  The Epidemic-Type Aftershock Sequences (ETAS) model and its variants
effectively capture the space-time clustering of seismicity, setting the
standard for earthquake forecasting. Accurate unbiased ETAS calibration is thus
crucial. But we identify three sources of bias, (i) boundary effects, (ii)
finite-size effects, and (iii) censorship, which are often overlooked or
misinterpreted, causing errors in seismic analysis and predictions. By
employing an ETAS model variant with variable spatial background rates, we
propose a method to correct for these biases, focusing on the branching ratio
n, a key indicator of earthquake triggering potential. Our approach quantifies
the variation in the apparent branching ratio (napp) with increased cut-off
magnitude (Mco) above the optimal cut-off (Mcobest). The napp(Mco) function
yields insights superior to traditional point estimates. We validate our method
using synthetic earthquake catalogs, accurately recovering the true branching
ratio (ntrue) after correcting biases with napp(Mco). Additionally, our method
introduces a refined estimation of the minimum triggering magnitude (m0), a
crucial parameter in the ETAS model. Applying our framework to the earthquake
catalogs of California, New Zealand, and the China Seismic Experimental Site
(CSES) in Sichuan and Yunnan provinces, we find that seismicity hovers away
from the critical point, nc = 1, remaining distinctly subcritical, however with
values tending to be larger than recent reports that do not consider the above
biases. It is interesting that, m0 is found around 4 for California, 3 for New
Zealand and 2 for CSES, suggesting that many small triggered earthquakes may
not be fertile. Understanding seismicity's critical state significantly
enhances our comprehension of seismic patterns, aftershock predictability, and
informs earthquake risk mitigation and management strategies.
