Two empirical double-corner frequency source spectra and their source physics implications

The best-known part of Brune’s (1970) spectral model is the single corner f^–2 source spectrum. However, Brune noted that a more realistic heterogeneous rupture would have a source displacement spectrum with a low-frequency f^–0 segment proportional to seismic momentM₀, a segment with f^–2 high-frequency decay, and an intermediate f^–1 branch that connects the two. This “f^–0–f^–1–f^–2” shape source spectrum features two corner frequencies f_C1 and f_C2>f_C1. Brune (1970) associated the emergence of the f_C2 with the partial stress drop over a fault considering a rupture with heterogeneous stress release. Here we introduce two double-corner source spectral models JA19 and JA19_2S for 3.3≤M≤7.3, constrained by stochastic modeling the mean PGA and mean PGV of the NGA West-2 database. JA19 is self-similar. Its two corner frequencies f_C1 and f_C2 scale with moment magnitude (M) as (1) log(f_C1(M))=1.754– 0.5M and (2) log(f_C2(M)) = 3.250 – 0.5M. We find that relation (1) is consistent with the known self-similar scaling relations of the rupture duration (T_d) where T_d= 1/(πf_C1). Relation (2) may reflect scaling relation of the average rise time (T_R), whereT_R ~0.8/f_C2. Stochastic simulations using JA19 cannot reproduce the sharp change in magnitude dependence of PGA and PGV at M5.3, suggesting a breakdown of self-similarity. To model this change, JA19_2S is found by perturbing the f_C1 scaling relationship in JA19. For JA19_2S: log(f_C1(M)) = 1.474 – 0.415M  for M≤5.3 and log(f_C1(M)) = 2.375 – 0.585M for M>5.3. In both models the relation f_C2/f_C1>>1 applies. Seismic radiated energy scales withM₀²f_C1²f_C2. The ratiof_C2/f_C1 scales not only with the ratio of effective stress drop and static stress drop as Brune (1970) pointed out but also with the fault aspect ratio. The source spectral shape “f^–0–f^–1–f^–2”, originally proposed by Brune (1970), provides a bridge to reconcile the known scaling relationships in source duration, static stress drop, seismic radiated energy, fault aspect ratio, and ground motion parameters within acceptable uncertainties. It also explains why the stress parameter would generally be larger than static stress drop which is related to the lower corner frequency.