Evaluating the incompleteness magnitude using an unbiased estimate of the $b$ value

TL;DR

This paper presents a new method to estimate the incompleteness magnitude in earthquake catalogs by analyzing the bias in the Gutenberg-Richter $b$ value, accounting for deviations from the law and variability in magnitude distributions.

Contribution

The study introduces a novel procedure to determine the incompleteness magnitude using the dependence of $b$ on sample size and variability coefficient, validated on synthetic and real data.

Findings

The bias in $b$ is proportional to the square of the variability coefficient.

The method accurately estimates the incompleteness magnitude in synthetic catalogs.

Applied to real data, the method identifies $m_c$ in Southern California, Japan, and New Zealand.

Abstract

The evaluation of the $b$ value of the Gutenberg-Richter (GR) law, for a sample composed of $n$ earthquakes, presents a systematic positive bias $\delta b$ which is proportional to $1/n$, as already observed by Ogata \& Yamashina (1986). In this study we show how to incorporate in $\delta b$ the bias introduced by deviations from the GR law. More precisely we show that $\delta b$ is proportional to the square of the variability coefficient $CV$, defined as the ratio between {the standard deviation of the magnitude distribution and its mean value.} When the magnitude distribution follows the GR law $CV=1$ and this allows us to introduce a new procedure, based on the dependence of $b$ on $n$, which allows us to {identify} the incompleteness magnitude $m_c$ as the threshold magnitude leading to $CV=1$. The method is tested on synthetic catalogs and it is applied to estimate $m_c$ in Southern California, Japan and New Zealand.