Omori Law Research Articles

Market dynamics immediately before and after financial shocks: Quantifying the Omori, productivity, and Bath laws

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 2 yr period of 2001-2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude M≡log{10} V(T{c}) and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on M , both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

Network of earthquakes and recurrences therein

We quantify the correlation between earthquakes and use the same to distinguish between relevant causally connected earthquakes. Our correlation metric is a variation on the one introduced by Baiesi and Paczuski (Phys Rev E 69:066,106, 2004). A network of earthquakes is constructed, which is time-ordered and with links between the more correlated ones. Data pertaining to the California region has been used in the study. Recurrences to earthquakes are identified employing correlation thresholds to demarcate the most meaningful ones in each cluster. The distribution of recurrence lengths and recurrence times are analyzed subsequently to extract information about the complex dynamics. We find that the unimodal feature of recurrence lengths helps to associate typical rupture lengths with different magnitude earthquakes. The out-degree of the network shows a hub structure rooted on the large magnitude earthquakes. In-degree distribution is seen to be dependent on the density of events in the neighborhood. Power laws are also obtained with recurrence time distribution agreeing with the Omori law.

Epileptic seizures: Quakes of the brain?

A dynamical analogy supported by five scale-free statistics (the Gutenberg-Richter distribution of event sizes, the distribution of interevent intervals, the Omori and inverse Omori laws, and the conditional waiting time until the next event) is shown to exist between two classes of seizures ("focal" in humans and generalized in animals) and earthquakes. Increments in excitatory interneuronal coupling in animals expose the system's dependence on this parameter and its dynamical transmutability: moderate increases lead to power-law behavior of seizure energy and interevent times, while marked ones to scale-free (power-law) coextensive with characteristic scales and events. The coextensivity of power law and characteristic size regimes is predicted by models of coupled heterogeneous threshold oscillators of relaxation and underscores the role of coupling strength in shaping the dynamics of these systems.

A mechanism for spatial and temporal earthquake clustering

The Gutenberg‐Richter law states that the size‐frequency distribution of earthquakes follows a power law. This trend is usually justified using spring‐block models, where slips with the appropriate statistics of sizes have been numerically observed. However, prominent spatial and temporal clustering features of earthquakes, as those implied by the Omori law of aftershocks, are not accounted for by this kind of model unless they are complemented with ad hoc assumptions, such as stress recovery laws after slip events, or the phenomenological rate‐and‐state equations to describe friction. We show that when a mechanism of structural relaxation is incorporated into a spring‐block model, realistic earthquake patterns following the Gutenberg‐Richter and Omori laws are obtained. Moreover, features well known from laboratory friction experiments, such as velocity weakening and increase of static friction with contact time, appear as a consequence of the relaxational mechanism as well, without making any a priori assumptions on the velocity dependence of the friction force in the model. In this way, our model shows that a single physical mechanism may be a unifying concept behind the Gutenberg‐Richter and Omori laws and the rate‐and‐state equations of rock friction.

Sensitivity study of forecasted aftershock seismicity based on Coulomb stress calculation and rate‐ and state‐dependent frictional response

We use the Dieterich (1994) physics‐based approach to simulate the spatiotemporal evolution of seismicity caused by stress changes applied to an infinite population of nucleating patches modeled through a rate‐ and state‐dependent friction law. According to this model, seismicity rate changes depend on the amplitude of stress perturbation, the physical constitutive properties of faults (represented by the parameter Aσ), the stressing rate, and the background seismicity rate of the study area. In order to apply this model in a predictive manner, we need to understand the impact of physical model parameters and the correlations between them. First, we discuss different definitions of the reference seismicity rate and show their impact on the computed rate of earthquake production for the 1992 Landers earthquake sequence as a case study. Furthermore, we demonstrate that all model parameters are strongly correlated for physical and statistical reasons. We discuss this correlation, emphasizing that the estimations of the background seismicity rate, stressing rate, and Aσ are strongly correlated to reproduce the observed aftershock productivity. Our analytically derived relation demonstrates the impact of these model parameters on the Omori‐like aftershock decay: the c value and the productivity of the Omori law, implying a p value smaller than or equal to 1. Finally, we discuss an optimal strategy to constrain model parameters for near‐real‐time forecasts.

Realistic spatial and temporal earthquake distributions in a modified Olami-Feder-Christensen model

The Olami-Feder-Christensen model describes a limiting case of an elastic surface that slides on top of a substrate and is one of the simplest models that display some features observed in actual seismicity patterns. However, temporal and spatial correlations of real earthquakes are not correctly described by this model in its original form. I propose and study a modified version of the model, which includes a mechanism of structural relaxation. With this modification, realistic features of seismicity emerge, which are not obtained with the original version, mainly: aftershocks that cluster spatially around the slip surface of the main shock and follow the Omori law, and averaged frictional properties similar to those observed in rock friction, in particular the velocity-weakening effect. In addition, a Gutenberg-Richter law for the decaying of number of earthquakes with magnitude is obtained, with a decaying exponent within the range of experimentally observed values. Contrary to the original version of the model, a realistic value of the exponent appears without the necessity to fine tune any parameter.

