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In this appendix, we derive
, given by equation 6, in the
-domain. Using such an equation can avoid the process of mapping
from
depth to time and back.
The vertical two-way traveltime,
, is written as
| ![\begin{displaymath}
\tau(x,z) = \int_0^z \frac{2}{v_v(x,\zeta)} d\zeta,\end{displaymath}](img10.gif) |
(29) |
where z corresponds to depth.
Similarly,
| ![\begin{displaymath}
z(\tilde{x},\tau) = \frac{1}{2} \int_0^{\tau} v_v(\tilde{x},t) dt,\end{displaymath}](img83.gif) |
(30) |
where
corresponds to the new coordinate system.
Using the chain rule,
| ![\begin{displaymath}
\frac{\partial t}{\partial \tilde{x}} = \frac{\partial t}{\partial x} +
\frac{\partial t}{\partial z} \beta,\end{displaymath}](img84.gif) |
(31) |
where
extracted from equation (30) is given by
| ![\begin{displaymath}
\beta (\tilde{x},\tau)= \frac{\partial z}{\partial \tilde{x}...
...u} \frac{\partial v_v(\tilde{x},t)}{\partial \tilde{x}}\,\, dt,\end{displaymath}](img86.gif) |
(32) |
the partial derivative in
is
| ![\begin{displaymath}
\frac{\partial t}{\partial \tau} = \frac{v_v}{2} \frac{\partial t}{\partial z}.\end{displaymath}](img87.gif) |
(33) |
Therefore, the
transformation from (
,
) to (x, z) is governed
by the following Jacobian matrix in 2-D:
| ![\begin{displaymath}
J_c = \left(\begin{array}
{cc}
1& \beta \ 0& \frac{v_v}{2}\ \end{array}\right).\end{displaymath}](img88.gif) |
(34) |
The inverse of Jc is
| ![\begin{displaymath}
J_c^{-1} = \left(\begin{array}
{cc}
1& {\frac{-2\,\beta }{{v_v}}} \ 0& {\frac{2}{{v_v}}}\ \end{array}\right),\end{displaymath}](img89.gif) |
(35) |
which should equal the Jacobian matrix for the transformation from (x, z) to (
,
),
given by
| ![\begin{displaymath}
J = \left(\begin{array}
{cc}
1& \sigma\ 0& \frac{2}{v_v}\ \end{array}\right).\end{displaymath}](img90.gif) |
(36) |
As a result,
![\begin{displaymath}
\sigma(\tilde{x},\tau) = \frac{-2\,\beta }{{v_v}} =
\frac{...
...u} \frac{\partial v_v(\tilde{x},t)}{\partial \tilde{x}}\,\, dt,\end{displaymath}](img91.gif)
which is a convenient equation,
since we want to keep all fields, including velocity, in
coordinates.
B
Next: The amplitude transport equation
Up: VTI processing in inhomogeneous
Previous: REFERENCES
Stanford Exploration Project
9/12/2000