the notions of statistical mechanics and NESM, providing further insights into earthquake physics and fault growth processes.

Unfolding large-scale online collaborative human dynamics

Large-scale interacting human activities underlie all social and economic phenomena, but quantitative understanding of regular patterns and mechanism is very challenging and still rare. Self-organized online collaborative activities with a precise record of event timing provide unprecedented opportunity. Our empirical analysis of the history of millions of updates in Wikipedia shows a universal double-power-law distribution of time intervals between consecutive updates of an article. We then propose a generic model to unfold collaborative human activities into three modules: (i) individual behavior characterized by Poissonian initiation of an action, (ii) human interaction captured by a cascading response to previous actions with a power-law waiting time, and (iii) population growth due to the increasing number of interacting individuals. This unfolding allows us to obtain an analytical formula that is fully supported by the universal patterns in empirical data. Our modeling approaches reveal "simplicity" beyond complex interacting human activities.

Magnitude-dependent epidemic-type aftershock sequences model for earthquakes

We propose a version of the pure temporal epidemic type aftershock sequences (ETAS) model: the ETAS model with correlated magnitudes. As for the standard case, we assume the Gutenberg-Richter law to be the probability density for the magnitudes of the background events. Instead, the magnitude of the triggered shocks is assumed to be probabilistically dependent on that of the relative mother events. This probabilistic dependence is motivated by some recent works in the literature and by the results of a statistical analysis made on some seismic catalogs [Spassiani and Sebastiani, J. Geophys. Res. 121, 903 (2016)10.1002/2015JB012398]. On the basis of the experimental evidences obtained in the latter paper for the real catalogs, we theoretically derive the probability density function for the magnitudes of the triggered shocks proposed in Spassiani and Sebastiani and there used for the analysis of two simulated catalogs. To this aim, we impose a fundamental condition: averaging over all the magnitudes of the mother events, we must obtain again the Gutenberg-Richter law. This ensures the validity of this law at any event's generation when ignoring past seismicity. The ETAS model with correlated magnitudes is then theoretically analyzed here. In particular, we use the tool of the probability generating function and the Palm theory, in order to derive an approximation of the probability of zero events in a small time interval and to interpret the results in terms of the interevent time between consecutive shocks, the latter being a very useful random variable in the assessment of seismic hazard.

Universal bursty behavior in the air transportation system

Social activities display bursty behavior characterized by heavy-tailed interevent time distributions. We examine the bursty behavior of airplanes' arrivals in hub airports. The analysis indicates that the air transportation system universally follows a power-law interarrival time distribution with an exponent α=2.5 and an exponential cutoff. Moreover, we investigate the mechanism of this bursty behavior by introducing a simple model to describe it. In addition, we compare the extent of the hub-and-spoke structure and the burstiness of various airline networks in the system. Remarkably, the results suggest that the hub-and-spoke network of the system and the carriers' strategy to facilitate transit are the origins of this universality.

Estimating interevent time distributions from finite observation periods in communication networks

A diverse variety of processes-including recurrent disease episodes, neuron firing, and communication patterns among humans-can be described using interevent time (IET) distributions. Many such processes are ongoing, although event sequences are only available during a finite observation window. Because the observation time window is more likely to begin or end during long IETs than during short ones, the analysis of such data is susceptible to a bias induced by the finite observation period. In this paper, we illustrate how this length bias is born and how it can be corrected without assuming any particular shape for the IET distribution. To do this, we model event sequences using stationary renewal processes, and we formulate simple heuristics for determining the severity of the bias. To illustrate our results, we focus on the example of empirical communication networks, which are temporal networks that are constructed from communication events. The IET distributions of such systems guide efforts to build models of human behavior, and the variance of IETs is very important for estimating the spreading rate of information in networks of temporal interactions. We analyze several well-known data sets from the literature, and we find that the resulting bias can lead to systematic underestimates of the variance in the IET distributions and that correcting for the bias can lead to qualitatively different results for the tails of the IET distributions.

