Omori Law Research Articles

Critical point theory of earthquakes: Observation of correlated and cooperative behavior on earthquake fault systems

The critical point theory for earthquakes was originally proposed to explain the scaling relations observed in earthquakes, including the Gutenberg‐Richter frequency‐ magnitude relation and the Omori's law for aftershocks. In this model, main shocks, their foreshocks and aftershocks are all associated with the formation of a correlated, cooperative spatial region with high stress. Until now, only indirect evidence of the existence of these correlated regions has been reported. Here in this paper we present observations and analyses that allow us to directly map the high stress, spatially correlated regions preceding four major earthquakes, i.e. the 1992 Landers (California), 1995 Kobe (Japan), 1999 Chi‐Chi (Taiwan) and 1999 Hector Mine (California) earthquakes. We therefore conclude that the locations and extent of large main shocks and their immediate aftershocks can be determined from seismicity data taken prior to the main shocks, and provide additional evidence in support of the critical point theory for earthquakes.

A phenomenological model describing the flow of excited emission signals of a geophysical medium

A phenomenological model for emission responses of a geophysical medium is proposed for the description of the general variation patterns of seismic and acoustic emission signals from rock masses and loaded samples. In this model, based on simplified equations, the responses of the medium to pulsed vibrational and electric fields are examined together with the relaxation process; in the simplest particular case, the latter can be described by the Omori law.

Short-term market reaction after extreme price changes of liquid stocks

In our empirical study we examine the dynamics of the price evolution of liquid stocks after experiencing a large intra-day price change, using data from the NYSE and the NASDAQ. We find a significant reversal for both intra-day price decreases and increases. Volatility, volume and, in the case of the NYSE, the bid–ask spread, which increase sharply at the event, stay significantly high days afterwards. The decay of the volatility follows a power law in accordance with the `Omori law'. While on the NYSE the large widening of the bid–ask spread eliminates most of the profits that can be achieved by an outside investor, on the NASDAQ the bid–ask spread stays almost constant, yielding significant short-term profits. The results thus give an insight into the size and speed of the realization of an excess return for providing liquidity in a turbulent market.

Long‐range triggered earthquakes that continue after the wave train passes

Large earthquakes can trigger distant earthquakes in geothermal areas. Some triggered earthquakes happen while the surface waves pass through a site, but others occur hours or even days later. Does this prolonged seismicity require a special mechanism to store the stress from the seismic waves that differs from ordinary aftershock mechanisms? These questions have driven studies of long‐range triggering since the phenomenon's discovery. Here I attempt to answer the questions by examining the statistics of triggered sequences. Two separate observations are consistent with the prolonged sequences being simply local aftershocks of earthquakes triggered early in the wave train. First, the sequences obey Omori's Law over both short (1 hour) and longer (5 day) time intervals. Secondly, the number of observed triggered earthquakes in the first hour after the wave train can be predicted from the number of earthquakes triggered during the wave train. Even the very vigorous 10‐day triggering at Long Valley from the 1992 Landers Mw 7.3 earthquakes can be interpreted as the aftershocks of either a local M ≈ 4.1 earthquake or an equivalent combination of several smaller mainshocks. Therefore, long‐range triggering does not need to include a mechanism to produce sustained stresses other than the process that generates aftershocks of the earthquakes that occur while the wave train is passing.

Impact of the monetary crisis on statistical properties of the Jakarta and Kuala Lumpur stock exchange indices

Using the tools developed for statistical physics, we simultaneously analyze statistical properties of the Jakarta and Kuala Lumpur Stock Exchange indices. In spite of the small number of the data used in the analysis, the result still shows the universal behavior of complex systems previously found in the leading stock indices. We also analyze their properties before and after the crash caused by the monetary crisis. To locate the time position when the crash started we use the Omori law. We found that after the crash both stocks do not show a same statistical behavior. The impact of currency controls is observed in the distribution of the index returns.

