The main difficulty in extending seismic processing to anisotropic media is the recovery of anisotropic velocity fields from surface reflection data. We suggest carrying out velocity analysis for transversely isotropic (TI) media by inverting the dependence of P‐wave moveout velocities on the ray parameter. The inversion technique is based on the exact analytic equation for the normal‐moveout (NMO) velocity for dipping reflectors in anisotropic media. We show that P‐wave NMO velocity for dipping reflectors in homogeneous TI media with a vertical symmetry axis depends just on the zero‐dip value [Formula: see text] and a new effective parameter η that reduces to the difference between Thomsen parameters ε and δ in the limit of weak anisotropy. Our inversion procedure makes it possible to obtain η and reconstruct the NMO velocity as a function of ray parameter using moveout velocities for two different dips. Moreover, [Formula: see text] and η determine not only the NMO velocity, but also long‐spread (nonhyperbolic) P‐wave moveout for horizontal reflectors and the time‐migration impulse response. This means that inversion of dip‐moveout information allows one to perform all time‐processing steps in TI media using only surface P‐wave data. For elliptical anisotropy (ε = δ), isotropic time‐processing methods remain entirely valid. We show the performance of our velocity‐analysis method not only on synthetic, but also on field data from offshore Africa. Accurate time‐to‐depth conversion, however, requires that the vertical velocity [Formula: see text] be resolved independently. Unfortunately, it cannot be done using P‐wave surface moveout data alone, no matter how many dips are available. In some cases [Formula: see text] is known (e.g., from check shots or well logs); then the anisotropy parameters ε and δ can be found by inverting two P‐wave NMO velocities corresponding to a horizontal and a dipping reflector. If no well information is available, all three parameters ([Formula: see text], ε, and δ) can be obtained by combining our inversion results with shear‐wave information, such as the P‐SV or SV‐SV wave NMO velocities for a horizontal reflector. Generalization of the single‐layer NMO equation to layered anisotropic media with a dipping reflector provides a basis for extending anisotropic velocity analysis to vertically inhomogeneous media. We demonstrate how the influence of a stratified anisotropic overburden on moveout velocity can be stripped through a Dix‐type differentiation procedure.

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.

Although orthorhombic (or orthotropic) symmetry is believed to be common for fractured reservoirs, the difficulties in dealing with nine independent elastic constants have precluded this model from being used in seismology. A notation introduced in this work is designed to help make seismic inversion and processing for orthorhombic media more practical by simplifying the description of a wide range of seismic signatures. Taking advantage of the fact that the Christoffel equation has the same form in the symmetry planes of orthorhombic and transversely isotropic (TI) media, we can replace the stiffness coefficients by two vertical (P and S) velocities and seven dimensionless parameters that represent an extension of Thomsen's anisotropy coefficients to orthorhombic models. By design, this notation provides a uniform description of anisotropic media with both orthorhombic and TI symmetry.The dimensionless anisotropic parameters introduced here preserve all attractive features of Thomsen notation in treating wave propagation and performing 2-D

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Progress in seismic inversion and processing in anisotropic media depends on our ability to relate different seismic signatures to the anisotropic parameters. While the conventional notation (stiffness coefficients) is suitable for forward modeling, it is inconvenient in developing analytic insight into the influence of anisotropy on wave propagation. Here, a consistent description of P-wave signatures in transversely isotropic (TI) media with arbitrary strength of the anisotropy is given in terms of Thomsen notation. The influence of transverse isotropy on P-wave propagation is shown to be practically independent of the vertical S-wave velocity VS0 , even in models with strong velocity variations. Therefore, the contribution of transverse isotropy to P-wave kinematic and dynamic signatures is controlled by just two anisotropic parameters, e and 8, with the vertical velocity V, 0 being a scaling coefficient in homogeneous models. The distortions of reflection moveouts and amplitudes are not necessarily correlated with the magnitude of INTRODUCTION: CONVENTIONAL AND THOMSEN NOTATION

