Beyond Completeness: The Persistent Triggering Power of Earthquakes at Unobservable Small Magnitudes

We develop a novel Epidemic-Type Aftershock Sequence (ETAS) model with incorporating a magnitude-dependent transition from fertile to infertile at small magnitudes in order to resolve the ultraviolet (UV) divergence problem inherent to conventional ETAS models. Essentially all existing ETAS model versions impose a sharp cutoff m0, assuming earthquakes below this magnitude cannot trigger aftershocks. In contrast, our model replaces this abrupt threshold with a smooth S-shaped transition f_{m_0}\ \left(m\ -\ m_0\right)=\ \frac{e^{\delta\left(m-m_0\right)}}{1+e^{\delta\left(m-m_0\right)}} where the parameter δ controls the transition width between triggering and non-triggering regimes, and m0 marks its inflection point. This formulation captures the gradual decline in triggering capacity as earthquake magnitudes decrease, aligning with physical expectations of stress redistribution in heterogeneous crustal systems. By calibrating the model to high-resolution datasets—including AI-enhanced catalogs from Italy, California, China, New Zealand, and Iceland—we find no evidence for the lower branch of this S-shaped transition. Instead, δ values are found consistently small and estimatted m0 are below the completeness magnitude (mc) of even machine-learning-optimized catalogs. Simulations reveal that the existence of the inflection magnitude m0 becomes identifiable only when it approaches mc, manifesting as small values of δ. When m0≪mc, no discernible signal emerges, rationalising its absence in current observations. The lack of an observable transition of the smooth S-shaped transition f_{m_0}\ \left(m\ -\ m_0\right) to its lower plateau implies within the ETAS models that detected earthquakes are significantly driven by sub-threshold seismicity (m