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Recently A. Gusev suggested and numerically investigated the doublystochastic earthquake source model. The model is supposed to demonstrate thefollowing features in the far-field body waves: 1) the omega-squarehigh-frequency (HF) behavior of displacement spectra; 2) lack of thedirectivity effect in HF radiation; and 3) a stochastic nature of the HF signalcomponent. The model involves two stochastic elements: the local stress drop(SD) on a fault and the rupture time function (RT) with a linear dominantcomponent. The goal of the present study is to investigate the Gusev modeltheoretically and to find conditions for (1, 2) to be valid and stable relativeto receiver site. The models with smooth elements SD, RT are insufficient forthese purposes. Therefore SD and RT are treated as realizations of stochasticfields of the fractal type. The local smoothness of such fields ischaracterized by the fractional (Hurst) exponent H, 0 < H < 1. This allows usto consider a wide class of stochastic functions without regard to their globalspectral properties. We show that the omega-square behavior of the model isachieved approximately if the rupture time function is almost regular (H~1)while the stress drop is rough function of any index H. However, if the rupturefront is linear, the local stress drop has to be function of minimal smoothness(H~0). The situation with the directivity effect is more complicated: fordifferent RT models with the same fractal index, the effect may or may notoccur. The nature of the phenomenon is purely analytical. The main controllingfactor for the directivity is the degree of smoothness of the two dimensionaldistributions of RT random function. For this reason the directivity effect isunstable. This means that in practice the opposite conclusions relative to thestatistical significance of the directivity effect are possible

Stochastic earthquake source model: the omega-square hypothesis and the directivity effect