Multiple-Time Scaling and Universal Behavior of the Earthquake Interevent Time Distribution

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

Short- and Long-Term Earthquake Forecasts for California and Nevada

We present estimates of future earthquake rate density (probability per unit area, time, and magnitude) on a 0.1-degree grid for a region including California and Nevada, based only on data from past earthquakes. Our long-term forecast is not explicitly time-dependent, but it can be updated at any time to incorporate information from recent earthquakes. The present version, founded on several decades worth of data, is suitable for testing without updating over a five-year period as part of the experiment conducted by the Collaboratory for Study of Earthquake Predictability (CSEP). The short-term forecast is meant to be updated daily and tested against similar models by CSEP. The short-term forecast includes a fraction of our long-term one plus time-dependent contributions from all previous earthquakes. Those contributions decrease with time according to the Omori law: proportional to the reciprocal of the elapsed time. Both forecasts estimate rate density using a radially symmetric spatial smoothing kernel decreasing approximately as the reciprocal of the square of epicentral distance, weighted according to the magnitude of each past earthquake. We made two versions of both the long- and short-term forecasts, based on the Advanced National Seismic System (ANSS) and Preliminary Determinations of Epicenters (PDE) catalogs, respectively. The two versions are quite consistent, but for testing purposes we prefer those based on the ANSS catalog since it covers a longer time interval, is complete to a lower magnitude threshold and has more precise locations. Both forecasts apply to shallow earthquakes only (depth 25 km or less) and assume a tapered Gutenberg-Richter magnitude distribution extending to a lower threshold of 4.0.

The dynamics of the acoustic emission with water initiation

The long-term experiments on the biaxial compression of the models prepared from basalt sand, crushed limestone, and cement are carried out. The volume of the models exceeded 4 × 103 cm3. At different grades of load, water was infused into the model through the borehole. The volume of the infused water was less than 0.1% of the volume of the model itself. Water infusion resulted in the activation of the acoustic emission with a particular time lag. The initiation of the acoustic emission is connected with the local reduction in the strength and with the increase in the stresses near the metastable, which does not contradict the trigger mechanism. The hypocenters of the sources of acoustic signals initiated by water infusion lay in the vicinity of the boreholes. Within the time interval between weak events, pulses with higher energy appeared, which has an analogy with the swarm of earthquakes. The growth and extinction of the acoustic emission after the stepwise additional load and with the initiation by water infusion is significantly different. In the first case, the Omori law is obeyed. In the second case, the intensity of acoustic emission has a pronounced maximum. The equations of the kinetic concept of strength in solids make it possible to describe the dynamics of the acoustic emission caused by water infusion.

Statistical Properties of Mine Tremor Aftershocks

Mine tremors and their aftershocks pose a risk to mine workers in the deep gold mines of South Africa. The statistical properties of mine-tremor aftershocks were investigated as part of an endeavour to assess the hazard and manage the risk. Data from two gold mines in the Carletonville mining district were used in the analysis. Main shocks were aligned in space and time and the aftershock sequences stacked and analysed. The aftershocks were found to satisfy Gutenberg–Richter scaling, with a b value close to 1. Aftershock activity diminished with time in accordance with the modified Omori law, with p values close to 1. However, the relationship between the main shock and its biggest aftershock violated Baths law, with ΔM L ≈ 1.9 for main shocks with M L < 3 and increasing for main shocks with M L > 3. The aftershock density was found to fall-off with distance as r −1.3, suggesting triggering by dynamic stress.

A rate-state model for aftershocks triggered by dislocation on a rectangular fault: a review and new insights

We compute the static displacement, stress, strain and the Coulomb 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 perturbed by a stress change caused only by the first dislocation. The rate of triggered events in our simulations is exponentially proportional to the shear stress change, but the time at which the maximum rate begins to decrease is variable from fractions of hour for positive stress changes of the order of some MPa, up to more than a year for smaller stress changes. As a consequence, the final number of triggered events is proportional to the shear stress change. The model predicts that the total number of events triggered on a plane containing the fault is proportional to the 2/3 power of the seismic moment. Indeed, the total number of aftershocks produced on the fault plane scales in magnitude, M, as 10 M . Including the negative contribution of the stress-drop inside the source, we observe that the number of events inhibited on the fault is, at long term, nearly identical to the number of those induced outside, representing a sort of conservative natural rule. Considering its behavior in time, our model does not completely match the popular Omori Law; in fact it has been shown that the seismicity induced closely to the fault edges is intense but of short duration, while that expected at large distances (up to some tens times the fault dimensions) exhibits a much slower decay.

