Moveout approximations for P- and SV-waves in transversely isotropic media with vertical axis of symmetry

Veronique Farra & Ivan Psencik


We propose alternative expressions for the P- and SV-wave moveout formulae in VTI media based on the weak-anisotropy approximation. Our moveout formulae represent expansions with respect to small parameters, which are related to deviations of anisotropy from isotropy. First-order P-wave formulae depend on four parameters, two-way zero offset traveltime T0 related to the vertical velocity α, the depth H of the single horizontal reflector and two weak-anisotropy (WA) parameters ε and δ. The first-order SV-wave formulae depend on three parameters, again on T0 now related to the SV-wave vertical velocity β, depth H and the WA version of parameter σ. The second-order formulae are slightly more complicated. Both P- and SV-wave formulae depend on an additional parameter r, the ratio of the SV- and P-wave velocities. The SV-wave formula depends, in addition, on the WA parameter ε. Since the dependence of the moveout formulae on r is very weak, r can be specified as a typical SV- to P-wave velocity ratio and the number of parameters necessary to specify the second-order formulae is four for both waves. The formulae are relatively simple, highly accurate around zero offset and yield exact long offset asymptote. Their accuracy at intermediate offsets depends on deviations of ray and phase-velocity directions. The proposed formulae are also applicable in cases when the reflected ray is situated in a plane of symmetry of an orthorhombic medium, whose another symmetry plane is horizontal. This also includes any HTI medium with axis of symmetry in the plane containing the reflected ray.

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In: Seismic Waves in Complex 3-D Structures, Report 23, pp. 81-103, Dep. Geophys., Charles Univ., Prague, 2013.