Power Law Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering

We consider a general stochastic branching process, which is relevant to earthquakes as well as to many other systems, and we study the distributions of the total number of offsprings (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction. We apply our results to a branching model of triggered seismicity, the ETAS (epidemic-type aftershock sequence) model. The ETAS model 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 (``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 ~1/\mu^(1+gamma). For earthquakes, we expect such a power-law distribution of fertilities with gamma = b/alpha based on the Gutenberg-Richter magnitude distribution ~10^(-bm) and on the increase ~10^(alpha m) of the number of aftershocks with the mainshock magnitude m. We derive the asymptotic distributions p_r(r) and p_g(g) of the total number r of offsprings and of the total number g of generations until extinction following a mainshock. In the regime \gamma<2 relevant for earhquakes, for which the distribution of fertilities has an infinite variance, we find p_r(r)~1/r^(1+1/gamma) and p_g(g)~1/g^(1+1/(gamma -1)). These predictions are checked by numerical simulations.Comment: revtex, 12 pages, 2 ps figures. In press in Pure and Applied Geophysics (2004

Earthquake source parameters and fault kinematics in the Eastern California Shear Zone

Based on waveform data from a profile of aftershocks following the north-south trace of the June 28, 1992 Landers rupture across the Mojave desert, we construct a new velocity model for the Mojave region which features a thin, slow crust. Using this model, we obtain source parameters, including depth and duration, for each of the aftershocks in the profile, and in addition, any significant (M>3.7) Joshua Tree--Landers aftershock between April, 1992 and October, 1994 for which coherent TERRAscope data were available. In all, we determine source parameters and stress-drops for 45 significant (M_w > 4) earthquakes associated with the Joshua Tree and Landers sequences, using a waveform grid-search algorithm. Stress drops for these earthquakes appear to vary systematically with location, with respect to previous seismic activity, proximity to previous rupture (i.e., with respect to the Landers rupture), and with tectonic province. In general, for areas north of the Pinto Mountain fault, stress-drops of aftershocks located off the faults involved with the Landers rupture are higher than those located on the fault, with the exception of aftershocks on the newly recognized Kickapoo (Landers) fault. Stress drops are moderate south of the Pinto Mountain fault, where there is a history of seismic swarms but no single through-going fault. In contrast to aftershocks in the eastern Transverse ranges, and related to the 1992 Big Bear, California, sequence, Landers events show no clear relationship between stress-drop and depth. Instead, higher stress-drop aftershocks appear to correlate with activity on nascent faults, or those which experienced relatively small slip during mainshock rupture.Comment: 27 pages, 15 figures, to appear in Bull. Seism. Soc. A

Unified Scaling Law for Earthquakes

We show that the distribution of waiting times between earthquakes occurring in California obeys a simple unified scaling law valid from tens of seconds to tens of years, see Eq. (1) and Fig. 4. The short time clustering, commonly referred to as aftershocks, is nothing but the short time limit of the general hierarchical properties of earthquakes. There is no unique operational way of distinguishing between main shocks and aftershocks. In the unified law, the Gutenberg-Richter b-value, the exponent -1 of the Omori law for aftershocks, and the fractal dimension d_f of earthquakes appear as critical indices.Comment: 4 pages, 4 figure

Bath's law Derived from the Gutenberg-Richter law and from Aftershock Properties

The empirical Bath's law states that the average difference in magnitude between a mainshock and its largest aftershock is 1.2, regardless of the mainshock magnitude. Following Vere-Jones [1969] and Console et al. [2003], we show that the origin of Bath's law is to be found in the selection procedure used to define mainshocks and aftershocks rather than in any difference in the mechanisms controlling the magnitude of the mainshock and of the aftershocks. We use the ETAS model of seismicity, which provides a more realistic model of aftershocks, based on (i) a universal Gutenberg-Richter (GR) law for all earthquakes, and on (ii) the increase of the number of aftershocks with the mainshock magnitude. Using numerical simulations of the ETAS model, we show that this model is in good agreement with Bath's law in a certain range of the model parameters.Comment: major revisions, in press in Geophys. Res. Let

Sub-critical and Super-critical Regimes in Epidemic Models of Earthquake Aftershocks

We present an analytical solution and numerical tests of the epidemic-type aftershock (ETAS) model for aftershocks, which describes foreshocks, aftershocks and mainshocks on the same footing. The occurrence rate of aftershocks triggered by a single mainshock decreases with the time from the mainshock according to the modified Omori law K/(t+c)^p with p=1+theta. A mainshock at time t=0 triggers aftershocks according to the local Omori law, that in turn trigger their own aftershocks and so on. The effective branching parameter n, defined as the mean aftershock number triggered per event, controls the transition between a sub-critical regime n<1 to a super-critical regime n>1. In the sub-critical regime, we recover and document the crossover from an Omori exponent 1-theta for t<t* to 1+theta for t<t* found previously in [Sornette and Sornette, 1999a] for a special case of the ETAS model. In the super-critical regime n>1 and theta>0, we find a novel transition from an Omori decay law with exponent 1-theta fot t<t* to an explosive exponential increase of the seismicity rate fot t>t*. The case theta<0 yields an infinite n-value. In this case, we find another characteristic time tau controlling the crossover from an Omori law with exponent 1-theta for t<tau, similar to the local law, to an exponential increase at large times. These results can rationalize many of the stylized facts reported for aftershock and foreshock sequences, such as (i) the suggestion that a small p-value may be a precursor of a large earthquake, (ii) the relative seismic quiescence sometimes observed before large aftershocks, (iii) the positive correlation between b and p-values, (iv) the observation that great earthquakes are sometimes preceded by a decrease of b-value and (v) the acceleration of the seismicity preceding great earthquakes.Comment: Latex document of 41 pages + 6 eps figures + 1 tabl

Aftershock production rate of driven viscoelastic interfaces

We study analytically and by numerical simulations the statistics of the aftershocks generated after large avalanches in models of interface depinning that include viscoelastic relaxation effects. We find in all the analyzed cases that the decay law of aftershocks with time can be understood by considering the typical roughness of the interface and its evolution due to relaxation. In models where there is a single viscoelastic relaxation time there is an exponential decay of the number of aftershocks with time. In models in which viscoelastic relaxation is wave-vector dependent we typically find a power law dependence of the decay rate, compatible with the Omori law. The factors that determine the value of the decay exponent are analyzed

Importance of direct and indirect triggered seismicity

Using the simple ETAS branching model of seismicity, which assumes that each earthquake can trigger other earthquakes, we quantify the role played by the cascade of triggered seismicity in controlling the rate of aftershock decay as well as the overall level of seismicity in the presence of a constant external seismicity source. We show that, in this model, the fraction of earthquakes in the population that are aftershocks is equal to the fraction of aftershocks that are indirectly triggered and is given by the average number of triggered events per earthquake. Previous observations that a significant fraction of earthquakes are triggered earthquakes therefore imply that most aftershocks are indirectly triggered by the mainshock.Comment: Latex document of 17 pages + 2 postscript figure

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

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.Comment: 11 pages, 2 figures, 1 table. Dedicated to Francois Bardou (1968-2006