Downward-continuation migration

In migration by downward-continuation, the wavefield at depth $z+\Delta z$, $\mathcal W\left({z+\Delta z} \right)$, is obtained by phase-shift from the wavefield at depth z, $\mathcal W\left(z \right)$. 

$$\mathcal W\left({z+\Delta z} \right)= \mathcal W\left(z \right)e^{-i \k_z\Delta z}.$$ (1)

This equation corresponds to the analytical solution of the ordinary differential equation  

$$\mathcal W'(z) = -i\k_z\mathcal W(z),$$ (2)

where the ' sign represents a derivative with respect to the depth z.

We can consider that the depth wavenumber ($\k_z$) depends linearly, through a Taylor series expansion, on its value in the reference medium (${\k_z}_r$) and the laterally varying slowness in the depth interval from z to $z+\Delta z$, $s\left(x,y,z \right)$

$$\k_z\approx {\k_z}_r+ \left. \frac{d \k_z} {d s} \right\vert _{s=s_r}\left(s- s_r\right),$$ (3)

where s_r represents the constant slowness associated with the depth slab between the two depth intervals, and $\left. \frac{d \k_z} {d s} \right\vert _{s=s_r}$ represents the derivative of the depth wavenumber with respect to the reference slowness and which can be implemented in many different ways Sava (2000). The wavefield downward-continued through the background slowness $s_b\left(x,y,z \right)$ can, therefore, be written as

$$\mathcal W_b \left({z+\Delta z} \right)=\mathcal W\left(z \r... ...d s} \right\vert _{s=s_r}\left(s_b- s_r\right)\right]\Delta z},$$ (4)

from which we obtain that the full wavefield $\mathcal W\left({z+\Delta z} \right)$ depends on the background wavefield $\mathcal W_b \left({z+\Delta z} \right)$ through the relation

$$\mathcal W\left({z+\Delta z} \right)= \mathcal W_b \left({z+... ...ft. \frac{d \k_z} {d s} \right\vert _{s=s_r}\Delta s\Delta z},$$ (5)

where $\Delta s$ represents the difference between the true and background slownesses $\Delta s= s- s_b$.