Let be the wavefield at depth step *N* in the
-domain and let be the wavefield extrapolated from
depth step *N* to depth step *N*+1 using reference velocity *V*_{l}.
That is:

(1) |

Extended split-step adds a correction before the interpolation, the so-called ``thin lens term'':

where
Other methods, such as
pseudo-screen and Fourier finite-difference, increase the accuracy of the
result by adding high-order spatial derivatives to the computation of
the *k*_{zl} term Biondi (2004); Huang et al. (1999); Ristow and Ruhl (1994); Xie and Wu (1999). The more
accurate approximation of *k*_{zl} relaxes the need for a large number
of reference velocities such that with fewer reference velocities similar or
even better accuracies can be obtained compared with split-step.

The last step in either of these methods is to Fourier transform the interpolated wavefield to the - space. This wavefield will then be the input to the propagation at the next depth step.

It should be clear that the accuracy of these methods, especially PSPI and extended split-step, is directly related to the accuracy of the wavefield interpolation and the number and choice of the reference velocities.

5/3/2005