Omori Law Research Articles

Properties of foreshocks and aftershocks of the nonconservative self-organized critical Olami-Feder-Christensen model

Following Phys. Rev. Lett. 88, 238501 (2002)] who discovered aftershocks and foreshocks in the Olami-Feder-Christensen (OFC) discrete block-spring earthquake model, we investigate to what degree the simple toppling mechanism of this model is sufficient to account for the clustering of real seismicity in time and space. We find that synthetic catalogs generated by the OFC model share many properties of real seismicity at a qualitative level: Omori's law (aftershocks) and inverse Omori's law (foreshocks), increase of the number of aftershocks and of the aftershock zone size with the mainshock magnitude. There are, however, significant quantitative differences. The number of aftershocks per mainshock in the OFC model is smaller than in real seismicity, especially for large mainshocks. We find that foreshocks in the OFC catalogs can be in large part described by a simple model of triggered seismicity, such as the epidemic-type aftershock sequence (ETAS) model. But the properties of foreshocks in the OFC model depend on the mainshock magnitude, in qualitative agreement with the critical earthquake model and in disagreement with real seismicity and with the ETAS model.

Anomalous power law distribution of total lifetimes of branching processes: Application to earthquake aftershock sequences

We consider a general stochastic branching process, which is relevant to earthquakes, and study the distributions of global lifetimes of the branching processes. In the earthquake context, this amounts to the distribution of the total durations of aftershock sequences including aftershocks of arbitrary generation number. Our results extend previous results on the distribution of the total number of offspring (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction. We consider a branching model of triggered seismicity, the 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 fertilities mu is characterized by a power law approximately 1/ mu(1+gamma) and the bare Omori law for the memory of previous triggering mothers decays slowly as approximately 1/ t(1+theta;) , with 0<theta;<1 relevant for earthquakes. Using the tool of generating probability functions and a quasistatic approximation which is shown to be exact asymptotically for large durations, we show that the density distribution of total aftershock lifetimes scales as approximately 1/ t(1+theta;/gamma) when the average branching ratio is critical (n=1) . The coefficient 1<gamma=b/alpha<2 quantifies the interplay between the exponent b approximately 1 of the Gutenberg-Richter magnitude distribution approximately 10(-bm) and the increase approximately 10(alpham) of the number of aftershocks with mainshock magnitude m (productivity), with 0.5<alpha<1 . The renormalization of the bare Omori decay law approximately 1/ t(1+theta;) into approximately 1/ t(1+theta;/gamma) stems from the nonlinear amplification due to the heavy-tailed distribution of fertilities and the critical nature of the branching cascade process. In the subcritical case n<1 , the crossover from approximately 1/ t(1+theta;/gamma) at early times to approximately 1/ t(1+theta;) at longer times is described. More generally, our results apply to any stochastic branching process with a power-law distribution of offspring per parent and a long memory.

Coseismic and postseismic velocity changes measured by repeating earthquakes

Repeating earthquakes that rupture approximately the same fault patch and have nearly identical waveforms are a useful tool for measuring temporal changes in wave propagation in the Earth's crust. Since source and path effects are common to all earthquakes in a repeating earthquake sequence (multiplet), differences in their waveforms can be attributed to changes in the characteristics of the medium. We have identified over 20 multiplets containing between 5 and 40 repeating events in the aftershock zones of the 1989 Loma Prieta and 1984 Morgan Hill, California, earthquakes. Postmain shock events reveal delays of phases in the early S wave coda of as much as 0.2 s relative to premain shock events. The delay amounts to a path‐averaged coseismic velocity decrease of about 1.5% for P waves and 3.5% for S waves. Since most of the multiplets are aftershocks and follow Omori's law, we have excellent temporal sampling in the immediate postmain shock period. We find that the amplitude of the velocity decrease decays logarithmically in time following the main shock. In some cases it returns to the premain shock values, while in others it does not. Similar results are obtained for the Morgan Hill main shock. Because the fractional change in S wave velocity is greater than the fractional change in P wave velocity, it suggests that the opening or connection of fluid‐filled fractures is the underlying cause. The magnitude of the velocity change implies that low effective pressures are present in the source region of the velocity change. Our results suggest that the changes are predominantly near the stations and shallow, but we cannot exclude the possibility that changes occur at greater depth as well. If the variations are shallow, we may be detecting the lingering effects of nonlinearity during main shock strong ground motion. If the variations are deep, it suggests that pore pressures at seismogenic depths are high, which would likely play a key role in the earthquake process.