Relation between stress heterogeneity and aftershock rate in the rate‐and‐state model

We estimate the rate of aftershocks triggered by a heterogeneous stress change, using the rate‐and‐state model of Dieterich. We show that an exponential stress distribution Pτ(τ) ∼ exp(−τ/τ0) gives an Omori law decay of aftershocks with time ∼1/tp, with an exponent p = 1 − Aσn/τ0, where A is a parameter of the rate‐and‐state friction law and σn is the normal stress. Omori exponent p thus decreases if the stress “heterogeneity” τ0 decreases. We also invert the stress distribution Pτ(τ) from the seismicity rate R(t), assuming that the stress does not change with time. We apply this method to a synthetic stress map, using the (modified) scale invariant “k2” slip model (Herrero and Bernard). We generate synthetic aftershock catalogs from this stress change. The seismicity rate on the rupture area shows a huge increase at short times, even if the stress decreases on average. Aftershocks are clustered in the regions of low slip, but the spatial distribution is more diffuse than for a simple slip dislocation. Because the stress field is very heterogeneous, there are many patches of positive stress changes everywhere on the fault. This stochastic slip model gives a Gaussian stress distribution but nevertheless produces an aftershock rate which is very close to Omori's law, with an effective p ≤ 1, which increases slowly with time. We obtain a good estimation of the stress distribution for realistic catalogs when we constrain the shape of the distribution. However, there are probably other factors which also affect the temporal decay of aftershocks with time. In particular, heterogeneity of Aσn can also modify the parameters p and c of Omori's law. Finally, we show that stress shadows are very difficult to observe in a heterogeneous stress context.

Aftershock decay, productivity, and stress rates in Hawaii: Indicators of temperature and stress from magma sources

We examined dozens of aftershock sequences in Hawaii in terms of Gutenberg‐Richter and modified Omori law parameters. We studied p, the rate of aftershock decay; Ap, the aftershock productivity, defined as the observed divided by the expected number of aftershocks; and c, the time delay when aftershock rates begin to fall. We found that for earthquakes shallower than 20 km, p values >1.2 are near active magma centers. We associate this high decay rate with higher temperatures and faster stress relaxation near magma reservoirs. Deep earthquakes near Kilauea's inferred magma transport path show a range of p values, suggesting the absence of a large, deep magma reservoir. Aftershock productivity is >4.0 for flank earthquakes known to be triggered by intrusions but is normal (0.25 to 4.0) for isolated main shocks. We infer that continuing, post‐main shock stress from the intrusion adds to the main shock's stress step and causes higher Ap. High Ap in other zones suggests less obvious intrusions and pulsing magma pressure near Kilauea's feeding conduit. We calculate stress rates and stress rate changes from pre‐main shock and aftershock rates. Stress rate increased after many intrusions but decreased after large M7–8 earthquakes. Stress rates are highest in the seismically active volcano flanks and lowest in areas far from volcanic centers. We found sequences triggered by intrusions tend to have high Ap, high (>0.10 day) c values, a stress rate increase, and sometimes a peak in aftershock rate hours after the main shock. We interpret these values as indicating continuing intrusive stress after the main shock.

Physics of the Omori law: Inferences from interevent time distributions and pore pressure diffusion modeling

Empirical laws and statistics of earthquakes are valuable as a basis for a better understanding of the earthquake cycle. In this paper we focus on the postseismic phase and the physics of aftershock sequences. Using interevent time distributions for a catalogue of Icelandic seismicity, we infer that the parameter C 2 in the Omori law, often considered to represent incomplete detection of aftershocks, is at least in part related to the physics of the earthquake process. We investigate the role of postseismic pore pressure diffusion after two Icelandic earthquakes on the rate of aftershocks and what we can infer about the physical meaning of C 2 from the diffusion process. Using the Mohr–Coulomb failure criterion we obtain a rate of triggered points in our diffusion model that agrees with the modified Omori law, with a value of C 2 that is consistent with data. Our pore pressure diffusion model suggests that C 2 is related to the process of reducing high pore pressure gradients existing across a fault zone at short times after a main shock.

Modelling fundamental waiting time distributions for earthquake sequences

The distribution of waiting times between time-neighbouring events for a time series obeying the Omori law is examined theoretically and numerically with the aim of understanding the characteristics of these distributions, how these characteristics change (e.g. scale) with the parameters of the Omori series, and thus how empirical waiting time data may be correctly interpreted. It is found that the waiting time distribution, for a single Omori aftershock sequence, consists in general of two power law segments followed by a rapid decay at larger waiting times. The analyses are illustrated using real data from the SIL network on Iceland. This data often shows characteristics predominantly consistent with the Omori law, but there are significant exceptions. We conclude that waiting time distributions and related statistical analysis has meaningful potential for the analysis of earthquake data sets, as a step towards developing physical models of the earthquake process.

