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''). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. We develop the full theory in terms of generating functions for describing the space-time organization of earthquake sequences and develop several approximations to solve the equations. The calibration of the theory to the empirical observations shows that it is essential to augment the ETAS model by taking account of the pre-existing frozen heterogeneity of spontaneous earthquake sources. This seems natural in view of the complex multi-scale nature of fault networks, on which earthquakes nucleate. Our extended theory is able to account for the empirical observation satisfactorily. In particular, the adjustable parameters are determined by fitting the largest time window $dt=1000$ days and are then used as frozen in the formulas for other time scales, with very good agreement with the empirical data.
This paper presents an analysis of the distribution of the time $τ$ between two consecutive events in a stationary point process. The study is motivated by the discovery of a unified scaling law for $τ$ for the case of seismic events. It is shown that the unified law must necessarily be exponential. We discuss the parameterization of the empirical unified law and the physical meaning of the parameters involved.
A new non-parametric statistic is introduced for the characterization of deviations from power laws. It is tested on the distribution of seismic energies given by the Gutenberg-Richter law. Based on the two first statistical log-moments, it evaluates quantitatively the deviations of the distribution of scalar seismic moments from a power-like (Pareto) law. This statistic is close to zero for the Pareto law with arbitrary power index, and deviates from zero for any non-Pareto distribution. A version of this statistic for discrete distribution of quantified magnitudes is also given. A methodolog…
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 …
Earthquakes rank among the most destructive manifestations of the Earth's dynamics. Can they be predicted? This is often the first question students ask. To answer that right away: no, at present it is not possible to anticipate the date, site and magnitude of future seismic events. However, there does exist a general framework to describe observations related to earthquakes and understand the processes that lead to their occurrence: the seismic cycle. This chapter introduces the reader to the friction laws from a historical to state of the art perspective. It then deals with mechanical modell…
We establish the abstract and semi-abstract equations that translate the conservation law for a totally-filled, or totally-empty basin of arbitrary shape submitted to a SH plane seismic body wave. This law states that the input flux is equal to the sum of the scattered and absorbed fluxes, the latter being equal to zero when the filler medium is non-lossy. We show that the well-known, exact, solutions for the case of a semi-circular basin indeed satisfy this law. The latter can, and should, be employed to test the validity of any theoretical or numerical solution one might propose for a basin …
Statistical similarities between earthquakes and other systems that emit cracking noises have been explored in diverse contexts, ranging from materials science to financial and social systems. Such analogies give promise of a unified and universal theory for describing the complex responses of those systems. There are, however, very few attempts to simultaneously characterize the most fundamental seismic laws in such systems. Here we present a complete description of the Gutenberg-Richter law, the recurrence times, Omori's law, the productivity law, and Bath's law for the acoustic emissions th…
After a large earthquake, the likelihood of successive strong aftershocks needs to be estimated. Exploiting similarities with critical phenomena, we introduce a scaling law for the decay in time following a main shock of the expected number of aftershocks greater than a certain magnitude. Empirical results that support our scaling hypothesis are obtained from analyzing the record of earthquakes in California. The proposed form unifies the well-known Omori and Gutenberg-Richter laws of seismicity, together with other phenomenological observations. Our results substantially modify presently empl…
Scaling analysis of seismicity in the space-time-magnitude domain very often starts from the relation N(m,L)=a(L)*10**(-bm)*L**c for the rate of seismic events of magnitude M>m in an area of size L. There are some evidences in favor of multifractal property of seismic process. In this case the choice of the scale exponent 'c' is not unique. It is shown how different 'c''s are related to different types of spatial averaging applied to lambda(m, L) and what are the 'c''s for which the distributions of a(L) best agree for small L. Theoretical analysis is supplemented with an analysis of Californi…