Spatial and temporal variations of the aftershock sequences of the 1999 İzmit and Düzce earthquakes

In this study the properties of the aftershock sequences in the first five months after the 17 August 1999 Izmit and 12 November 1999 Duzce earthquakes are investigated. For this purpose, the b-value of frequency-magnitude distribution of earthquakes and temporal decay rate of aftershocks as described by the p-value in the modified Omori law are calculated. The data taken from the website of the KOERI consist of 1841 events for the Izmit and 913 events for the Duzce activity, respectively. The b-value for the Izmit sequence is found as 1.10±0.03 with Mc = 2.6 and b = 1.16 ± 0.05 for the Duzce sequence with Mc = 2.8. Both b-values are close to 1 and typical for aftershock sequences. The p-values are calculated as 0.86 ± 0.05 for the Izmit and p = 1.34 ± 0.09 for Duzce aftershocks, respectively. Small p-value found in Izmit sequence may be a result of the background seismicity of the aftershock region. Because of the lower p-value for the Izmit activity than that of the Duzce sequence, the aftershocks show a relatively slow decay rate in the Izmit earthquake. Considerable spatial variability is found, b-values ranging from 0.8 to 1.6, and p-values ranging from 0.4 to 1.4. The spatial variations of b and p-values of an aftershock sequence may associate with the rupture mechanism and geological structure of an earthquake area. A good agreement between the slip and b-value for the Duzce sequence are found whereas we could not observe a linear relation between these parameters for the Izmit sequence. After the Izmit earthquake, the larger b-values correlate with the regions where the lower stress changes occurred whereas the lower b-values are in the areas with the larger stress variations. Also, the lower and higher b-values are related to the higher and lower P-wave velocity in general. Regarding the geological aspect, the larger b and p-values for the two sequences are related to Holocene alluvium structures which have low velocity. On the other hand, after the both earthquakes, the larger p-values correlate well with the regions on which the maximum slip is located and the lower p-values are related to the lower slip regions. However, the material properties seem more effective than the stress changes and slip distributions in the b-value variations and the slip distribution is the most significant factor on p-values in the both sequences.

Short-Term Properties of Earthquake Catalogs and Models of Earthquake Source

I review the short-term properties of earthquake catalogs, in particular the time and size distributions and the completeness of the early part of aftershock sequences for strong, shallow earthquakes. I determine the parameters of the Omori and Gutenberg-Richter laws for aftershocks close in time to a mainshock. Aftershock sequences of large earthquakes in southern California (1952 Kern County, 1992 Joshua Tree-Landers-Big Bear sequence, 1994 Northridge, and 1999 Hector Mine), recorded in the CalTech catalog, are analyzed to demonstrate that at the beginning of these series, many small earthquakes are absent from the catalog. The number of missing earthquakes increases with the magnitude range of a catalog and for some data sets exceeds the number of aftershocks close to a mainshock listed in a catalog. Comparing global earthquake catalogs (Harvard Centroid Moment Tensor and Preliminary Determination of Epicenter) with local data sets indicates that the catalogs based on longer period waves miss many early aftershocks even when their magnitudes are well above the stated magnitude threshold. Such short-term incompleteness may introduce significant biases to the statistical analysis of the seismicity pattern, in particular for branching models of earthquake occurrence incorporating the Omori law. For such models the likelihood function strongly depends on close aftershocks. I review the techniques to alleviate this problem. Analyzing the source rupture process of several recent large earthquakes suggests that rupture propagation is highly inhomogeneous in space, time, and focal mechanism. These random variations in the rupture process can be viewed as an extension of the aftershock stochastic generating mechanism toward the origin time of a mainshock. I review various models of the earthquake rupture process and suggest that fractal distributions of microevents in time, space, and focal mechanism constitute the development of an earthquake. The final identification of an individual earthquake depends on both objective and subjective factors. Manuscript received 21 May 2003.