Cumulative Benioff strain-release, modified Omori's law and transient behaviour of rocks

Irreversible thermodynamic theories with internal state variables can be used to derive a general constitutive law for both transient and steady-state behaviours of rocks. This constitutive law can represent the concepts of damage and damage evolution in either the fibre-bundle model or continuum damage mechanics. We have previously proposed an empirically based constitutive law for both the transient and steady-state behaviours of rocks ultimately derived from laboratory experimental data. We show here that this law is concordant with the general constitutive law derived from irreversible thermodynamic theories, and that the relaxation modulus has a temporal power–law that depends on a structural fractal property of rocks. Our constitutive law predicts forms for the cumulative Benioff strain-release for precursory seismic activations and the modified Omori's laws of aftershocks, both aspects of the temporal fractal properties of seismicity. These seismic properties can also be derived by the fibre-bundle model or continuum damage mechanics. Our model suggests that these time-scale invariant processes of seismicity may be regulated by the fractal structures of crustal rocks.

Can Damage Mechanics Explain Temporal Scaling Laws in Brittle Fracture and Seismicity?

Time delays associated with processes leading to a failure or stress relaxation in materials and earthquakes are studied in terms of continuum damage mechanics. Damage mechanics is a quasi-empirical approach that describes inelastic irreversible phenomena in the deformation of solids. When a rock sample is loaded, there is generally a time delay before the rock fails. This period is characterized by the occurrence and coalescence of microcracks which radiate acoustic signals of broad amplitudes. These acoustic emission events have been shown to exhibit power-law scaling as they increase in intensity prior to a rupture. In case of seismogenic processes in the Earth's brittle crust, all earthquakes are followed by an aftershock sequence. A universal feature of aftershocks is that their rate decays in time according to the modified Omori's law, a power-law decay. In this paper a model of continuum damage mechanics in which damage (microcracking) starts to develop when the applied stress exceeds a prescribed yield stress (a material parameter) is introduced to explain both laboratory experiments and systematic temporal variations in seismicity.

Converting displacement dynamics into creep in block media

The possibility of converting displacement dynamics into a creep regime in deformed block media is corroborated by the results of numerical simulation. The model is based on the dynamics of one tectonic platform moving on another. The proposed model describes the characteristic features in the behavior of geological media quite well; in particular, it reproduces the earthquake statistics (as described by the Gutenberg-Richter and Omori laws). An analysis showed that local, relatively low-energy actions could transform the block dynamics from the stick-slip to creep regime. This implies that the statistics of seismic energy evolution can be qualitatively modified using a series of local, periodic, and relatively weak actions so that the average strength of seismic impacts significantly decreases. In this way, it is possible to “suppress” strong earthquakes. The results justify good prospects for a concept previously proposed by the authors, according to which the initiation of displacements in fault-block media can provide a mechanism for releasing local elastic energy and, in particular, decreasing seismic danger.

Earthquake activity related to seismic cycles in a model for a heterogeneous strike-slip fault

We investigate the evolution of seismicity within large earthquake cycles in a model of a discrete strike-slip fault in elastic solid. The model dynamics is governed by realistic boundary conditions consisting of constant velocity motion of regions around the fault, static/kinetic friction and dislocation creep along the fault, and 3D elastic stress transfer. The fault consists of brittle parts which fail during earthquakes and undergo small creep deformation between events, and aseismic creep cells which are characterized by high ongoing creep motion. This mixture of brittle and creep cells is found to generate realistic aftershock sequences which follow the modified Omori law and scale with the mainshock size. Furthermore, we find that the distribution of interevent times of the simulated earthquakes is in good agreement with observations. The temporal occurrence, however, is magnitude-dependent; in particular, the small events are clustered in time, whereas the largest earthquakes occur quasiperiodically. Averaging the seismicity before several large earthquakes, we observe an increase of activity and a broadening scaling range of magnitudes when the time of the next mainshock is approached. These results are characteristics of a critical point behavior. The presence of critical point dynamics is further supported by the evolution of the stress field in the model, which is compatible with the observation of accelerating moment release in natural fault systems.

