582 research outputs found
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
Brittle creep, damage and time to failure in rocks
International audienceWe propose a numerical model based on static fatigue laws in order to model the time-dependent damage and deformation of rocks under creep. An empirical relation between time to failure and applied stress is used to simulate the behavior of each element of our finite element model. We review available data on creep experiments in order to study how the material properties and the loading conditions control the failure time. The main parameter that controls the failure time is the applied stress. Two commonly used models, an exponential tfexp (bs/s0) and a power law function tfsb0 fit the data as well. These time-to-failure laws are used at the scale of each element to simulate its damage as a function of its stress history. An element is damaged by decreasing its Young's modulus to simulate the effect of increasing crack density at smaller scales. Elastic interactions between elements and heterogeneity of the mechanical properties lead to the emergence of a complex macroscopic behavior, which is richer than the elementary one. In particular, we observe primary and tertiary creep regimes associated respectively with a power law decay and increase of the rate of strain, damage event and energy release. Our model produces a power law distribution of damage event sizes, with an average size that increases with time as a power law until macroscopic failure. Damage localization emerges at the transition between primary and tertiary creep, when damage rate starts accelerating. The final state of the simulation shows highly damaged bands, similar to shear bands observed in laboratory experiments. The thickness and the orientation of these bands depend on the applied stress. This model thus reproduces many properties of rock creep, which were previously not modeled simultaneously
Power Law Distributions of Seismic Rates
We report an empirical determination of the probability density functions
of the number of earthquakes in finite space-time
windows for the California catalog. We find a stable power law tail
with exponent for all
space ( to km) and time intervals (0.1 to 1000
days). These observations, as well as the non-universal dependence on
space-time windows for all different space-time windows simultaneously, are
explained by solving one of the most used reference model in seismology (ETAS),
which assumes that each earthquake can trigger other earthquakes. The data
imposes that active seismic regions are Cauchy-like fractals, whose exponent
is well-constrained by the seismic rate data.Comment: 5 pages with 1 figur
Properties of Foreshocks and Aftershocks of the Non-Conservative SOC Olami-Feder-Christensen Model: Triggered or Critical Earthquakes?
Following Hergarten and Neugebauer [2002] who discovered aftershock and
foreshock sequences 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 properties of earthquake
clustering in time and space. Our main finding is that synthetic catalogs
generated by the OFC model share practically all properties of real seismicity
at a qualitative level, with however significant quantitative differences. We
find that OFC catalogs can be in large part described by the concept of
triggered seismicity but the properties of foreshocks depend on the mainshock
magnitude, in qualitative agreement with the critical earthquake model and in
disagreement with simple models of triggered seismicity such as the Epidemic
Type Aftershock Sequence (ETAS) model [Ogata, 1988]. Many other features of OFC
catalogs can be reproduced with the ETAS model with a weaker clustering than
real seismicity, i.e. for a very small average number of triggered earthquakes
of first generation per mother-earthquake.Comment: revtex, 19 pages, 8 eps figure
Vere-Jones' Self-Similar Branching Model
Motivated by its potential application to earthquake statistics, we study the
exactly self-similar branching process introduced recently by Vere-Jones, which
extends the ETAS class of conditional branching point-processes of triggered
seismicity. One of the main ingredient of Vere-Jones' model is that the power
law distribution of magnitudes m' of daughters of first-generation of a mother
of magnitude m has two branches m'm with
exponent beta+d, where beta and d are two positive parameters. We predict that
the distribution of magnitudes of events triggered by a mother of magnitude
over all generations has also two branches m'm
with exponent beta+h, with h= d \sqrt{1-s}, where s is the fraction of
triggered events. This corresponds to a renormalization of the exponent d into
h by the hierarchy of successive generations of triggered events. The empirical
absence of such two-branched distributions implies, if this model is seriously
considered, that the earth is close to criticality (s close to 1) so that beta
- h \approx \beta + h \approx \beta. We also find that, for a significant part
of the parameter space, the distribution of magnitudes over a full catalog
summed over an average steady flow of spontaneous sources (immigrants)
reproduces the distribution of the spontaneous sources and is blind to the
exponents beta, d of the distribution of triggered events.Comment: 13 page + 3 eps figure
Adaptively Smoothed Seismicity Earthquake Forecasts for Italy
We present a model for estimating the probabilities of future earthquakes of
magnitudes m > 4.95 in Italy. The model, a slightly modified version of the one
proposed for California by Helmstetter et al. (2007) and Werner et al. (2010),
approximates seismicity by a spatially heterogeneous, temporally homogeneous
Poisson point process. The temporal, spatial and magnitude dimensions are
entirely decoupled. Magnitudes are independently and identically distributed
according to a tapered Gutenberg-Richter magnitude distribution. We estimated
the spatial distribution of future seismicity by smoothing the locations of
past earthquakes listed in two Italian catalogs: a short instrumental catalog
and a longer instrumental and historical catalog. The bandwidth of the adaptive
spatial kernel is estimated by optimizing the predictive power of the kernel
estimate of the spatial earthquake density in retrospective forecasts. When
available and trustworthy, we used small earthquakes m>2.95 to illuminate
active fault structures and likely future epicenters. By calibrating the model
on two catalogs of different duration to create two forecasts, we intend to
quantify the loss (or gain) of predictability incurred when only a short but
recent data record is available. Both forecasts, scaled to five and ten years,
were submitted to the Italian prospective forecasting experiment of the global
Collaboratory for the Study of Earthquake Predictability (CSEP). An earlier
forecast from the model was submitted by Helmstetter et al. (2007) to the
Regional Earthquake Likelihood Model (RELM) experiment in California, and, with
over half of the five-year experiment over, the forecast performs better than
its competitors.Comment: revised manuscript. 22 pages, 3 figures, 2 table
Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes
We present the first exact analysis of some of the temporal properties of
multivariate self-excited Hawkes conditional Poisson processes, which
constitute powerful representations of a large variety of systems with bursty
events, for which past activity triggers future activity. The term
"multivariate" refers to the property that events come in different types, with
possibly different intra- and inter-triggering abilities. We develop the
general formalism of the multivariate generating moment function for the
cumulative number of first-generation and of all generation events triggered by
a given mother event (the "shock") as a function of the current time . This
corresponds to studying the response function of the process. A variety of
different systems have been analyzed. In particular, for systems in which
triggering between events of different types proceeds through a one-dimension
directed or symmetric chain of influence in type space, we report a novel
hierarchy of intermediate asymptotic power law decays of the rate of triggered events as a function of the
distance of the events to the initial shock in the type space, where for the relevant long-memory processes characterizing many natural
and social systems. The richness of the generated time dynamics comes from the
cascades of intermediate events of possibly different kinds, unfolding via a
kind of inter-breeding genealogy.Comment: 40 pages, 8 figure
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