Xiaoning Yang, Thorne Lay, Xiao-Bi Xie, and Michael S. Thorne
Geometric spreading of P _n and S _n in a spherical Earth model
Bulletin of the Seismological Society of America (December 2007), 97(6):2053-2065

Abstract:
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 (super n ₃ (f)) /r ₀ ](r ₀ /r) (super n ₁ (f)log(r ₀ /r)+n ₂ (f)) and n _i (f) = n _i1 [log(f/f ₀ )] ² +n _i2 log(f/f ₀ )+n _i3 , where i = 1, 2, or 3; r is epicentral distance; f is frequency; r ₀ = 1 km; and f ₀ = 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.