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

Abstract

AbstractThe 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 t2 ‐ x2‐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 t2 ‐ x2‐curves and the true velocities of a transversely isotropic layer, we approximate t2 ‐ x2‐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 t2 ‐ x2‐curves. For P‐waves the effect of transverse isotropy is that the t2 ‐ x2‐curves are convex upwards; for SV‐waves the curves are convex downwards. For SH‐waves transverse isotropy has no effect on curvature.