Saichev A, Sornette D
Mathematical Department, Nizhny Novgorod State University, Gagarin prosp. 23, Nizhny Novgorod 603950, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 May;71(5 Pt 2):056127. doi: 10.1103/PhysRevE.71.056127. Epub 2005 May 31.
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.
利用触发地震活动的流行型余震序列（ETAS）分支模型，我们应用生成概率函数的形式体系，精确计算了所有世代中主震震级与其最大余震震级之间的平均差值。通过经验发现，这种平均震级差值与主震震级无关，且等于1.2，这是一种被称为巴思定律的普遍行为。我们的理论表明，巴思定律仅在足够接近ETAS分支过程的临界状态时成立。考虑到巴思常数在1.2左右的误差范围为±0.1，我们对巴思定律的精确解析处理为生产率指数α和分支比n提供了新的约束：0.9 ≈ < α ≤ 1。我们提出了一种基于最大余震震级的古登堡-里希特分布的预测重整化来测量α的方法。我们还引入了“前震的第二条巴思定律”：一次主震最终成为前震的概率不取决于其震级ρ。
