A scaling law for the spectra of large earthquakes

Abstract An ω-squared model suggested by Hanks and McGuire has had impressive success in predicting measures of strong ground motions over a broad range in moment magnitude. There remains the question of how the ground motion spectra should scale for earthquakes large enough to rupture the entire width of a seismogenic zone, i.e., earthquakes for which the usual similarity principles do not apply. The scaling law proposed in this paper takes as its fundamental postulate that the high-frequency level of the acceleration spectrum is proportional to the square root of the rupture area for all earthquakes, not just those for which the similarity principles apply. The spectrum corresponding to the proposed scaling law is controlled by two corner frequencies, one inversely proportional to the rupture length and the other inversely proportional to the rupture width. The proposed law agrees with the Hanks-McGuire model below the value of moment corresponding to rupture of the whole width of the seismogenic zone. A stochastic kinematic model is devised for the purpose of making ground motion predictions corresponding to the proposed law. The spectrum of the stochastic kinematic model can be written in terms of the product of two factors, one controlled by the corner frequency related to rupture length and the other by the corner frequency related to rupture width. Above their respective corner frequencies both factors are proportional to frequency raised to a power. The two exponents are chosen in order to make the spectrum correspond to the proposed law, but it is noteworthy that both exponents can easily be justified on other grounds. The exponent in the factor controlled by the corner frequency related to rupture width corresponds to the square root singularity in slip velocity exhibited by Kostrov9s solution for a constant stress relaxation propagating at constant velocity. The exponent in the factor controlled by the corner frequency related to rupture length has the value implied by the well-established proportionality between average dislocation and rupture length, on the assumption that dislocation is a random function of position along the fault. Predictions of peak horizontal acceleration, velocity, and response spectra corresponding to the proposed law are made with the stochastic kinematic model and also using random vibration theory. The two sets of predictions show good agreement. The predictions are consistent with strong-motion data for shallow earthquakes in western North America, within the scatter of the data and the uncertainty of the parameters of the proposed law. The predictions differ by more than a factor of two from a simple extrapolation of the Hanks-McGuire model only at moment magnitudes more than one unit greater than the critical magnitude corresponding to rupture of the entire width of the seismogenic zone. It is likely that few if any such earthquakes ever occur.