Geometric spreading of Pn and Sn 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 Pn and Sn geometric spreading in a spherical Earth model impedes our ability to characterize Earth properties including anelasticity. We conduct numerical simulations to quantify Pn and Sn geometric spreading in a spherical Earth model with constant mantle-lid velocities. Based on our simulation results, we present new empirical Pn and Sn geometric-spreading models in the form G(r,f)=[10n3(f)/r0](r0/r)n1(f)log(r0/r)+n2(f) and ni(f)=ni1[log(f/f0)]2+ni2log(f/f0)+ni3, 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 nij by fitting the model to computed Pn and Sn 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 Pn attenuation, obtaining more reasonable results compared to using a standard power-law model. Our new Pn and Sn geometric-spreading models provide generally applicable reference behavior for spherical Earth models with constant uppermost-mantle velocities.

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