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The $b$-value in earthquake magnitude-frequency distribution quantifies therelative frequency of large versus small earthquakes. Monitoring its evolutioncould provide fundamental insights into temporal variations of stress ondifferent fault patches. However, genuine $b$-value changes are often difficultto distinguish from artificial ones induced by temporal variations of thedetection threshold. A highly innovative and effective solution to this issuehas recently been proposed by van der Elst (2021) through the b-positivemethod, which is based on analyzing only the positive differences in magnitudebetween successive earthquakes. Here, we provide support to the robustness ofthe method, largely unaffected by detection issues due to the properties ofconditional probability. However, we show that the b-positive method becomesless efficient when earthquakes below the threshold are reported, leading tothe paradoxical behavior that it is more efficient when the catalog is moreincomplete. Thus, we propose the b-more-incomplete method, where the b-methodis applied only after artificially filtering the instrumental catalog to bemore incomplete. We also present other modifications of the b-method, such asthe b-more-positive method, and demonstrate when these approaches can beefficient in managing time-independent incompleteness present when the seismicnetwork is sparse. We provide analytical and numerical results and apply themethods to fore-mainshock sequences investigated by van der Elst (2021) forvalidation. The results support the observed small changes in $b$-value asgenuine foreshock features.

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