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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*,
*m*_{0} is the arbitrary location of a fixed scatter point,
*h* is the half-offset,
*T*_{0} is the one-way zero-offset traveltime from the normal to the constant-time ellipse
at (*x* = (*m* - *m*_{0}),*y*) to *b*, and *T* is the time required to traverse the path *SPR*.
Note that in this case, both *m*_{0} and *m* are referenced to the same fixed origin
while *x* = *m* - *m*_{0} 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