Nonhyperbolic reflection moveout for horizontal transverse isotropy

Abstract

The transversely isotropic model with a horizontal axis of symmetry (HTI) has been used extensively in studies of shear-wave splitting to describe fractured formations with a single system of parallel vertical penny-shaped cracks. Here, we present an analytic description of longspread reflection moveout in horizontally layered HTI media with arbitrary strength of anisotropy. The hyperbolic moveout equation parameterized by the exact normal-moveout (NMO) velocity is sufficiently accurate for P-waves on conventional-length spreads (close to the reflector depth), although the NMO velocity is not, in general, usable for converting time to depth. However, the influence of anisotropy leads to the deviation of the moveout curve from a hyperbola with increasing spread length, even in a single-layer model. To account for nonhyperbolic moveout, we have derived an exact expression for the azimuthally dependent quartic term of the Taylor series traveltime expansion [t 2 (x 2)] valid for any pure mode in an HTI layer. The quartic moveout coefficient and the NMO velocity are then substituted into the nonhyperbolic moveout equation of Tsvankin and Thomsen, originally designed for vertical transverse isotropy (VTI). Numerical examples for media with both moderate and uncommonly strong non-hyperbolic moveout show that this equation accurately describes azimuthally dependent P-wave reflection traveltimes in an HTI layer, even for spread lengths twice as large as the reflector depth. In multilayered HTI media, the NMO velocity and the quartic moveout coefficient reflect the influence of layering as well as azimuthal anisotropy. We show that the conventional Dix equation for NMO velocity remains entirely valid for any azimuth in HTI media if the group-velocity vectors (rays) for data in a commonmidpoint (CMP) gather do not deviate from the vertical incidence plane. Although this condition is not exactly satisfied in the presence of azimuthal velocity variations, rms averaging of the interval NMO velocities represents a good approximation for models with moderate azimuthal anisotropy. Furthermore, the quartic moveout coefficient for multilayered HTI media can also be calculated with acceptable accuracy using the known averaging equations for vertical transverse isotropy. This allows us to extend the nonhyperbolic moveout equation to horizontally stratified media composed of any combination of isotropic, VTI, and HTI layers. In addition to providing analytic insight into the behavior of nonhyperbolic moveout, these results can be used in modeling and inversion of reflection traveltimes in azimuthally anisotropic media.

Azimuthal τ-panalysis in anisotropic media

“…In the x–t domain, the truncated (three terms) version of is generally used (). For HTI media, Tsvankin & Thomsen (1994) modified to obtain where A 2 is defined as the NMO velocity for the HTI medium and A 4 is a complex term (a combination of several anisotropic parameters) responsible for non‐hyperbolic nature of the traveltime curve of the HTI medium (Al‐Dajani & Tsvankin 1998). The other parameter, A, is a function of ‘Thomsen style’ parameters (Tsvankin, 1997) that depends on the nature of the seismic wave ( P , SH or SV ) used to generate the seismogram.…”

Non‐hyperbolic moveout inversion of wide‐azimuth P‐wave data for orthorhombic media

“…The averaging equation for the interval quartic coefficients A 4 is more complicated and, in principle, can be obtained from the exact expression for A 4 presented by Pech et al (2003). Al‐Dajani and Tsvankin (1998), however, showed that if azimuthal anisotropy is not severe, a close approximation for A 4 is provided by averaging the interval quartic coefficients for each azimuth using the VTI equations of Tsvankin and Thomsen (1994) and Tsvankin (2005). Such an averaging operation results in the effective η parameter that has the same azimuthal dependence as that in for a single orthorhombic layer.…”

“…Even for azimuthally anisotropic and laterally heterogeneous media, the NMO velocity as a function of azimuth is described by a simple quadratic form and usually traces out an ellipse in the horizontal plane (Grechka and Tsvankin 1998a). The quartic coefficient A 4 for horizontally‐layered HTI (TI with a horizontal symmetry axis) and orthorhombic media was derived by Al‐Dajani and Tsvankin (1998) and Al‐Dajani et al (1998), who also showed that the Tsvankin–Thomsen equation remains accurate even in the presence of pronounced azimuthal anisotropy. Pech, Tsvankin and Grechka (2003) gave a more general analytic description of A 4 , valid for arbitrary anisotropy and heterogeneity, that was applied by Pech and Tsvankin (2004) to P‐wave non‐hyperbolic moveout in dipping orthorhombic layers.…”

Moveout approximation for horizontal transversely isotropic and vertical transversely isotropic layered medium. Part I: 1D ray propagation‡

“…Within a given horizontal transversely isotropic (HTI) layer, the fractures are considered aligned with a specific azimuth (e.g., Bakulin, Grechka and Tsvankin 2000a,b). In this study we extend the existing work on moveout approximation in HTI layered medium (e.g., Al‐Dajani and Tsvankin 1998; Grechka and Tsvankin 1999), accounting for the deviation of the ray velocity from the phase velocity plane at each HTI layer. Consequently, the resulting relationships include the direction of the phase velocity, described by its zenith angle, θ phs (angle between the phase velocity and the vertical axis) and azimuth angle, ϕ phs .…”