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Fortunately, we can convert this common-midpoint transform (9) into
an equivalent common-source transform (6).
Let us make two additional Fourier transforms over spatial dimensions
of s and y for the spatial frequencies ks and ky:
| ![\begin{eqnarray}
\tilde {\tilde S} (k_s, p_s, f_s) &=&
\int \exp( -i 2 \pi k_s ...
...k_s s )
\exp( i 2 \pi f_s p_s h ) \tilde d (s,r=s+h,f=f_s) dh ds\end{eqnarray}](img17.gif) |
|
| (10) |
and
| ![\begin{eqnarray}
&\tilde {\tilde Y}& (k_y, p_y, f_y)
= \int \exp( -i 2 \pi k_y ...
...exp( i 2 \pi f_y p_y h ) \tilde d (s=y-h/2,r=y+h/2,f=f_y ) dh dy .\end{eqnarray}](img18.gif) |
|
| (11) |
To place the second integral (11) in the form of the
first (10), we should change the variables of integration
from h and y to h and s. (The Jacobian of this transformation
is
.) Substituting y=s+h/2 we get
| ![\begin{eqnarray}
&\tilde {\tilde Y}& (k_y, p_y, f_y) \nonumber\\ &=& \int \int \...
...tilde {\tilde S} ( k_s = k_y , p_s = p_y - k_y/2f_y , f_s = f_y) .\end{eqnarray}](img20.gif) |
|
| |
| |
| (12) |
Thus, a two-dimensional stretch of the midpoint-gather transform
becomes equivalent to the source-gather transform.
For a given dip over offset in a midpoint gather
py, we can identify a dip over midpoint
| ![\begin{displaymath}
- k_y/f_y =
\left. {\partial \tau_y \over \partial y}\right\vert _{p_y} =
\left. {\partial t \over \partial y}\right\vert _h .\end{displaymath}](img21.gif) |
(13) |
The adjustment of ps = py - ky/2fy subtracts
half of this midpoint dip from the offset dip.
With a careful application of the chain rule, and carefully
distinguishing partial derivatives, we could arrive at the
same result
| ![\begin{displaymath}
\left. {\partial t \over \partial h}\right\vert _s =
\left....
...1 \over 2}
\left. {\partial t \over \partial y}\right\vert _h .\end{displaymath}](img22.gif) |
(14) |
Next: ACKNOWLEDGEMENT
Up: NOTES FROM TIEMAN's SEMINAR
Previous: The midpoint gather
Stanford Exploration Project
11/12/1997