b-more-incomplete and b-more-positive: Insights on A Robust Estimator of Magnitude Distribution

The $b$-value in earthquake magnitude-frequency distribution quantifies the relative frequency of large versus small earthquakes. Monitoring its evolution could provide fundamental insights into temporal variations of stress on different fault patches. However, genuine $b$-value changes are often difficult to distinguish from artificial ones induced by temporal variations of the detection threshold.A highly innovative and effective solution to this issue has recently been proposed by van der Elst (2021) by means of the b-positive estimator, which is based on analyzing only the positive differences in magnitude between successive earthquakes.Here, we demonstrate the robustness of the estimator, which remains largely unaffected by detection issues due to the properties of conditional probability. We illustrate that this robustness can be further improved by considering positive differences in magnitude, not only between successive earthquakes but also between different pairs of earthquakes. This generalized approach, defined as the “b-more-positive estimator,” enhances efficiency by providing a precise estimate of the $b$-value while including a larger number of earthquakes from an incomplete catalog. However, our analysis reveals that the accuracy of the $b$ estimators diminishes when earthquakes below the completeness threshold are included in the catalog. This leads to the paradoxical observation that greater efficiency is achieved when the catalog is more incomplete. To address this, we introduce the “b-more-incomplete estimator”, where the b-more-positive estimator is applied only after artificially filtering the instrumental catalog to make it more incomplete. Our findings show the superior efficiency of the b-more-incomplete method.