Multifractal Omori law for earthquake triggering: new tests on the California, Japan and worldwide catalogues

SUMMARY The Multifractal Stress-Activated 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 time decay of the rate of aftershocks is a linear increasing function p(M) = a 0 M + b0 of the main shock magnitude M. We previously reported empirical support for this prediction, using the Southern California Earthquake Center (SCEC) catalogue. Here, we confirm this observation using an updated, longer version of the same catalogue, as well as new methods to estimate p. One of this methods is the newly defined Scaling Function Analysis (SFA), adapted from the wavelet transform. This method is able to measure a mathematical singularity (hence a p-value), erasing the possible regular part of a time-series. The SFA also proves particularly efficient to reveal the coexistence and superposition of several types of relaxation laws (typical Omori sequences and short-lived swarms sequences) which can be mixed within the same catalogue. Another new method consists in monitoring the largest aftershock magnitude observed in successive time intervals, and thus shortcuts the problem of missing events with small magnitudes in aftershock catalogues. The same methods are used on data from the worldwide Harvard Centroid Moment Tensor (CMT) catalogue and show results compatible with those of Southern California. For the Japan Meteorological Agency (JMA) catalogue, we still observe a linear dependence of p on M ,b ut with a smaller slope. The SFA shows however that results for this catalogue may be biased by numerous swarm sequences, despite our efforts to remove them before the analysis.

Strain waves, earthquakes, slow earthquakes, and afterslip in the framework of the Frenkel-Kontorova model

The one-dimensional Frenkel-Kontorova (FK) model, well known from the theory of dislocations in crystal materials, is applied to the simulation of the process of nonelastic stress propagation along transform faults. Dynamic parameters of plate boundary earthquakes as well as slow earthquakes and afterslip are quantitatively described, including propagation velocity along the strike, plate boundary velocity during and after the strike, stress drop, displacement, extent of the rupture zone, and spatiotemporal distribution of stress and strain. The three fundamental speeds of plate movement, earthquake migration, and seismic waves are shown to be connected in framework of the continuum FK model. The magnitude of the strain wave velocity is a strong (almost exponential) function of accumulated stress or strain. It changes from a few km/s during earthquakes to a few dozen km per day, month, or year during afterslip and interearthquake periods. Results of the earthquake parameter calculation based on real data are in reasonable agreement with measured values. The distributions of aftershocks in this model are consistent with the Omori law for temporal distribution and a 1/r for the spatial distributions.

Correlated earthquakes in a self-organized model

Abstract. Motivated by the fact that empirical time series of earthquakes exhibit long-range correlations in space and time and the Gutenberg-Richter distribution of magnitudes, we propose a simple fault model that can account for these types of scale-invariance. It is an avalanching process that displays power-laws in the event sizes, in the epicenter distances as well as in the waiting-time distributions, and also aftershock rates obeying a generalized Omori law. We thus confirm that there is a relation between temporal and spatial clustering of the activity in this kind of models. The fluctuating boundaries of possible slipping areas show that the size of the largest possible earthquake is not always maximal, and the average correlation length is a fraction of the system size. This suggests that there is a concrete alternative to the extreme interpretation of self-organized criticality as a process in which every small event can cascade to an arbitrary large one: the new picture includes fluctuating domains of coherent stress field as part of the global self-organization. Moreover, this picture can be more easily compared with other scenarios discussing fluctuating correlations lengths in seismicity.

Violation of the scaling relation and non-Markovian nature of earthquake aftershocks

Abstract The statistical properties of earthquake aftershocks are studied. The scaling relation for 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.

Empirical study of recent Chinese stock market

We investigate the statistical properties of the empirical data taken from the Chinese stock market during the time period from January, 2006 to July, 2007. By using the methods of detrended fluctuation analysis (DFA) and calculating correlation coefficients, we acquire the evidence of strong correlations among different stock types, stock index, stock volume turnover, A share (B share) seat number, and GDP per capita. In addition, we study the behavior of “volatility”, which is now defined as the difference between the new account numbers for two consecutive days. It is shown that the empirical power-law of the number of aftershock events exceeding the selected threshold is analogous to the Omori law originally observed in geophysics. Furthermore, we find that the cumulative distributions of stock return, trade volume and trade number are all exponential-like, which does not belong to the universality class of such distributions found by Xavier Gabaix et al. [Xavier Gabaix, Parameswaran Gopikrishnan, Vasiliki Plerou, H. Eugene Stanley, Nature, 423 (2003)] for major western markets. Through the comparison, we draw a conclusion that regardless of developed stock markets or emerging ones, “cubic law of returns” is valid only in the long-term absolute return, and in the short-term one, the distributions are exponential-like. Specifically, the distributions of both trade volume and trade number display distinct decaying behaviors in two separate regimes. Lastly, the scaling behavior of the relation is analyzed between dispersion and the mean monthly trade value for each administrative area in China.

