We discuss various statistical distributions of earthquake numbers. Previously we derived several discrete distributions to describe earthquake numbers for the branching model of earthquake occurrence: these distributions are the Poisson, geometric, logarithmic, and the negative binomial (NBD). The theoretical model is the `birth and immigration' population process. The first three distributions above can be considered special cases of the NBD. In particular, a point branching process along the magnitude (or log seismic moment) axis with independent events (immigrants) explains the magnitude/moment-frequency relation and the NBD of earthquake counts in large time/space windows, as well as the dependence of the NBD parameters on the magnitude threshold (magnitude of an earthquake catalog completeness). We discuss applying these distributions, especially the NBD, to approximate event numbers in earthquake catalogs. There are many different representations of the NBD. Most can be traced either to the Pascal distribution or to the mixture of the Poisson distribution with the gamma law. We discuss advantages and drawbacks of both representations for statistical analysis of earthquake catalogs. We also consider applying the NBD to earthquake forecasts and describe the limits of the application for the given equations. In contrast to the one-parameter Poisson distribution so widely used to describe earthquake occurrence, the NBD has two parameters. The second parameter can be used to characterize clustering or over-dispersion of a process. We determine the parameter values and their uncertainties for several local and global catalogs, and their subdivisions in various time intervals, magnitude thresholds, spatial windows, and tectonic categories.
We consider a branching model of triggered seismicity, the ETAS (epidemic-type aftershock sequence) model which assumes that each earthquake can trigger other earthquakes (``aftershocks''). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. Due to the large fluctuations of the number of aftershocks triggered directly by any earthquake (``productivity'' or ``fertility''), there is a large variability of the total number of aftershocks from one sequence to another, for the same mainshock magnitude. We study the regime where the distribution of f…
We discuss various methods used to investigate the geometric complexity of earthquakes and earthquake faults, based both on a point-source representation and the study of interrelations between earthquake focal mechanisms. We briefly review the seismic moment tensor formalism and discuss in some detail the representation of double-couple (DC) earthquake sources by normalized quaternions. Non-DC earthquake sources like the CLVD focal mechanism are also considered. We obtain the characterization of the earthquake complex source caused by summation of disoriented DC sources. We show that commonly…
Several recent works point out that the crowd of small unobservable earthquakes (with magnitudes below the detection threshold $m_d$) may play a significant and perhaps dominant role in triggering future seismicity. Using the ETAS branching model of triggered seismicity, we apply the formalism of generating probability functions to investigate how the statistical properties of observable earthquakes differ from the statistics of all events. The ETAS (epidemic-type aftershock sequence) model assumes that each earthquake can trigger other earthquakes (``aftershocks''). An aftershock sequence res…
We explore in depth the validity of a recently proposed scaling law for earthquake interevent time distributions in the case of the Southern California, using the waveform cross-correlation catalog of Shearer et al. Two statistical tests are used: on the one hand, the standard two-sample Kolmogorov-Smirnov test is in agreement with the scaling of the distributions. On the other hand, the one-sample Kolmogorov-Smirnov statistic complemented with Monte Carlo simulation of the inter-event times, as done by Clauset et al., supports the validity of the gamma distribution as a simple model of the sc…
We study the structural similarity of earthquake networks constructed from seismic catalogs of different geographical regions. A hierarchical clustering of underlying undirected earthquake networks is shown using Jensen-Shannon divergence in graph spectra. The directed nature of links indicates that each earthquake network is strongly connected, which motivates us to study the directed version statistically. Our statistical analysis of each earthquake region identifies the hub regions. We calculate the conditional probability of the forthcoming occurrences of earthquakes in each region. The co…
The impact of including postcursors in the null hypothesis test is discussed. Unequal prediction probabilities can be included in the null hypothesis test using a generalization of the central limit theorem. A test for determining the enhancement factor over random chance is given. The seismic earthquake signal may preferentially precede earthquakes even if the VAN methodology fails to forecast the earthquakes. We formulate a statistical test for this possibility.