We establish the equations which translate a conservation law for the problem of the seismic response of an above-ground structure (e.g., building, hill or mountain) of arbitrary shape and inquire whether both the implicit (formal) and explicit (numerical) solutions for the response obey this law for the case of a cylindrical, rectangular protuberance. Both the low-order (poor approximations of the response) as well as higher-order (supposedly better approximations) turn out to satisfy the conservation of flux relation, which means that the satisfaction of this relation is a necessary, but not…
We introduce a shear experiment that quantitatively reproduces the main laws of seismicity. By continuously and slowly shearing a compressed monolayer of disks in a ring-like geometry, our system delivers events of frictional failures with energies following a Gutenberg-Richter law. Moreover foreshocks and aftershocks are described by Omori laws and inter-event times also follow exactly the same distribution as real earthquakes, showing the existence of memory of past events. Other features of real earthquakes qualitatively reproduced in our system are both the existence of a quiescence preced…
It has been observed that the earthquake events possess short-term memory, i.e. that events occurring in a particular location are dependent on the short history of that location. We conduct an analysis to see whether real-time earthquake data also possess long-term memory and, if so, whether such autocorrelations depend on the size of earthquakes within close spatiotemporal proximity. We analyze the seismic waveform database recorded by 64 stations in Japan, including the 2011 "Great East Japan Earthquake", one of the five most powerful earthquakes ever recorded which resulted in a tsunami an…
In their paper (Europhys. Lett., 71 (2005) 1036), Carbone, Sorriso-Valvo, Harabaglia and Guerra showed that "unified scaling law" for conventional waiting times of earthquakes claimed by Bak et al. (Phys. Rev. Lett., 88 (2002) 178501) is actually not universal. Here, instead of the conventional time, the concept of the internal time termed the event time is considered for seismicity. It is shown that, in contrast to the conventional waiting time, the waiting event time obeys a power law. This implies the existence of temporal long-range correlations in terms of the event time with no sharp dec…
We report an empirical determination of the probability density functions $P_{\text{data}}(r)$ of the number $r$ of earthquakes in finite space-time windows for the California catalog. We find a stable power law tail $P_{\text{data}}(r) \sim 1/r^{1+μ}$ with exponent $μ\approx 1.6$ for all space ($5 \times 5$ to $20 \times 20$ km$^2$) and time intervals (0.1 to 1000 days). These observations, as well as the non-universal dependence on space-time windows for all different space-time windows simultaneously, are explained by solving one of the most used reference model in seismology (ETAS), which …
It is well accepted that, at the global scale, the Gutenberg-Richter (GR) law describing the distribution of earthquake magnitude or seismic moment has to be modified at the tail to properly account for the most extreme events. It is debated, though, how much additional time of earthquake recording will be necessary to properly constrain this tail. Using the global CMT catalog, we study how three modifications of the GR law that incorporate a corner-value parameter are compatible with the size of the largest observed earthquake in a given time window. Current data lead to a rather large range …
The unified scaling law for earthquakes, proposed by Bak, Christensen, Danon and Scanlon, is shown to hold worldwide, as well as for areas as diverse as Japan, New Zealand, Spain or New Madrid. The scaling functions that account for the rescaled recurrence-time probability densities show a power-law behavior for long times, with a universal exponent about (minus) 2.2. Another decreasing power law governs short times, but with an exponent that may change from one area to another. This is in contrast with a spatially independent, time-homogenized version of Bak et al's procedure, which seems to …
Using the ETAS branching model of triggered seismicity, we apply the formalism of generating probability functions to calculate exactly the average difference between the magnitude of a mainshock and the magnitude of its largest aftershock over all generations. This average magnitude difference is found empirically to be independent of the mainshock magnitude and equal to 1.2, a universal behavior known as Bath's law. Our theory shows that Bath's law holds only sufficiently close to the critical regime of the ETAS branching process. Allowing for error bars +- 0.1 for Bath's constant value arou…
Earthquakes are complex phenomena characterized by some fundamental seismic laws. However, under the framework of brittle fracture in disordered material, these universal scaling behaviors, especially for critical exponents, are still far from understood. In this Letter, we study avalanches in compression of a nonbrittle material, the ductile metallic glass foam. In addition to elasticity and disorder, more complex inelastic effects can be also introduced by strain localization in cellular structures. We show that different from brittle fracture, the additional shear-banding interactions resul…
Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）；Observatoire de la Côte d’Azur, Université Côte d’Azur（蔚蓝海岸天文台，蔚蓝海岸大学）；CNRS, IRD, Géoazur（法国国家科学研究中心、海外发展研究院、Géoazur）；University of California, Riverside（加州大学河滨分校）
The aftershock productivity law, first described by Utsu in 1970, is an exponential function of the form K=K0.exp(αM) where K is the number of aftershocks, M the mainshock magnitude, and α the productivity parameter. The Utsu law remains empirical in nature although it has also been retrieved in static stress simulations. Here, we explain this law based on Solid Seismicity, a geometrical theory of seismicity where seismicity patterns are described by mathematical expressions obtained from geometric operations on a permanent static stress field. We recover the exponential form but with a break …
Based on several empirical evidence, a series of papers has advocated the concept that seismicity prior to a large earthquake can be understood in terms of the statistical physics of a critical phase transition. In this model, the cumulative Benioff strain (BS) increases as a power-law time-to-failure before the final event. This power law reflects a kind of scale invariance with respect to the distance to the critical point. A few years ago, on the basis of a fit of the cumulative BS released prior to the 1989 Loma Prieta earthquake, Sornette and Sammis [1995] proposed that this scale invaria…
The optimal scaling problem for the time t(LxL) between two successive events in a seismogenic cell of size L is considered. The quantity t(LxL) is defined for a random cell of a grid covering a seismic region G. We solve that problem in terms of a multifractal characteristic of epicenters in G known as the tau-function or generalized fractal dimensions; the solution depends on the type of cell randomization. Our theoretical deductions are corroborated by California seismicity with magnitude M>2. In other words, the population of waiting time distributions for L = 10-100 km provides positive i…
Using the standard ETAS model of triggered seismicity, we present a rigorous theoretical analysis of the main statistical properties of temporal clusters, defined as the group of events triggered by a given main shock of fixed magnitude m that occurred at the origin of time, at times larger than some present time t. Using the technology of generating probability function (GPF), we derive the explicit expressions for the GPF of the number of future offsprings in a given temporal seismic cluster, defining, in particular, the statistics of the cluster's duration and the cluster's offsprings maxim…
We report on an extensive characterization of the cracking noise produced by charcoal samples when dampened with ethanol. We argue that the evaporation of ethanol causes transient and irregularly distributed internal stresses that promote the fragmentation of the samples and mimic some situations found in mining processes. The results show that, in general, the most fundamental seismic laws ruling earthquakes (Gutenberg-Richter law, unified scaling law for the recurrence times, Omori's law, productivity law and Bath's law) hold under the conditions of the experiment. Some discrepancies were al…
Crustal deformation due to the 2016 earthquake sequence in Kumamoto, Japan, that culminated in a preceding earthquake of magnitude M6.5 and a subsequent M7.3 earthquake 28 hours later, caused stress perturbation on and around the causative Futagawa-Hinagu fault zone. Monitoring changes in seismicity pattern along this zone plays a role in understanding the process before and after major earthquakes. For this purpose, stress-dependent laws in statistical seismology can be used: the Gutenberg-Richter frequency-size law and the Omori-Utsu aftershock-decay law. We review the results obtained by us…
This paper presents an explanation of a possible mechanism underlying the shape of the universal curve of Scaling Law for Earthquake Recurrence Time Distributions. The presented simple stochastic cellular automaton model is reproducing the gamma distribution fit with the proper value of the parameter $γ$ characterizing Earth's seismicity and also imitates a deviation from the fit at the short interevent times, as observed in real data. Thus the model suggests an explanation of the universal pattern of rescaled Earthquake Recurrence Time Distributions in terms of combinatorial rules for accumul…
We propose a simple theory for the ``universal'' scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of inter-event times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforces the view t…
We show that Bak et al.'s unified scaling law cannot be trivially explained by a single Omori aftershock sequence. On the contrary, the spatial heterogeneity of seismicity plays a fundamental role in the unified scaling law, which implies that the law provides a way to quantify seismic patterns in space.
