Tariq Alkhalifah

An acoustic wave equation for anisotropic media

A wave equation, derived using the acoustic medium assumption for P-waves in transversely isotropic (TI) media with a vertical symmetry axis (VTI media), yields a good kinematic approximation to the familiar elastic wave equation for VTI media. The wavefield solutions obtained using this VTI acoustic wave equation are free of shear waves, which significantly reduces the computation time compared to the elastic wavefield solutions for exploding‐reflector type applications. From this VTI acoustic wave equation, the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a TI medium are derived. These equations, based on the acoustic assumption (shear wave velocity = 0), are much simpler than their elastic counterparts, yet they yield an accurate description of traveltimes and geometrical amplitudes. Numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex TI models.

Acoustic approximations for processing in transversely isotropic media

When transversely isotropic (VTI) media with vertical symmetry axes are characterized using the zero‐dip normal moveout (NMO) velocity [Vnmo(0)] 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 (VP0 and VS0, respectively). The independence on VP0 and VS0 is well within the limits of seismic accuracy, even for relatively strong anisotropy. The dependency on VP0 and VS0 reduces even further as the ratio VS0/VP0 decreases. In fact, for VS0=0, all time‐related processing depends exactly on only Vnmo(0) and η. This fortunate dependence on two parameters is demonstrated here through analytical derivations of time‐related processing equations in terms of Vnmo(0) and η. The time‐migration dispersion relation, the NMO velocity for dipping events, and the ray‐tracing equations extracted by setting VS0=0 (i.e., by considering VTI as acoustic) not only de...

An acoustic wave equation for orthorhombic anisotropy

Using a dispersion relation derived under the acoustic medium assumption, I obtain an acoustic wave equation for orthorhombic media. Although an acoustic wave equation does not strictly describe a wave in anisotropic media, it accurately describes the kinematics of P-waves. The orthorhombic acoustic wave equation, unlike the transversely isotropic one, is a sixth-order equation with three sets of complex conjugate solutions. Only one set of these solutions are perturbations of the familiar acoustic wavefield solution for isotropic media for incoming and outgoing P-waves and, thus, are of interest here. The other two sets of solutions are simply the result of this artificially derived sixth-order equation.

Gaussian beam depth migration for anisotropic media

Gaussian beam migration (GBM), as it is implemented today, efficiently handles isotropic inhomogeneous media. The approach is based on the solution of the wave equation in ray‐centered coordinates. Here, I extend the method to work for 2-D migration in generally anisotropic inhomogeneous media. Extension of the Gaussian‐beam method from isotropic to anisotropic media involves modification of the kinematics and dynamics in the required ray tracing. While the accuracy of the paraxial expansion for anisotropic media is comparable to that for isotropic media, ray tracing in anisotropic media is much slower than that in isotropic media. However, because ray tracing is just a small portion of the computation in GBM, the increased computational effort in general anisotropic GBM is typically only about 40%. Application of this method to synthetic examples shows successful migration in inhomogeneous, transversely isotropic media for reflector dips up to and beyond 90°. Further applications to synthetic data of lay...

Migration error in transversely isotropic media

Most migration algorithms today are based on the assumption that the earth is isotropic, an approximation that is often not valid and thus can lead to position errors on migrated images. Here, we compute curves of such position error as a function of reflector dip for transversely isotropic (TI) media characterized by Thomsen’s anisotropy parameters δ and e. Depending on whether the migration velocity is derived from stacking velocity or vertical root‐mean‐square (rms) velocity, we find quite contrary sensitivities of the error behavior to the values of δ and e. Likewise error‐versus‐dip behavior depends in a complicated way on vertical velocity gradient and vertical time, as well as orientation of the symmetry axis. Moreover, error behavior is dependent on just how δ and e vary with depth. In addition to presenting such error curves, we show migrations of synthetic data that exemplify the mispositioning that results from ignoring anisotropy for P‐wave data. When migration is done using velocities derived...

