SWAG -- TABLE OF CONTENTS

SWAG -- TABLE OF CONTENTS

Traveltime approximations for transversely isotropic media with an inhomogeneous background [pdf 512K]
Tariq Alkhalifah
A transversely isotropic model with a tilted symmetry axis (TI) is regarded as one of the most effective approximations to the Earth subsurface, especially for imaging purposes. However, we commonly utilize this model by setting the axis of symmetry normal to the reflector. This assumption may be accurate in many places, but deviations from this assumption will cause errors in the wavefield description. Using perturbation theory and Taylor's series, I expand the solutions of the eikonal equation for 2D transversely isotropic media with respect to the independent parameter $$, the angle the tilt of the axis of symmetry makes with the vertical, in a generally inhomogeneous TI background with a vertical axis of symmetry (VTI). I do an additional expansion in terms of the independent (anellipticity) parameter $$ in a generally inhomogeneous elliptically anisotropic background medium. These new TI traveltime solutions are given by expansions in $$ and $$ with coefficients extracted from solving linear first-order partial differential equations. Pade approximations are used to enhance the accuracy of the representation by predicting the behavior of the higher-order terms of the expansion. A simplification of the expansion for homogenous media provides nonhyperbolic moveout descriptions of the traveltime for TI models that are more accurate than other recently derived approximations. In addition, for 3D media, I develop traveltime approximations using Taylor's series type of expansions in the azimuth of the axis of symmetry. The coefficients of all these expansions can also provide us with the medium sensitivity gradients (Jacobian) for nonlinear tomographic-based inversion for the tilt in the symmetry axis.
Angle gathers in wave-equation imaging for transversely isotropic media [pdf 2.5M]
Tariq Alkhalifah and Sergey Fomel
In recent years, wave-equation imaged data are often presented in common-image angle-domain gathers as a decomposition in scattering angle at the reflector, which provide a natural access to analyzing migration velocities and amplitudes. In the case of anisotropic media, the importance of angle gathers is enhanced by the need to properly estimate multiple anisotropic parameters for a proper representation of the medium. We extract angle gathers for each downward-continuation step from converting offset-frequency planes into angle-frequency planes simultaneously with applying the imaging condition in a transversely isotropic 2with a vertical symmetry axis (VTI) medium. The analytic equations, though cumbersome, are exact within the framework of the acoustic approximation. They are also easily programmable and show that angle gather mapping in the case of anisotropic media differs from its isotropic counterpart, with the difference depending mainly on the strength of anisotropy. Synthetic examples demonstrate the importance of including anisotropy in the angle gather generation as mapping of the energy is negatively altered otherwise. In the case of a titled axis of symmetry (TTI), the same VTI formulation is applicable but requires a rotation of the wavenumbers.
A transversely isotropic medium with a tilted symmetry axis normal to the reflector [pdf 648K]
Tariq Alkhalifah and Paul Sava
The computational tools for imaging in transversely isotropic media with tilted axes of symmetry (TTI) are complex and in most cases do not have an explicit closed-form representation. As discussed in this paper, developing such tools for a TTI medium with tilt constrained to be normal to the reflector dip (DTI) reduces their complexity and allows for closed-form representations. We show that, for the homogeneous case zero-offset migration in such a medium can be performed using an isotropic operator scaled by the velocity of the medium in the tilt direction. We also show that, for the nonzero-offset case, the reflection angle is always equal to the incidence angle, and thus, the velocities for the source and receiver waves at the reflection point are equal and explicitly dependent on the reflection angle. This fact allows us to develop explicit representations for angle decomposition as well as moveout formulas for analysis of extended images obtained by wave-equation migration. Although setting the tilt normal to the reflector dip may not be valid everywhere (i.e., salt flanks), it can be used in the process of velocity model building where such constrains are useful and typically used.
Acoustic wavefield evolution as function of source location perturbation [pdf 596K]
Tariq Alkhalifah
The wavefield is typically simulated for seismic exploration applications through solving the wave equation for a specific seismic source location. The direct relation between the form (or shape) of the wavefield and the source location can provide insights useful for velocity estimation and interpolation. As a result, I derive partial differential equations that relate changes in the wavefield shape to perturbations in the source location, especially along the Earth's surface. These partial differential equations have the same structure as the wave equation with a source function that depends on the background (original source) wavefield. The similarity in form implies that we can use familiar numerical methods to solve the perturbation equations, including finite difference and downward continuation. In fact, we can use the same Green's function to solve the wave equation and its source perturbations by simply incorporating source functions derived from the background field. The solutions of the perturbation equations represent the coefficients of a Taylor's series type expansion of the wavefield as a function of source location. As a result, we can speed up the wavefield calculation as we approximate the wavefield shape for sources in the vicinity of the original source. The new formula introduces changes to the background wavefield only in the presence of lateral velocity variation or in general terms velocity variations in the perturbation direction. The approach is demonstrated on the smoothed Marmousi model. Another form of the perturbation partial differential wave equation is independent of direct velocity derivatives, and thus, provide possibilities for wavefield continuation in complex media. The caveat here is that the medium complexity information is embedded in the wavefield and thus the wavefield shape evolution as a function of shift in the velocity or source can be extracted from the background wavefield and produce wavefield shapes for nearby sources.
An eikonal based formulation for traveltime perturbation with respect to the source location [pdf 836K]
Tariq Alkhalifah and Sergey Fomel
Traveltime calculations amount to solving the nonlinear eikonal equation for a given source location. We analyze the relationship between the eikonal solution and its perturbations with respect to the source location and develop a partial differential equation that relates the traveltime field for one source location to that for a nearby source. This linear first-order equation in one form depends on lateral changes in velocity and in another form is independent of the velocity field and relies on second-order derivatives of the original traveltime field. For stable finite-difference calculations, this requires the velocity field to be smooth in a sense similar to ray-tracing requirements. Our formulation for traveltime perturbation formulation has several potential applications, such that fast traveltime calculation by source-location perturbation, velocity-independent interpolation including datuming, and velocity estimation. Additionally, higher-order expansions provide parameters necessary for Gaussian-beam computations.
Wavefield extrapolation in pseudodepth domain [pdf 5.7M]
Xuxin Ma and Tariq Alkhalifah
Wavefields are commonly computed in the Cartesian coordinate frame. Its efficiency is inherently limited due to spatial oversampling in deep layers, where the velocity is high and wavelengths are long. To alleviate this computational waste due to uneven wavelength sampling, we convert the vertical axis of the conventional domain from depth to vertical time or pseudo depth. This creates a nonorthognal Riemannian coordinate system. Both isotropic and anisotropic wavefields can be extrapolated in the new coordinate frame with improved efficiency and good consistency with Cartesian domain extrapolation results. Prestack depth migrations are also evaluated based on the wavefield extrapolation in the pseudodepth domain.
Automatic traveltime picking using the instantaneous traveltime [pdf 1.7M]
Christos Saragiotis, Tariq Alkhalifah, and Sergey Fomel
Event picking is used in many steps of seismic processing. We present an automatic event picking method that is based on a new attribute of seismic signals, the instantaneous traveltime. The calculation of the instantaneous traveltime consists of two separate but interrelated stages. First, a trace is mapped onto the time-frequency domain. Then the time-frequency representation is mapped back onto the time domain by an appropriate operation. The computed instantaneous traveltime equals the recording time at those instances at which there is a seismic event, a feature that is used to pick the events. We analyze the concept of the instantaneous traveltime and demonstrate the application of our automatic picking method on dynamite and Vibroseis field data.

SWAG -- TABLE OF CONTENTS