Following are brief discussions on the functional forms of items in equation 1 of Earthquake Ground Motion and Response Spectral Analysis
Source Function \(F_{src}(M_0,f)\):
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).
Geometrical Spreading Function \(F_{geo}(R)\):
\(F_{geo}(R)\) quantifies the decay of seismic waves amplitudes with distance. For distances less than about 70-80 km, \(F_{geo}(R)\) is typically formulated as \(1/R\). Beyond that, surface waves dominate the GM, and the decay rate with distance often is formulated as \(1/\sqrt{R}\). However, in regions with well-defined crustal boundaries, it is observed that within the distance range of about 70-130 km, the arrivals of wide-angle reflected waves from Moho tend to counterbalance the GM amplitude decay with distance. In general, all GMPEs determine the functional details of \(F_{geo}(R)\) empirically using the recorded ground motions and elaborate statistical processing techniques to decouple the impacts of different parameters on GMs.
Regional Attenuation Function \(F_{attn}(R,f)\):
The amplitudes of seismic waves decay with distance due to the geometrical spreading and also due to the inelastic or internal friction of rocks during vibration that causes seismic energy loss. At any location, this energy loss often is expressed as
$$\frac{1}{Q(\omega)}=-\frac{\Delta E}{E}…(4)$$
where \(E\) is peak stored energy in the volume of rock, \(\Delta(E)\) is energy loss during one cycle of vibration, and \(\omega\) is the natural frequency of vibration. It can be shown that after many cycles of vibration the seismic amplitude can be estimated as
$$A(t)=A_0*e^{-\frac{\omega t}{2Q}}…(5)$$
The \(Q\) in Equation 5 is known as the temporal quality factor. However, seismologists are interested in measuring and formulating inelastic seismic wave decay with distance. For that, the spatial \(Q\) and the amplitude decay with distance is formulated as
$$A(x)=A_0*e^{-\frac{\omega x}{2cQ}}…(6)$$
where \(x\) is distance and \(c\) is the seismic wave velocity.
