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The DSR equation in shot-geophone space

Let the geophones descend a distance $ dz_g $ into the earth. The change of the travel time of the observed upcoming wave will be

{{\partial t} \over {\partial z}_g}  \eq \
-  \sqrt{ {1...
... }  -\
\left( {\partial t \over \partial g}   \right)^2  }
\end{displaymath} (8)

Suppose the shots had been let off at depth $ dz_s $ instead of at $z = 0$. Likewise then,
{{\partial t} \over {\partial z}_s}  \eq \
-  \sqrt{ {1...
... }  -\
\left( {\partial t \over \partial s}   \right)^2  }
\end{displaymath} (9)

Both (9.8) and (9.9) require minus signs because the travel time decreases as either geophones or shots move down.

Simultaneously downward project both the shots and geophones by an identical vertical amount $dz=dz_g = dz_s$. The travel-time change is the sum of (9.8) and (9.9), namely,

dt  \eq \
{{\partial t} \over {\partial z}_g }  dz_g  +...
...artial t} \
\over {\partial z}_s} \right)  dz       \
\end{displaymath} (10)

{\partial t \over \partial z}  \eq \
-  \left(  \sqrt{ {...
...ft( {\partial t \over \partial s}   \right)^2}
 }  \right)
\end{displaymath} (11)

This expression for ${\partial t}/{\partial z} $ may be substituted into equation (9.4):
{ \partial U \over \partial z }   =    +  \left( \sq...
...   \right)^2 \
}  \right)  { \partial U \over \partial t }
\end{displaymath} (12)

Three-dimensional Fourier transformation converts upcoming wave data $u(t,s,g)$ to $U( \omega , k_s , k_g )$. Expressing equation (9.12) in Fourier space gives

{ \partial U \over \partial z } \eq - i \omega  \left[  ...
... - \left(
{ k_s \over \omega }  \right)^2  }  \right]  U
\end{displaymath} (13)

Recall the origin of the two square roots in equation (9.13). One is the cosine of the arrival angle at the geophones divided by the velocity at the geophones. The other is the cosine of the takeoff angle at the shots divided by the velocity at the shots. With the wisdom of previous chapters we know how to go into the lateral space domain by replacing $i k_g$ by $\partial / \partial g$ and $i k_s$ by $\partial / \partial s$. To incorporate lateral velocity variation $v(x)$, the velocity at the shot location must be distinguished from the velocity at the geophone location. Thus,

{ \partial U \over \partial z } \eq
\left[  \sqrt{ \lef...
...2 -\
{\partial^2  \over \partial s^2}   }  \right]  U
\end{displaymath} (14)

Equation (9.14) is known as the double-square-root (DSR) equation in shot-geophone space. It might be more descriptive to call it the survey-sinking equation since it pushes geophones and shots downward together. Recalling the section on splitting and full separation we realize that the two square-root operators are commutative ($v(s)$ commutes with $\partial / \partial g$), so it is completely equivalent to downward continue shots and geophones separately or together. This equation will produce waves for the rays that are found on zero-offset sections but are absent from the exploding-reflector model.

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