Regional Attenuation Function: \(F_{attn}(R,f)\). See Earthquake Ground Motion and Response Spectral Analysis:
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.
High frequency decay function: \(F_{\kappa}(f)\). See Earthquake Ground Motion and Response Spectral Analysis:
Most earthquakes exhibit faster decay of their high-frequency spectral displacement than the theory predicts. This is often attributed to the crustal and/or path-dependent effects, although some seismologists consider that to be a source-related phenomenon. For simulation, the effect is formulated by a frequency-dependent and distance independent function:
$$F_{\kappa}(f)=e^{-\kappa \pi f}…(7)$$
The value of \(\kappa\) is estimated based on regional earthquake ground motion analysis. Regions with older and competent rocks, like the central US, estimate lower \(\kappa\) values compared to regions younger with less competent rocks, like the western US.