John G. Anderson
Seismic energy and stress-drop parameters for a composite source model
Bulletin of the Seismological Society of America (February 1997), 87(1):85-96

Abstract:
This article examines relationships among radiated energy and several stress-drop parameters that are used to describe earthquake faulting. This is done in the context of a composite source model that has been quite successful in its ability to reproduce statistical characteristics of strong-motion accelerograms. The main feature of the composite source model is a superposition of subevents with a fractal distribution of sizes, but all with the same subevent stress drop (Delta sigma (sub delta ) ) that is independent of the static stress drop (Delta sigma _s ). In the model, Delta sigma (sub delta ) is intended to represent the effective dynamic stress, and it does this well when Delta sigma (sub delta ) >2Delta sigma _s . The radiated energy in the S wave is E ^CS _S = 0.233 C _E (Delta sigma _d /mu ) M ₀ , where M ₀ is the seismic moment of the earthquake, mu is shear modulus, and C _E is a dimensionless parameter that equals unity when Delta sigma _d >2Delta sigma _s . The apparent stress (sigma _a ) is sigma _a = 0.243 C _E Delta sigma _d . The effective stress is sigma _e nearly equal C0.44C _E Delta sigma _d . The Orowan stress drop (Delta sigma ₀ ) is Delta sigma ₀ = 0.486 Delta sigma _d . The root-mean-square (rms) stress drop (Delta sigma _rms ) is Delta sigma _rms = Delta sigma _d I (super 1/2) (sub theta ) M ₀ /M _OS (R _max ) (super 1/2) (f _c /f ₀ ) (super 1/2) , where f ₀ is corner frequency of the earthquake, M _OS (R _max ) and f _c are the moment and corner frequency of the largest subevent, and I (super 1/2) (sub theta ) is a dimensionless constant approximately equal to 1.7. Finally, the Savage-Wood ratio (SWR) is given by SWR nearly equal C _E Delta sigma _d /2Delta sigma _s . These results clarify the relationships among all of these stress parameters in the context of a complex fault, showing the critical role of the subevent stress drop. They also provide an additional tool for energy, stress, and Savage-Wood ratio estimation. Since the process of modeling strong motion with the composite source uses realistic Green's functions, estimates of energy and stress parameters using this model are expected to have a good correction for wave propagation.