The distribution of inter-occurrence time between seismic events is a quantity of great interest in seismic risk assessment. We evaluate this distribution for different models of earthquakes occurrence and follow two distinct approaches: The non homogeneous Poissonian and the non Poissonian one. In all cases we obtain either a power law or a power law damped by an exponential factor behaviour. This feature of the distribution makes impossible any prediction of earthquakes occurrence. Nevertheless it suggests the interpretation of the earthquake occurrence phenomenon as due to some non-linear d…
In models of triggered seismicity and in their inversion with empirical data, the detection threshold m_d is commonly equated to the magnitude m_0 of the smallest triggering earthquake. This unjustified assumption neglects the possibility of shocks below the detection threshold triggering observable events. We introduce a formalism that distinguishes between the detection threshold m_d and the minimum triggering earthquake m_0 < m_d. By considering the branching structure of one complete cascade of triggered events, we derive the apparent branching ratio n_a (which is the apparent fraction of …
Testing the global earthquake catalogue for indications of non-Poissonian attributes has been an area of intense research, especially since the 2011 Tohoku earthquake. The usual approach is to test statistically for the hypothesis that the global earthquake catalogue is well explained by a Poissonian process. In this paper we analyse one aspect of this problem which has been disregarded by the literature: the power of such tests to detect non-Poissonian features if they existed; that is, the probability of type II statistical errors. We argue that the low frequency of large events and the brev…
We propose that the widely observed and universal Gutenberg-Richter relation is a mathematical consequence of the critical branching nature of earthquake process in a brittle fracture environment. These arguments, though preliminary, are confirmed by recent investigations of the seismic moment distribution in global earthquake catalogs and by the results on the distribution in crystals of dislocation avalanche sizes. We consider possible systematic and random errors in determining earthquake size, especially its seismic moment. These effects increase the estimate of the parameter beta of the p…
Smartphone-based earthquake early warning systems (EEWS) are emerging as a complementary solution to classic EEWS based on expensive scientific-grade instruments. Smartphone-based systems, however, are characterized by a highly dynamic network geometry and by noisy measurements. Thus the need to control the probability of false alarm and the probability of missed detection. This paper proposes a statistical approach based on the maximum likelihood method to address this challenge and to jointly estimate in near real-time earthquake parameters like epicentre and depth. The approach is tested us…
The physics of earthquake triggering together with simple assumptions of self-similarity impose the existence of a minimum magnitude m0 below which earthquakes do not trigger other earthquakes. Noting that the magnitude md of completeness of seismic catalogs has no reason to be the same as the magnitude m0 of the smallest triggering earthquake, we use quantitative fits and maximum likelihood inversions of observed aftershock sequences as well as Baath's law, compare with ETAS model predictions and thereby constrain the value of m0. We show that the branching ratio $n$ (average number of trigge…
We study statistical properties of the number of large earthquakes over the past century. We analyze the cumulative distribution of the number of earthquakes with magnitude larger than threshold M in time interval T, and quantify the statistical significance of these results by simulating a large number of synthetic random catalogs. We find that in general, the earthquake record cannot be distinguished from a process that is random in time. This conclusion holds whether aftershocks are removed or not, except at magnitudes below M = 7.3. At long time intervals (T = 2-5 years), we find that stat…
Smartphone-based earthquake early warning systems implemented by citizen science initiatives are characterized by a significant variability in their smartphone network geometry. This has an direct impact on the earthquake detection capability and performance of the system. Here, a simulation framework based on the Monte Carlo method is implemented for making inference on relevant quantities of the earthquake detection such as the detection distance from the epicentre, the detection delay and the warning time for people exposed to high ground shaking levels. The framework is applied to Haiti, w…
