Estimation of the Frequency-Magnitude Gutenberg-Richter b-Value without Making Assumptions on Levels of Completeness

A pronounced weakness of the classic Aki-Utsu b-value estimator is its heavy dependence on the level of magnitude completeness mc, of which the assessment itself is no trivial task. Two new estimators for frequency-magnitude Gutenberg-Richter b-value that are not dependent on the level of completeness are proposed in the present work. Monte Carlo simulations show that the proposed estimators by the method of moments (MM) and the maximum-likelihood (ML) procedures are especially effective when the incomplete frequency-magnitude distribution is modeled by the three-parameter gamma distribution, which curves gradually and has only one maximum; that is, it belongs to the distributions of category IV, according to the classification by Mignan (2012). In some instances, the MM estimator is comparable with its ML counterpart and is significantly easier to calculate. In instances where the applied sample of earthquake magnitudes is complete, the newly derived MM and ML b-value estimators take the form of the classic Aki (1965) and Utsu (1965) solutions. For the purpose of illustration, the new procedures were applied for estimating the b-value in the Ceres-Tulbagh area, the most seismically active (tectonic) area of South Africa.