Source Function: \(F_{src}(M_0,f)\). See Earthquake Ground Motion and Response Spectral Analysis:
The far field, i.e. far enough from the rupture area that the rupture details can be characterized in a simple functional form, source displacement spectrum is often formulated as
$$F_{src}=\frac{C*M_0}{1+{(f/f_0)}^2}…(2)$$
where \(C\) is a region dependent constant, \(M_0\) is seismic moment, and \(f_0\) is the corner frequency. The corner frequency characterizes the shape of the source radiation spectrum, which depends upon the rupture dimension and the average shear stress drop (\(\Delta\sigma)\) over the rupture area. For a constant stress drop, the corner frequency is often formulated as
$$f_0=K_s*{\beta}_s*{(\frac{\Delta\sigma}{M_0})}^{1/3}…(3)$$
where \(K_s\) is a rupture geometry-dependent constant and \(\beta_s\) is the shear wave velocity near the rupture area. According to Equation 2, the displacement spectra decay faster with increasing frequency at frequencies larger than the corner frequency (see Figure 2). It should be noted that corner frequency increases with increasing stress drop, which results in stronger amplitudes for high-frequency seismic radiation (see Figure 2).