The largest events in earthquake clusters and their relationship to foreshocks

The space-time epidemic-type aftershock sequence (ETAS) model is a stochastic branch process in which earthquake activity is classified into background and clustering components and each earthquake triggers other earthquakes independently according to some rules. Its conditional intensity can be written as λ(t,x,y)=μ(x,y)+ A eα(mi-mc) g(t-ti) f(x-xi, y-yi; mi) and magnitude distribution for all the event is s(m)=β \ee -β(m-mc), for m not less than magnitude threshold mc. This paper gives the probability distributions associated with the largest event in a cluster, i.e., the probability that an event of magnitude m has no offspring greater than m' is ζ(m,m')=exp [{-κ(m) F(m')}], where F(m')=1- β/α [A F(m')] β/α[Γ(-β/α, A F(m'))-Γ(-β/α, A F(m') \eeα(m'-mc))] represents the cumulative probability function for the largest event in an arbitrary earthquake cluster. When the process is subcritical (critical, or supercritical) and m' → +∞, F(m') s(m') (F(m') e-α (m'-mc) or F(m') Const), leading to ζ(m, m') exp [- A' eα m - β m'] ( ζ(m, m') exp [-Ae-α (m - m')] or ζ(m, m') exp [-C e-α (m - mc)]. We define a foreshock by a background event that has at least one descendant with a larger magnitude. Thus, the probability of a background earthquake of magnitude m to be a foreshocks is H(m)= 1-ζ(m, m). We have also derived magnitude distributions of foreshocks and non-foreshock earthquakes. To verify these theoretical results, the JMA (Japan Meteorological Agency) earthquake catalog is analyzed, and we found no discrepancy between the analytic results and the inversion results by using the stochastic reconstruction method.