# Dictionary:Double-square-root equation

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The traveltime surface of a point diffractor in offset space where raypaths are straight and the source and receiver are not coincident; see Figure D-20b.

Figure D-20. DMO. (a) Depth section showing the updip movement of the reflecting point for an offset geophone for constant velocity; Δ=(h2/D)cosξsinξ, where ξ is the dip (Levin, 1971)[1]. To avoid reflection point smearing, an offset trace should be gathered with the updip zero-offset trace at a distance G=(–h2/D)sinξ, but such a gather is not hyperbolic; the DMO correction makes this gather hyperbolic. (b) A diffraction in location-offset space, a Cheops pyramid, is not a hyperboloid. (c) Applying NMO changes the Cheops pyramid into a saddle-shaped surface. (d) Applying DMO along with NMO yields data that can be stacked without reflection-point smear. (e) NMO corrects for the time delay on an offset trace assuming horizontality, DMO moves the data to the correct zero-offset 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 (source-geophone distance), it is the surface t(x,y)

${\displaystyle t={\sqrt {\left({\frac {h}{2V}}\right)^{2}+\left({\frac {m+{\frac {y}{2}}}{V}}\right)^{2}}}+{\sqrt {\left({\frac {h}{2V}}\right)^{2}+\left({\frac {m-{\frac {y}{2}}}{V}}\right)^{2}}}}$,

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 single-square-root equation for a zero-offset (CMP) section,

${\displaystyle t={\sqrt {\left({\frac {h}{V}}\right)^{2}+\left({\frac {2m}{V}}\right)^{2}}}}$.

DMO processing transforms a Cheops pyramid so that a cylindrical hyperbola (see Figure D-20d) 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 x-axis, NMO can be applied correctly.

## References

1. Levin, F. K., 1971, Apparent velocity from dipping interface reflections: Geophysics, 36: 510–516.
2. Deregowski, S. M., 1986, What is DMO: First Break, 4, No. 7, 7–24.