Correlated bursts and the role of memory range

Inhomogeneous temporal processes in natural and social phenomena have been described by bursts that are rapidly occurring events within short time periods alternating with long periods of low activity. In addition to the analysis of heavy-tailed interevent time distributions, higher-order correlations between interevent times, called correlated bursts, have been studied only recently. As the underlying mechanism behind such correlated bursts is far from being fully understood, we devise a simple model for correlated bursts using a self-exciting point process with a variable range of memory. Whether a new event occurs is stochastically determined by a memory function that is the sum of decaying memories of past events. In order to incorporate the noise and/or limited memory capacity of systems, we apply two memory loss mechanisms: a fixed number or a variable number of memories. By analysis and numerical simulations, we find that too much memory effect may lead to a Poissonian process, implying that there exists an intermediate range of memory effect to generate correlated bursts comparable to empirical findings. Our conclusions provide a deeper understanding of how long-range memory affects correlated bursts.

Minima of the fluctuations of the order parameter of global seismicity

It has been recently shown [N. V. Sarlis, Phys. Rev. E 84, 022101 (2011) and N. V. Sarlis and S.-R. G. Christopoulos, Chaos 22, 023123 (2012)] that earthquakes of magnitude M greater or equal to 7 are globally correlated. Such correlations were identified by studying the variance κ1 of natural time which has been proposed as an order parameter for seismicity. Here, we study the fluctuations of this order parameter using the Global Centroid Moment Tensor catalog for a magnitude threshold Mthres = 5.0 and focus on its behavior before major earthquakes. Natural time analysis reveals that distinct minima of the fluctuations of the order parameter of seismicity appear within almost five and a half months on average before all major earthquakes of magnitude larger than 8.4. This phenomenon corroborates the recent finding [N. V. Sarlis et al., Proc. Natl. Acad. Sci. U.S.A. 110, 13734 (2013)] that similar minima of the seismicity order parameter fluctuations had preceded all major shallow earthquakes in Japan. Moreover, on the basis of these minima a statistically significant binary prediction method for earthquakes of magnitude larger than 8.4 with hit rate 100% and false alarm rate 6.67% is suggested.

Finite-size effects on return interval distributions for weakest-link-scaling systems

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and the fracture strength of quasibrittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. Our analysis employs the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the κ-Weibull distribution. The upper tail of the κ-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the κ-Weibull distribution decreases linearly after a waiting time τ(c) ∝ n(1/m), where m is the Weibull modulus and n is the system size in terms of representative volume elements. We conduct statistical analysis of experimental data and simulations which show that the κ Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we demonstrate that the κ-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

Statistical similarity between the compression of a porous material and earthquakes

It has long been stated that there are profound analogies between fracture experiments and earthquakes; however, few works attempt a complete characterization of the parallels between these so separate phenomena. We study the acoustic emission events produced during the compression of Vycor (SiO(2)). The Gutenberg-Richter law, the modified Omori's law, and the law of aftershock productivity hold for a minimum of 5 decades, are independent of the compression rate, and keep stationary for all the duration of the experiments. The waiting-time distribution fulfills a unified scaling law with a power-law exponent close to 2.45 for long times, which is explained in terms of the temporal variations of the activity rate.

Earthquake interevent time distribution for induced micro-, nano-, and picoseismicity

We examine the temporal statistics of micro-, nano-, and picoseismicity induced by mining as well as by long-term fluid injection. Specifically, we analyze catalogs of seismic events recorded at the Mponeng deep gold mine, South Africa, and at the German deep drilling site. We show that the distribution of time intervals between successive earthquakes is form invariant between the different catalogs. In particular, the distribution can be described by the same scaling function recently established for tectonic seismicity and acoustic emissions from laboratory rock fracture. Thus, our findings bridge the energy gap between those two cases and provide clear evidence that these temporal features of seismicity are independent of the energy scales of the events and whether they are of tectonic or induced origin.

