The size-frequency distribution of many geological and geophysical variables, in relation to fractures, faulting and seismicity, is well described by a statistical distribution of the power law type which is characterized by its exponent. For earthquake magnitudes, the exponent is the well-known b-parameter of the Gutenberg-Richter scaling law. In this paper we:•provide a strict statistical derivation of the distribution law of earthquake magnitudes,•show that the maximum likelihood estimator of the b-parameter is unbiased,•demonstrate that the maximum likelihood estimator is invariant to the value chosen as the minimum magnitude threshold in so far as it is larger than the magnitude of completeness of the earthquake catalogue, and•provide a new estimator based on the minimization of the Kolmogorov-Smirnov statistic and provide a strategy for detecting and mapping the spatio-temporal variation of the b-parameter in seismic swarms.

中文翻译：

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Gutenberg-Richter b 值的推断：新见解和结果


许多地质和地球物理变量的大小-频率分布，与裂缝、断层和地震活动有关，可以用幂律类型的统计分布很好地描述，该统计分布以其指数为特征。对于地震震级，指数是众所周知的古腾堡-里氏标度定律的 b 参数。在本文中，我们：•提供了地震震级分布规律的严格统计推导，•表明 b 参数的最大似然估计量是无偏的，•证明最大似然估计量对于选择作为最小震级阈值的值是不变的，只要它大于地震目录的完备性震级， 以及•提供基于 Kolmogorov-Smirnov 统计量最小化的新估计器，并提供一种用于检测和绘制地震群中 b 参数的时空变化的策略。