Coseismic slip resolution and post-seismic relaxation time of the 1999 Chi-Chi, Taiwan, earthquake as constrained by geological observations, geodetic measurements and seismicity

SUMMARY We analysed the surface displacements estimated by Global Positioning System (GPS) mea- surements and Systeme Probatoire d'Observation de la Terre (SPOT) image correlation and the seismicity change induced by the Mw =7.6 1999 September 20 Chi-Chi earthquake. We first modelled the seismogenic zone for which we introduced a realistic geologically constrained ramp and flat geometry, and estimated by joint inversion of the GPS and SPOT data the co- seismic slip distribution, stressing the resolution of the model parameters. The fair adjustment indicates that the ramp and flat geometry is consistent with the geodetic measurement. The cal- culated coseismic slip increases both towards the surface and northwards with maximum slip of approximately 15 m, in good agreement with the observed surface rupture and previously ob- tained slip models based on seismic waveforms. This distribution may be represented to a good approximation by a linear increase of slip with a constant azimuth in the plate convergence direction. Special emphasis is given to the analysis of the model resolution, which sharply decreases with depth in spite of the exceptionally well distributed and high-quality inverted data. This resolution decrease, expected for any slip distribution based on surface observations only, led us then to examine the seismicity as independent data to test our model following the static stress triggering hypothesis. The model is accurately defined in terms of Coulomb stress change in the central part of the rupture where the fault geometry is well constrained by seismic profile. In contrast, at the southern and northern ends it is less appropriate, however the static stress triggering approach is not precise enough to determine whether fault geometry or slip distribution is less suitable. We thus conclude that the analysis of seismicity in terms of Coulomb stress change might not allow us to examine small-scale features of seismogenic zones. Finally, we discuss post-seismic relaxation from the temporal variation of seismicity rate. In contrast to the relaxation time of 0.23-0.90 yr inferred from post-seismic permanent GPS measurements, we found that the decay of aftershock sequence follows Omori's law with an 8-14 yr relaxation time. We suggest that this discrepancy results from the coexistence of several different processes, not all associated with the production of aftershocks.

Scale-free networks of earthquakes and aftershocks.

We propose a metric to quantify correlations between 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 predetermined space-time windows. In the simplest network construction, each earthquake receives an incoming link from its most correlated predecessor. The number of aftershocks for any event, identified by its outgoing links, is found to be scale free with exponent gamma=2.0(1). The original Omori law with p=1 emerges as a robust feature of seismicity, holding up to years even for aftershock sequences initiated by intermediate magnitude events. The broad distribution of distances between earthquakes and their linked aftershocks suggests that aftershock collection with fixed space windows is not appropriate.

A generalized Omori's law for earthquake aftershock decay

Earthquake aftershock sequences have been found to approximately satisfy three empirical scaling relations: i) the Gutenberg‐Richter frequency‐magnitude scaling, ii) Båth's law for the difference in the magnitude of a mainshock and its largest aftershock, and iii) the modified Omori's law for the temporal decay of aftershock rates. The three laws are incorporated to give a generalized Omori's law for aftershock decay rates that depend on several parameters specific for each given seismogenic region. It is shown that the characteristic time c, first introduced in the modified Omori's law, is no longer a constant but scales with a lower magnitude cutoff and a mainshock magnitude. The generalized Omori's law is tested against earthquake catalogs for the aftershock sequences of the Landers, Northridge, Hector Mine, and San Simeon earthquakes.