In statistical seismology, the Epidemic Type Aftershocks Sequence (ETAS) model is a branching process used world-wide to forecast earthquake intensity rates and reproduce many statistical features observed in seismicity catalogs. In this paper, we describe a fractional differential equation that governs the earthquake intensity rate of the pure temporal ETAS model by using the Caputo fractional derivative and we solve it analytically. We highlight that the tools and special functions of fractional calculus simplify the classical methods employed to obtain the intensity rate and let us describe…
Spatiotemporal clustering of earthquake events is a generally-established fact, and is important for designing models and assessment techniques in seismicity. Here, we investigate how this behavior can manifest in the statistical distributions of inter-event distances and times between earthquakes from different regional catalogs. We complement the analysis of previous authors (Touati et al., PRL 102, 168501 (2009)) and observe histograms best described by a superposition of two component distributions for "short" and "long" distances and times. Our results quantify the spatiotemporal clusteri…
In order to clarify how the statistical properties of earthquakes depend on the constitutive law characterizing the stick-slip dynamics, we make an extensive numerical simulation of the one-dimensional spring-block model with the rate- and state-dependent friction law. Both the magnitude distribution and the recurrence-time distribution are studied with varying the constitutive parameters characterizing the model. While a continuous spectrum of seismic events from smaller to larger magnitudes is obtained, earthquakes described by this model turn out to possess pronounced ``characteristic'' fea…
We consider three questions related to the 2011 Tohoku mega-earthquake: (1) Why was the event size so grossly under-estimated? (2) How should we evaluate the chances of giant earthquakes in subduction zones? and (3) What is the repeat time for magnitude 9 earthquakes off the Tohoku coast? The "maximum earthquake size" is often guessed from the available history of earthquakes, a method known for its significant downward bias. There are two quantitative methods for estimating the maximum magnitude in any region: a statistical analysis of the available earthquake record, and the moment conservat…
We report remarkable similarities in the output signal of two distinct out-of- equilibrium physical systems - earthquakes and the intermittent acoustic noise emitted by crum- pled plastic sheets - Biaxially Oriented Polypropylene (BOPP) films. We show that both signals share several statistical properties including the distribution of energy, distribution of energy in- crements for distinct time scales, distribution of return intervals and correlations in the magnitude and sign of energy increments. This analogy is consistent with the concept of universality in com- plex systems and could prov…
The distribution of seismic moment is of capital interest to evaluate earthquake hazard, in particular regarding the most extreme events. We make use of likelihood-ratio tests to compare the simple Gutenberg-Richter power-law distribution with two statistical models that incorporate an exponential tail: the so-called tapered Gutenberg-Richter and the truncated gamma, when fitted to the global CMT earthquake catalog. The outcome is that the truncated gamma model outperforms the other two models. If simulated samples of the truncated gamma are reshuffled in order to mimic the time occurrence of …
The ETAS model is widely employed to model the spatio-temporal distribution of earthquakes, generally using spatially invariant parameters. We propose an efficient method for the estimation of spatially varying parameters, using the Expectation-Maximization (EM) algorithm and spatial Voronoi tessellation ensembles. We use the Bayesian Information Criterion (BIC) to rank inverted models given their likelihood and complexity and select the best models to finally compute an ensemble model at any location. Using a synthetic catalog, we also check that the proposed method correctly inverts the know…
We report an empirical determination of the probability density functions P(r) of the number r of earthquakes in finite space-time windows for the California catalog, over fixed spatial boxes 5 x 5 km^2 and time intervals dt =1, 10, 100 and 1000 days. We find a stable power law tail P(r) ~ 1/r^{1+mu} with exponent mu \approx 1.6 for all time intervals. These observations are explained by a simple stochastic branching process previously studied by many authors, the ETAS (epidemic-type aftershock sequence) model which assumes that each earthquake can trigger other earthquakes (``aftershocks''). …
The ranking problem of earthquake forecasts is considered. We formulate simple statistical requirements to forecasting quality measure R and analyze some R-ranking methods on this basis, in particular, the pari-mutuel gambling method by Zechar&Zhuang (2014).