Fertility heterogeneity as a mechanism for power law distributions of recurrence times

We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent α, where the fertility of an event is the number of triggered events of first generation, on the probability distribution function (PDF) f(τ) of the recurrence times τ between successive events. The other input of the model is an exponential law quantifying the PDF of waiting times between an event and its first generation triggered events, whose characteristic time scale is taken as our time unit. At short-time scales, we discover two intermediate power law asymptotics, f(τ)~τ(-(2-α)) for τ<<τ(c) and f(τ)~τ(-α) for τ(c)<<τ<<1, where τ(c) is associated with the self-excited cascades of triggered events. For 1<<τ<<1/ν, we find a constant plateau f(τ)=/~const, while at long times, 1/ν</~τ, f(τ)=/~e(-ντ) has an exponential tail controlled by the arrival rate ν of exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.

Scaling behavior of the earthquake intertime distribution: influence of large shocks and time scales in the Omori law

We present a study of the earthquake intertime distribution D(Δt) for a California catalog in temporal periods of short duration T. We compare experimental results with theoretical predictions and analytical approximate solutions. For the majority of intervals, rescaling intertimes by the average rate leads to collapse of the distributions D(Δt) on a universal curve, whose functional form is well fitted by a Gamma distribution. The remaining intervals, exhibiting a more complex D(Δt), are all characterized by the presence of large shocks. These results can be understood in terms of the relevance of the ratio between the characteristic time c in the Omori law and T: Intervals with Gamma-like behavior are indeed characterized by a vanishing c/T. The above features are also investigated by means of numerical simulations of the Epidemic Type Aftershock Sequence (ETAS) model. This study shows that collapse of D(Δt) is also observed in numerical catalogs; however, the fit with a Gamma distribution is possible only assuming that c depends on the main-shock magnitude m. This result confirms that the dependence of c on m, previously observed for m>6 main shocks, extends also to small m>2.

Natural time analysis of the Centennial Earthquake Catalog

By using the most recent version (1900-2007) of the Centennial Earthquake Catalog, we examine the properties of the global seismicity. Natural time analysis reveals that the fluctuations of the order parameter κ(1) of seismicity exhibit for at least three orders of magnitude a characteristic feature similar to that of the order parameter for other equilibrium or non-equilibrium critical systems-including self-organized critical systems. Moreover, we find non-trivial magnitude correlations for earthquakes of magnitude greater than or equal to 7.

Universal features of correlated bursty behaviour

Inhomogeneous temporal processes, like those appearing in human communications, neuron spike trains, and seismic signals, consist of high-activity bursty intervals alternating with long low-activity periods. In recent studies such bursty behavior has been characterized by a fat-tailed inter-event time distribution, while temporal correlations were measured by the autocorrelation function. However, these characteristic functions are not capable to fully characterize temporally correlated heterogenous behavior. Here we show that the distribution of the number of events in a bursty period serves as a good indicator of the dependencies, leading to the universal observation of power-law distribution for a broad class of phenomena. We find that the correlations in these quite different systems can be commonly interpreted by memory effects and described by a simple phenomenological model, which displays temporal behavior qualitatively similar to that in real systems.

How does an ice block assembly melt?

The melting of an assembly of ice blocks contained in a vertical cylinder and under an unidirectional load was investigated. The total volume occupied by the ice blocks and the volume of ice were simultaneously measured which allowed one to determine the volume fraction of the ice in the cylinder. While the ice volume continuously decreases, sudden breakdowns of the total volume were observed. Large reorganizations of the whole assembly occur. However, the maximal volume fraction found just after a large reorganization decreased with time. In addition, the modifications of the pile structure were investigated using an x-ray tomography imaging before and after one collapse. As the packing is better ordered along the walls, we suggest that the motion of the piston is governed by the layer of ice blocks located along the container wall. This layer was modeled by a two-dimensional assembly of disks. The model supports the idea that the geometrical frustrations explain the dynamics of the successive reorganization due to the shrinkage of the grains. Finally, numerical simulations allow one to conclude that the dynamics of the melting of the ice blocks is governed (i) by the confinement effect which induces defects in the packing and (ii) by the low friction between the ice blocks.