Universal mean moment rate profiles of earthquake ruptures

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 spatiotemporal 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 exponents that have large error bars. Universal shape profiles contain crucial information on the underlying dynamics in a variety of systems. As in magnetic systems, we find that our analysis for earthquakes provides a good overall agreement between theory and observations, but with a potential discrepancy in one particular universal scaling function for moment rates. We primarily use mean field theory for the theoretical analysis, since it has been shown to be in the same universality class as the full three-dimensional version of the model (up to logarithmic corrections). The results point to the existence of deep connections between the physics of avalanches in different systems.

Correlation between the parameters of the aftershock rate equation: Implications for the forecasting of future sequences

We analyzed the correlations among the parameters of the Reasenberg and Jones [Reasenberg, P.A., Jones, L.M., 1989. Earthquake hazard after a mainshock in California, Science 243, 1173–1176] 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 also found significant correlations between p and c but, different from some previous works, not between p and b. We verified that assuming a coefficient for mainshock magnitude α ≈ 2/3 b (instead of b) removes the correlation between a and b and improves the ability to forecast the behavior of Italian sequences occurred from 1997 to 2003 on the basis of average parameters estimated from sequences occurred from 1981 to 1996. This assumption well agrees with direct α estimates made in the framework of an epidemic type model (ETAS) from the data of some large Italian sequences. Our results suggest a modification of the original Reasenberg and Jones (1989) formulation leading to predict lower rates (and probabilities) for stronger mainshocks and conversely higher rates for weaker ones. We also inferred that the correlation of a with p and c might be the consequence of the trade-off between the two parameters of the modified Omori's law. In this case the correlation can be partially removed by renormalizing the time-dependent part of the rate equation. Finally, the absence of correlation between p and b, observed for all the examined datasets, indicates that such correlation, previously inferred from theoretical considerations and empirical results in some regions, does not represent a common property of aftershock sequences in different part of the world.

Lindmanet al.Reply:

In seismology, an important goal is to attain a better understanding of the earthquake process. In this study of the physics of aftershock generation, I couple statistical analysis with modelling of physical processes in the postseismic period. I present a theoretical formulation for the distribution of interevent times for aftershock sequences obeying the empirically well established Omori law. As opposed to claims by other authors, this work demonstrates that the duration of the time interval between two successive earthquakes cannot be used to identify whether or not they belong to the same aftershock sequence or occur as a result of the same underlying process. This implies that a proper understanding of earthquake interevent time distributions is necessary before conclusions regarding the physics of the earthquake process are drawn.In a discussion of self-organised criticality (SOC) in relation to empirical laws in seismology, I find that Omori's law for aftershocks cannot be used as evidence for the theory of SOC. Instead, I consider that the occurrence of aftershocks in accordance with Omori's law is a result of a physical process that can be modelled and understood.I analyse characteristic features in the spatiotemporal distribution of aftershocks in the south Iceland seismic zone, following the two M6.5 June 2000 earthquakes and a M4.5 earthquake in September, 1999. These features include an initially constant aftershock rate, whose duration is larger following a larger main shock, and a subsequent power law decay that is interrupted by distinct and temporary deviations in terms of rate increases and decreases. Based on pore pressure diffusion modelling, I interpret these features in terms of main shock initiated diffusion processes. I conclude that thorough data analysis and physics-based modelling are essential components in attempts to improve our understanding of processes governing the occurrence of earthquakes.

Physical and stochastic models of earthquake clustering

The phenomenon of earthquake clustering, i.e., the increase of occurrence probability for seismic events close in space and time to other previous earthquakes, has been modeled both by statistical and physical processes. From a statistical viewpoint the so-called epidemic model (ETAS) introduced by Ogata in 1988 and its variations have become fairly well known in the seismological community. Tests on real seismicity and comparison with a plain time-independent Poissonian model through likelihood-based methods have reliably proved their validity. On the other hand, in the last decade many papers have been published on the so-called Coulomb stress change principle, based on the theory of elasticity, showing qualitatively that an increase of the Coulomb stress in a given area is usually associated with an increase of seismic activity. More specifically, the rate-and-state theory developed by Dieterich in the ′90s has been able to give a physical justification to the phenomenon known as Omori law. According to this law, a mainshock is followed by a series of aftershocks whose frequency decreases in time as an inverse power law. In this study we give an outline of the above-mentioned stochastic and physical models, and build up an approach by which these models can be merged in a single algorithm and statistically tested. The application to the seismicity of Japan from 1970 to 2003 shows that the new model incorporating the physical concept of the rate-and-state theory performs not worse than the purely stochastic model with two free parameters only. The numerical results obtained in these applications are related to physical characters of the model as the stress change produced by an earthquake close to its edges and to the A and σ parameters of the rate-and-state constitutive law.

