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The existence of seismic discontinuities within the continental upper mantle has long been recognized, with more recent studies often indicating an association with elastic anisotropy. Their near-vertical sampling renders teleseismic P and S waves suitable for characterization of mantle discontinuities, but computationally efficient methods of calculating synthetic seismograms are required for structures that exhibit lateral variability. We consider lithospheric models consisting of planar, homogeneous anisotropic layers with arbitrary dip. We adopt the traveltime equation of Diebold for dipping, plane-layered media as the basis for a high-frequency asymptotic method that does not require ray tracing. Traveltimes of plane waves in anisotropic media are calculated from simple analytic formulae involving the depths of layers beneath a station and the vertical components of phase slowness within the layers. We …