Is the Number of Foreshocks Underestimated? The Role of Declustering Symmetry
The physical origin of foreshocks remains one of the major unresolved problems in statistical seismology. In bottom-up (BU) triggering models, foreshocks are interpreted as ordinary earthquakes that stochastically trigger a larger event through cascading interactions, as in the Epidemic-Type Aftershock Sequence (ETAS) model. In contrast, top-down (TD) models interpret foreshocks as manifestations of a preparatory process preceding the mainshock. This ambiguity has important implications not only for earthquake forecasting, but also for declustering itself, since the inferred number of foreshocks may depend on the assumed physical mechanism governing seismicity. Here we investigate the detectability of foreshocks using synthetic catalogues representing two limiting physical scenarios: purely bottom-up triggering and mixed preparatory-triggering dynamics. These cases are implemented through the ETAS and ETAFS models, respectively. We compare the classical nearest-neighbor (NN) declustering metric with a reverse nearest-neighbor metric (rNN) introduced here. While the standard NN metric scales the correlation metric to the magnitude of the earlier event, rNN scales it with the magnitude of the later event. We show that rNN is mathematically equivalent to applying the standard nearest-neighbor metric to the time-reversed catalog, establishing a direct symmetry between the two approaches. Our simulations reveal [...]