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The chain rule of differentiation leads to the equality
| ![\begin{displaymath}
p_x = \frac{\partial t}{\partial x} = -p_{\tau}
\frac{\partial \tau}{\partial x},\end{displaymath}](img57.gif) |
(24) |
where
. It is
convenient to transform equality (A-1) to the form
| ![\begin{displaymath}
\frac{\partial \tau}{\partial x} = -\frac{p_x}{p_{\tau}}.\end{displaymath}](img59.gif) |
(25) |
Using the expression for
from the main text, we can
write equation (A-2) as a quadratic polynomial in px2
as follows
where
![\begin{displaymath}
a=-2v^2 \eta,\end{displaymath}](img60.gif)
![\begin{displaymath}
b= (\frac{\partial \tau}{\partial x})^2 v^2 (1+2 \eta)+1,\end{displaymath}](img61.gif)
and
![\begin{displaymath}
c=-(\frac{\partial \tau}{\partial x})^2.\end{displaymath}](img62.gif)
Since
can be small (as small as zero for isotropic media),
we use the following form of solution to the quadratic equation
![\begin{displaymath}
p_x^2 = \frac{2 c}{-b \pm \sqrt{b^2-4ac}}\end{displaymath}](img63.gif)
Press et al. (1992). This form does not go to infinity as
approaches
. We choose the solution with the negative sign in front of the
square root, because this solution complies with the isotropic result
when
is equal to zero.
Next: Linearized approximations
Up: Alkhalifah and Fomel: Anisotropy
Previous: REFERENCES
Stanford Exploration Project
11/11/1997