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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.
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
We report an empirical determination of the probability density functions $P_{text{data}}(r)$ of the number $r$ of earthquakes in finite space-time windows for the California catalog. We find a stable power law tail $P_{text{data}}(r) sim 1/r^{1+mu}$
We report an empirical determination of the probability density functions P(r) of the number r of earthquakes in finite space-time windows for the California catalog, over fixed spatial boxes 5 x 5 km^2 and time intervals dt =1, 10, 100 and 1000 days
We analyze the transformation properties of Faraday law in an empty space and its relationship with Maxwell equations. In our analysis we express the Faraday law via the four-potential of electromagnetic field and the field of four-velocity, defined
The driving concept behind one of the most successful statistical forecasting models, the ETAS model, has been that the seismicity is driven by spontaneously occurring background earthquakes that cascade into multitudes of triggered earthquakes. In n