Migration is the process of downward continuation of the
wavefield plus the application of the imaging condition (*t*=0). The downward
continuation process can be achieved in the Fourier domain
Gazdag (1978); Stolt (1978). The phase-shift method extrapolates
the wavefield downwards with , where
the *dispersion relation of the
scalar wave equation* defines *k*_{z} Claerbout (1985):

(1) |

The use of (1) for calculating *k*_{z} obscures two
aspects of it: first, which of the two square roots is
intended, and second, what happens when
(evanescent waves).
Claerbout (1999a) emphasizes the fact that for coding and
theoretical work it is necessary to define *k*_{z} for both
positive and negative , and for all *k*_{x} values.

The inclusion of the damping factor can solve the ambiguity
in the *k*_{z} selection. This damping is traditionally included
as a cutoff frequency; Claerbout (1999a) defines a function that includes this cutoff frequency as:

(2) |

This function (2) has a positive real part (Figure 1), which implies that we can extrapolate waves safely with . However, since the damping factor has been included as a cutoff frequency we can lose frequency information with an inappropriate choice of .

fradan
Function (2) for an
. Claerbout (1999a)
Figure 1 |

A theoretical redefinition of *R*=*ik*_{z} that incorporates causality
and viscosity concepts will not only solve for the ambiguity of the
*k*_{z} selection but also preserve the frequency content of
our original data.

4/29/2001