Omori Law describes the evolution of the aftershocks of a strong earthquake. Established at the end of the century before last, it is characterized by the beauty of its form, quite definite clarity, as a result of which it still attracts considerable attention of the geophysical community. In recent years, we have accumulated considerable experience in studying the Omori law by theoretical and experimental methods. This paper summarizes the results of our study briefly. The main attention is focused on the phenomenological theory of aftershocks, the foundations of which were laid by Fusakichi Omori. Keywords: earthquake, source deactivation, logistic equation, nonlinear diffusion equation, Omori epoch, round-the-world echo, mirror triad.
One hundred years ago, Fusakichi Omori died. Our paper is dedicated to his memory. Omori made an outstanding contribution to the physics of earthquakes. In 1894 he formulated the law of aftershock evolution. Omori's Law states that after the main shock of an earthquake, the frequency of aftershocks decreases hyperbolically with time. In this paper, we briefly describe one of the directions of modern aftershock research. Keywords: earthquake, mainshock, aftershock, evolution equation, deactivation coefficient, inverse problem, trigger, proper time.
The paper is based on the report read by the author on October 24, 2018 at the meeting of the Scientific Council of the Institute of Earth Physics of the Russian Academy of Sciences. The report was dedicated to the 150th anniversary of the outstanding Japanese seismologist Fusakichi Omori. As is known, Omori established the first empirical law of the earthquakes physics, bearing his name. The Omori law states that the frequency of aftershocks on average decreases hyperbolically over the time. Three versions of Omori law are described briefly. The recent version allows to poses the inverse prob…
The earthquake source after the main shock can theoretically be represented as a black box without an entrance. At the output, there is a signal in the form of aftershocks, the frequency of which decreases on average with time according to the Omori law. The task of the researcher is to evaluate the structure, state and mechanism of functioning of a dynamic system simulating the earthquake source based on the output signal. In this paper, we outline an approach to a partial solution of this general problem. Omori's law is presented as a differential equation of aftershock evolution. An inverse…
A spatio-temporal version of the well-known Omori-Utsu law of aftershock sequences is proposed. This 'diffusive Omori-Utsu law' satisfies a nonlinear partial differential equation (PDE). A similarity reduction is obtained that reduces the PDE to an ordinary differential equation (ODE). A nonzero constant solution of this ODE leads to the usual Omori-Utsu law. An exact and explicit similarity solution is found that corresponds to the original Omori law. An initial value problem for the 'diffusive Omori-Utsu law' is also considered, and whose spatio-temporal dynamics are described by bounding fu…
The presented paper is devoted to the search for mathematical basis for describing the aftershock evolution of strong earthquakes. We consider the experimental facts and heuristic arguments that allow to make a choice and to focus on the nonlinear diffusion equation as the master equation. Analysis of the master equation indicates that, apparently, the selected mathematical basis makes it possible to simulate two important properties of the aftershock evolution known from the experiment. We are talking about the Omori law and the slow propagation of aftershocks from the epicenter of the main s…
Recently, simple differential models of aftershocks have attracted increased attention of researchers at the Institute of Physics of the Earth, Russian Academy of Sciences for their remarkable properties. The shortened Bernoulli equation allows us to introduce the concept of the source deactivation factor and phenomenologically describe the nonstationarity of the geological environment in the earthquake source, "cooling down" after the main shock. The Verhulst logistic equation makes it possible to take into account the transition of the aftershock flow to the background seismicity mode. The K…
A modified Carlson-Langer model for earthquakes is proposed, which includes nonlinear visco-elasticity. Several aftershocks are generated after the main shock owing to the damping of the additional visco-elastic force. Both the Gutenberg-Richter law and Omori's law are reproduced in a numerical simulation of the modified Carlson-Langer model on a critical percolation cluster of a square lattice.
