Asymptotic pseudounitary stacking operators

2. Zero-offset migration

In the case of zero-offset migration, Gritsenko's formula simplifies the true-amplitude migration weighting function (46) to the form

$$\widehat{w}_{ZO}(y;z,x) = {{2^m}\over{\left(2 \pi\right)^{m/2}}} {{\cos{\alpha(y)}} \over {v(y)}}\;.$$ (49)

In a constant-velocity medium, one can accomplish the true-amplitude zero-offset migration by premultiplying the recorded zero-offset seismic section by the factor $\left(v \over 2 \right)^{m-1} \left(t \over 2\right)^{m/2}$ [which corresponds at the stationary point to the geometric spreading $R(x,y)$ ] and downward continuation according to formula (40) with the effective velocity $v/2$ (Hubral et al., 1991; Goldin, 1987). This conclusion is in agreement with the analogous result of Born inversion (Bleistein et al., 1985), though derived from a different viewpoint.

In the zero-offset case, the pseudo-unitary forward operator reduces to downward pseudo-unitary continuation with a velocity of $v/2$ .