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Extensions to the Move-out Formulae

The move-out formulae used for both pre-migration velocity analysis and residual velocity analysis are typically simple hyperbolic equations which are functions of the time t0 at the apex of the hyperbola and the half-offset h of a trace. For NMO velocity analysis, the travel-time t to a reflection is expressed by:
t^2 = t_0^2 + \frac{4h^2}{V_{RMS}^2}.\end{displaymath} (1)
For residual velocity analysis, the equation is essentially the same. By convention, the use of a residual parameter r in the numerator rather than the denominator allows perfectly flat gathers to correspond to a residual of zero:

t2 = t02 + 4h2r.


It is implicit in these equations that all of the data used for the analysis have the same midpoint. Bins are usually chosen to be small enough that post-stack reflections are not aliased. Unmigrated data with midpoints at opposite ends of a bin are not likely to have significantly different t0 times for a single reflection. For migrated data, the midpoints of all traces in a bin are by definition identical. But when super-gathers are formed, it is likely that the apex of dipping reflections from a trace at one end of the super-gather may be shifted from the apex in a trace at the other end. To correct for this problem, the solution is to extend the hyperbolic move-out equation to move the apex of the reflection up or down along dip. In order to accomplish this, the equation becomes a function of the vector distance $\mathbf{\Delta m}$ of a trace midpoint from the center of the gather and the apparent vector dip $\mathbf{\sigma}$ as determined from dip estimation on the stack section. In the case of NMO velocities, dip is estimated on an unmigrated stack, and in the case of residual velocities, it is estimated on a migrated stack. With these extensions, the move-out formulae become:
t^2 = (t_0^2 + \mathbf{\sigma\cdot\Delta m}) + \frac{4h^2}{V_{RMS}^2}\end{displaymath} (3)
for NMO velocity analysis and
t^2 = (t_0^2 + \mathbf{\sigma\cdot\Delta m}) + 4h^2r\end{displaymath} (4)
for residual velocity analysis.

In this formulation, the only change to the equation is a shift in the apexes of travel-time hyperbolas. For the case of residual velocity analysis, this is the final form of the equation. For the NMO velocities, it is possible to use knowledge of the dips for an additional correction. The typical goal for pre-migration velocity analysis is to obtain velocities suitable for time migration. These velocities should be the zero-dip velocities obtained after dip move-out (DMO), not dip-dependant NMO velocities. DMO corrects the shape of reflections for all possible dips, but velocities are chosen from semblance panels, picks are typically made for strong individual reflections. Such reflections usually have well-defined dips that are identified by the dip scan. With knowledge of this dip, it is possible to make an explicit dip correction to the velocity term. It is convenient to convert the dip $\mathbf{\sigma}$ to azimuth $\alpha$ and dip angle $\beta$, where azimuth is the map-view angle and $\beta$ is defined as:
\beta = \sin^{-1}\left(\frac{V\cdot\Vert\mathbf{\sigma}\Vert}{2}\right)\end{displaymath} (5)
The dip and azimuth variables extend the move-out formula with the familiar dip correction to velocity:
t^2 = (t_0 + \mathbf{\sigma\cdot\Delta m})^2 + \frac{4\cdot ...
 ...h_y)^2+(\sin\alpha\cdot h_x+\cos\alpha\cdot h_y)^2}{V_{RMS}^2}.\end{displaymath} (6)
For offsets in the direction parallel to the dip, the velocities are decreased while the velocities for ``strike'' offsets remain unchanged. Because the same dip corrections can be achieved by applying DMO to the data prior to velocity analysis, this additional correction is neglected in the data examples, which instead focus on the affects of shifting t0 up or down within super-gathers.