Theory review

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 $\exp({ik_{z}z})$, where the dispersion relation of the scalar wave equation defines k_z Claerbout (1985):

$$\begin{eqnarray} k_{x}^{2} + k_{z}^{2} \ \ = \ \ \frac{\omega^{2}}{v^{2}}, \nonu... ...\ k_{z} \ \ = \ \ \pm \sqrt{\frac{\omega^{2}}{v^{2}} - k_{x}^{2}}.\end{eqnarray}$$ (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 $k_{x}^{2} \gt \frac{\omega^{2}}{v^{2}}$ (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 $\omega$, 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 $R=ik_{z}(\omega,k_{x})$that includes this cutoff frequency as:

 

$$R \ \ = \ \ ik_{z} \ \ = \ \ \sqrt{({-i\omega+\epsilon})^2 + (vk_x)^2}.$$ (2)

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

 

fradan Figure 1 Function (2) for an $\epsilon = 0.1$. Claerbout (1999a)

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.