Next: Lateral velocity variation
Up: WAVE-EXTRAPOLATION EQUATIONS
Previous: Depth-variable velocity
It is often convenient to arrange the calculation of a wave
to remove the effect of overall translation,
thereby making the wave appear to ``stand still.''
This subject, wave retardation,
will be examined more thoroughly in chapter
.
Meanwhile, it is easy enough to introduce
the time shift t0 of a vertically propagating wave
in a hypothetical medium of velocity
, namely,
| ![\begin{displaymath}
t_0 \eq \int_0^z \ { dz \over \bar v (z) }\end{displaymath}](img44.gif) |
(14) |
A time delay t0 in
the time domain corresponds to multiplication
by
in the
-domain.
Thus, the wave pressure P is related
to the time-shifted mathematical variable Q by
| ![\begin{displaymath}
P(z, \omega ) \eq
Q(z, \omega ) \ \exp
\left( \ i \omega \int_0^z \ {dz \over \bar v (z)} \ \right)\end{displaymath}](img46.gif) |
(15) |
which is a generalization of equation (3)
to depth-variable velocity.
(Equations (15) and (17) apply in both x- and
kx-space).
Differentiating with respect to z gives
| ![\begin{eqnarray}
{\partial P \over \partial z } &=& { \partial Q \over \partial ...
...over \partial z} \ +\
{ i \omega \over \bar v (z) } \ \right) \ Q\end{eqnarray}](img47.gif) |
(16) |
| (17) |
Next, substitute (15) and (17)
into Table
.3
to obtain the retarded equations in Table
.4.
Table 4:
Retarded form of phase-shift equations.
|
|
|
![$5^\circ$](img28.gif) |
zero |
![$+\ i\omega \left( \displaystyle {1\over v} -
{\strut 1\over\overline{v}(z)} \right) Q$](img49.gif) |
|
|
|
|
|
|
![$15^\circ$](img12.gif) |
![$\displaystyle {\strut\partial Q\over
\partial z} \eq - \,i\, {\displaystyle
{\strut v k_x^2\over 2\omega}} \ Q$](img50.gif) |
![$+\ i\omega \left( \displaystyle {1\over v} -
{\strut 1\over\overline{v}(z)} \right) Q$](img49.gif) |
|
|
|
|
|
|
![$45^\circ$](img15.gif) |
![$\displaystyle {\strut\partial Q\over
\partial z} \eq - \,i\, {\displaystyle
...
...over\displaystyle
{2\,{\omega\over v} - {\strut v k_x^2
\over 2\omega}}} \ Q$](img51.gif) |
![$+\ i\omega \left( \displaystyle {1\over v} -
{\strut 1\over\overline{v}(z)} \right) Q$](img49.gif) |
|
|
|
|
|
|
general |
diffraction |
+ thin lens |
|
|
|
Next: Lateral velocity variation
Up: WAVE-EXTRAPOLATION EQUATIONS
Previous: Depth-variable velocity
Stanford Exploration Project
10/31/1997