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The kinematic
-continuation equation (19) corresponds
to the following linear fourth-order dynamic equation
| ![\begin{displaymath}
\frac{\partial^4 P}{\partial t^3 \, \partial \eta} +
v^2 \...
...artial \eta} +
t v^4 \frac{\partial^4 P}{\partial x^4} = 0 \;,\end{displaymath}](img41.gif) |
(21) |
where the t coordinate refers to the vertical traveltime
, and
is the migrated image, parameterized in the anisotropy
parameter
. To find the correspondence between equations
(19) and (21), it is sufficient to apply a
ray-theoretical model of the image
| ![\begin{displaymath}
P (t, x, \eta) = A (x, \eta) f (t - \tau (x, \eta))\end{displaymath}](img43.gif) |
(22) |
as a trial solution to (21). Here the surface
is the anisotropy continuation ``wavefront'' - the image of a
reflector for the corresponding value of
, and the function A
is the amplitude. Substituting the trial solution into the partial
differential equation (21) and considering only the terms
with the highest asymptotic order (those containing the fourth-order
derivative of the wavelet f), we arrive at the kinematic equation
(19). The next asymptotic order (the third-order derivatives
of f) gives us the linear partial differential equation of the
amplitude transport, as follows:
| ![\begin{displaymath}
\left(
1 + v^2 \tau_x^2 \right) \frac{\partial A}{\partial...
...\eta \tau_{xx}
- 6 v^2 \tau \tau_x^2 \tau_{xx} \right) = 0 \;.\end{displaymath}](img45.gif) |
(23) |
We can see that when the reflector is flat (
and
), equation (23) reduces to the equality
![\begin{displaymath}
\frac{\partial A}{\partial \eta} = 0\;,\end{displaymath}](img48.gif)
and the amplitude remains unchanged for different
. This is of
course a reasonable behavior in the case of a flat reflector. It
doesn't guarantee though that the amplitudes, defined by
(23), behave equally well for dipping and curved
reflectors. The amplitude behavior may be altered by adding low-order
terms to equation (21). According to the ray theory, such
terms can influence the amplitude behavior, but do not change the
kinematics of the wave propagation.
An appropriate initial-value condition for equation (21) is
the result of isotropic migration that corresponds to the
section in the
domain. In practice, the initial-value
problem can be solved by a finite-difference technique.
Next: synthetic test
Up: Ordinary differential equation representation:
Previous: Ordinary differential equation representation:
Stanford Exploration Project
9/12/2000