The field of study of complex systems considers that the dynamics of complex systems are founded on universal principles that may be used to describe a great variety of scientific and technological approaches of different types of natural, artificial, and social systems. Several authors have suggested that earthquake dynamics and the dynamics of economic (financial) systems can be analyzed within similar mathematical frameworks. We apply concepts of the nonextensive statistical physics, on time-series data of observable manifestations of the underlying complex processes ending up to these diff…
The field of study of complex systems considers that the dynamics of complex systems are founded on universal principles that may be used to describe a great variety of scientific and technological approaches of different types of natural, artificial, and social systems. Several authors have suggested that earthquake dynamics and neurodynamics can be analyzed within similar mathematical frameworks. Recently, authors have shown that a dynamical analogy supported by scale-free statistics exists between seizures and earthquakes, analysing populations of different seizures and earthquakes, respect…
Recent observation studies have revealed that earthquakes are classified into several different categories. Each category might be characterized by the unique statistical feature in the time series, but the present understanding is still limited due to their nonlinear and nonstationary nature. Here we utilize complex network theory to shed new light on the statistical properties of earthquake time series. We investigate two kinds of time series, which are magnitude and inter-event time (IET), for three different categories of earthquakes: regular earthquakes, earthquake swarms, and tectonic tr…
Predicting earthquakes is of the utmost importance, especially to those countries of high risk, and although much effort has been made, it has yet to be realised. Nevertheless, there is a paucity of statistical approaches in seismic studies to the extent that an old theory is believed without verification. Seismic records of time and magnitude in Japan were analysed by exploratory data analysis (EDA). EDA is a parametric statistical approach based on the characteristics of data and is suitable for data-driven investigations. The distribution style of each dataset was determined, and the import…
Forecasts of the focal mechanisms of future earthquakes are important for seismic hazard estimates and Coulomb stress and other models of earthquake occurrence. Here we report on a high-resolution global forecast of earthquake rate density as a function of location, magnitude, and focal mechanism. In previous publications we reported forecasts of 0.5 degree spatial resolution, covering the latitude range magnitude, and focal mechanism. In previous publications we reported forecasts of 0.5 degree spatial resolution, covering the latitude range from -75 to +75 degrees, based on the Global Centra…
We illustrate how a simple statistical model can describe the quasiperiodic occurrence of large earthquakes. The model idealizes the loading of elastic energy in a seismic fault by the stochastic filling of a box. The emptying of the box after it is full is analogous to the generation of a large earthquake in which the fault relaxes after having been loaded to its failure threshold. The duration of the filling process is analogous to the seismic cycle, the time interval between two successive large earthquakes in a particular fault. The simplicity of the model enables us to derive the statisti…
The driving concept behind one of the most successful statistical forecasting models, the ETAS model, has been that the seismicity is driven by spontaneously occurring background earthquakes that cascade into multitudes of triggered earthquakes. In nearly all generalizations of the ETAS model, the magnitudes of the background and the triggered earthquakes are assumed to follow Gutenberg-Richter law with the same exponent (\b{eta}-value). Furthermore, the magnitudes of the triggered earthquakes are always assumed to be independent of the magnitude of the triggering earthquake. Using an EM algor…
The paper by Taroni et al. (2016) considers results of forward prediction of Italian strong earthquakes by CN algorithm with the declared intent of providing "a careful assessment of CN prediction performances... using standard testing procedures". Given the very limited number of target events within each region, however, the considered situation is non statistical, and a priori it is clear that the standard statistical methods are not effective here. The attempt to replace the standard approaches by Pari-mutuel Gambling Score (PGS) method leads to almost complete loss of information about pr…