Temporal Changes of Shallow Seismic Velocity Around the Karadere-Düzce Branch of the North Anatolian Fault and Strong Ground Motion

We analyze temporal variations of seismic velocity along the Karadere-Duzce branch of the north Anatolian fault using seismograms generated by repeating earthquake clusters in the aftershock zones of the 1999 Mw7.4 Izmit and Mw7.1 Duzce earthquakes. The analysis employs 36 sets of highly repeating earthquakes, each containing 4–18 events. The events in each cluster are relocated by detailed multi-step analysis and are likely to rupture approximately the same fault patch at different times. The decay rates of the repeating events in individual clusters are compatible with the Omori's law for the decay rate of regional aftershocks. A sliding window waveform cross-correlation technique is used to measure travel time differences and evolving decorrelation in waveforms generated by each set of the repeating events. We find clear step-like delays in the direct S and early S-coda waves (sharp seismic velocity reduction) immediately after the Duzce main shock, followed by gradual logarithmic-type recoveries. A gradual increase of seismic velocities is also observed before the Duzce main shock, probably reflecting post-seismic recovery from the earlier Izmit main shock. The temporal behavior is similar at each station for clusters at various source locations, indicating that the temporal changes of material properties occur in the top most portion of the crust. The effects are most prominent at stations situated in the immediate vicinity of the recently ruptured fault zones, and generally decrease with normal distance from the fault. A strong correlation between the co-seismic delays and intensities of the strong ground motion generated by the Duzce main shock implies that the radiated seismic waves produced the velocity reductions in the shallow material.

Analysis of aftershocks in a lithospheric model with seismogenic zone governed by damage rheology

We perform analytical and numerical studies of aftershock sequences following abrupt steps of strain in a rheologically layered model of the lithosphere. The model consists of a weak sedimentary layer, over a seismogenic zone governed by a viscoelastic damage rheology, underlain by a viscoelastic upper mantle. The damage rheology accounts for fundamental irreversible aspects of brittle rock deformation and is constrained by laboratory data of fracture and friction experiments. A 1-D version of the viscoelastic damage rheology leads to an exponential analytical solution for aftershock rates. The corresponding solution for a 3-D volume is expected to be sum of exponentials. The exponential solution depends primarily on a material parameter R given by the ratio of timescale for damage increase to timescale for accumulation of gradual inelastic deformation, and to a lesser extent on the initial damage and a threshold strain state for material degradation. The parameter R is also inversely proportional to the degree of seismic coupling across the fault. Simplifying the governing equations leads to a solution following the modified Omori power-law decay with an analytical exponent p= 1. In addition, the results associated with the general exponential expression can be fitted for various values of R with the modified Omori law. The same holds for the decay rates of aftershocks simulated numerically using the 3-D layered lithospheric model. The results indicate that low R values (e.g. R≤ 1) corresponding to cold brittle material produce long Omori-type aftershock sequences with high event productivity, while high R values (e.g. R≥ 5) corresponding to hot viscous material produce short diffuse response with low event productivity. The frequency-size statistics of aftershocks simulated in 3-D cases with low R values follow the Gutenberg–Richter power law relation, while events simulated for high R values are concentrated in a narrow magnitude range. Increasing thickness of the weak sedimentary cover produces results that are similar to those associated with higher R values. Increasing the assumed geothermal gradient reduces the depth extent of the simulated earthquakes. The magnitude of the largest simulated aftershocks is compatible with the Båth law for a range of values of a dynamic damage-weakening parameter. The results provide a physical basis for interpreting the main observed features of aftershock sequences in terms of basic structural and material properties.

Universality in Solar Flare and Earthquake Occurrence

Earthquakes and solar flares are phenomena involving huge and rapid releases of energy characterized by complex temporal occurrence. By analyzing available experimental catalogs, we show that the stochastic processes underlying these apparently different phenomena have universal properties. Namely, both problems exhibit the same distributions of sizes, interoccurrence times, and the same temporal clustering: We find after flare sequences with power law temporal correlations as the Omori law for seismic sequences. The observed universality suggests a common approach to the interpretation of both phenomena in terms of the same driving physical mechanism.