Variability of earthquake nucleation in continuum models of rate‐and‐state faults and implications for aftershock rates

Using two continuum models of rate‐and‐state faults, one with a weaker patch and the other with rheological transition from steady state velocity‐weakening to velocity‐strengthening friction, we simulate several scenarios of spontaneous earthquake nucleation plausible for natural faults, investigate their response to static shear stress steps, and infer the corresponding aftershock rates. Overall, nucleation processes at weaker patches behave similarly to theories based on spring‐slider models, with some notable deviations. In particular, nucleation and aftershock rates are affected by normal stress heterogeneity in the nucleation zone. Nucleation processes at rheological transitions behave differently, producing complex slip velocity histories, nonmonotonic responses to static stress changes, and aftershock rates with pronounced peaks and seismic quiescence. For such processes, positive stress steps sometimes delay nucleation of seismic events by inducing aseismic transients that relieve stress and postpone seismic slip. Superposition of the complex aftershock response for spatially variable stress changes results in Omori's law for a period of time followed by seismic quiescence. Such behavior was observed at the base of the seismogenic zone near the 1984 Morgan Hill earthquake. We show that the computed aftershock rates are linked to unperturbed slip velocity evolution in the nucleation zone and construct simplified analytical scenarios that explain some features of the response. The qualitative differences that we find between the two nucleation models indicate that aftershock response of rate‐and‐state faults to static stress changes would depend on the conditions under which nucleation occurs on natural faults and may be different from predictions based on spring‐slider models.

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

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

Exponential decline of aftershocks of the M7.9 1868 great Kau earthquake, Hawaii, through the 20th century

The remarkable catalog of Hawaiian earthquakes going back to the 1820s is based on missionary diaries, newspaper accounts, and instrumental records and spans the great M7.9 Kau earthquake of April 1868 and its aftershock sequence. The earthquake record since 1868 defines a smooth curve complete to M5.2 of the declining rate into the 21st century, after five short volcanic swarms are removed. A single aftershock curve fits the earthquake record, even with numerous M6 and 7 main shocks and eruptions. The timing of some moderate earthquakes may be controlled by magmatic stresses, but their overall long‐term rate reflects one of aftershocks of the Kau earthquake. The 1868 earthquake is, therefore, the largest and most controlling stress event in the 19th and 20th centuries. We fit both the modified Omori (power law) and stretched exponential (SE) functions to the earthquakes. We found that the modified Omori law is a good fit to the M ≥ 5.2 earthquake rate for the first 10 years or so and the more rapidly declining SE function fits better thereafter, as supported by three statistical tests. The switch to exponential decay suggests that a possible change in aftershock physics may occur from rate and state fault friction, with no change in the stress rate, to viscoelastic stress relaxation. The 61‐year exponential decay constant is at the upper end of the range of geodetic relaxation times seen after other global earthquakes. Modeling deformation in Hawaii is beyond the scope of this paper, but a simple interpretation of the decay suggests an effective viscosity of 1019 to 1020 Pa s pertains in the volcanic spreading of Hawaii's flanks. The rapid decline in earthquake rate poses questions for seismic hazard estimates in an area that is cited as one of the most hazardous in the United States.

The earthquakes on 12 April 1998 and 12 July 2004 in Krn Mountains (NW Slovenia) – time distribution of aftershocks and magnitude-frequency relation

Earthquakes on 12 April 1998 (ML=5.6) and 12. July 2004 (ML =4.9) in Krn Mountains were followed by extensive aftershock sequences with several thousand of shocks which lasted for more than one year. In 400 days the first earthquake was followed by 104 and the second one by 89 aftershocks with ML≥2,0. Both strongest aftershocks had magnitudes smaller for 1.4 and 1.3 with respect to the mainshocks, which is slightly more than published values (1.2). Analysis of time distribution of aftershocks with maximum likelihood estimate has shown that the parameters of modified Omori's law, which describes the hyperbolic decay of aftershock activity with time, are very similar. This corresponds to the fact that both earthquakes occurred in the same hypocentral area. The value of p parameter is around 1.02. Aftershocks in 1998 clearly show secondary aftershock sequence which started with strongest aftershock on 6 May 1998 (ML=4.2), but there was no secondary sequence in 2004. Analysis of magnitude- frequency (Gutenberg-Richter's) relation with least squares and maximum likelihood estimate has given the value of b parameter for 1998 aftershock sequence between 0.77 and 0.83 and for 2004 sequence between 0.97 and 0.98. This means that the first earthquake was followed by more stronger aftershocks. Obtained parameters of modified Omori's law and Gutenberg-Richter's relation are in good agreement with values published for similar aftershock sequences.