Frequency-magnitude distributions, and their associated uncertainties, are of key importance in statistical seismology. When fitting these distributions, the assumption of Gaussian residuals is invalid since event numbers are both discrete and of unequal variance. In general, the observed number in any given magnitude range is described by a binomial distribution which, given a large total number of events of all magnitudes, approximates to a Poisson distribution for a sufficiently small probability associated with that range. In this paper, we examine four earthquake catalogues: New Zealand (…
We modify the new method for the statistical estimation of the tail distribution of earthquake seismic moments introduced by Pisarenko et al. [2009] and apply it to the earthquake catalog of Japan (1923-2007). The method is based on the two main limit theorems of the theory of extreme values and on the derived duality between the Generalized Pareto Distribution (GPD) and Generalized Extreme Value distribution (GEV). We obtain the distribution of maximum earthquake magnitudes in future time intervals of arbitrary duration tau. This distribution can be characterized by its quantile Qq(tau) at an…
We develop a new method for the statistical esitmation of the tail of the distribution of earthquake sizes recorded in the Worldwide Harvard catalog of seismic moments converted to mW-magnitudes (1977-2004 and 1977-2006). We show that using the set of maximum magnitudes (the set of T-maxima) in windows of duration T days provides a significant improvement over existing methods, in particular (i) by minimizing the negative impact of time-clustering of foreshock / main shock /aftershock sequences in the estimation of the tail of the magnitude distribution, and (ii) by providing via a simulation …
I study a recently proposed statistical model of earthquake dynamics that incorporates aging as a fundamental ingredient. The model is known to generate earthquake sequences that quantitatively reproduce the spatial and temporal clustering of events observed in actual seismic patterns. The aim of the present work is to investigate if this model can give support to the empirical evidence that earthquakes can be triggered by transient small perturbations, particularly by the passing of seismic waves originated in events occurring in far geographical locations. The effect of seismic waves is inco…
It has been long stated that there are profound analogies between fracture experiments and earthquakes; however, few works attempt a complete characterization of the parallelisms between these so separate phenomena. We study the Acoustic Emission events produced during the compression of Vycor (SiO2). The Gutenberg-Richter law, the modified Omori's law, and the law of aftershock productivity are found to 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 …
In this paper, we used the Global Catalog of the National Earthquake Information Center US Geological Survey (NEIC USGS) for analysis of the magnitude-frequency distribution of earthquakes. We selected the unimodal part of the distribution and proposed an empirical formula that approximates this part in a wide range of magnitudes. The formula may be useful in studying the problems seismology, in which the expected effect is manifested in relatively weak earthquakes. The paper will be sent to the journal Geodynamics and Tectonophysics (ISSN 2078-502X). Keywords: planetary seismicity, Gutenberg-…
In our paper published earlier we discussed forecasts of earthquake focal mechanism and ways to test the forecast efficiency. Several verification methods were proposed, but they were based on ad-hoc, empirical assumptions, thus their performance is questionable. In this work we apply a conventional likelihood method to measure a skill of forecast. The advantage of such an approach is that earthquake rate prediction can in principle be adequately combined with focal mechanism forecast, if both are based on the likelihood scores, resulting in a general forecast optimization. To calculate the li…
We consider statistical analysis of double couple (DC) earthquake focal mechanism orientation. The symmetry of DC changes with its geometrical properties, and the number of 3-D rotations one DC source can be transformed into another depends on its symmetry. Four rotations exist in a general case of DC with the nodal-plane ambiguity, two transformations if the fault plane is known, and one rotation if the sides of the fault plane are known. The symmetry of rotated objects is extensively analyzed in statistical material texture studies, and we apply their results to analyzing DC orientation. We …
Recent studies in the literature have introduced a new approach to earthquake forecasting based on representing the space-time patterns of localized seismicity by a time-dependent system state vector in a real-valued Hilbert space and deducing information about future space-time fluctuations from the phase angle of the state vector. While the success rate of this Pattern Informatics (PI) method has been encouraging, the method is still in its infancy. Procedural analysis, statistical testing, parameter sensitivity investigation and optimization all still need to be performed. In this paper, we…
