We can write equations (2) and (1) for a given reference velocity () as
If we express the frequency from equation (3) as
and introduce it in equation (1), we obtain the common-azimuth residual migration equations
Note that for 2-D prestack data, equations (5) reduce to the 2-D prestack form Sava (1999):
which, furthermore, reduces to the well-known Stolt 2-D post-stack residual migration equation Stolt (1996) :
As for the 2-D prestack data, the common-azimuth residual migration is velocity independent; that is, we need not make any assumption about the actual values of the velocities for the reference and improved migration, but only about their ratio. In this way, we can take an image and residually migrate it without knowing what velocity model has been used to image it in the first place.