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
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
interpolation including datuming, and
velocity estimation. Additionally, higher-order
expansions provide parameters necessary for Gaussian-beam
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.