Moveout approximations for P- and SV-waves in VTI media

Abstract

We tested alternative expressions for the P- and SV-wave moveout formulas in VTI media based on the weak-anisotropy (WA) approximation. Our moveout formulas represent expansions with respect to small parameters, which are related to deviations of anisotropy from isotropy. First-order P-wave formulas depend on four parameters, two-way zero-offset traveltime [Formula: see text] related to the vertical velocity [Formula: see text], the depth [Formula: see text] of the single horizontal reflector, and two WA parameters [Formula: see text] and [Formula: see text]. The first-order SV-wave formulas depend on three parameters, again on [Formula: see text], now related to the SV-wave vertical velocity [Formula: see text], depth [Formula: see text], and the WA version of parameter [Formula: see text]. The second-order formulas are slightly more complicated. The P- and SV-wave formulas depend on an additional parameter [Formula: see text], the ratio of the SV- and P-wave vertical velocities. The SV-wave formula depends, in addition, on the WA parameter [Formula: see text]. Because the dependence of the moveout formulas on [Formula: see text] is very weak, [Formula: see text] can be specified as a typical SV- to P-wave velocity ratio, and the number of parameters necessary to specify the second-order formulas is four for both waves. The formulas are relatively simple, highly accurate around zero offset, and yield an exact long-offset asymptote. Their accuracy at intermediate offsets depends on deviations of ray- and phase-velocity directions. These formulas are also applicable in cases in which the reflected ray is situated in a plane of symmetry of an orthorhombic medium, whose other symmetry plane is horizontal. This also includes any HTI medium with an axis of symmetry in the plane containing the reflected ray.