The decay rate of aftershocks has been modeled as a power law since the pioneering work of Omori in the late nineteenth century. Considered the second most fundamental empirical law after the Gutenberg-Richter relationship, the power law paradigm has rarely been challenged by the seismological community. By taking a view of aftershock research not biased by prior conceptions of Omori power law decay and by applying statistical methods recommended in applied mathematics, I show that all aftershock sequences tested in three regional earthquake catalogs (Southern and Northern California, Taiwan) …
A part of the seismic time series, in which the Omori law for temporal pattern of aftershocks holds, is refereed to as the Omori regime. Here the properties of correlation between earthquake events both inside and outside of the Omori regime are studied by analysis of the data taken in southern California. It is found that inside of the Omori regime correlation exhibits the aging phenomenon, in marked contrast to the fact that no aging is observed outside of the Omori regime. The scaling nature of aging is also discussed and the form of the function responsible for it is presented. The results…
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…
The known Omori law is presented in the form of differential equation that describes the evolution of the aftershock activity. This equation is derived hypothetically with taking into account deactivation of the faults in epicentral zone of the main shock. A generalization of the Omori law is proposed.
This paper is devoted to the memory of the outstanding Japanese scientist. In 1896, Fusakichi Omori discovered the law of the aftershocks evolution that bears his name. We represent the Omori law in the form of a differential equation. This allows us to take into account the non-stationarity of rocks in the earthquake source, which "cools down" after the main shock.
A recent model predicting Omori's law giving the number of aftershocks per unit time following an earthquake involves a differential equation analogous to the Friedmann equation of cosmology. The before-shock phase is analogous to an accelerating universe approaching a Big Rip, the main shock to the Big Rip singularity, and the aftershock to a contracting universe. The analogy provides some physical intuition and Lagrangian and Hamiltonian formulations for Omori's law and its generalizations.
This paper is devoted to the theory of aftershocks. The history of discovery of the Omori law is briefly described, the initial formulation of the law is given in the form of an algebraic formula describing the decrease in the frequency of aftershocks over time. An important generalization of the Oiori formula which is widely used in modern seismology is presented. The generalized law is also expressed by an algebraic formula, but it contains an additional parameter, which makes it possible to more flexibly approximate the observational data. The alternative approach to the theoretical descrip…
We evaluated the efficiency of various models in describing the time decay of aftershock rate of 47 simple sequences occurred in California (37) from 1933 to 2004 and in Italy (10) from 1976 to 2004. We compared the models by the corrected Akaike Information Criterion (AICc) and the Bayesian Information Criterion (BIC), both based on the log-likelihood function but also including a penalty term that takes into account the number of independent observations and of free parameters of each model. We compared the performance of different models by varying the starting time Ts and the minimum magni…
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…
Strong deep earthquakes occurred in the Sea of Okhotsk, May 24, 2013 at 05:44 UTC. The magnitude of the earthquake is M = 8.3, the depth of the hypocenter is 629 km. The earthquake was accompanied by the deep-focus aftershocks. The paper is devoted to the study of the space-time properties of aftershocks. An estimate is made of the deactivation coefficient of the earthquake source, which "cools down" after the main shock. The modulation of aftershocks by toroidal oscillations of the Earth was discovered. The questions about the round-the-world seismic echo and the ribbed structure of the spati…
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…
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.
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 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…
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 conf…
The analysis of the classical and mirror triads of the sequence of earthquakes has been carried out in order to find the equations of evolution of foreshocks and aftershocks. The differential equation with cubic (quadratic) nonlinearity has been proposed to describe the activity of foreshocks (aftershocks). The solution of the governing equation for foreshocks indicates the explosive instability of rocks, leading to the main rupture and the main shock of the earthquake. It is suggested that explosive instability leads to a first-order phase transition, in which the main parameters of the earth…
We report new tests on the Taiwan earthquake catalog of the prediction by the Multifractal Stress Activation (MSA) model that the p-value of the Omori law for the rate of aftershocks following a mainshock is an increasing function of its magnitude Mm. This effort is motivated by the quest to validate this crucial prediction of the MSA model and to investigate its possible dependence on local tectonic conditions. With careful attention to the long-term as well as short-term time-dependent magnitude completeness of the Taiwan catalog, and with the use of three different declustering techniques, …
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…
