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

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 (H1) 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 (H0). 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

💡 Research Summary

The paper provides a rigorous theoretical examination of the doubly stochastic earthquake source model originally proposed by A. Gusev, focusing on two hallmark features observed in far‑field body‑wave recordings at high frequencies: (1) the ω‑square (ω²) decay of displacement spectra and (2) the apparent absence of rupture directivity in the high‑frequency (HF) radiation. Gusev’s model introduces two random components – a spatially variable stress drop (SD) and a rupture‑time function (RT) that contains a dominant linear term – and demonstrates these features numerically. The present study asks under what mathematical conditions these phenomena are stable with respect to receiver location and model realization.

Fractal formulation of the random fields
Instead of assuming globally smooth functions, the authors treat SD and RT as realizations of fractal stochastic fields. The local regularity of each field is quantified by a Hurst (fractional) exponent H (0 < H < 1). This framework allows a very broad class of functions, ranging from almost differentiable (H ≈ 1) to extremely rough (H ≈ 0), without imposing any specific global spectral shape.

Analysis of the ω‑square hypothesis
By inserting the stochastic representation of SD and RT into the far‑field displacement integral and performing a high‑frequency asymptotic expansion, the authors derive the conditions for a ω⁻² spectral tail. The key result is that the ω‑square behavior is obtained when the rupture‑time field is “almost regular”, i.e., its Hurst exponent is close to one. In this regime the phase term i ω RT varies linearly with ω, and the resulting integral yields the classic ω⁻² decay. The stress‑drop field, by contrast, can be arbitrarily rough; its Hurst exponent does not affect the ω‑square asymptotics. A special case arises when the rupture front is strictly linear: then the stress‑drop must be at the minimal smoothness limit (H ≈ 0) for the ω‑square law to hold, highlighting a subtle interplay between front geometry and stress‑drop roughness.

Directivity effect and its instability
The directivity effect refers to the anisotropic amplification of HF radiation in the direction of rupture propagation. The authors show that the presence or absence of directivity cannot be inferred from the Hurst exponent alone. Different realizations of RT that share the same H value may either exhibit a strong directivity signature or none at all. The decisive factor is the smoothness of the joint two‑dimensional probability distribution of RT and SD. If this joint distribution contains sharp gradients or localized irregularities, the phase cancellations that normally suppress directivity are incomplete, and a measurable directivity emerges. Conversely, when the joint distribution is smoothly varying, the phase contributions cancel, and the directivity effect disappears. Because the joint distribution is highly sensitive to small changes in the underlying random fields, the directivity effect is fundamentally unstable: small variations in model parameters or in the receiver’s position can flip the outcome. This analytical insight explains why empirical studies sometimes report statistically significant directivity and sometimes do not.

Numerical experiments
To validate the analytical predictions, synthetic earthquakes are generated with prescribed Hurst exponents for SD and RT, and with both linear and mildly curved rupture fronts. Spectral analyses confirm that only the cases with HRT ≈ 1 produce a clear ω⁻² tail, regardless of the HSD value. Directivity is observed only when the joint PDF of RT and SD possesses pronounced local irregularities; otherwise the HF radiation is isotropic.

Implications and conclusions
The study concludes that the ω‑square high‑frequency decay is robust provided the rupture‑time field is sufficiently smooth, while the stress‑drop roughness is largely irrelevant. In contrast, the directivity effect is governed by the detailed two‑dimensional smoothness of the combined random fields and is therefore intrinsically unstable. Practically, this means that statistical tests of directivity based on limited data sets may yield contradictory results, and that realistic stochastic source models must incorporate fractal descriptions of both SD and RT to capture the full range of observed HF behavior. The work underscores the necessity of fractal stochastic modeling in seismology and offers a clear analytical framework for interpreting high‑frequency earthquake spectra and directivity observations.