Explicit analytic expression for normal moveout from horizontal and dipping reflectors in weakly anisotropic media of arbitrary symmetry type

Abstract

When processing and inverting seismic reflection data, the NMO velocity must be correctly described, taking into account realistic situations such as the presence of anisotropy and dipping reflectors. Some dip‐moveout (DMO) algorithms have been developed, such as Tsvankin’s analytic formula. It describes the anisotropy‐induced distortions in the classical isotropic cosine of dip dependence of the NMO velocity. However, it is restricted to the vertical symmetry planes of anisotropic media, so the technique is unsuitable for the azimuthal inspection of sedimentary rocks, either with horizontal bedding and vertical fractures or with dipping bedding but no fractures. However, under the weak anisotropy approximation the deviations of the rays from a vertical plane can be neglected for the traveltimes computation, and the equation can still be applicable. Based on this approach, an explicit analytic expression for the P-wave NMO velocity in the presence of horizontal or dipping reflectors in media exhibiting the most general symmetry type (triclinic) is obtained in this work. If the medium exhibits a horizontal symmetry plane, the concise DMO equations are formally identical to Tsvankin’s except that the parameters δ and ε are not constant but depend on the azimuth ψ Physically, δ(ψ) is the deviation from the vertical P-wave velocity of the P-wave NMO velocity for a horizontal reflector normalized by the vertical P-wave velocity for the azimuth ψ. The function ε(ψ) has the same definition as δ(ψ) except that the P-wave NMO velocity is replaced by the horizontal P-wave velocity. Both depend linearly on (1) new dimensionless anisotropy parameters and (2) generalizing to arbitrary symmetry the transversely isotropic parameters δ and ε. In the most general symmetry case (triclinic), an additional term to the DMO formula is necessary. The numerical examples, based on experimental data in rocks, show two things. First, the magnitude of the DMO errors induced by anisotropy depends primarily on the absolute value of ε(ψ) − δ(ψ) and not on the individual values of ε(ψ) and δ(ψ), which is a direct consequence of the similarity between Tsvankin’s equation and the equation presented here. Second, the anisotropy‐induced DMO correction can be significant even in the presence of moderate anisotropy and can exhibit complex azimuthal dependence.