We propose a new metric to quantify the correlation between any two earthquakes. The metric consists of a product involving the time interval and spatial distance between two events, as well as the magnitude of the first one. According to this metric, events typically are strongly correlated to only one or a few preceding ones. Thus a classification of events as foreshocks, main shocks or aftershocks emerges automatically without imposing predefined space-time windows. To construct a network, each earthquake receives an incoming link from its most correlated predecessor. The number of aftersho…
Whether aftershocks originate directly from the mainshock and surrounding stress environment or from afterslip dynamics is crucial to the understanding of the nature of aftershocks. We build on a classical description of the fault and creeping regions as two blocks connected elastically, subject to different friction laws. We show analytically that, upon introduction of variability in the fault plane's static friction threshold, a non trivial stick-slip dynamics ensues. In particular we support the hypothesis \citep{PA04} that the aftershock occurrence rate is proportional to the afterslip rat…
We analyzed the correlations among the parameters of the Reasenberg and Jones (1989) formula describing the aftershock rate after a mainshock as a function of time and magnitude, on the basis of parameter estimates made in previous works for New Zealand, Italy and California. For all of three datasets we found that the magnitude-independent productivity a is significantly correlated with the b-value of the Gutenberg-Richter law and, in some cases, with parameters p and c of the modified Omori's law. We argued that the correlation between a and b can be ascribed to an inappropriate definition o…
The recently proposed discrete scale invariance and its associated log-periodicity are an elaboration of the concept of scale invariance in which the system is scale invariant only under powers of specific values of the magnification factor. We report on the discovery of a novel mechanism for such log-periodicity relying solely on the manipulation of data. This ``synthetic'' scenario for log-periodicity relies on two steps: (1) the fact that approximately logarithmic sampling in time corresponds to uniform sampling in the logarithm of time; and (2) a low-pass-filtering step, as occurs in const…
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 invoke a metric to quantify the correlation between any two earthquakes. This provides a simple and straightforward alternative to using space-time windows to detect aftershock sequences and obviates the need to distinguish main shocks from aftershocks. Directed networks of earthquakes are constructed by placing a link, directed from the past to the future, between pairs of events that are strongly correlated. Each link has a weight giving the relative strength of correlation such that the sum over the incoming links to any node equals unity for aftershocks, or zero if the event had no corr…
Earthquakes are complex physical processes driven by the stick-slip motion of a sliding fault. After the main quake, a series of aftershocks typically follows. These are loosely defined as events that follow a given event and occur within prescribed space-time windows. In seismology, it is however impossible to establish a causal relation and the popular Nearest-Neighbor metric is commonly used to distinguish aftershocks from independent events. Here we employ a model for earthquake dynamics, previously shown to be able to correctly reproduce the phenomenology of earthquakes, and a technique t…
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 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…
Earthquake Aftershock Forecasting using Conditional Generative Models
Department of Earth and Planetary Science, University of California, Berkeley（加州大学伯克利分校地球与行星科学系）；Berkeley Seismological Laboratory, University of California, Berkeley（加州大学伯克利分校伯克利地震实验室）
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 statistical properties of avalanches in a dissipative particulate system under slow shear are investigated using molecular dynamics simulations. It is found that the magnitude-frequency distribution obeys the Gutenberg-Richter law only in the proximity of a critical density and that the exponent is sensitive to the minute changes in density. It is also found that aftershocks occur in this system with a decay rate that follows the Modified Omori law. We show that the exponent of the magnitude-frequency distribution and the time constant of the Modified Omori law are decreasing functions of …
Phenomenology is the unity of principles and methods of studying the essence of phenomena. This paper is a concise review of recent works in which the phenomenological ideas of physics are used to analyze earthquakes. An example of a phenomenological theory is thermodynamics. Maxwell's electrodynamics is also a perfect example of a phenomenological theory. The phenomenology of earthquakes has not yet reached such a level. So far, we have reached the status of prolegomena for the future phenomenology of earthquakes. The current state can be characterized as preliminary. Nevertheless, in this pa…
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 aftershocks of first generation that it triggers, on the probability distribution function (pdf) f(τ) of the recurrence times τbetween successive events. The other input of the model is an exponential Omori l…
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…
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 …
Earthquake phenomenology exhibits a number of power law distributions including the Gutenberg-Richter frequency-size statistics and the Omori law for aftershock decay rates. In search for a basic model that renders correct predictions on long spatio-temporal scales, we discuss results associated with a heterogeneous fault with long range stress-transfer interactions. To better understand earthquake dynamics we focus on faults with Gutenberg-Richter like earthquake statistics and develop two universal scaling functions as a stronger test of the theory against observations than mere scaling expo…