The understanding of long-distance relations between seismic activities has for long been of interest to seismologists and geologists. In this paper we have used data from the world-wide earthquake catalog for the period between 1972 and 2011 to generate a network of sites around the world for earthquakes with magnitude m $\geq$ 4.5 in the Richter scale. After the network construction, we have analyzed the results under two viewpoints. Firstly, in contrast to previous works, which have considered just small areas, we showed that the best fitting for networks of seismic events is not a pure pow…
We analyse the statistical pattern of seismicity before a 1-2 103 m3 chalk cliff collapse on the Normandie ocean shore, Western France. We show that a power law acceleration of seismicity rate and energy in both 40 Hz-1.5 kHz and 2 Hz-10kHz frequency range, is defined on 3 order of magnitude, within 2 hours from the collapse time. Simultaneously, the average size of the seismic events increases toward the time to failure. These in-situ results are derived from the only station located within one rupture length distance from the rock fall rupture plane. They mimic the "critical point" like beha…
We report the analysis of the statistics of the phase fluctuations in the coda of earthquakes recorded during a temporary experiment deployed at Pinyon Flats Observatory, California. The practical measurement of the phase is discussed and the main pitfalls are underlined. For large values, the experimental distributions of the phase first, second and third derivatives obey universal power-law decays whose exponents are remarkably well predicted by circular Gaussian statistics. For small values, these distributions are flat. The details of the transition between the plateau and the power-law be…
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…
Time-history deformation analyses of upstream-raised tailings dams use seismic records as input data. Such records must be representative of the in-situ seismicity in terms of a wide range of intensity measures (IMs) including peak ground acceleration (PGA), Arias intensity (AI), cumulative absolute velocity (CAV), source-to-site distance, duration, among others. No single IM is a sufficient descriptor of a given seismic demand (e.g. crest settlement) because different records, all of them compliant with any IM, can produce a very wide range of results from insignificant damage to global failu…
The theory about the brittle failures of multiple locked patches in a seismogenic fault system developed by us since 2010 is introduced in the present study. It is stated by the theory that the progressive failures of locked patch result in the occurrence of earthquakes due to fault movement, where the major earthquakes occurred at its volume expansion and peak strength points are referred to as characteristic ones. We analyze the seismogenic law of characteristic earthquakes in the Islamabad-Kathmandu seismic zone that extends along Himalayan seismic belt, and assess the seismicity trend of t…
The Collaboratory for the Study of Earthquake Predictability (CSEP) aims to prospectively test time-dependent earthquake probability forecasts on their consistency with observations. To compete, time-dependent seismicity models are calibrated on earthquake catalog data. But catalogs contain much observational uncertainty. We study the impact of magnitude uncertainties on rate estimates in clustering models, on their forecasts and on their evaluation by CSEP's consistency tests. First, we quantify magnitude uncertainties. We find that magnitude uncertainty is more heavy-tailed than a Gaussian, …
We address several questions on the behavior of a numerical model recently introduced to study seismic phenomena, that includes relaxation in the plates as a key ingredient. We make an analysis of the scaling of the largest events with system size, and show that when parameters are appropriately interpreted, the typical size of the largest events scale as the system size, without the necessity to tune any parameter. Secondly, we show that the temporal activity in the model is inherently non-stationary, and obtain from here justification and support for the concept of a "seismic cycle" in the t…
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…
We base our study on the statistical analysis of the Rigan earthquake 2010 December 20, which consists of estimating the earthquake network by means of virtual seismometer technique, and also considering the avalanche-type dynamics on top of this complex network.The virtual seismometer complex network shows power-law degree distribution with the exponent $γ=2.3\pm 0.2$. Our findings show that the seismic activity is strongly intermittent, and have a \textit{cyclic shape} as is seen in the natural situations, which is main finding of this study. The branching ratio inside and between avalanches…
Despite the enormous efforts towards searching for precursors, no precursors have exhibited real predictive power with respect to an earthquake thus far. Seismogenic locked segments that can accumulate adequate strain energy to cause major earthquakes are very heterogeneous and less brittle; progressive failures of the locked segments with these properties can produce an interesting seismic phenomenon: a characteristic earthquake and a sequence of smaller subsequent earthquakes (pre-shocks) always arise prior to another characteristic earthquake within a well-defined seismic zone and its curre…
The distribution of the magnitudes of seismic events is generally assumed to be independent on past seismicity. However, by considering events in causal relation, for example mother-daughter, it seems natural to assume that the magnitude of a daughter event is conditionally dependent on the one of the corresponding mother event. In order to find experimental evidence supporting this hypothesis, we analyze different catalogs, both real and simulated, in two different ways. From each catalog, we obtain the law of triggered events' magnitude by kernel density. The results obtained show that the d…