Velocity analysis and imaging in transversely isotropic media: Methodology and a case study

Physicists and mathematicians, for many years, have studied the intricacies and complexities of how elastic waves propagate in anisotropic media (media in which velocity varies with direction of propagation). Moveover, over the years, a select few visionary exploration geophysicists (e.g., G. Postma, F. Levin, and K. Helbig) have been telling us that the earth’s subsurface is not isotropic, so processing that fails to take anisotropy into account should yield biased estimates of depth and subsurface velocity. Nevertheless, it is only in this past decade that the anisotropic character of the earth’s subsurface has been seriously studied and treated. During that time, most emphasis has been focused on the influence of anisotropy on the behavior of shear waves (e.g. Crampin 1985), a common belief being that departures of medium properties from isotropic waves were of second order for P-waves.

Scanning anisotropy parameters in complex media

Parameter estimation in an inhomogeneous anisotropic medium offers many challenges; chief among them is the trade-off between inhomogeneity and anisotropy. It is especially hard to estimate the anisotropy anellipticity parameter η in complex media. Using perturbation theory and Taylor’s series, I have expanded the solutions of the anisotropic eikonal equation for transversely isotropic (TI) media with a vertical symmetry axis (VTI) in terms of the independent parameter η from a generally inhomogeneous elliptically anisotropic medium background. This new VTI traveltime solution is based on a set of precomputed perturbations extracted from solving linear partial differential equations. The traveltimes obtained from these equations serve as the coefficients of a Taylor-type expansion of the total traveltime in terms of η. Shanks transform is used to predict the transient behavior of the expansion and improve its accuracy using fewer terms. A homogeneous medium simplification of the expansion provides classic...

The offset‐midpoint traveltime pyramid in transversely isotropic media

Prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an offset‐midpoint traveltime equation, Cheop’s pyramid equivalent equation for VTI media. The derivation of such an equation for VTI media requires approximations that pertain to high frequency and weak anisotropy. Yet the resultant offset‐midpoint traveltime equation for VTI media is highly accurate for even strong anisotropy. It is also strictly dependent on two parameters: NMO velocity and the anisotropy parameter, η. It reduces to the exact offset‐midpoint traveltime equation for isotropic media when η = 0. In vertically inhomogeneous media, the NMO velocity and η parameters in the offset‐midpoint traveltime equation are replaced by their effective values: the velocity is replaced by the rms velocity and η is given by a more complicated equation that includes summation of the fourth power of velocity.

Efficient synthetic-seismogram generation in transversely isotopic, inhomogeneous media

I develop an efficient modeling technique for transversely isotropic, inhomogeneous media using a mix of analytical equations and numerical calculations. The analytic equation for the raypath in a factorized transversely isotropic (FTI) media with linear velocity variation, derived by Shearer and Chapman, is used to trace rays between two points. In addition, I derive an analytical equation for geometrical spreading in FTI media that aids in preserving program efficiency; however, the traveltimes are calculated numerically. I then generalize the method to treat general transversely isotropic (TI) media that are not factorized anisotropic inhomogeneous by perturbing the FTI traveltimes, following the perturbation ideas of Cervený and Filho. A Kirchhoff‐summation‐based program relying on Trorey’s diffraction method is used to generate synthetic seismograms for such a medium. For the type of velocity models treated, the program is much more efficient than finite‐difference or general ray‐trace modeling techn...

Seismic data processing in vertically inhomogeneous TI media

The first and most important step in processing data in transversely isotropic (TI) media for which velocities vary with depth is parameter estimation. The multilayer normal‐moveout (NMO) equation for a dipping reflector provides the basis for extending the TI velocity analysis of Alkhalifah and Tsvankin to vertically inhomogeneous media. This NMO equation is based on a root‐mean‐square (rms) average of interval NMO velocities that correspond to a single ray parameter, that of the dipping event. Therefore, interval NMO velocities [including the normal‐moveout velocity for horizontal events, Vnmo(0)] can be extracted from the stacking velocities using a Dix‐type differentiation procedure. On the other hand, η, which is a key combination of Thomsens parameters that time‐related processing relies on, is extracted from the interval NMO velocities using a homogeneous inversion within each layer. Time migration, like dip moveout, depends on the same two parameters in vertically inhomogeneous media, namely Vnmo...