Geometrical Spreading Function: \(F_{geo}(R)\). 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.
\(F_{geo}(R)\) quantifies the decay of seismic waves amplitudes with distance. In general, the functional details of \(F_{geo}(R)\) are empirically determined through the development of regional ground motion prediction equations (GMPEs), using the recording of ground motions of past earthquakes and elaborate statistical processing techniques that decouples the impacts of different source and path related parameters on GMs. The general practice is that for distances less than about 70-80 km, \(F_{geo}(R)\) is typically formulated as \(1/R\) which reflect the decay of the body waves with distance. Beyond that, surface waves dominate the GM, and the decay rate with distance often is formulated as \(1/\sqrt{R}\).
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 to create larger than expected long-period GM levels.