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 kz Claerbout (1985):
The use of (1) for calculating kz 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 kz for both positive and negative , and for all kx values.
The inclusion of the damping factor can solve the ambiguity in the kz selection. This damping is traditionally included as a cutoff frequency; Claerbout (1999a) defines a function that includes this cutoff frequency as:
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 .
Figure 1 Function (2) for an . Claerbout (1999a)
A theoretical redefinition of R=ikz that incorporates causality and viscosity concepts will not only solve for the ambiguity of the kz selection but also preserve the frequency content of our original data.