Biomedical subjects

A Saichev

Publications and source records attributed to A Saichev.

6 recordsLinked to original sources

PubMed2006

"Universal" distribution of interearthquake times explained.

We propose a simple theory for the "universal" scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks, and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of interevent times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforce the view that triggering earthquakes by other earthquakes is a key physical mechanism to understand seismicity.

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PubMed2006

Generic multifractality in exponentials of long memory processes.

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent phi+1/2, where phi>0. This generalizes previous studies performed only with phi=0(with a truncation at an integral scale) by showing that multifractality holds over a remarkably large range of dimensionless scales for phi>0. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation phi from 1/2 and of another parameter sigma2 embodying information on the short-range amplitude of the memory kernel, the ultraviolet cutoff ("viscous") scale, and the variance of the white-noise innovations. In these processes, both a viscous scale and an integral scale naturally appear, bracketing the "inertial" scaling regime. We exhibit a surprisingly good collapse of the multifractal spectra zeta(q) on a universal scaling function, which enables us to derive high-order multifractal exponents from the small-order values and also obtain a given multifractal spectrum zeta(q) by different combinations of phi and sigma2.

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PubMed2005

Vere-Jones' self-similar branching model.

Motivated by its potential application to earthquake statistics as well as for its intrinsic interest in the theory of branching processes, we study the exactly self-similar branching process introduced recently by Vere-Jones. This model extends the ETAS class of conditional self-excited branching point-processes of triggered seismicity by removing the problematic need for a minimum (as well as maximum) earthquake size. To make the theory convergent without the need for the usual ultraviolet and infrared cutoffs, the distribution of magnitudes m' of daughters of first-generation of a mother of magnitude m has two branches m < m' with exponent beta - d and m' > m with exponent beta + d, where beta and d are two positive parameters. We investigate the condition and nature of the subcritical, critical, and supercritical regime in this and in an extended version interpolating smoothly between several models. We predict that the distribution of magnitudes of events triggered by a mother of magnitude m over all generations has also two branches m' < m with exponent and with exponent beta - h, with h=d squareroot of (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. 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 with a single branch and is blind to the exponents beta, d of the distribution of triggered events. Since the distribution of earthquake magnitudes is usually obtained with catalogs including many sequences, we conclude that the two branches of the distribution of aftershocks are not directly observable and the model is compatible with real seismic catalogs. In summary, the exactly self-similar Vere-Jones model provides an attractive new approach to model triggered seismicity, which alleviates delicate questions on the role of magnitude cutoffs in other non-self-similar models. The new prediction concerning two branches in the distribution of magnitudes of aftershocks could be tested with recently introduced stochastic reconstruction methods, tailored to disentangle the different triggered sequences.

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PubMed2005

Distribution of the largest aftershocks in branching models of triggered seismicity: theory of the universal Båth law.

Using the epidemic-type aftershock sequence (ETAS) branching model of triggered seismicity, we apply the formalism of generating probability functions to calculate exactly the average difference between the magnitude of a mainshock and the magnitude of its largest aftershock over all generations. This average magnitude difference is found empirically to be independent of the mainshock magnitude and equal to 1.2, a universal behavior known as Båth's law. Our theory shows that Båth's law holds only sufficiently close to the critical regime of the ETAS branching process. Allowing for error bars +/- 0.1 for Båth's constant value around 1.2, our exact analytical treatment of Båth's law provides new constraints on the productivity exponent alpha and the branching ratio n: 0.9 approximately < alpha < or =1. We propose a method for measuring alpha based on the predicted renormalization of the Gutenberg-Richter distribution of the magnitudes of the largest aftershock. We also introduce the "second Båth law for foreshocks:" the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude rho.

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PubMed2005

Andrade, Omori, and time-to-failure laws from thermal noise in material rupture.

Using a simple mean-field rupture model with quenched disorder in the presence of thermal fluctuations introduced by S. Ciliberto et al., we provide an analytical theory of three ubiquitous empirical observations obtained in creep (constant applied stress) experiments: the initial Andrade-like and Omori-like 1/t decay of the rate of deformation and of fiber ruptures and the 1/( tc-t) critical time-to-failure behavior of acoustic emissions just prior to the macroscopic rupture. The lifetime of the material is controlled by a thermally activated Arrhenius nucleation process, describing the crossover between these two regimes, as shown by S. Ciliberto et al. Thus tiny thermal fluctuations may actually play an essential role in macroscopic deformation and rupture processes at room temperature. We also discover a reentrant dependence of the lifetime as a function of the amount of quenched disorder.

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PubMed2004

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.

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