1,639 research outputs found
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 , the average
clustering ratio , and the productivity exponent times the -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
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