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 around 1.2, our exact analytical treatment of Bath's law provides new constraints on the productivity exponent alpha and the branching ratio n: $0.9 <= alpha <= 1$ and 0.8 <= n <= 1. We propose a novel method for measuring alpha based on the predicted renormalization of the Gutenberg-Richter distribution of the magnitudes of the largest aftershock. We also introduce the ``second Bath's law for foreshocks: the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude.
Using the simple ETAS branching model of seismicity, which assumes that each earthquake can trigger other earthquakes, we quantify the role played by the cascade of triggered seismicity in controlling the rate of aftershock decay as well as the overall level of seismicity in the presence of a constant external seismicity source. We show that, in this model, the fraction of earthquakes in the population that are aftershocks is equal to the fraction of aftershocks that are indirectly triggered and is given by the average number of triggered events per earthquake. Previous observations that a sig…
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…
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 …
Earthquake aftershock identification is closely related to the question "Are aftershocks different from the rest of earthquakes?" We give a positive answer to this question and introduce a general statistical procedure for clustering analysis of seismicity that can be used, in particular, for aftershock detection. The proposed approach expands the analysis of Baiesi and Paczuski [PRE, 69, 066106 (2004)] based on the space-time-magnitude nearest-neighbor distance $η$ between earthquakes. We show that for a homogeneous Poisson marked point field with exponential marks, the distance $η$ has Weibu…
Currently, one of the best performing and most popular earthquake forecasting models rely on the working hypothesis that: "locations of past background earthquakes reveal the probable location of future seismicity". As an alternative, we present a class of smoothed seismicity models (SSMs) based on the principles of the Epidemic Type Aftershock Sequence (ETAS) model, which forecast the location, time and magnitude of all future earthquakes using the estimates of the background seismicity rate and the rates of future aftershocks of all generations. Using the Californian earthquake catalog, we f…
The empirical Bath's law states that the average difference in magnitude between a mainshock and its largest aftershock is 1.2, regardless of the mainshock magnitude. Following Vere-Jones [1969] and Console et al. [2003], we show that the origin of Bath's law is to be found in the selection procedure used to define mainshocks and aftershocks rather than in any difference in the mechanisms controlling the magnitude of the mainshock and of the aftershocks. We use the ETAS model of seismicity, which provides a more realistic model of aftershocks, based on (i) a universal Gutenberg-Richter (GR) la…
Any periodic variations of earthquake occurrence rates in response to small, known, periodic stress variations provide important opportunities to learn about the earthquake nucleation process. Yet, reliable detection of earthquake periodicity is complicated by the presence of earthquake clustering due to aftershocks and foreshocks. Existing methods for detecting periodicity in an earthquake catalogue typically require the prior removal of these clustered events. Declustering is a highly uncertain process, so declustering methods are inherently non-unique. Incorrect declustering may remove some…
The spatio-temporal properties of seismicity give us incisive insight into the stress state evolution and fault structures of the crust. Empirical models based on self-exciting point-processes continue to provide an important tool for analyzing seismicity, given the epistemic uncertainty associated with physical models. In particular, the epidemic-type aftershock sequence (ETAS) model acts as a reference model for studying seismicity catalogs. The traditional ETAS model uses simple parametric definitions for the background rate of triggering-independent seismicity. This reduces the effectivene…
Earthquakes and aftershock sequences follow several empirical scaling laws: One of these laws is Bath's law for the magnitude of the largest aftershock. In this work, Modified Form of Bath's Law and its application to KOERI data have been studied. Bath's law states that the differences in magnitudes between mainshocks and their largest detected aftershocks are approximately constant, independent of the magnitudes of mainshocks and it is about 1.2. In the modified form of Bath's law for a given mainshock we get the inferred largest aftershock of this mainshock by using an extrapolation of the G…
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 …
The development of new earthquake forecasting models is often motivated by one of the following complementary goals: to gain new insights into the governing physics and to produce improved forecasts quantified by objective metrics. Often, one comes at the cost of the other. Here, we propose a question-driven ensemble (QDE) modeling approach to address both goals. We first describe flexible ETAS models in which we relax the assumptions of parametrically defined aftershock productivity and background earthquake rates during model calibration. Instead, both productivity and background rates are c…
We estimate the rate of aftershocks triggered by a heterogeneous stress change, using the rate-and-state model of Dieterich [1994].We show that an exponential stress distribution Pt(au) ~exp(-tautau_0) gives an Omori law decay of aftershocks with time ~1/t^p, with an exponent p=1-A sigma_n/tau_0, where A is a parameter of the rate-and-state friction law, and σ_n the normal stress. Omori exponent p thus decreases if the stress "heterogeneity" tau_0 decreases. We also invert the stress distribution P(tau) from the seismicity rate R(t), assuming that the stress does not change with time. We apply…
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…
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…
The productivity of a magnitude $m$ event can be characterized in term of triggered events of magnitude above $m-Δ$: it is the number of direct "descendants" $ν_Δ$ and the number of all "descendants" $V_Δ$. There is evidence in favour of the discrete exponential distribution for both $ν_Δ$ and $V_Δ$ with a dominant magnitude $m$ (the case of aftershock cluster). We consider the general Epidemic Type Aftershock Sequence (ETAS) model adapted to any distribution of $ν_Δ$. It turns out that the branching structure of the model excludes the possibility of having exponential distributions for both p…
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…
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…
Seismicity and faulting within the Earth 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 …
The observation of foreshocks preceding large earthquakes and the suggestion that foreshocks have specific properties that may be used to distinguish them from other earthquakes have raised the hope that large earthquakes may be predictable. Among proposed anomalous properties are the larger proportion than normal of large versus small foreshocks, the power law acceleration of seismicity rate as a function of time to the mainshock and the spatial migration of foreshocks toward the mainshock, when averaging over many sequences. Using Southern California seismicity, we show that these properties…
We propose and study a modified version of the Olami-Feder-Christiensen model of seismicity, that includes a mechanism of structural relaxation. We obtain realistic features of seismicity that are not obtained with the original version, mainly: aftershocks that obey the Omori law and cluster spatially around the slip surface of the main shock, and averaged frictional properties qualitatively similar to those observed in rock friction, in particular the velocity weakening effect.
