Interevent time distribution in seismicity: a theoretical approach

Forecasting ability of a multi-renewal seismicity model for Italy

The inter-event time, IET, is sometimes used as a basis for prediction of large earthquakes. It is the case when theoretical analysis of prediction is possible. Quite recently a specific IET- model was suggested for dynamic probabilistic prediction of M > 5.5 events in Italy . In this study we analyze both some aspects of the statistical estimation of the model and its predictive ability. We find that more or less effective prediction is possible within 4 out of 34 seismotectonic zones where seismicity rate or clustering of events is relatively high. We show that, in the framework of the model…

Utsu aftershock productivity law explained from geometric operations on the permanent static stress field of mainshocks

The aftershock productivity law, first described by Utsu in 1970, is an exponential function of the form K=K0.exp(αM) where K is the number of aftershocks, M the mainshock magnitude, and α the productivity parameter. The Utsu law remains empirical in nature although it has also been retrieved in static stress simulations. Here, we explain this law based on Solid Seismicity, a geometrical theory of seismicity where seismicity patterns are described by mathematical expressions obtained from geometric operations on a permanent static stress field. We recover the exponential form but with a break …

Text Classification of the Precursory Accelerating Seismicity Corpus: Inference on some Theoretical Trends in Earthquake Predictability Research from 1988 to 2018

Text analytics based on supervised machine learning classifiers has shown great promise in a multitude of domains, but has yet to be applied to Seismology. We test various standard models (Naive Bayes, k-Nearest Neighbors, Support Vector Machines, and Random Forests) on a seismological corpus of 100 articles related to the topic of precursory accelerating seismicity, spanning from 1988 to 2010. This corpus was labelled in Mignan (2011) with the precursor whether explained by critical processes (i.e., cascade triggering) or by other processes (such as signature of main fault loading). We invest…

Productivity within the ETAS seismicity model

The productivity of a magnitude $m$ event can be characterized in term of triggered events of magnitude above $m-Δ$: it is the number of direct "descendants" $ν_Δ$ and the number of all "descendants" $V_Δ$. There is evidence in favour of the discrete exponential distribution for both $ν_Δ$ and $V_Δ$ with a dominant magnitude $m$ (the case of aftershock cluster). We consider the general Epidemic Type Aftershock Sequence (ETAS) model adapted to any distribution of $ν_Δ$. It turns out that the branching structure of the model excludes the possibility of having exponential distributions for both p…

流行病型地震模型中的余震数量

Number of aftershocks in epidemic-type seismicity models

本文研究了流行病型地震模型 ETAS(F) 中余震数量的极限分布，发现在次临界情况下极限分布极其简单，为测试 F 分布类型、Bath 定律及聚类概念提供了理论依据。

Self-similar aftershock rates

In many important systems exhibiting crackling noise --- intermittent avalanche-like relaxation response with power-law and, thus, self-similar distributed event sizes --- the "laws" for the rate of activity after large events are not consistent with the overall self-similar behavior expected on theoretical grounds. This is in particular true for the case of seismicity and a satisfying solution to this paradox has remained outstanding. Here, we propose a generalized description of the aftershock rates which is both self-similar and consistent with all other known self-similar features. Compari…

Statistics of seismic cluster durations

Using the standard ETAS model of triggered seismicity, we present a rigorous theoretical analysis of the main statistical properties of temporal clusters, defined as the group of events triggered by a given main shock of fixed magnitude m that occurred at the origin of time, at times larger than some present time t. Using the technology of generating probability function (GPF), we derive the explicit expressions for the GPF of the number of future offsprings in a given temporal seismic cluster, defining, in particular, the statistics of the cluster's duration and the cluster's offsprings maxim…

Random stress and Omori's law

We consider two statistical regularities that were used to explain Omori's law of the aftershock rate decay: the Levy and Inverse Gaussian (IGD) distributions. These distributions are thought to describe stress behavior influenced by various random factors: post-earthquake stress time history is described by a Brownian motion. Both distributions decay to zero for time intervals close to zero. But this feature contradicts the high immediate aftershock level according to Omori's law. We propose that these statistical distributions are influenced by the power-law stress distribution near the eart…

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 em…