Geometric Spreading of P_n and S_n in a Spherical Earth Model

Geometric spreading of P_n and S_n waves in a spherical Earth model is different than that of classical headwaves and is frequency dependent. The behavior cannot be fully represented by a frequency-independent power-law model, as is commonly assumed. The lack of an accurate representation of P_n and S_n geometric spreading in a spherical Earth model impedes our ability to characterize Earth properties including anelasticity. We conduct numerical simulations to quantify P_n and S_n geometric spreading in a spherical Earth model with constant mantle-lid velocities. Based on our simulation results, we present new empirical P_n and S_n geometric-spreading models in the form G(r,f)=[10^n₃(f)/r₀](r₀/r)^{n₁(f)log(r₀/r)+n₂(f)} and n_i(f)=n_i1[log(f/f₀)]²+n_i2log(f/f₀)+n_i3, where i=1, 2, or 3; r is epicentral distance; f is frequency; r₀=1 km; and f₀=1 Hz. We derive values of coefficients n_ij by fitting the model to computed P_n and S_n amplitudes for a spherical Earth model having a 40-km-thick crust, generic values of P and S velocities, and a constant-velocity uppermost mantle. We apply the new spreading model to observed data in Eurasia to estimate average P_n attenuation, obtaining more reasonable results compared to using a standard power-law model. Our new P_n and S_n geometric-spreading models provide generally applicable reference behavior for spherical Earth models with constant uppermost-mantle velocities.