Declustering aims to divide earthquake catalogs into independent events (mainshocks), and dependent (clustered) events, and is an integral component of many seismicity studies, including seismic hazard assessment. We assess the effect of declustering on the frequency-magnitude distribution of mainshocks. In particular, we examine the dependence of the b-value of declustered catalogs on the choice of declustering approach and algorithm-specific parameters. Using the catalog of earthquakes in California since 1980, we show that the b-value decreases by up to 30% due to declustering with respect …
We study how a stress perturbation generated by a mainshock affects a population of faults obeying a rate-state friction law. Depending on the model parameters and on the initial state, the fault exhibits aftershocks, slow earthquakes, or decaying afterslip. We found several regimes with slip rate decaying as a power-law of time, with different characteristic times and exponents. The complexity of the model makes it unrealistic to invert for the friction law parameters from afterslip data. We modeled afterslip measurements for the Southern California Superstition Hills earthquake using the com…
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…
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…
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…
It is natural to call the trinity of foreshocks, main shock and aftershocks a classical triad. The magnitude of the main shock is always greater than the maximum magnitudes of foreshocks and aftershocks. In the classical triad, the number of foreshocks is less than the number of aftershocks. In many cases, foreshocks are completely absent even before very strong earthquakes. In this work, we found a rare but extremely interesting species of triad (we called it the mirror triad), in which the number of foreshocks is greater than the number of aftershocks at the interval of 24 h before and 24 h …
We present the "condensation" method that exploits the heterogeneity of the probability distribution functions (PDF) of event locations to improve the spatial information content of seismic catalogs. The method reduces the size of seismic catalogs while improving the access to the spatial information content of seismic catalogs. The PDFs of events are first ranked by decreasing location errors and then successively condensed onto better located and lower variance event PDFs. The obtained condensed catalog attributes different weights to each event, providing an optimal spatial representation w…
We find the static displacement, stress, strain and the modified Columb failure stress produced in an elastic medium by a finite size rectangular fault after its dislocation with uniform stress drop but a non uniform dislocation on the source. The time-dependent rate of triggered earthquakes is estimated by a rate-state model applied to a uniformly distributed population of faults whose equilibrium is perturbated by a stress change caused only by the first dislocation. The rate of triggered events in our simulations is exponentially proportional to the stress change, but the time at which the …
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…
Three fundamental laws of the physics of earthquakes, bearing the names of their discoverers Omori, Gutenberg, Richter and Bath, are widely used in original, review, monographic, and encyclopedic literature. In this paper, we have tried to look at laws from an unusual point of view. The idea behind our approach is as follows. Each of the three laws was established in the course of work on solving a specific research problem, and the result was expressed in a specific mathematical form. We asked ourselves whether it is possible to modify the statement of the problem, and/or modify the form of e…
We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock T_{c} to be the time for which the market volatility V(T_{c}) has a peak that exceeds a predetermined threshold. The cascade of high volatility "aftershocks" triggered by the "main shock" is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws --- the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the two-year period 2001-2002 at the 1-minute time res…
We consider two statistical regularities that were used to explain Omori's law of the aftershock rate decay: the Levy and Inverse Gaussian (IGD) distributions. These distributions are thought to describe stress behavior influenced by various random factors: post-earthquake stress time history is described by a Brownian motion. Both distributions decay to zero for time intervals close to zero. But this feature contradicts the high immediate aftershock level according to Omori's law. We propose that these statistical distributions are influenced by the power-law stress distribution near the eart…
The Multifractal Stress-Activated (MSA) model is a statistical model of triggered seismicity based on mechanical and thermodynamic principles. It predicts that, above a triggering magnitude cut-off $M_0$, the exponent $p$ of the Omori law for the seismic decay of aftershocks is a linear increasing function $p(M) =a M+b$ of the main shock magnitude $M$. We previously reported empirical support for this prediction, using the Southern California SCEC catalog. Here, we confirm this law using an updated, longer version of the same catalog, as well as new methods to estimate $p$. One of this methods…
The complete part of the earthquake frequency-magnitude distribution (FMD), above completeness magnitude mc, is well described by the Gutenberg-Richter law. The parameter mc however varies in space due to the seismic network configuration, yielding a convoluted FMD shape below max(mc). This paper investigates the shape of the generalized FMD (GFMD), which may be described as a mixture of elemental FMDs (eFMDs) defined as asymmetric Laplace distributions of mode mc [Mignan, 2012, https://doi.org/10.1029/2012JB009347]. An asymmetric Laplace mixture model (GFMD- ALMM) is thus proposed with its pa…
