  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) through the b-positive
method, which is based on analyzing only the positive differences in magnitude
between successive earthquakes. Here, we provide support to the robustness of
the method, largely unaffected by detection issues due to the properties of
conditional probability. However, we show that the b-positive method becomes
less efficient when earthquakes below the threshold are reported, leading to
the paradoxical behavior that it is more efficient when the catalog is more
incomplete. Thus, we propose the b-more-incomplete method, where the b-method
is applied only after artificially filtering the instrumental catalog to be
more incomplete. We also present other modifications of the b-method, such as
the b-more-positive method, and demonstrate when these approaches can be
efficient in managing time-independent incompleteness present when the seismic
network is sparse. We provide analytical and numerical results and apply the
methods to fore-mainshock sequences investigated by van der Elst (2021) for
validation. The results support the observed small changes in $b$-value as
genuine foreshock features.
