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The DSR equation in midpoint-offset space

By converting the DSR equation to midpoint-offset space we will be able to identify the familiar zero-offset migration part along with corrections for offset. The transformation between (g,s) recording parameters and (y,h) interpretation parameters is
      \begin{eqnarray}
y\ \ &=&\ \ {g \ +\ s \over 2 }
\\ h\ \ &=&\ \ {g \ -\ s \over 2 }\end{eqnarray} (37)
(38)
Travel time t may be parameterized in (g,s)-space or (y,h)-space. Differential relations for this conversion are given by the chain rule for derivatives:
      \begin{eqnarray}
{\partial t \over \partial g} \ \ \ \ &=&\ \ \ \ 
{\partial t \...
 ... \over \partial y} \ -\ 
{\partial t \over \partial h } \, \right)\end{eqnarray} (39)
(40)

Having seen how stepouts transform from shot-geophone space to midpoint-offset space, let us next see that spatial frequencies transform in much the same way. Clearly, data could be transformed from (s,g)-space to (y,h)-space with (37) and (38) and then Fourier transformed to ( ky , kh )-space. The question is then, what form would the double-square-root equation (35) take in terms of the spatial frequencies ( ky , kh )? Define the seismic data field in either coordinate system as  
 \begin{displaymath}
U ( s, g )\ \eq \ U' ( y , h )\end{displaymath} (41)
This introduces a new mathematical function U' with the same physical meaning as U but, like a computer subroutine or function call, with a different subscript look-up procedure for (y,h) than for (s,g). Applying the chain rule for partial differentiation to (41) gives
      \begin{eqnarray}
{ \partial U \over \partial s} \ \ \ \ &=&\ \ \ \ 
{ \partial y...
 ...{ \partial h \over \partial g}\ { \partial U' \over \partial h \ }\end{eqnarray} (42)
(43)
and utilizing (37) and (38) gives
      \begin{eqnarray}
{ \partial U \over \partial s }\ \ \ \ &=&\ \ \ \ 
{1 \over 2 }...
 ...partial y \, }\ +\ 
{ \partial U' \over \partial h \, } \, \right)\end{eqnarray} (44)
(45)
In Fourier transform space where $ \partial / \partial x $ transforms to i kx, equations (44) and (45), when i and U = U' are cancelled, become
      \begin{eqnarray}
k_s\ \ \ \ &=&\ \ \ \ {1 \over 2 }\ ( k_y\ -\ k_h )
\\ k_g\ \ \ \ &=&\ \ \ \ {1 \over 2 }\ ( k_y\ +\ k_h )\end{eqnarray} (46)
(47)
Equations (46) and (47) are Fourier representations of (44) and (45). Substituting (46) and (47) into (35) achieves the main purpose of this section, which is to get the double-square-root migration equation into midpoint-offset coordinates:  
 \begin{displaymath}
{\partial\ \over \partial z} \ U\ \ =\ \ -\,i \, 
{\omega \o...
 ..._y \,-\, v k_h \over 2\,\omega } \, \right)^2
\ } \ \right] \ U\end{displaymath} (48)

Equation (48) is the takeoff point for many kinds of common-midpoint seismogram analyses. Some convenient definitions that simplify its appearance are
            \begin{eqnarray}
G\ \ \ \ &=&\ \ \ \ { v\ k_g \over \omega }
\\ S\ \ \ \ &=&\ \ ...
 ...over 2\ \omega }
\\ H\ \ \ \ &=&\ \ \ \ { v\ k_h \over 2\ \omega }\end{eqnarray} (49)
(50)
(51)
(52)
Chapter [*] showed that the quantity $v \, k_x / \omega $ can be interpreted as the angle of a wave. Thus the new definitions S and G are the sines of the takeoff angle and of the arrival angle of a ray. When these sines are at their limits of $ \pm 1 $ they refer to the steepest possible slopes in (s,t)- or (g,t)-space. Likewise, Y may be interpreted as the dip of the data as seen on a seismic section. The quantity H refers to stepout observed on a common-midpoint gather. With these definitions (48) becomes slightly less cluttered:  
 \begin{displaymath}
\begin{tabular}
{\vert c\vert} \hline
 \\  $\displaystyle {\...
 ...\sqrt{1-(Y-H)^2} \ \right) U$\space \\  \\  \hline\end{tabular}\end{displaymath} (53)

Most present-day before-stack migration procedures can be interpreted through equation (53). Further analysis of it will explain the limitations of conventional processing procedures as well as suggest improvements in the procedures.

EXERCISES:

  1. Adapt equation (48) to allow for a difference in velocity between the shot and the geophone.
  2. Adapt equation (48) to allow for downgoing pressure waves and upcoming shear waves.

previous up next print clean
Next: THE MEANING OF THE Up: SURVEY SINKING WITH THE Previous: The DSR equation in
Stanford Exploration Project
10/31/1997