Nonhyperbolic reflection moveout in anisotropic media

Abstract

The standard hyperbolic approximation for reflection moveouts in layered media is accurate only for relatively short spreads, even if the layers are isotropic. Velocity anisotropy may significantly enhance deviations from hyperbolic moveout. Nonhyperbolic analysis in anisotropic media is also important because conventional hyperbolic moveout processing on short spreads is insufficient to recover the true vertical velocity (hence the depth). We present analytic and numerical analysis of the combined influence of vertical transverse isotropy and layering on long‐spread reflection moveouts. Qualitative description of nonhyperbolic moveout on “intermediate” spreads (offset‐to‐depth ratio x/z < 1.7–2) is given in terms of the exact fourth‐order Taylor series expansion for P, SV, and P‐SV traveltime curves, valid for multilayered transversely isotropic media with arbitrary strength of anisotropy. We use this expansion to provide an analytic explanation for deviations from hyperbolic moveout, such as the strongly nonhyperbolic SV‐moveout observed numerically in the case where δ < ε. With this expansion, we also show that the weak anisotropy approximation becomes inadequate (to describe nonhyperbolic moveout) for surprisingly small values of the anisotropies δ and ε. However, the fourth‐order Taylor series rapidly loses numerical accuracy with increasing offset. We suggest a new, more general analytical approximation, and test it against several transversely isotropic models. For P‐waves, this moveout equation remains numerically accurate even for substantial anisotropy and large offsets. This approximation provides a fast and effective way to estimate the behavior of long‐spread moveouts for layered anisotropic models.

Traveltime approximations and parameter estimation for orthorhombic media

“…and Thomsen, 1994;Alkhalifah and Tsvankin, 1995) to an orthorhombic layer by considering azimuthally varying parameters as follows:…”

“…Ray tracing (Červený, 2001) or discretizing the eikonal equation using finite difference (Vidale, 1990;van Trier and Symes, 1991) allow to obtain the traveltimes needed in many imaging applications. Although finite-difference methods are very efficient to solve the eikonal equation of an isotropic medium, using such methods for anisotropic eikonal equations is a challenge.…”

Approximate Conditions for the Off-Axis Triplication in Transversely Isotropic Media

“…Condition (4) can also be expressed using the weak-anisotropy parameters σ and ε or σ and δ, which represent alternative parameterizations of transverse isotropy (Thomsen, 1986;Tsvankin and Thomsen, 1994): …”

Nonhyperbolic Reflection Moveout for Orthorhombic Media

“…They also introduced a normalization factor for the quartic term that ensures the convergence of the Taylor series traveltime expansion at infinitely large horizontal offsets for VTI media. The reflection moveout expression of Tsvankin and Thomsen (1994) will serve as a basis for our study of nonhyperbolic reflection moveout in azimuthally anisotropic media.…”

“…Hake et al (1984) derived the quartic Taylor series term A 4 of t 2 − x 2 reflection-moveout curves for pure modes in TI media with a vertical axis of symmetry. Tsvankin and Thomsen (1994) recasted the quartic term of Hake et al (1984) in a more compact form using Thomsen's (1986) notation. They also introduced a normalization factor for the quartic term that ensures the convergence of the Taylor series traveltime expansion at infinitely large horizontal offsets for VTI media.…”