Magnitude correlations in global seismicity

By employing natural time analysis, we analyze the worldwide seismicity and study the existence of correlations between earthquake magnitudes. We find that global seismicity exhibits nontrivial magnitude correlations for earthquake magnitudes greater than M(w) 6.5.

Fractality of eroded coastlines of correlated landscapes

Using numerical simulations of a simple sea-coast mechanical erosion model, we investigate the effect of spatial long-range correlations in the lithology of coastal landscapes on the fractal behavior of the corresponding coastlines. In the model, the resistance of a coast section to erosion depends on the local lithology configuration as well as on the number of neighboring sea sides. For weak sea forces, the sea is trapped by the coastline and the eroding process stops after some time. For strong sea forces erosion is perpetual. The transition between these two regimes takes place at a critical sea force, characterized by a fractal coastline front. For uncorrelated landscapes, we obtain, at the critical value, a fractal dimension D=1.33, which is consistent with the dimension of the accessible external perimeter of the spanning cluster in two-dimensional percolation. For sea forces above the critical value, our results indicate that the coastline is self-affine and belongs to the Kardar-Parisi-Zhang universality class. In the case of landscapes generated with power-law spatial long-range correlations, the coastline fractal dimension changes continuously with the Hurst exponent H, decreasing from D=1.34 to 1.04, for H=0 and 1, respectively. This nonuniversal behavior is compatible with the multitude of fractal dimensions found for real coastlines.

Distribution of first-return times in correlated stationary signals

We present an analytical expression for the first return time (FRT) probability density function of a stationary correlated signal. Precisely, we start by considering a stationary discrete-time Ornstein-Uhlenbeck (OU) process with exponential decaying correlation function. The first return time distribution for this process is derived by adopting a well-known formalism typically used in the study of the FRT statistics for nonstationary diffusive processes. Then, by a subordination approach, we treat the case of a stationary process with power-law tail correlation function and diverging correlation time. We numerically test our findings, obtaining in both cases a good agreement with the analytical results. We notice that neither in the standard OU nor in the subordinated case a simple form of waiting time statistics, like stretched-exponential or similar, can be obtained while it is apparent that long time transient may shadow the final asymptotic behavior.

Optimal observation time window for forecasting the next earthquake

We report that the accuracy of predicting the occurrence time of the next earthquake is significantly enhanced by observing the latest rate of earthquake occurrences. The observation period that minimizes the temporal uncertainty of the next occurrence is on the order of 10 hours. This result is independent of the threshold magnitude and is consistent across different geographic areas. This time scale is much shorter than the months or years that have previously been considered characteristic of seismic activities.

Nonextensivity and natural time: The case of seismicity

Nonextensive statistical mechanics, pioneered by Tsallis, has recently achieved a generalization of the Gutenberg-Richter law for earthquakes. This remarkable generalization is combined here with natural time analysis, which enables the distinction of two origins of self-similarity, i.e., the process' memory and the process' increments infinite variance. By using also detrended fluctuation analysis for the detection of long-range temporal correlations, we demonstrate the existence of both temporal and magnitude correlations in real seismic data of California and Japan. Natural time analysis reveals that the nonextensivity parameter q , in contrast to some published claims, cannot be considered as a measure of temporal organization, but the Tsallis formulation does achieve a satisfactory description of real seismic data for Japan for q=1.66 when supplemented by long-range temporal correlations.

