Image perturbation by WEMVA

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

$$\mathcal W_{z+\Delta z}= \mathcal W_{z} e^{-i \k_z\Delta z},$$ (1)

where the depth wavenumber $\k_z$ depends linearly through a Taylor series expansion on its value in the reference medium (${\k_z}^o$) and the slowness in the depth interval from z to $z+\Delta z$, $s_o\left(z \right)$ and $s\left(x,y,z \right)$:

$$\k_z\approx {\k_z}^o+ \left. \frac{d \k_z} {d s} \right\vert _{s=s_o}\Delta s,$$ (2)

where, by definition, $\Delta s=s-s_o$.

If we denote the wavefield downward continued through the reference velocity as $\mathcal W_{z+\Delta z}^o$

$$\mathcal W_{z+\Delta z}^o=\mathcal W_{z} e^{-i {\k_z}^o\Delta z},$$ (3)

we obtain

$$\mathcal W_{z+\Delta z}= \mathcal W_{z+\Delta z}^oe^{-i \left. \frac{d \k_z} {d s} \right\vert _{s=s_o}\Delta s\Delta z}.$$ (4)

The Born approximation linearizes the phase-shift exponential $\left(e^x\approx 1+x \right)$, such that we can write

$$\mathcal W_{z+\Delta z}\approx \mathcal W_{z+\Delta z}^o-i \... ...eft. \frac{d \k_z} {d s} \right\vert _{s=s_o}\Delta s\Delta z.$$ (5)

Therefore, at any particular depth level, the wavefield perturbation $\Delta \mathcal W$ is

$$\Delta \mathcal W\approx -i\Delta z\mathcal W\left. \frac{d \k_z} {d s} \right\vert _{s=s_o}\Delta s,$$ (6)

which we can also write as  

$$\fbox {$ \displaystyle \Delta \mathcal W\approx \frac{d \ma... ...k_z}\left. \frac{d \k_z} {d s} \right\vert _{s=s_o}\Delta s.$}$$ (7)

The image perturbation is simply obtained from the wavefield perturbation by summation over frequencies:

$$\Delta \mathcal R= \sum_\omega \Delta \mathcal W.$$ (8)