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.

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When transversely isotropic (VTI) media with vertical symmetry axes are characterized using the zero‐dip normal moveout (NMO) velocity [[Formula: see text]] and the anisotropy parameter ηinstead of Thomsen’s parameters, time‐related processing [moveout correction, dip moveout (DMO), and time migration] become nearly independent of the vertical P- and S-wave velocities ([Formula: see text] and [Formula: see text], respectively). The independence on [Formula: see text] and [Formula: see text] is well within the limits of seismic accuracy, even for relatively strong anisotropy. The dependency on [Formula: see text] and [Formula: see text] reduces even further as the ratio [Formula: see text] decreases. In fact, for [Formula: see text], all time‐related processing depends exactly on only [Formula: see text] and η. This fortunate dependence on two parameters is demonstrated here through analytical derivations of time‐related processing equations in terms of [Formula: see text] and η. The time‐migration dispersion relation, the NMO velocity for dipping events, and the ray‐tracing equations extracted by setting [Formula: see text] (i.e., by considering VTI as acoustic) not only depend solely on [Formula: see text] and η but are much simpler than the counterpart expressions for elastic media. Errors attributed to this use of the acoustic assumption are small and may be neglected. Therefore, as in isotropic media, the acoustic model arising from setting [Formula: see text], although not exactly true for VTI media, can serve as a useful approximation to the elastic model for the kinematics of P-wave data. This approximation can boost the efficiency of imaging and DMO programs for VTI media as well as simplify their description.

A wave equation, derived under the acoustic medium assumption for P-waves in transversely isotropic media with a vertical symmetry axis (VTI media), though physically impossible, yields good kinematic approximation to the familiar elastic wave equation for VTI media. The VTI acoustic wave equation is fourth-order and has two sets of complex conjugate solutions. One set of solutions is just perturbations of the familiar acoustic wavefield solutions for isotropic media for incoming and outgoing waves. The second set describes an unwanted wave type that propagates at speeds slower than the P-wave for the positive anisotropy parameter, , and grows exponentially, becoming unstable, for negative values of. Luckily, most values corresponding to anisotropies in the subsurface have positive values which is in the stability range of the acoustic equation. Placing the source in an isotropic layer, a common occurrence in marine surveys where the water layer is isotropic, eliminates most of the energy of this additional wave type. Numerical examples prove the usefulness of this acoustic equation in simulating wave propagation in complex models.

In multiparameter full-waveform inversion (FWI) and specifically one describing the anisotropic behavior of the medium, it is essential that we have an understanding of the parameter resolution possibilities and limits. Because the imaging kernel is at the heart of the inversion engine (the model update), we drew our development and choice of parameters from what we have experienced in imaging seismic data in anisotropic media. In representing the most common (firstorder influence and gravity induced) acoustic anisotropy, specifically, a transversely isotropic medium with a vertical symmetry direction (VTI), with the P-wave normal moveout velocity, anisotropy parameters δ, and η, we obtained a perturbation radiation pattern that has limited trade-off between the parameters. Because δ is weakly resolvable from the kinematics of P-wave propagation, we can use it to play the role that density plays in improving the data fit for an imperfect physical model that ignores the elastic nature of the earth. An FWI scheme that starts from diving waves would benefit from representing the acoustic VTI model with the P-wave horizontal velocity, η, and ε. In this representation, the diving waves will help us first resolve the horizontal velocity and then reflections, if the nonlinearity is properly handled, could help us resolve η, and ε could help improve the amplitude fit (instead of the density). The model update wavenumber for acoustic anisotropic FWI is very similar to that for the isotropic case, which is mainly dependent on the scattering angle and frequency.