Analytic study of the geometrical spreading of P-waves in a layered transversely isotropic medium with a vertical symmetry axis

An analytical formula for geometrical spreading is derived for a horizontally layered transversely isotropic medium with a vertical symmetry axis (VTI). With this expression, geometrical spreading can be determined using only the anisotropy parameters in the first layer, the traveltime derivatives, and the source-receiver offset. Explicit, numerically feasible expressions for geometrical spreading are obtained for special cases of transverse isotropy (weak anisotropy and elliptic anisotropy). Geometrical spreading can be calculated for transversly isotropic (TI) media by using picked traveltimes of primary nonhyperbolic P-wave reflections without having to know the actual parameters in the deeper subsurface; no ray tracing is needed. Synthetic examples verify the algorithm and show that it is numerically feasible for calculation of geometrical spreading. For media with a few (4-5) layers, relative errors in the computed geometrical spreading remain less than 0.5% for offset/depth ratios less than 1.0. Errors that change with offset are attributed to inaccuracy in the expression used for nonhyberbolic moveout. Geometrical spreading is most sensitive to errors in NMO velocity, followed by errors in zero-offset reflection time, followed by errors in anisotropy of the surface layer. New relations between group and phase velocities and between group and phase angles are shown in appendices.