REVIEW AND A MODEL OF PRE-SEISMIC ELECTROMAGNETIC EMISSIONS IN TERMS OF FRACTAL ELECTRODYNAMICS

We introduce a new model of the generation of pre-seismic electromagnetic emissions, which explains the observed phenomenology in terms of its geometry and fractal electrodynamics. Accumulated evidence indicates that an earthquake can be viewed as a critical phenomenon culminating in a large event that corresponds to a type of critical point. The principle feature of criticality is the fractal organization in both space and time. Earthquakes display a complex spatio-temporal behavior: in addition to the regularity in the rate of occurrence (e.g. Gutenberg-Richter law, Omori law), the spatial distribution of epicenters is fractal and earthquakes occur on a fractal structure of faults. Thus, the hypothesis that the fault develops as a fractal is reasonable. A mounting body of laboratory evidence suggests that micro-fracturing of rocks are associated with the appearance of spontaneous charge production and transient electromagnetic emissions (EME). The emitting, diffusing and recombination charge accompanying the micro-fracturing, can act as current generated during the crack opening. In this view, an active crack or rupture, can be simulated by a "radiating element." The idea is that a fractal geo-antenna (FGA) can be formed as an array of line elements having a fractal distribution on the ground surface as the critical point is approached. We test this idea in terms of fractal electrodynamics: we argue that the precursory VLF-VHF EM signals associated with recent earthquakes in Greece are governed by characteristics (e.g. scaling laws, temporal evolution of the spectrum content, broad band spectrum region and accelerating emission rate) predicted by fractal electrodynamics.

A damage model for the continuum rheology of the upper continental crust

In orogenic zones it is often considered appropriate to use a continuum rheology to model the deformation of the upper continental crust. In this paper we derive an applicable rheology utilizing fiber-bundle and continuum-damage models. We show that the results are identical and yield a power relation between stress σ and strain rate ε of the form σ= ε ρ−1 . We constrain the applicable values of ρ utilizing Omori's law for the decay of aftershocks and conclude that ρ−1 is in the range of 5–15. With this strong nonlinear viscous rheology the behavior of the deforming upper crust approaches that of a perfect-plastic material.

Dynamics of a financial market index after a crash

We discuss the statistical properties of index returns in a financial market just after a major market crash. The observed non-stationary behavior of index returns is characterized in terms of the exceedances over a given threshold. This characterization is analogous to the Omori law originally observed in geophysics. By performing numerical simulations and theoretical modelling, we show that the non-linear behavior observed in real market crashes cannot be described by a GARCH(1,1) model. We also show that the time evolution of the Value at Risk observed just after a major crash is described by a power-law function lacking a typical scale.

Statistical similarities between internetquakes and earthquakes

The temporal behavior of the Internet is studied by performing the Ping experiment. Analyzing the time series data of the round-trip time of Ping signals, similarities between the Internet and earthquake as complex systems are discussed. Sudden drastic changes of the Internet time series are regarded as main shocks or catastrophes, referred to here as “internetquakes”. By analogy with earthquakes, magnitude of the internetquakes is appropriately defined. Then, two known empirical power laws for earthquakes, the Omori law for the temporal pattern for aftershocks and the Gutenberg–Richter law for the relationship between frequency and magnitude, are found to hold also for the Internet. Thus, the internet shares with earthquakes common scale-invariant features in its temporal behaviors.

Postseismic relaxation driven by brittle creep: A possible mechanism to reconcile geodetic measurements and the decay rate of aftershocks, application to the Chi‐Chi earthquake, Taiwan

We evaluate the effect of coseismic stress changes on the fault slip at midcrustal depth, assuming a velocity‐strengthening brittle creep rheology. We show that this model can help reconcile the time evolution of afterslip, as measured from geodesy, with aftershocks decay. We propose an analytical expression for slip of the brittle creeping fault zone (BCFZ) that applies to any dynamic or static stress perturbation, including shear stress and normal stress changes. The model predicts an initial logarithmic increase of slip with time. Postseismic slip rate decays over a characteristic time tr = that does not depend on the amplitude of the stress perturbation, and it asymptotically joins the long‐term creep imposed by interseismic stress buildup . Given that the seismicity rate might be considered proportional to the sliding velocity of the BCFZ, the model predicts a decay rate of aftershocks that follows Omori's law, with a mathematical formalism identical to that of Dieterich [1994] although based on a different mechanical rationale. Our model also differs from Dieterich's model in that it requires that aftershock sequences and deep afterslip, as constrained from geodetic measurements, should follow the same temporal evolution. We test this for the 1999 Chi‐Chi earthquake, Mw = 7.6 and find that both sets of data are consistent with a model of afterslip due to the response of the BCFZ. The inferred relaxation time tr = 8.5 years requires a value for a = ∂μ/∂log(V) (μ being the coefficient of friction) in the range between 1.3 10−3 and 10−2.

