Most bulk elastic media are weakly anisotropic. The equations governing weak anisotropy are much simpler than those governing strong anisotropy, and they are much easier to grasp intuitively. These equations indicate that a certain anisotropic parameter (denoted δ) controls most anisotropic phenomena of importance in exploration geophysics, some of which are nonnegligible even when the anisotropy is weak. The critical parameter δ is an awkward combination of elastic parameters, a combination which is totally independent of horizontal velocity and which may be either positive or negative in natural contexts.

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.

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Recent surveys have shown that azimuthal anisotropy (due most plausibly to aligned fractures) has an important effect on seismic shear waves. Previous work had discussed these effects on VSP data; the same effects are seen in surface recording of reflections at small to moderate angles of incidence. The anisotropic effects one different polarization components of vertically traveling shear waves permit the recognition and estimation of very small degrees of azimuthal anisotropy (of order 2 1 percent), as in an interferometer. Anisotropic effects on traveltime yield estimates of anisotropy which are averages over large depth intervals. Often, raw field data must be corrected for these effects before the reflectors may be imaged; two variations of a rotational algorithm to determine the "principal time series" are derived. Anisotropic effects on moveout lead to abnormal moveout unless the survey line is parallel to the fractures. Anisotropic effects on reflection amplitude permit the recognition and estimation of anisotropy (hence fracture intensity) differences at the reflecting horizon, i.e., with high vertical resolution,

Converted-wave processing is more critically dependent on physical assumptions concerning rock velocities than is pure-mode processing, because not only moveout but also the offset of the imaged point itself depend upon the physical parameters of the medium. Hence, unrealistic assumptions of homogeneity and isotropy are more critical than for pure-mode propagation, where the image-point offset is determined geometrically rather than physically. In layered anisotropic media, an effective velocity ratio yeff -yz /yo (where yo -Vp /VS is the ratio of average vertical velocities and y2 is the corresponding ratio of short-spread moveout velocities) governs most of the behavior of the conversion-point offset. These ratios can be constructed from P-wave and converted-wave data if an approximate correlation is established between corresponding reflection events. Acquisition designs based naively on yo instead of yeff can Manuscript

All theoretical expressions which relate the characteristics of saturated aligned cracks to the associated elastic anisotropy are restricted in some important way, for example to the case of stiff pore fluids, or of the absence of equant porosity, or of a moderately high frequency band. Because of these restrictions, previous theory is not suitable for application to the upper crust, where the pore fluid is brine (1(, = KJ2O), the equant porosity is often substantial (do > 0.t;, and the frequency band is sonic to seismic. This work removes these particular restrictions, recognizing in the process an important mechanisrn of dispersion. A notable feature of these more general expressions is their insensitivity, at low frequency, to the aspect ratio of the cracks; only the crack density is critical. An important conclusion of this more general model is that many insights previously achieved, concerning the shear-wave splitting due to vertical aligned saturated cracks, are sustained. However, conclusions on crack orientation or crack aspect ratio, which were derived from P-wave data or from shear-wave 'critical angles', ffiay need to be reconsidered. Further, the non-linear coupling between pores and cracks, due to pressure equalization effects, means that the (linear) Schoenberg-Muir calculus may not be applied to such systems. The theory receives strong support from recent data by Rathore et al. on artificial samples with controlled crack geometry. lntroduction This work concerns the theory of the effect of a set of aligned circular cracks upon the elasticity of the resulting composite material. Of course, the primary effect is a reduction in certain of the elastic moduli, so that the resulting composite material is elastically anisotropic. Assuming that the matrix material is homogeneous and isotropic, the composite material is clearly transversely isotropic, with its symmetry axis lying perpendicular to the flat faces ofthe cracks. At issue is the dependence of the anisotropy upon angle, upon crack density, upon crack shape, upon stiffness of I Paper presented at the 53rd EAEG meeting,

In anisotropic media, the short-spread stacking velocity is generally different from the root-mean-square vertical velocity. The influence of anisotropy makes it impossible to recover the vertical velocity (or the reflector depth) using hyperbolic moveout analysis on short-spread, common-midpoint (CMP) gathers, even if both P-and S-waves are recorded.Hence, we examine the feasibility of inverting longspread (nonhyperbolic) reflection moveouts for parameters of transversely isotropic media with a vertical symmetry axis. One possible solution is to recover the quartic term of the Taylor series expansion for t 2 -X 2 curves for P-and SV-waves, and to use it to determine the anisotropy. However, this procedure turns out to be unstable because of the ambiguity in the joint inversion of intermediate-spread (i.e., spreads of about 1.5 times the reflector depth) P and SV moveManuscript received by the

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