A transversely isotropic medium with a vertical symmetry axis (VTI) is the most commonly used anisotropic model. This model is generally attributed to fine layering in sedimentary basins. One of the first nonhyperbolic approximations for P-wave reflection traveltimes in VTI was proposed by Muir and Dellinger 1985 and further developed by Dellinger et al. 1993. In a classic paper, Thomsen 1986 developed a weak anisotropy approximation for describing the transversely isotropic model. Tsvankin and Thomsen 1994 used the weak anisotropy assumption to approximate nonhyperbolic reflection moveout in VTI media.
We start this paper with a brief overview of the weak anisotropy approximation and use this approximation in the following sections for analytical derivations. First, we consider the case of a vertically heterogeneous anisotropic layer. For this case, the three-parameter approximation suggested by Tsvankin and Thomsen 1994 is compared with the shifted hyperbola approximation Castle (1988); Malovichko (1978); Sword (1987); de Bazelaire (1988). The second case is a homogeneous anisotropic medium with a curved reflector. In this case, we analyze the cumulative effect of anisotropy, reflector dip, and reflector curvature and develop an appropriate three-parameter approximation of the reflection moveout. Third, we consider the case of a weak lateral heterogeneity. We show that with an appropriate choice of the lateral velocity variation, it can can mimic the effect of transverse isotropy on nonhyperbolic moveout. In conclusion, discuss possible practical applications of the theory.