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The TI eikonal and expansion in $\theta $

In VTI media, the eikonal equation (Alkhalifah, 1998) in the acoustic approximation has the form:

$\displaystyle {v^2} (1+2 \eta) \,{\left(\left(\frac{\partial \tau}{\partial x}\...
...\right)^2 +
\left(\frac{\partial \tau}{\partial y}\right)^2 \right)} \right)=1,$     (1)

where $\tau (x,y,z)$ is the traveltime (eikonal) measured from the source to a point with the coordinates $(x, y, z)$, and $v_{t}$ and $v$ are the velocity and NMO velocity (= $v_{t} \sqrt{1+2 \delta}$), respectively, described with respect to the symmetry direction at that point. To formulate a well-posed initial-value problem for equation 1, it is sufficient to specify $\tau$ at some closed surface and to choose one of the two solutions: the wave going from or toward the source. The level of nonlinearity in this quartic (in terms of $\tau$) equation is higher than that for the isotropic or elliptically anisotropic eikonal equations. This results in much more complicated finite-difference approximations of the VTI eikonal equation.

For a tilted TI medium, the traveltime derivatives in equation 1 are taken with respect to the tilt direction, and thus, we have to rotate the derivatives in equation 1 using the following Jacobian in 3D:

\cos\phi \cos\theta & \sin\phi \c...
...\theta & -\sin\phi \sin\theta & \cos\theta
\end{displaymath} (2)

to obtain an eikonal equation corresponding to the conventional computational coordinates governed by the acquisition surface. In equation 2, $\theta $ is the angle of the symmetry axis measured from the vertical and $\phi$ corresponds to the azimuth of the vertical plane that contains the symmetry axis measured from the $x$-axis (the axis of the source-receiver direction). Setting initially $\phi$=0, for simplicity, allows us to obtain the eikonal equation for 2D TI media given by:
    $\displaystyle {v^2} (1+2 \eta) \,{\left(\cos\theta \frac{\partial \tau}{\partial x} + \sin\theta \frac{\partial \tau}{\partial z}\right)^2 } +$  
    $\displaystyle {{{v_t}}^2}\,{\left( \cos\theta \frac{\partial \tau}{\partial z}-...
...{\partial x} +\sin\theta \frac{\partial \tau}{\partial z} \right)^2} \right)=1.$ (3)

The full 3D version of this equation is stated in Appendix D.

Figure 1.
A schematic plot showing the relation between a background traveltime field for $\theta $=0 and that when $\theta $ is larger than zero. The round dot at the top of the $\theta =0$ plane represents a source.
[pdf] [png] [xfig]

Solving equation 3 numerically requires solving a quartic equation (instead of the quadratic in the isotropic and elliptical anisotropic case) at each computational step. Alternatively, it can be solved using perturbation theory (Bender and Orszag, 1978) by approximating equation 3 with a series of simpler linear equations. Considering $\theta $ constant and small, we can represent the traveltime solution as a series expansion in $\theta $. This will result in a solution that is globally representative in the space domain and, despite the approximation of small $\theta $, the accuracy for even large $\theta $, as we will see later, is high. The constant-$\theta $ assumption assumes a factorized medium (Alkhalifah, 1995) in $\theta $ (useful for smooth $\theta $ estimation applications). However, all other velocities and parameters including $v_t$, $v$ (or $\delta$) and $\eta $ are allowed to vary. Figure 1 illustrates the concept of the global expansion as we predict the traveltime for any $\theta $ from its behavior at $\theta =0$ for the full traveltime field using, in this case, a quadratic approximation. Specifically, we substitute the following trial solution,

\tau(x,z) \approx \tau_0(x,z) +\tau_1(x,z) \sin\theta+ \tau_2(x,z) \sin^{2}\theta,
\end{displaymath} (4)

where $\tau_0$, $\tau_1$, and $\tau_2$ are coefficients of the expansion with dimensions of traveltime, into the eikonal equation 3. For practical purposes, I consider here only three terms of the expansion. As a result (as shown in Appendix A), $\tau_0$ satisfies the eikonal equation for VTI anisotropy, while $\tau_1$ and $\tau_2$ satisfy linear first-order partial differential equations having the following general form (see Appendix A):
{v^2} (1+2 \eta) \,{\frac{\partial \tau_{0}}{\partial x} \fr...
...l x} \frac{\partial \tau_{i}}{\partial z}} \right) =
\end{displaymath} (5)

with $i=1,2$. The functions $f_i(x,z)$ get more complicated for larger $i$ and depend on terms that can be evaluated only sequentially. Therefore, these linear partial differential equations must be solved in the order of increasing $i$ starting with $i=1$.

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