Recurrence interval analysis of trading volumes

We study the statistical properties of the recurrence intervals τ between successive trading volumes exceeding a certain threshold q. The recurrence interval analysis is carried out for the 20 liquid Chinese stocks covering a period from January 2000 to May 2009, and two Chinese indices from January 2003 to April 2009. Similar to the recurrence interval distribution of the price returns, the tail of the recurrence interval distribution of the trading volumes follows a power-law scaling, and the results are verified by the goodness-of-fit tests using the Kolmogorov-Smirnov (KS) statistic, the weighted KS statistic and the Cramér-von Mises criterion. The measurements of the conditional probability distribution and the detrended fluctuation function show that both short-term and long-term memory effects exist in the recurrence intervals between trading volumes. We further study the relationship between trading volumes and price returns based on the recurrence interval analysis method. It is found that large trading volumes are more likely to occur following large price returns, and the comovement between trading volumes and price returns is more pronounced for large trading volumes.

Multiple-time scaling and universal behavior of the earthquake interevent time distribution

The interevent time distribution characterizes the temporal occurrence in seismic catalogs. Universal scaling properties of this distribution have been evidenced for entire catalogs and seismic sequences. Recently, these universal features have been questioned and some criticisms have been raised. We investigate the existence of universal scaling properties by analyzing a Californian catalog and by means of numerical simulations of an epidemic-type model. We show that the interevent time distribution exhibits a universal behavior over the entire temporal range if four characteristic times are taken into account. The above analysis allows us to identify the scaling form leading to universal behavior and explains the observed deviations. Furthermore, it provides a tool to identify the dependence on the mainshock magnitude of the c parameter that fixes the onset of the power law decay in the Omori law.

Multiplicative cascades and seismicity in natural time

Natural time chi enables the distinction of two origins of self-similarity, i.e., the process memory and the process increments infinite variance. Employing multiplicative cascades in natural time, the most probable value of the variance kappa(1)(is identical to chi(2)-chi(2))is explicitly related with the parameter b of the Gutenberg-Richter law of randomly shuffled earthquake data. Moreover, the existence of temporal and magnitude correlations is studied in the original earthquake data. Magnitude correlations are larger for closer in time earthquakes, when the maximum interoccurrence time varies from half a day to 1 min.

Self-similarity of waiting times in fracture systems

Experimental and numerical results are presented for a fracture experiment carried out on a fiber-reinforced element under flexural loading, and a statistical analysis is performed for acoustic emission waiting-time distributions. By an optimization procedure, a recently proposed scaling law describing these distributions for different event magnitude scales is confirmed by both experimental and numerical data, thus reinforcing the idea that fracture of heterogeneous materials has scaling properties similar to those found for earthquakes. Analysis of the different scaling parameters obtained for experimental and numerical data leads us to formulate the hypothesis that the type of scaling function obtained depends on the level of correlation among fracture events in the system.

Limits of declustering methods for disentangling exogenous from endogenous events in time series with foreshocks, main shocks, and aftershocks

Many time series in natural and social sciences can be seen as resulting from an interplay between exogenous influences and an endogenous organization. We use a simple epidemic-type aftershock model of events occurring sequentially, in which future events are influenced (partially triggered) by past events to ask the question of how well can one disentangle the exogenous events from the endogenous ones. We apply both model-dependent and model-independent stochastic declustering methods to reconstruct the tree of ancestry and estimate key parameters. In contrast with previously reported positive results, we have to conclude that declustered catalogs are rather unreliable for the synthetic catalogs that we have investigated, which contains of the order of thousands of events, typical of realistic applications. The estimated rates of exogenous events suffer from large errors. The branching ratio n, quantifying the fraction of events that have been triggered by previous events, is also badly estimated in general from declustered catalogs. We find, however, that the errors tend to be smaller and perhaps acceptable in some cases for small triggering efficiency and branching ratios. The high level of randomness together with the long memory makes the stochastic reconstruction of trees of ancestry and the estimation of the key parameters perhaps intrinsically unreliable for long-memory processes. For shorter memories (larger "bare" Omori exponent), the results improve significantly.

