Conclusions

Nonhyperbolic reflection moveout of P-waves is an important indicator of anisotropy. However, its correct interpretation is impossible without taking other factors into account. In this paper, we have considered three other important factors: vertical heterogeneity, curvature of the reflector, and lateral heterogeneity. Each of these three factors can have an effect on nonhyperbolic behavior of the reflection moveout comparable with the effect of anisotropy. In particular, vertical heterogeneity produces a depth-variant anisotropic pattern, different from the pattern of VTI media. In the isotropic case, this pattern is reasonably well approximated by the shifted hyperbola formula. In the case of a VTI vertically heterogeneous medium, the parameters of anisotropy should be replaced with their effective values. For the case of a curved reflector in a homogeneous VTI medium, we have developed an approximation based on the Taylor series expansion of the traveltime with both the reflector curvature and the anisotropic parameters entering the nonhyperbolic term. In the case of a lateral heterogeneity, virtually any effectively anisotropic effect can be created.

The theoretical results of this paper are directly applicable for modeling nonhyperbolic moveouts. Particularly attractive in this context are the general formulas connecting the reflection traveltime derivatives with the traveltime derivatives of a direct wave. For smooth velocity models, these formulas may reduce the problem of tracing a family of reflected rays to the problem of tracing one central ray. Practical estimation and inversion of nonhyperbolic moveout is a different and more difficult problem. Nevertheless, the theoretical guidelines provided by the analytical theory are helpful for a correct formulation of the inversion problem. They show us explicitly what parameters of the medium we may hope to extract from the kinematics of P-wave seismic reflection data.