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## First definition: MZO by Fourier transform

Both Hale and Zhang separate the NMO correction from the DMO correction. However for variable velocity media this can not be done, and in order to generalize the NMO DMO operation the migration to zero-offset (MZO) concept has to be used. Migration to zero-offset is the process that transforms a nonzero-offset (constant offset) section into a zero-offset section. In constant velocity media MZO = NMO+DMO.

MZOdip
Figure 1
Geometry for a dipping reflector in a constant velocity medium. The zero-offset ray JR and the nonzero-offset ray bounce at the same reflection point R. The dipping angle is .

Taking into account the geometry in Figure the MZO correction in constant velocity media is
 (1)
where h is the half offset, t0 is twice the zero-offset travel time from the reflection point to the surface, v is the velocity of the medium and th is the constant-offset travel time.

The MZO correction for a given dip is
 (2)
To clarify the sign convention for the spatial coordinate in equation (2), we can notice that the y-axis is oriented to the left, and the angles are positive if the reflector dips toward the right and negative if the reflector dips toward the left. In Figure the angle is negative. This can be also derived from the equation

where the sign of dy0 determines the sign of the angle .

The differentials of the new variables are
 (3)

The 2-D Fourier transform of the zero-offset field is
 (4)
Following Hale's (1984) technique, replace the variables t0 and y0 in equation (4) with their expression in equation (2). Fortunately, it is not necessary to express explicitly th=th(t0,y0) and y=y(t0,y0), though the respective dependencies are assumed.

After replacing the variables y0 and t0 in equation (4), the migration to zero-offset transformation becomes
 (5)
Note that by replacing the quantity

in equation (5), we get the same phase as the one obtained by Hale and Zhang, but the Jacobian differs from Zhang's Jacobian by a factor
 (6)
which corresponds to the Jacobian of the NMO transformation.

Next: Second definition: MZO extracted Up: Introduction Previous: Introduction
Stanford Exploration Project
11/17/1997