THREE‐TERM TAYLOR SERIES FOR t² ‐ x²‐CURVES OF P‐ AND S‐WAVES OVER LAYERED TRANSVERSELY ISOTROPIC GROUND*

A bstract The arrival‐time curve of a reflection from a horizontal interface, beneath a homogeneous isotropic layer, is a hyperbola in the x ‐ t ‐domain. If the subsurface is one‐dimensionally inhomogeneous (horizontally layered), or if some or all of the layers are transversely isotropic with vertical axis of symmetry, the statement is no longer strictly true, though the arrival‐time curves are still hyperbola‐like. In the case of transverse isotropy, however, classical interpretation of these curves fails. Interval velocities calculated from t 2 ‐ x 2 ‐curves do not always approximate vertical velocities and therefore cannot be used to calculate depths of reflectors. To study the relationship between velocities calculated from t 2 ‐ x 2 ‐curves and the true velocities of a transversely isotropic layer, we approximate t 2 ‐ x 2 ‐curves over a vertically inhomogeneous transversely isotropic medium by a three‐term Taylor series and calculate expressions for these terms as a function of the elastic parameters. It is shown that both inhomogeneity and transverse isotropy affect slope and curvature of t 2 ‐ x 2 ‐curves. For P‐waves the effect of transverse isotropy is that the t 2 ‐ x 2 ‐curves are convex upwards; for SV‐waves the curves are convex downwards. For SH‐waves transverse isotropy has no effect on curvature.