Introduction

Hyperbolic approximation of common-midpoint traveltime curves (reflection moveouts) plays an important role in conventional seismic data processing and interpretation. The hyperbolic formula is exact for homogeneous isotropic media with a plane reflector. Deviations from this simple model result in deviations of the true reflection moveout from the hyperbolic approximation. If the nonhyperbolicity is large enough, we may want to take it into account to correct the errors of conventional processing or to obtain additional information about the medium. One of the important causes of nonhyperbolicity is the seismic anisotropy found in a variety of geological environments. The three other important causes are vertical and lateral heterogeneity and the reflector curvature. Even if nonhyperbolic moveout is not caused by anisotropy, we may consider its presence as evidence of an effective anisotropy. However, in order to provide a correct interpretation, it is important to distinguish among the different kinds of effects. In this paper, we analyze the situations when the effect of anisotropy couples with one of the other three effects. We provide a theoretical description of these effects and compare their influence on P-wave reflection moveouts.

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.