Nonhyperbolic reflection moveout of $P$-waves: An overview and comparison of reasons

Introduction

The hyperbolic approximation of $P$-wave reflection traveltimes in common-midpoint gathers plays an important role in conventional seismic data processing and interpretation. It is well known that hyperbolic moveout gives exact traveltimes for homogeneous isotropic or elliptically anisotropic media overlaying a plane dipping reflector. Deviations from this simple model generally cause departure from hyperbolic moveout. If the nonhyperbolicity is measurable, we can take it into account to correct errors in conventional processing or to obtain additional information about the medium. To achieve this, however, it is important to know what causes the $P$-wave moveouts to be nonhyperbolic. Although seismic anisotropy is one possible reason, it is not always the dominant one; others include the vertical or lateral heterogeneity and reflector curvature. In this paper, we give a theoretical description of $P$-wave reflection traveltimes in different models and compare the behavior and degree of nonhyperbolic moveout caused by various reasons.

A transversely isotropic model with a vertical symmetry axis (VTI medium) is the most commonly used anisotropic model for sedimentary basins, where the deviation from isotropy is usually attributed to some combination of fine layering and inherent anisotropy of shales. One of the first nonhyperbolic approximations for the $P$-wave reflection traveltimes in VTI media was proposed by Muir and Dellinger (1985) and further developed by Dellinger et al. (1993). Thomsen (1986) introduced a convenient parameterization of VTI media that was used by Tsvankin and Thomsen (1994) to describe nonhyperbolic reflection moveouts.

We begin with an overview of the weak-anisotropy approximation for $P$-wave velocities in VTI media and use it for analytic derivations throughout the paper. First, we consider a vertically heterogeneous anisotropic layer. For this model, we compare the three-parameter approximation for the $P$-wave traveltimes suggested by Tsvankin and Thomsen (1994) with the shifted hyperbola (Malovichko, 1978; Castle, 1988; de Bazelaire, 1988). Next, we examine $P$-wave moveout in VTI media above a curved reflector. We analyze the cumulative action of anisotropy, reflector dip, and reflector curvature, and develop an appropriate three-parameter representation for the reflection moveout. Finally, we consider models characterized by weak lateral heterogeneity and show that it can mimic the influence of transverse isotropy on nonhyperbolic moveout.

Nonhyperbolic reflection moveout of $P$-waves: An overview and comparison of reasons