  Recently A. Gusev suggested and numerically investigated the doubly
stochastic earthquake source model. The model is supposed to demonstrate the
following features in the far-field body waves: 1) the omega-square
high-frequency (HF) behavior of displacement spectra; 2) lack of the
directivity effect in HF radiation; and 3) a stochastic nature of the HF signal
component. The model involves two stochastic elements: the local stress drop
(SD) on a fault and the rupture time function (RT) with a linear dominant
component. The goal of the present study is to investigate the Gusev model
theoretically and to find conditions for (1, 2) to be valid and stable relative
to receiver site. The models with smooth elements SD, RT are insufficient for
these purposes. Therefore SD and RT are treated as realizations of stochastic
fields of the fractal type. The local smoothness of such fields is
characterized by the fractional (Hurst) exponent H, 0 < H < 1. This allows us
to consider a wide class of stochastic functions without regard to their global
spectral properties. We show that the omega-square behavior of the model is
achieved 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 rupture
front is linear, the local stress drop has to be function of minimal smoothness
(H~0). The situation with the directivity effect is more complicated: for
different RT models with the same fractal index, the effect may or may not
occur. The nature of the phenomenon is purely analytical. The main controlling
factor for the directivity is the degree of smoothness of the two dimensional
distributions of RT random function. For this reason the directivity effect is
unstable. This means that in practice the opposite conclusions relative to the
statistical significance of the directivity effect are possible
