The omega-square hypothesis for the seismic source

TL;DR

This paper investigates the omega-square hypothesis in seismic sources, analyzing the source time function to explain the observed spectral decay in ground displacement.

Contribution

It identifies specific local features of the source time function that produce the omega-square spectral behavior in seismic ground motion.

Findings

f exhibits a local inverse-square-root behavior near the rupture front

f is bounded near the rupture front with slight roughness

The analysis helps understand omega-square spectral behavior in source models

Abstract

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.