Doubly Stochastic Earthquake Source Model: Omega-Square Spectrum and Low High-Frequency Directivity Revealed by Numerical Experiments

Since its formulation in 1967-1970, the classical omega (-2) model of earthquake source spectrum awaits a consistent theoretical foundation. To obtain one, stochastic elements are incorporated both into the final structure of the fault and into the mode of rupture propagation. The main components of the proposed doubly stochastic model are: (1) the Andrews's concept, that local stress drop over a fault is a random self-similar field; (2) the concept of rupture with running slip pulse, after Heaton; (3) the hypothesis that a rupture front is a tortuous, multiply connected (lacy) fractal polyline that occupies a strip of finite width close to the slip-pulse width; and (4) the assumption that the propagation distance of fault-guided, mostly Rayleigh waves from a failing spot on a fault is determined by the slip-pulse width. Waveforms produced by this model are determined based on the fault asperity failure model after Das and Kostrov. Properties of the model are studied by numerical experiments. At high frequency, simulated source spectra behave as omega (-2), and acceleration spectra are flat. Their level, at a given seismic moment and rms stress drop, is inversely related to the relative width of the slip pulse. When this width is relatively low, a well-defined second corner frequency (lower cutoff of acceleration spectrum) is seen. The model shows clear dependence of propagation-related directivity on frequency. Between the first and the second corner frequency, amplitude spectra are strongly enhanced for the forward direction; whereas, above the second corner frequency, directivity is significantly reduced. Still, it is not inhibited totally, suggesting incomplete incoherence of the simulated radiator at high frequencies.