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Apex correction due to LMO shift

The LMO operation leads to vertical time shifts in the data that incorrectly position the scattering point horizontally. Figure [*] illustrates the kinematics of the LMO operation. In Figure [*], a light gray line connects the energy scattered from the discontinuous structure shown in Figure [*] with the associated source point on the incident wavefront. As illustrated in Figure [*]b, after the LMO operation the scattering point must be shifted horizontally so that it falls beneath the corresponding source point on the source wavefront. This is a requirement of zero-offset migration. Noting that the relationship between the horizontal change in apex location, ${\rm d}x_{apex}$, as a function of depth, ${\rm d}z$ is,
\begin{displaymath}
{\rm d}x_{apex} = {\rm tan} \theta_s {\rm d}z,\end{displaymath} (15)
the required horizontal shift operator is derived by evaluating a Fourier transform (${\cal F}$) of the vertical component of velocity from depth z to the surface,
\begin{displaymath}
{\cal F} \left( \delta(x-x_0) \right) ={\cal F} \left( \delt...
 ..._x \int_0^z {\rm d}z \frac {p
 v_s}{\sqrt{1-p^2v_s^2}} \right).\end{displaymath} (16)
Note that equation (16) is dependent solely on the v=v(z) velocity profile and can be applied either pre- or post-migration. I have chosen to apply this shift post-migration (Figure [*])

 
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Figure 3
Diagram of LMO transform to approximate equivalent zero-offset section. a) Before LMO correction hyperbola is symmetric about the scattering point. The light gray line links points of scattered energy to the location on the wavefront causing scattering. b) After the LMO correction the hyperbola and `Imaging box' are skewed. In-filled area illustrates the travel-time error introduced by LMO operation. Note that the hyperbola apex has zero error. The tilted light gray line shows the non-centered location of apex hyperbola. This line needs to be straighted out through a horizontal shift so that it falls beneath the scattering point.
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next up previous print clean
Next: Velocities Up: Methodology Previous: Double square root equations
Stanford Exploration Project
7/8/2003