The 1997 Kagoshima (Japan) earthquake doublet: A quantitative analysis of aftershock rate changes

We quantitatively map relative rate changes for the aftershock sequence following the second mainshock of the 1997 earthquake doublet (MW = 6.1, MW = 6.0) in the Kagoshima province (Japan). Using the spatial distribution of the modified Omori law parameters for aftershocks that occurred during the 47.8 days between the two mainshocks, we forecast the aftershock activity in the next 50 days and compare it to the actually observed rates. The relative rate change map reveals four regions with statistically significant relative rate changes ‐ three negative and one positive. “Our analysis suggests that the coseismic rate changes for off‐fault aftershocks could be explained by changes in static stress. However, to explain the activation and deactivation of on‐fault seismicity, other mechanism such as unusual crustal properties and the presence of abundant crustal fluids are required.”

Financial earthquakes, aftershocks and scaling in emerging stock markets

This paper provides evidence for scaling laws in emerging stock markets. Estimated parameters using different definitions of volatility show that the empirical scaling law in every stock market is a power law. This power law holds from 2 to 240 business days (almost 1 year). The scaling parameter in these economies changes after a change in the definition of volatility. This finding indicates that the stock returns may have a multifractal nature. Another scaling property of stock returns is examined by relating the time after a main shock to the number of aftershocks per unit time. The empirical findings show that after a major fall in the stock returns, the stock market volatility above a certain threshold shows a power law decay, described by Omori's law.

Aging and scaling of earthquake aftershocks

A part of the seismic time series, in which the Omori law for temporal pattern of aftershocks holds, is referred 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 (with respect to the label of the aftershock, not with respect to time), 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 seem to indicate that complex fault–fault interactions for aftershocks may be governed by glassy dynamics.

Predictability in the Epidemic‐Type Aftershock Sequence model of interacting triggered seismicity

As part of an effort to develop a systematic methodology for earthquake forecasting, we use a simple model of seismicity on the basis of interacting events which may trigger a cascade of earthquakes, known as the Epidemic‐Type Aftershock Sequence model (ETAS). The ETAS model is constructed on a bare (unrenormalized) Omori law, the Gutenberg‐Richter law, and the idea that large events trigger more numerous aftershocks. For simplicity, we do not use the information on the spatial location of earthquakes and work only in the time domain. We demonstrate the essential 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 offer an analytical approach to account for the yet unobserved triggered seismicity adapted to the problem of forecasting future seismic rates at varying horizons from the present. Tests presented on synthetic catalogs validate strongly the importance of taking into account all the cascades of still unobserved triggered events in order to predict correctly the future level of seismicity beyond a few minutes. We find a strong predictability if one accepts to predict only a small fraction of the large‐magnitude targets. Specifically, we find a prediction gain (defined as the ratio of the fraction of predicted events over the fraction of time in alarms) equal to 21 for a fraction of alarm of 1%, a target magnitude M ≥ 6, an update time of 0.5 days between two predictions, and for realistic parameters of the ETAS model. However, the probability gains degrade fast when one attempts to predict a larger fraction of the targets. This is because a significant fraction of events remain uncorrelated from past seismicity. This delineates the fundamental limits underlying forecasting skills, stemming from an intrinsic stochastic component in these interacting triggered seismicity models. Quantitatively, the fundamental limits of predictability found here are only lower bounds of the true values corresponding to the full information on the spatial location of earthquakes.

