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 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.
Slant stacking transforms seismic data, recorded as a function of source-receiver offset and traveltime, into the domain of intercept time r and ray parameter p . The shape of the 7-p-curves thus obtained is closely related to the slowness surfaces of the layers. A layerstripping operation in the z-p-domain removes all effects of the layers above the target layer. The resulting curve is equal to the slowness surface of the layer except for a scaling factor containing the thickness and dip of the layer. The slowness surface is a characteristic surface for anisotropic media. This makes the r-p-domain very suitable for detecting and describing anisotropic layers. The relationship between the shape of r-p-curves, the slowness surfaces, and the geometry of the layers is derived. Synthetic 7-p-curves calculated with the reflectivity method show some difficulties that can arise in determining the shape of the curves and in applying the stripping operation. It is shown that the effects of vertical inhomogeneity on the interpretation of r-p-curves in terms of anisotropy are small.
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