This entry in the Encyclopedia of Complexity and Systems Science, Springer present a summary of some of the concepts and calculational tools that have been developed in attempts to apply statistical physics approaches to seismology. We summarize the leading theoretical physical models of the space-time organization of earthquakes. We present a general discussion and several examples of the new metrics proposed by statistical physicists, underlining their strengths and weaknesses. The entry concludes by briefly outlining future directions. The presentation is organized as follows. I Glossary II…
By using the M>=5 global earthquake data for Jan. 1950 to Dec. 2015, we performed statistical analyses for the parameters magnitude, time, and depth on a yearly scale. The magnitude spectrum, which is the earthquake number accumulated at different magnitudes, had an exponential distribution. For the first time, we report a very significant characteristic of the sinusoidal periodic variation in the spectral index. The cycle of the sine function fitting was 30.98 years. The concept of annual equivalent total magnitude (AETM) of total released energy for each year was introduced and the trend var…
The statistical properties of the seismic time series data in southern California are studies. In particular, the calm time intervals, which are the time intervals between successive significant earthquakes above the fixed threshold value of magnitude, are analyzed. It is found that the system undergoes a series of transition between stationary states, whose cumulative waiting time distributions all obey the Zipf-Mandelbrot law, exhibiting a new scale free nature of earthquakes. Dependencies of the two parameters characterizing the Zipf-Mandelbrot distribution on the threshold are also discuss…
Event counts are a powerful diagnostic for earthquake analysis. We report on a bimodal distribution of earthquake event counts in the U.S. since 2013. The new peak is about a magnitude M1 with a distribution very similar to that of induced earthquakes in Groningen, The Netherlands, characterized by exponential growth in event count since 2001. The latter shows a doubling time of 6.24 years with a relatively constant rate of land subsidence. We model its internal shear stresses as a function of dimensionless curvature and illustrate the resulting exponential growth in a tabletop crack formation…
The statistical properties of earthquake aftershocks are studied. The scaling relation for the exponents of the Omori law and the power-law calm time distribution (i.e., the interoccurrence time distribution), which is valid if a sequence of aftershocks is a singular Markovian process, is carefully examined. Data analysis shows significant violation of the scaling relation, implying the non-Markovian nature of aftershocks.
Statistical properties of earthquakes are studied both by the analysis of real earthquake catalog of Japan and by numerical computer simulations of the spring-block model in both one and two dimensions. Particular attention is paid to the spatiotemporal correlations of earthquakes, e.g., the recurrence-time distribution or the time evolution before and after the mainshock of seismic distribution functions, including the magnitude distribution and the spatial seismic distribution. Certain eminent features of the spatiotemporal correlations, e.g., foreshocks, aftershocks, swarms and doughnut-lik…
The data analysis of aftershock events of L Aquila earthquake in Apennines following the main 6.3 Mw event of April 6, 2009 has been carried out by standard statistical geophysical tools. The results show the heterogeneity of seismic activity in five different geographical sub-regions indicated by anomalous critical indices of power law distributions: the exponents of the Omori law, the b values of Gutenberg-Richter magnitude-frequency distribution, and the distribution of waiting times. The heterogeneous distribution of dynamic stress and a different morphology in the five sub-regions has bee…
We present a generic and powerful approach to study the statistics of extreme phenomena (meteorology, finance, biology...) that we apply to the statistical estimation of the tail of the distribution of earthquake sizes. The chief innovation is to combine the two main limit theorems of Extreme Value Theory (EVT) that allow us to derive the distribution of T-maxima (maximum magnitude occurring in sequential time intervals of duration T) for arbitrary T. We propose a method for the estimation of the unknown parameters involved in the two limit theorems corresponding to the Generalized Extreme Val…
Coulomb-stress theory has been used for years in seismology to understand how earthquakes trigger each other. Whenever an earthquake occurs, the stress field changes, and places with positive increases are brought closer to failure. Earthquake models that relate earthquake rates and Coulomb stress after a main event, such as the rate-and-state model, assume that the magnitude distribution of earthquakes is not affected by the change in the Coulomb stress. By using different slip models, we calculate the change in Coulomb stress in the fault plane for every aftershock after the Landers event (C…