Refining earthquake clustering models

Assuming that earthquakes are the realization of a stochastic point process and that the magnitude distribution of all earthquakes is described by the Gutenberg‐Richter law with a constant b value, we model the occurrence rate density of earthquakes in space and time by means of an epidemic model. The occurrence rate density is computed by the sum of two terms, one representing the independent, or spontaneous activity, and the other representing the activity induced by previous earthquakes. While the first term depends only on space, the second one is factored into three terms that include the magnitude, time, and location, respectively, of the past earthquakes. In this paper we use the modified Omori law for the time term, focusing our investigation on the magnitude and space terms. We formulate two different hypotheses for each of them, and we find the respective maximum likelihood parameters on the basis of the catalog of instrumental seismicity recorded in Italy from 1987 to 2000. The comparison of the respective likelihood computed for the seismicity recorded in 2001 provides a way for choosing the best model. The confidence level of our choice is then assessed by means of a Monte Carlo simulation on the various hypotheses. Our study shows that an inverse power density function is more reliable than a normal density function for the space distribution and that the hypothesis of scale invariance of aftershock productivity with respect to magnitude can be rejected with high confidence level. The final model is suitable for computing earthquake occurrence probability in real circumstances.

A Stochastic Two-node Stress Transfer Model Reproducing Omori's Law

— We present an alternative to the epidemic type aftershock sequence (ETAS) model of Ogata (1988). The continuous time two-node network stress release/transfer Markov model is able to reproduce the (modified) Omori law for aftershock frequencies. One node (denoted by A) is loaded by external tectonic forces at a constant rate, with `events' (main shocks) occurring at random instances with risk given by a function of the `stress level' at the node. Each event is a random (negative) jump of the stress level, and adds (or removes) a random amount of stress to the second node (B), which experiences `events' in a similar way, but with another risk function (of the stress level at that node only). When that risk function satisfies certain simple conditions (it may, in particular, be exponential), the frequency of jumps (aftershocks) at node B, in the absence of any new events at node A, follows Omori's law \((\prpto (c + t)^{-1})\) for aftershock squenc es. When node B is allowed tectonic input, which may be negative, i.e., aseismic slip, the frequency of events takes on a decay form that parallels the constitutive law derived by Dieterich (1994), which fits very well to the modified Omori law. We illustrate the model by fitting it to aftershock data from California post-1973, and from the Valparaiso earthquake of March 3 1985.

The Colima, Mexico, Earthquake (MW 5.3) of 7 March 2000: Seismic Activity Along the Southern Colima Rift

The 7 March 2000 earthquake ( M W 5.3), located within the southern Colima rift in southwestern Mexico, is the largest crustal earthquake ever recorded within the state of Colima, Mexico. The mainshock was located at 8 km depth, and the aftershocks are distributed at depths between 5 and 10 km. Aftershock distribution follows a normal decay as measured by the modified Omori's law, with a p value of 0.96. Both aftershock locations and focal mechanisms (determined for the mainshock and some of the aftershocks) delineate a fault dipping toward the northwest and striking west-southwest. This fault segment is parallel to the southern end of the Tamazula fault, which ends at Manzanillo Bay. Regional waveform modeling for the mainshock situates the earthquake on a normal fault with a significant left-lateral strike-slip component on a plane striking 260° and dipping toward the northwest at 46°. We use a small aftershock with similar focal parameters as the mainshock to deconvolve P and S waves recorded at the permanent station COIG. From the deconvolution we obtained a source duration of 0.75 sec from the P waves and 0.9 sec from the S waves for the main event and a stress drop of 320 bars. The aftershock area is much larger than the area measured from the apparent source time function. From the aftershock area and the seismic moment obtained by Harvard University (Centroid Moment Tensor Catalog), we compute a static stress drop of 20 bars. Fault-plane solutions from the mainshock and some of the aftershocks indicate that the northern, and larger, segment of the Tamazula fault is perpendicular to the extensional axis (directed east-southeast). This study confirms the change in the direction of the extensional axis of the southern Colima rift with respect to the northern Colima rift (extensional axis oriented almost east-west). Here we show that the Tamazula fault zone, at least its eastern segment, is active, and a re-evaluation of the seismic hazard posed by this fault to the state of Colima and the cities of Colima and Manzanillo is urgent. Manuscript received 13 September 2002.