Apparent Clustering and Apparent Background Earthquakes Biased by Undetected Seismicity

In models of triggered seismicity and in their inversion with empirical data, the detection threshold m_d is commonly equated to the magnitude m_0 of the smallest triggering earthquake. This unjustified assumption neglects the possibility of shocks below the detection threshold triggering observable events. We introduce a formalism that distinguishes between the detection threshold m_d and the minimum triggering earthquake m_0 < m_d. By considering the branching structure of one complete cascade of triggered events, we derive the apparent branching ratio n_a (which is the apparent fraction of aftershocks in a given catalog) and the apparent background source S_a that are observed when only the structure above the detection threshold m_d is known due to the presence of smaller undetected events that are capable of triggering larger events. If earthquake triggering is controlled in large part by the smallest magnitudes as several recent analyses have shown, this implies that previous estimates of the clustering parameters may significantly underestimate the true values: for instance, an observed fraction of 55% of aftershocks is renormalized into a true value of 75% of triggered events.Comment: 12 pages; incl. 6 Figures, AGU styl

Is Earthquake Triggering Driven by Small Earthquakes?

Using a catalog of seismicity for Southern California, we measure how the number of triggered earthquakes increases with the earthquake magnitude. The trade-off between this relation and the distribution of earthquake magnitudes controls the relative role of small compared to large earthquakes. We show that seismicity triggering is driven by the smallest earthquakes, which trigger fewer events than larger earthquakes, but which are much more numerous. We propose that the non-trivial scaling of the number of triggered earthquakes emerges from the fractal spatial distribution of seismicity.Comment: 5 pages, 2 figure

Are Aftershocks of Large Californian Earthquakes Diffusing?

We analyze 21 aftershock sequences of California to test for evidence of space-time diffusion. Aftershock diffusion may result from stress diffusion and is also predicted by any mechanism of stress weakening. Here, we test an alternative mechanism to explain aftershock diffusion, based on multiple cascades of triggering. In order to characterize aftershock diffusion, we develop two methods, one based on a suitable time and space windowing, the other using a wavelet transform adapted to the removal of background seismicity. Both methods confirm that diffusion of seismic activity is very weak, much weaker than reported in previous studies. A possible mechanism explaining the weakness of observed diffusion is the effect of geometry, including the localization of aftershocks on a fractal fault network and the impact of extended rupture lengths which control the typical distances of interaction between earthquakes.Comment: latex file of 34 pages, 15 postscript figures, minor revision. In press in J. Geophys. Re

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

Intercluster Correlation in Seismicity

Mega et al.(cond-mat/0212529) proposed to use the ``diffusion entropy'' (DE) method to demonstrate that the distribution of time intervals between a large earthquake (the mainshock of a given seismic sequence) and the next one does not obey Poisson statistics. We have performed synthetic tests which show that the DE is unable to detect correlations between clusters, thus negating the claimed possibility of detecting an intercluster correlation. We also show that the LR model, proposed by Mega et al. to reproduce inter-cluster correlation, is insufficient to account for the correlation observed in the data.Comment: Comment on Mega et al., Phys. Rev. Lett. 90. 188501 (2003) (cond-mat/0212529

A note on fractional linear pure birth and pure death processes in epidemic models

In this note we highlight the role of fractional linear birth and linear death processes recently studied in \citet{sakhno} and \citet{pol}, in relation to epidemic models with empirical power law distribution of the events. Taking inspiration from a formal analogy between the equation of self consistency of the epidemic type aftershock sequences (ETAS) model, and the fractional differential equation describing the mean value of fractional linear growth processes, we show some interesting applications of fractional modelling to study \textit{ab initio} epidemic processes without the assumption of any empirical distribution. We also show that, in the frame of fractional modelling, subcritical regimes can be linked to linear fractional death processes and supercritical regimes to linear fractional birth processes. Moreover we discuss a simple toy model to underline the possible application of these stochastic growth models to more general epidemic phenomena such as tumoral growth

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

Theory of Earthquake Recurrence Times

The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions (Corral, 2003) and when averaged over multiple regions (Bak et al.,2002). These unified scaling laws are characterized by intermediate power law asymptotics. On the other hand, Molchan (2005) has presented a mathematical proof that, if such a universal law exists, it is necessarily an exponential, in obvious contradiction with the data. First, we generalize Molchan's argument to show that an approximate unified law can be found which is compatible with the empirical observations when incorporating the impact of the Omori law of earthquake triggering. We then develop the full theory of the statistics of inter-event times in the framework of the ETAS model of triggered seismicity and show that the empirical observations can be fully explained. Our theoretical expression fits well the empirical statistics over the whole range of recurrence times, accounting for different regimes by using only the physics of triggering quantified by Omori's law. The description of the statistics of recurrence times over multiple regions requires an additional subtle statistical derivation that maps the fractal geometry of earthquake epicenters onto the distribution of the average seismic rates in multiple regions. This yields a prediction in excellent agreement with the empirical data for reasonable values of the fractal dimension $d \approx 1.8$, the average clustering ratio $n \approx 0.9$, and the productivity exponent $\alpha \approx 0.9$ times the $b$-value of the Gutenberg-Richter law.Comment: 30 pages + 13 figure

Predictability in the ETAS 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 based on 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 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. 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.Comment: Latex file of 20 pages + 15 eps figures + 2 tables, in press in J. Geophys. Re