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Register a new userRenormalization of the ETAS branching model of triggered seismicity from total to observable seismicity
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Several recent works point out that the crowd of small unobservable earthquakes (with magnitudes below the detection threshold $m_d$) may play a significant and perhaps dominant role in triggering future seismicity. Using the ETAS branching model of triggered seismicity, we apply the formalism of generating probability functions to investigate how the statistical properties of observable earthquakes differ from the statistics of all events. The ETAS (epidemic-type aftershock sequence) 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. The triggering efficiency of earthquakes is assumed to vanish below a lower magnitude limit $m_0$, in order to ensure the convergence of the theory and may reflect the physics of state-and-velocity frictional rupture. We show that, to a good approximation, the ETAS model is renormalized onto itself under what amounts to a decimation procedure $m_0 to m_d$, with just a renormalization of the branching ratio from $n$ to an effective value $n(m_d)$. Our present analysis thus confirms, for the full statistical properties, the results obtained previously by one of us and Werner, based solely on the average seismic rates (the first-order moment of the statistics). However, our analysis also demonstrates that this renormalization is not exact, as there are small corrections which can be systematically calculated, in terms of additional contributions that can be mapped onto a different branching model (a new relevant direction in the language of the renormalization group).
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Using the 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 Baths law. Our theory shows that Baths law holds only sufficiently close to the critical regime of the ETAS branching process. Allowing for error bars +- 0.1 for Baths constant value around 1.2, our exact analytical treatment of Baths law provides new constraints on the productivity exponent alpha and the branching ratio n: $0.9 <= alpha <= 1$ and 0.8 <= n <= 1. We propose a novel 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 Baths law for foreshocks: the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude.
This entry in the Encyclopedia of Complexity and Systems Science, Springer present a summary of some of the concepts and calculational tools that have been developed in attempts to apply statistical physics approaches to seismology. We summarize the leading theoretical physical models of the space-time organization of earthquakes. We present a general discussion and several examples of the new metrics proposed by statistical physicists, underlining their strengths and weaknesses. The entry concludes by briefly outlining future directions. The presentation is organized as follows. I Glossary II Definition and Importance of the Subject III Introduction IV Concepts and Calculational Tools IV.1 Renormalization, Scaling and the Role of Small Earthquakes in Models of Triggered Seismicity IV.2 Universality IV.3 Intermittent Periodicity and Chaos IV.4 Turbulence IV.5 Self-Organized Criticality V Competing mechanisms and models V.1 Roots of complexity in seismicity: dynamics or heterogeneity? V.2 Critical earthquakes V.3 Spinodal decomposition V.4 Dynamics, stress interaction and thermal fluctuation effects VI Empirical studies of seismicity inspired by statistical physics VI.1 Early successes and latter subsequent challenges VI.2 Entropy method for the distribution of time intervals between mainshocks VI.3 Scaling of the PDF of Waiting Times VI.4 Scaling of the PDF of Distances Between Subsequent Earthquakes VI.5 The Network Approach VII Future Directions
The standard paradigm to describe seismicity induced by fluid injection is to apply nonlinear diffusion dynamics in a poroelastic medium. I show that the spatiotemporal behaviour and rate evolution of induced seismicity can, instead, be expressed by geometric operations on a static stress field produced by volume change at depth. I obtain laws similar in form to the ones derived from poroelasticity while requiring a lower description length. Although fluid flow is known to occur in the ground, it is not pertinent to the behaviour of induced seismicity. The proposed model is equivalent to the static stress model for tectonic foreshocks generated by the Non- Critical Precursory Accelerating Seismicity Theory. This study hence verifies the explanatory power of this theory outside of its original scope.
We report a similarity of fluctuations in equilibrium critical phenomena and non-equilibrium systems, which is based on the concept of natural time. The world-wide seismicity as well as that of San Andreas fault system and Japan are analyzed. An order parameter is chosen and its fluctuations relative to the standard deviation of the distribution are studied. We find that the scaled distributions fall on the same curve, which interestingly exhibits, over four orders of magnitude, features similar to those in several equilibrium critical phenomena (e.g., 2D Ising model) as well as in non-equilibrium systems (e.g., 3D turbulent flow).
Seismicity and faulting within the Earth crust are characterized by many scaling laws that are usually interpreted as qualifying the existence of underlying physical mechanisms associated with some kind of criticality in the sense of phase transitions. Using an augmented Epidemic-Type Aftershock Sequence (ETAS) model that accounts for the spatial variability of the background rates $mu(x,y)$, we present a direct quantitative test of criticality. We calibrate the model to the ANSS catalog of the entire globe, the region around California, and the Geonet catalog for the region around New Zealand using an extended Expectation-Maximization (EM) algorithm including the determination of $mu(x,y)$. We demonstrate that the criticality reported in previous studies is spurious and can be attributed to a systematic upward bias in the calibration of the branching ratio of the ETAS model, when not accounting correctly for spatial variability. We validate the version of the ETAS model which possesses a space varying background rate $mu(x,y)$ by performing pseudo prospective forecasting tests. The non-criticality of seismicity has major implications for the prediction of large events.
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