Origin and nonuniversality of the earthquake interevent time distribution

Many authors have modeled regional earthquake interevent times using a gamma distribution, whereby data collapse occurs under a simple rescaling of the data from different regions or time periods. We show, using earthquake data and simulations, that the distribution is fundamentally a bimodal mixture distribution dominated by correlated aftershocks at short waiting times and independent events at longer times. The much-discussed power-law segment often arises as a crossover between these two. We explain the variation of the distribution with region size and show that it is not universal.

Return interval distribution of extreme events and long-term memory

The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long-range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long-range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long-range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.

Missing data in aftershock sequences: explaining the deviations from scaling laws

In this paper we extend the branching aftershock sequence model to study the role of missing data at short times and small amplitudes after a mainshock. We apply this model, which contains three parameters characterizing the missing data, to the magnitude and temporal statistics of four aftershock sequences in California. We find that the observed time-dependent deviations of the frequency-magnitude scaling from the Gutenberg-Richter power law dependency can be described quantitatively by the model. We also show that, for the same set of parameters, the model is able to explain quantitatively the observed magnitude-dependent deviations of the temporal decay of aftershocks from Omori's law. In addition, we show that the same sets of data can also reproduce quite well the various functional forms of the probability density functions of the return times between consecutive events with magnitudes above a prescribed threshold, as well as the violation of scaling at short and intermediate time scales.

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.

Scaling of sound emission energy and fracture behavior of cellular solid foods

A detailed study was performed of the fracture behavior of toasted rusk rolls, a cellular solid food, at different water activities and morphologies. We find that the energies of the emitted sound pulses follow Gutenberg-Richter power laws with characteristic exponents b ~ 1.5 . The scaling exponents varied only within a range of 0.2 when the method of fracture, humidity, or morphology was changed. However, differences in b were observed, indicating nonuniversal behavior, that seems to be related to morphology and water activity. Also, power law scaling behavior was observed for the waiting time distributions with an exponent a ~ 1.9.

Influence of time and space correlations on earthquake magnitude

A crucial point in the debate on the feasibility of earthquake predictions is the dependence of an earthquake magnitude from past seismicity. Indeed, while clustering in time and space is widely accepted, much more questionable is the existence of magnitude correlations. The standard approach generally assumes that magnitudes are independent and therefore in principle unpredictable. Here we show the existence of clustering in magnitude: earthquakes occur with higher probability close in time, space, and magnitude to previous events. More precisely, the next earthquake tends to have a magnitude similar but smaller than the previous one. A dynamical scaling relation between magnitude, time, and space distances reproduces the complex pattern of magnitude, spatial, and temporal correlations observed in experimental seismic catalogs.

Interoccurrence time statistics in the two-dimensional Burridge-Knopoff earthquake model

We have numerically investigated statistical properties of the so-called interoccurrence time or the waiting time, i.e., the time interval between successive earthquakes, based on the two-dimensional (2D) spring-block (Burridge-Knopoff) model, selecting the velocity-weakening property as the constitutive friction law. The statistical properties of frequency distribution and the cumulative distribution of the interoccurrence time are discussed by tuning the dynamical parameters, namely, a stiffness and frictional property of a fault. We optimize these model parameters to reproduce the interoccurrence time statistics in nature; the frequency and cumulative distribution can be described by the power law and Zipf-Mandelbrot type power law, respectively. In an optimal case, the b value of the Gutenberg-Richter law and the ratio of wave propagation velocity are in agreement with those derived from real earthquakes. As the threshold of magnitude is increased, the interoccurrence time distribution tends to follow an exponential distribution. Hence it is suggested that a temporal sequence of earthquakes, aside from small-magnitude events, is a Poisson process, which is observed in nature. We found that the interoccurrence time statistics derived from the 2D BK (original) model can efficiently reproduce that of real earthquakes, so that the model can be recognized as a realistic one in view of interoccurrence time statistics.