Geometric spreading of Pb>nb> and Sb>nb> 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 Pb>nb> and Sb>nb> geometric spreading in a spherical Earth model impedes our ability to characterize Earth properties including anelasticity. We conduct numerical simulations to quantify Pb>nb> and Sb>nb> geometric spreading in a spherical Earth model with constant mantle-lid velocities. Based on our simulation results, we present new empirical Pb>nb> and Sb>nb> geometric-spreading models in the form G(r,f)=[10nb>3b>(f)/rb>0b>](rb>0b>/r)nb>1b>(f)log(rb>0b>/r)+nb>2b>(f) and nb>ib>(f)=nb>i1b>[log(f/fb>0b>)]2+nb>i2b>log(f/fb>0b>)+nb>i3b>, where i=1, 2, or 3; r is epicentral distance; f is frequency; rb>0b>=1 km; and fb>0b>=1 Hz. We derive values of coefficients nb>ijb> by fitting the model to computed Pb>nb> and Sb>nb> 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 Pb>nb> attenuation, obtaining more reasonable results compared to using a standard power-law model. Our new Pb>nb> and Sb>nb> geometric-spreading models provide generally applicable reference behavior for spherical Earth models with constant uppermost-mantle velocities.
