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WEMVA theory

The wavefield constructed by downward continuation from the surface to depth z, $\mathcal U\left(\omega\right)$ is  
 \begin{displaymath}
\mathcal U= \mathcal De^{ i \sum_z \Phi_z} \;,\end{displaymath} (1)
where $\mathcal D$ is the data at the surface and $\Phi_z$ is the complex phase shift at one depth level. We can write the phase $\Phi_z$ at every depth level as a Taylor expansion around a reference medium of slowness so
   \begin{eqnarray}
\Phi_z &=& {\Phi_z}_o + \left. \frac{d\Phi_z}{ds} \right\vert _{s={s_o}} \Delta s
\\  &=& {\Phi_z}_o + \Delta \Phi_z \;.\end{eqnarray} (2)
(3)

If we plug equation (2) in equation (1) we can write the following expression for the wavefield $\mathcal U$:
\begin{displaymath}
\mathcal U= \mathcal U_oe^{i \sum_z \Delta \Phi_z} \;,\end{displaymath} (4)
where $\mathcal U_o$ corresponds to the background slowness so, and $\mathcal U$ corresponds to an arbitrary spatially varying slowness $s={s_o}+\Delta s$.

We can define a wavefield perturbation at depth z by the expression
\begin{eqnarray}
\Delta \mathcal U&=& \mathcal U- \mathcal U_o
\\  &=& \mathcal U_o\left[e^{i \sum_z \Delta \Phi_z } -1\right]\end{eqnarray} (5)
(6)
or, if we use the notation $\Delta \Phi= \sum_z \Delta \Phi_z$ 
 \begin{displaymath}
\Delta \mathcal U= \mathcal U_o\left[e^{i \Delta \Phi} -1\right]\;.\end{displaymath} (7)

In general, we can compute a wavefield perturbation $\Delta \mathcal U\left(\omega, z \right)$by applying a non-linear operator ${\bf L}$ which depends on the background wavefield $\mathcal U_o\left(\omega, z \right)$ to a slowness perturbation $\Delta s\left(z \right)$, according to equation (7):  
 \begin{displaymath}
\Delta \mathcal U= {\bf L}\left(\mathcal U_o\right)\left[\Delta s\right]\;.\end{displaymath} (8)