The traveltime surface of a point diffractor in offset space where raypaths are straight and the source and receiver are not coincident; see Figure D20b.
Figure
D20.
DMO. (a) Depth section showing the updip movement of the reflecting point for an offset geophone for constant velocity; Δ=(
h^{2}/
D)cosξsinξ, where ξ is the dip (Levin, 1971)
^{[1]}. To avoid reflection point smearing, an offset trace should be gathered with the updip zerooffset trace at a distance G=(–
h^{2}/
D)sinξ, but such a gather is not hyperbolic; the DMO correction makes this gather hyperbolic.
(b) A diffraction in locationoffset space, a
Cheops pyramid, is not a hyperboloid.
(c) Applying NMO changes the Cheops pyramid into a saddleshaped surface.
(d) Applying DMO along with NMO yields data that can be stacked without reflectionpoint smear.
(e) NMO corrects for the time delay on an offset trace assuming horizontality, DMO moves the data to the correct zerooffset trace for a dipping reflection, and migration further moves it to its subsurface location. (After Deregowski, 1986)
^{[2]}
If x=midpoint location and y=offset (sourcegeophone distance), it is the surface t(x,y)
$t={\sqrt {\left({\frac {h}{2V}}\right)^{2}+\left({\frac {m+{\frac {y}{2}}}{V}}\right)^{2}}}+{\sqrt {\left({\frac {h}{2V}}\right)^{2}+\left({\frac {m{\frac {y}{2}}}{V}}\right)^{2}}}$,
where m=inline distance from diffracting point to the midpoint and h its depth. The surface is called a Cheops pyramid (ke’ ops). This equation contrasts with the hyperbolic singlesquareroot equation for a zerooffset (CMP) section,
$t={\sqrt {\left({\frac {h}{V}}\right)^{2}+\left({\frac {2m}{V}}\right)^{2}}}$.
DMO processing transforms a Cheops pyramid so that a cylindrical hyperbola (see Figure D20d) is obtained after normal moveout correction with the correct velocity. After a transformation y=Ut that constitutes slicing the pyramid by radial planes containing the xaxis, NMO can be applied correctly.
References
 ↑ Levin, F. K., 1971, Apparent velocity from dipping interface reflections: Geophysics, 36: 510–516.
 ↑ Deregowski, S. M., 1986, What is DMO: First Break, 4, No. 7, 7–24.
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