We show that seismic waiting time distributions in California and Iceland have many features in common as, for example, a power-law decay with exponent $α\approx 1.1$ for intermediate and with exponent $γ\approx 0.6$ for short waiting times. While the transition point between these two regimes scales proportionally with the size of the considered area, the full distribution is not universal and depends in a non-trivial way on the geological area under consideration and its size. This is due to the spatial distribution of epicenters which does \emph{not} form a simple mono-fractal. Yet, the dep…
We study the solution of the system of equations describing the dynamical evolution of spontaneous ruptures generated in a prestressed elastic-gravitational deforming body and governed by rate and state friction laws. We propose an iterative coupling scheme based on a weak formulation with nonlinear interior boundary conditions, both for continuous time and with implicit discretization (backward Euler) in time. We regularize the problem by introducing viscosity. This guarantees the convergence of the scheme for solutions of the regularized problems in both cases. We also make precise the condi…
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…
The empirical Bath's law is derived from the magnitude-difference statistical distribution of earthquake pairs. The pair distribution related to earthquake correlations is presented. The single-event distribution of dynamically correlated earthquakes is derived, by means of the geometric-growth model of energy accumulation in the focal region. The dynamical correlations may account, at least partially, for the roll-off effect in the Gutenberg-Richter distributions. The seismic activity which accompanies a main shock, including both the aftershocks and the foreshocks, can be viewed as fluctuati…
In the present paper, we analyze the signatures of long-range persistence in seismic sequences along Circum-Pacific subduction zones, from Chile to Kermadec, extracted from the National Earthquake Information Center (NEIC) catalog. This region, known as the Pacific Ring of Fire, is the world's most active fault line, containing about 90$\%$ of the world's earthquakes. We used the classical rescaled range ($R/S$) analysis to estimate the long-term persistence signals derived from a scaling parameter called the Hurst exponent, $H$. We measured the referred exponent and obtained values of $H>0.5$…
Nearly all aspects of earthquake rupture are controlled by the friction along the fault that progressively increases with tectonic forcing, but in general cannot be directly measured. We show that fault friction can be determined at any time, from the continuous seismic signal. In a classic laboratory experiment of repeating earthquakes, we find that the seismic signal follows a specific pattern with respect to fault friction, allowing us to determine the fault's position within its failure cycle. Using machine learning, we show that instantaneous statistical characteristics of the seismic sig…
Discovery of a new law for three dimensional spatial distance between the foci of successive earthquakes is reported. Analyzing the seismic data taken between 1984 and 2001 in southern California, it is found that the cumulative distribution of the distances follows the modified Zipf-Mandelbrot law, showing complexity of geometry of the events.
We propose a ``multifractal stress activation'' model combining thermally activated rupture and long memory stress relaxation, which predicts that seismic decay rates after mainshocks follow the Omori law $\sim 1/t^p$ with exponents $p$ linearly increasing with the magnitude $M_L$ of the mainshock and the inverse temperature. We carefully test this prediction on earthquake sequences in the Southern California Earthquake catalog: we find power law relaxations of seismic sequences triggered by mainshocks with exponents $p$ increasing with the mainshock magnitude by approximately $0.1-0.15$ for e…
Our understanding of earthquakes is based on the theory of plate tectonics. Earthquake dynamics is the study of the interactions of plates (solid disjoint parts of the lithosphere) which produce seismic activity. Over the last about fifty years many models have come up which try to simulate seismic activity by mimicking plate plate interactions. The validity of a given model is subject to the compliance of the synthetic seismic activity it produces to the well known empirical laws which describe the statistical features of observed seismic activity. Here we present a review of two such models …
Aftershock sequences are of particular interest in seismic research since they may condition seismic activity in a given region over long time spans. While they are typically identified with periods of enhanced seismic activity after a large earthquake as characterized by the Omori law, our knowledge of the spatiotemporal correlations between events in an aftershock sequence is limited. Here, we study the spatiotemporal correlations of two aftershock sequences form California (Parkfield and Hector Mine) using the recently introduced concept of "recurrent" events. We find that both sequences ha…
Brittle fractures of inhomogeneous materials like rocks, concrete, or ceramics are of two types: Nominally brittle and driven by the propagation of a single dominant crack or quasi-brittle and resulting from the accumulation of many microcracks. The latter goes along with acoustic noise, whose analysis has revealed that events form aftershock sequences obeying characteristic laws reminiscent of those in seismology. Yet, their origin lacks explanation. Here we show that such a statistical organization is not only specific to the multi-cracking situations of quasi-brittle failure and seismology,…
Statistical properties of the one-dimensional spring-block (Burridge-Knopoff) model of earthquakes obeying the rate and state dependent friction law are studied by extensive computer simulations. The quantities computed include the magnitude distribution, the rupture-length distribution, the mainshock recurrence-time distribution, the seismic time correlations before and after the mainshock, the mean slip amount, and the mean stress drop at the mainshock, etc. Events of the model can be classified into two distinct categories. One tends to be unilateral with its epicenter located at the rim of…