The simplest effective model of a formation containing a single fracture system is transversely isotropic with a horizontal symmetry axis (HTI). Reflection seismic signatures in HTI media, such as NMO velocity and amplitude variation with offset (AVO) gradient, can be conveniently described by the Thomsen‐type anisotropic parameters [Formula: see text] [Formula: see text] and [Formula: see text] Here, we use the linear slip theory of Schoenberg and co‐workers and the models developed by Hudson and Thomsen for pennyshaped cracks to relate the anisotropic parameters to the physical properties of the fracture network and to devise fracture characterization procedures based on surface seismic measurements. Concise expressions for [Formula: see text] [Formula: see text] and [Formula: see text] linearized in the crack density, show a substantial difference between the values of the anisotropic parameters for isolated fluid‐filled and dry (gas‐filled) penny‐sh aped cracks. While the dry‐crack model is close to elliptical with [Formula: see text] for thin fluid‐filled cracks [Formula: see text] and the absolute value of [Formula: see text] for typical [Formula: see text] ratios in the background is close to the crack density. The parameters [Formula: see text] and [Formula: see text] for models with partial saturation or hydraulically connected cracks and pores always lie between the values for dry and isolated fluid‐filled cracks. We also demonstrate that all possible pairs of [Formula: see text] and [Formula: see text] occupy a relatively narrow triangular area in the [Formula: see text] [Formula: see text]plane, which can be used to identify the fracture‐induced HTI model from seismic data. The parameter [Formula: see text] along with the fracture orientation, can be obtained from the P-wave NMO ellipse for a horizontal reflector. Given [Formula: see text] the NMO velocity of a dipping event or nonhyperbolic moveout can be inverted for [Formula: see text] The remaining anisotropic coefficient, [Formula: see text] can be determined from the constraint on the parameters of vertically fractured HTI media if an estimate of the [Formula: see text] ratio is available. Alternatively, it is possible to find [Formula: see text] by combining the NMO ellipse for horizontal events with the azimuthal variation of the P-wave AVO gradient. Also, we present a concise approximation for the AVO gradient of converted (PS) modes and show that all three relevant anisotropic coefficients of HTI media can be determined by the joint inversion of the AVO gradients or NMO velocities of P- and PS-waves. For purposes of evaluating the properties of the fractures, it is convenient to recalculate the anisotropic coefficients into the normal [Formula: see text] and tangential [Formula: see text] weaknesses of the fracture system. If the HTI model results from penny‐shaped cracks, [Formula: see text] gives a direct estimate of the crack density and the ratio [Formula: see text] is a sensitive indicator of fluid saturation. However, while there is a substantial difference between the values of [Formula: see text] for isolated fluid‐filled cracks and dry cracks, interpretation of intermediate values of [Formula: see text] for porous rocks requires accounting for the hydraulic interaction between cracks and pores.

We present a new equation for normal-moveout (NMO) velocity that describes azimuthally dependent reflection traveltimes of pure modes from both horizontal and dipping reflectors in arbitrary anisotropic inhomogeneous media. With the exception of anomalous areas such as those where common-midpoint (CMP) reflection time decreases with offset, the azimuthal variation of NMO velocity represents an ellipse in the horizontal plane, with the orientation of the axes determined by the properties of the medium and the direction of the reflector normal. In general, a minimum of three azimuthal measurements is necessary to reconstruct the best-fit ellipse and obtain NMO velocity in all azimuthal directions. This result provides a simple way to correct for the azimuthal variation in stacking velocity often observed in 3-D surveys. Even more importantly, analytic expressions for the parameters of the NMO ellipse can be used in the inversion of moveout data for the anisotropic coefficients of the medium.For homogeneous transversely isotropic media with a vertical axis of symmetry (VTI media), our equation for azimuthally dependent NMO velocity from dipping