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# The Velocity Independent DMO Equation

In the early 1980's G. H. F. Gardner and colleagues at the University of Houston showed that it was possible to DMO-correct recorded data in a velocity-independent manner. The fundamental basis for Gardner's method for velocity-independent DMO is the recognition that the double-square-root equation can be converted into a single-square-root equation by a radial-plane-diffraction process. After application of DMO, a second diffraction process, which takes place on constant time-slices completes the prestack migration step.

Consider a constant velocity medium with velocity v. With reference to Figure , the double-square root (DSR) equation is:
 (1)
Here m is the midpoint between source S and receiver R, m0 is the arbitrary location of a fixed scatter point, h is the half-offset, T0 is the one-way zero-offset traveltime from the normal to the constant-time ellipse at (x = (m - m0),y) to b, and T is the time required to traverse the path SPR. Note that in this case, both m0 and m are referenced to the same fixed origin while x = m - m0 is relative to m.

Fig1
Figure 1
Depth section showing constant-time ellipse for the DSR equation.

The ellipse in Figure is the locus of all points (x,y) for which T is constant. It will be convenient to write (1) in the form:

 (2)

Following Forel and Gardner (1988), let
 (3)
and
 (4)
Then
 (5)

 (6)
and the distance from b to P satisfies
 (7)
Since
 (8)
one has
 (9)
This is the basic result from which DMO will be derived.

Next: Velocity-Independent DMO Up: Bednar: EOM vs. DMO-PSI Previous: Introduction
Stanford Exploration Project
10/10/1997