INTRODUCTION

Downward continuation and imaging in the isotropic case (Gazdag, 1978) can be written as  

$$M(z,k_x)=\int d\omega e^{i k_z z} P(\omega,k_x),$$ (1)

where  

$$k_z = -{\rm sign} (\omega) \sqrt{{\omega^2 \over v^2}- k_x^2}.$$ (2)

In equation (1), $P(\omega,k_x)$ represents the Fourier transform of the seismic field p(t,x) recorded at the surface, following Claerbout's (1985) sign convention

$$P(\omega,k_x)= \int dt \; e^{i\omega t} \int dx e^{-ik_xx} p(t,x),$$

and v represents half the velocity, as used in the exploding reflectors model.

We can rewrite equation (1) as time migration replacing the depth steps by equivalent time steps $\tau$: 

$$M(\tau,k_x)=\int d\omega e^{i k_{\tau} \tau} P(\omega,k_x).$$ (3)

where we define

$$\left \{ \begin{array} {lcl} \tau & = & {z \over v} \\ \\ k_... ...\rm sign}(\omega)\sqrt{\omega^2- v^2 k_x^2}.\end{array}\right .$$

For a constant velocity medium, Stolt (1978) transforms the integral in $\omega$ using a Fourier transform, which can be computed rapidly via a Fast Fourier Transform (FFT) algorithm  

$$M(\tau,k_x)=\int dk_{\tau} e^{i k_{\tau} \tau} J(k_{\tau},k_x) P(\omega(k_{\tau},k_x),k_x),$$ (4)

where $J(k_{\tau},k_x)$ represents the Jacobian of the transformation from $\omega$ to $k_{\tau}$

$$J(k_{\tau},k_x)={{d\omega} \over {dk_{\tau}}} = {{k_{\tau}} \over {\sqrt{k_{\tau}^2+v^2k_x^2}}}$$

and $P(\omega(k_{\tau},k_x),k_x)$ represents the initial data as function of the new variable $k_{\tau}$.