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.

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.

The hypothesis of alternation leads to the idea of immunity after local disaster which, notwithstanding it sounds reasonable, it has been frequently rejected by objective testing. More generally the estimate of the occurrence probability of the next big shock on the basis of the time delay from the last earthquake still represents a big challenge. The problem is that this issue cannot be addressed only on the basis of historical catalogs which contain to few well documented big shocks and decades of future observations appear necessary. On the other hand, recent results have shown that important insights can be obtained from the spatial organization of aftershocks and its relationship to the mainshock slip profile. Here we address this issue by monitoring the stress evolution together with the occurrence of big shocks and their aftershocks in a physical model where the seismic fault is described as an elastic layer embedded in a ductile medium. The model reproduces all relevant statistical features of earthquake occurrence and allows us to perform accurate testing of the hypothesis of alternation and its consequences, particularly on the side of aftershock spatial patterns. We demonstrate that the hypothesis of characteristic earthquakes is not valid but that is possible to achieve insights on the time until the next big shock on the basis of the percentage of aftershocks occurring inside the high slip contour of the mainshock.