The omega-square hypothesis for the seismic source

The omega-square hypothesis assumes that the ground displacement u(t) in the far-field zone decays as the inverse square of frequency in the range ~1-30 Hz. This empirical fact remains theoretically unjustified. Our analysis of the problem is based on an integral representation of u(t) in terms of the source time function f and on the spectrum analysis of local features in f. The goal is to select the local features that will enable one to generate the omega-square behavior of u(t) on a large set of receivers. We found two appropriate fragments of f : first , f exhibits a local inverse-square-root behavior near the rupture front where the frontal surface is piecewise smooth (but not smooth and not rough ); and second, f is bounded near the rupture front where the frontal surface has a slight roughness (the Hurst parameter near 1). These facts can be useful for understanding the omega-square spectral behavior of the kinematic source models.