Stable determination of (Brune) stress drops

The Brune stress drop, σ_B, is generally calculated from two spectral observables: Ω₀, the zero-frequency level of the far-field displacement amplitude spectrum, and v_c, the corner frequency of that spectrum. σ_B takes the form

where h contains kinematic factors. In some recent studies, other estimates of the stress drop, such as the dynamic stress drop and the apparent stress, are found to be more stable and/or reliable than σ_B. We find that these apparent shortcomings of σ_B can be remedied if we replace the dependence of σ_B on v_c as a directly determined spectral observable with a dependence on a parameter which can be determined more reliably and which appears linearly (instead of as the third power) in the expression for σ_B. This observable is J, the integral of the square of the ground velocity, which, by Parseval's theorem, is the second moment of the power spectrum. For spectra from broadband seismographs, the determination of J is totally objective; it is an integral for which the integrand approaches zero at both limits. For Brune's model, v_c³ = qJ/Ω₀² with q = 1/(2π³), and equation (a) becomes

If the P-wave contribution to the radiated energy as well as the angular dependence of J are neglected, the right-hand side of equation (b) is a constant multiple of the apparent stress. Hence, studies which show that apparent stress determinations provide more robust estimates of the stress drop than σ_B determined through equation (a) support evaluating σ_B through equation (b).