Parametric Modeling of Spatial Variability
The simplest approximation for the modeling of the wave passage effect (Eq. 3.3), which is also commonly used in simulations (Chapters 7 and 8), is through Eqs. 2.73 and 2.74, repeated here for convenience:
ϑ(ξ, ω) = −ω(c · ξ )
|c |2 = − ωξ
where c reﬂects the apparent propagation of the motions, and the last equality in the above expression is valid if the direction of propagation of the waveforms and the direction of the station separation vector coincide. When a non-dispersive body wave dominates the analyzed time window of the motions, c can be considered as constant over the frequency range of the wave (Section 2.4.1). Past this frequency range, however, where scattered energy dominates, c is no longer a constant, as has been elaborated upon in Sections 2.4.1 and 2.4.4 and illustrated in Figs. 2.15 and 2.16. Additionally, when more waves than one arrive at the array from different directions, the apparent propagation of the motions cannot be approximated by Eq. 3.4. Early studies on spatial variability, e.g., , evaluated the “gross” apparent propagation velocity of a single wave dominating the motions by estimating the relative arrival time delays, τ0, of the motions at the array stations with respect to a reference station through alignment (Sections 2.3.2 and 2.4.1), and ﬁtting a straight line to the ξ/τ0 data. More rigorously, the direction of propagation of the incoming waves and an estimate of their apparent propagation velocity at each frequency can be obtained through frequency-wavenumber analyses that are described in Section 4.1. Phase estimates will not be addressed further in this chapter, which concentrates on the parameterization of the power spectral density and the lagged coherency of the motions.