Mc-Lilliefors: a completeness magnitude that complies with the exponential-like Gutenberg–Richter relation

This upload is associated with the publication:

Herrmann, M. and W. Marzocchi (2020). Inconsistencies and Lurking Pitfalls in the Magnitude–Frequency Distribution of High-Resolution Earthquake Catalogs. Seismological Research Letters, 92(2A). doi: 10.1785/0220200337

in which we explore high-resolution earthquake catalogs and analyze whether their magnitude distribution complies with the exponential-like Gutenberg–Richter (GR) relation. (Spoiler: those catalogs do not preserve the exponential-like magnitude distribution, which characterizes ordinary catalogs, toward low magnitudes.)

Specificially, we estimate a lower magnitude cutoff, or completeness magnitude, M_c, that complies with the exponential-like distribution of the Gutenberg–Richter relation. Note that an exponential-like distribution above M_c is a necessary and sufficient condition to calculate the b-value of the Gutenberg–Richter relation (otherwise the physical meaning of the b-value becomes questionable). We use the Lilliefors test [Lilliefors 1969]¹ as a statistical goodness-of-fit test with the exponential distribution to determine the lowest magnitude cutoff above which the magnitude is exponentially distributed, which we call M_c^Lilliefors. In other words, above M_c^Lilliefors, the magnitude–frequency distribution is consistent with the exponential-like Gutenberg–Richter relation.

The Jupyter notebook demo-Lilliefors-test.ipynb (can also be viewed at nbviewer.jupyter.org to render the interactive plot) demonstrates the principle of our method for one example catalog by making use of the Python code in mc_lilliefors.py (the class McLilliefors).

Please find more information, applications, and findings in our paper. The code repository to this upload is located at gitlab.com/marcus.herrmann/mc-lilliefors, which is at a more recent state and may receive further updates in the future.

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Footnote [1]

Lilliefors, H. W. (1969). On the Kolmogorov-Smirnov Test for the Exponential Distribution with Mean Unknown. Journal of the American Statistical Association, 64(325), 387–389. doi: 10.2307/2283748

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Copyright © 2020 Marcus Herrmann · Warner Marzocchi — Università degli Studi di Napoli 'Federico II'

Licensed under the European Union Public Licence (EUPL-1.2-or-later), the first European Free/Open Source Software (F/OSS) license. It is available in all official languages of the EU.