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In this appendix, we study the following medium:
![\begin{displaymath}
w (z,x) = w_0 - 2 q_x x = w_0 - 2 \mathbf{q} \cdot \mathbf{x}\;,
\end{displaymath}](img196.png) |
(55) |
where
.
Similarly to the constant velocity gradient medium, it is convenient to write down the ray-tracing system in
the form (Cervený, 2001)
![\begin{displaymath}
\left\{ \begin{array}{lcl}
d \mathbf{x} / d \sigma & = & \ma...
... d \sigma = \mathbf{p} \cdot \mathbf{p}\;.
\end{array} \right.
\end{displaymath}](img198.png) |
(56) |
Given equation D-1,
and thus
.
After integration over
, equation D-2 becomes
![\begin{displaymath}
\left\{ \begin{array}{lcl}
\mathbf{x} & = & \mathbf{x_0} + \...
...2 + \vert \mathbf{q} \vert^2 \sigma^3/3\;.
\end{array} \right.
\end{displaymath}](img202.png) |
(57) |
For a particular image ray
![\begin{displaymath}
\left\{ \begin{array}{lcl}
\mathbf{x_0} & = & [0, x_0]^T\;, ...
...0 - 2 q_x x_0}, 0]^T\;, \\
t_0 & = & t\;,
\end{array} \right.
\end{displaymath}](img203.png) |
(58) |
the equation for
in D-3 simplifies to
![\begin{displaymath}
\left\{ \begin{array}{lcl}
x = x_0 - q_x \sigma^2/2\;, \\
z = \sigma \sqrt{w_0 - 2 q_x x_0}\;.
\end{array} \right.
\end{displaymath}](img204.png) |
(59) |
Solving equation D-5 for
as a function of
and
![\begin{displaymath}
\sigma (z,x) = \left[ \frac{(w_0 - 2 q_x x)
- \sqrt{(w_0 - 2 q_x x)^2 - 4 q_x^2 z^2}}{2 q_x^2}
\right]^{\frac{1}{2}}\;.
\end{displaymath}](img205.png) |
(60) |
Combining equations D-3 through D-6, we find
According to equation 4, D-7 can give rise to the geometrical spreading:
![\begin{displaymath}
Q^2 (z,x) =
\frac{2 [(w_0 - 2 q_x x)^2 - 4 q_x^2 z^2]}
{(w_0...
...- 2 q_x x + \sqrt{(w_0 - 2 q_x x)^2 - 4 q_x^2 z^2} \right]}\;.
\end{displaymath}](img212.png) |
(63) |
It is more convenient to express equations D-1 and D-9 in
and
instead
of directly in
and
:
![\begin{displaymath}
w (\sigma,x_0) = w_0 - 2 q_x x_0 + q_x^2 \sigma^2\;,
\end{displaymath}](img213.png) |
(64) |
![\begin{displaymath}
Q^2 (\sigma,x_0) = 1 +
q_x^2 \sigma^2 \left( \frac{1}{w_0 - ...
...x x_0} - \frac{4}{w_0 - 2 q_x x_0 + q_x^2 \sigma^2} \right)\;,
\end{displaymath}](img214.png) |
(65) |
where we must resolve
. This is done by revisiting equation D-8. For
given
and
,
is the root of a depressed cubic function of the following form:
![\begin{displaymath}
\sigma^3 + \frac{3 (w_0 - 2 q_x x_0)}{q_x^2} \sigma - \frac{3}{q_x^2} t_0 = 0\;.
\end{displaymath}](img216.png) |
(66) |
After some algebraic manipulations, we find
Finally, inserting equations D-10 and D-11 into equation 3 results
in the Dix velocity:
![\begin{displaymath}
v_d (t_0,x_0) = \frac{\sqrt{w_0 - 2 q_x x_0}}{w_0 - 2 q_x x_0 - q_x^2 \sigma^2}\;.
\end{displaymath}](img221.png) |
(68) |
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Next: Bibliography
Up: Li & Fomel: Time-to-depth
Previous: Appendix C: Analytical expressions
2015-03-25