The above-described corrections consist of two
parts: one that is applied at the *z*=0 boundary, and one that is
applied in the propagation operator. The boundary condition correction
depends only on the velocity at the surface, while the propagation
correction takes the entire interval velocity model into account and
is directly proportional to its vertical gradient, *v*_{z}, vanishing
where this becomes zero. As a result, the effect of the boundary
condition correction can be tested in isolation, using constant velocity
models. Figure 1 in Vlad et al. (2003)
shows that the Zhang boundary condition correction improves the phase
and brings the
amplitudes very close to the ones computed analytically, especially in
the case of the mixed-domain
implementation. Valenciano et al. (2004) show the
effect of applying it to propagation through the Marmousi model. They also present a more
intuitive interpretation of the boundary condition correction, and an
alternative method for computing it.

Vlad et al. (2003) show in their Figure 2 that the
propagation operator correction is equivalent to the WKBJ one
in a *v*(*z*) medium. They also apply it to the *v*(*x*,*z*) model presented in
Figure 3 of their paper. However, the results (their Figure 4) did not show a
visible effect of the correction. In the first section that follows,
we explain why that was the case for that very particular
velocity model, and we quantitatively show that the propagation
operator correction does have visible effects. In the last section of
this paper, we discuss the application of the Zhang corrections to the
linearized downward continuation operator used in wave equation
migration velocity analysis.

5/23/2004