The UV-divergence problem in statistical seismology: insights from an ETAS model with smoothed minimum triggering magnitude

Earthquakes inherently trigger other earthquakes, a process central to the spatio-temporal organization of seismicity. When an earthquake occurs, stress redistributes in the brittle lithosphere, promoting aftershocks in areas of increased stress while inhibiting them in stress shadows. The Gutenberg-Richter law describes the distribution of earthquake magnitudes as an exponential function, resulting in cascades of nearly infinitely many small-energy events following any quake. This poses the ultraviolet (UV) divergence problem, where the triggering potential of these tiny events may not diminish sufficiently fast with decreasing magnitude, potentially escalating into large collective triggering intensities due to the smallest earthquakes. Additionally, undetected seismicity complicates the estimation of key physical metrics, such as clustering coefficients and the branching ratio (Sornette and Werner, 2005). In the standard ETAS model of statistical seismicity, the ultraviolet (UV) problem is addressed by assuming the existence of a minimum triggering magnitude, m0, below which earthquakes are considered too small to trigger others. Despite its critical physical implications, no research has been able to test for the existence and measure m0 reliably (see however Li et al., 2024 for a new technique in this direction). If m0 does not exist, forecasting earthquake occurrence from past seismic data becomes fundamentally unreliable, as every stress perturbation could potentially cascade into an avalanche leading to a mainshock. Moreover, disregarding the existence of m0 and defaulting to the magnitude mc of completeness introduces biases in estimations, effectively sweeping the problem under the rug and creating a critical gap in seismic modeling. We implement an ETAS model incorporating magnitude-dependent triggering power specifically designed to address this issue. Specifically, we propose a new ETAS model variant that replaces the sharp magnitude threshold m0 with a smooth S-shaped function. This function reflects the realistic assumption that the ability to trigger earthquakes decreases gradually with decreasing magnitude, rather than being an all-or-nothing property. The S-shaped function is characterized by two parameters: a central value m0 and a transition range md, which represents the magnitude interval over which the triggering propensity diminishes. Our statistical calibration of various datasets generally finds md, to be large (10 or more), indicating that m0 lies below the typical magnitude of completeness used in seismic catalogs. Our analysis of both long-term regional datasets and high-quality, AI-enhanced seismic catalogs fails to detect any significant m0, which is thus likely smaller than the standard completeness threshold mc of instrumental catalogs and, at best, close to mc of machine-learning-based datasets. Our findings suggest that much of the underlying dynamics responsible for detected seismic events remain unobservable. Earthquakes that cannot be detectable by existing networks may contribute to triggering future earthquakes of possibly large magnitudes. References Li J., Wu Z., Zhuang J., Jiang C., Sornette, D.; 2024: Revisiting Seismicity Criticality: A New Framework for Bias Correction of Statistical Seismology Model Calibrations, under review, available at https://doi.org/10.48550/arXiv.2404.16374. Sornette D., Werner M. J.; 2005: Apparent clustering and apparent background earthquakes biased by undetected seismicity. J. Geophys. Res. Solid Earth 110(B9), B09303, doi:10.1029/2005JB00362