WEMVA theory

The wavefield constructed by downward continuation from the surface to depth z, $\mathcal U\left(\omega\right)$ is  

$$\mathcal U= \mathcal De^{ i \sum_z \Phi_z} \;,$$ (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 s_o

$$\begin{eqnarray} \Phi_z &=& {\Phi_z}_o + \left. \frac{d\Phi_z}{ds} \right\vert _{s={s_o}} \Delta s \\ &=& {\Phi_z}_o + \Delta \Phi_z \;.\end{eqnarray}$$ (2) (3)

If we plug equation (2) in equation (1) we can write the following expression for the wavefield $\mathcal U$:

$$\mathcal U= \mathcal U_oe^{i \sum_z \Delta \Phi_z} \;,$$ (4)

where $\mathcal U_o$ corresponds to the background slowness s_o, 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 $\Delta \Phi= \sum_z \Delta \Phi_z$ 

$$\Delta \mathcal U= \mathcal U_o\left[e^{i \Delta \Phi} -1\right]\;.$$ (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):  

$$\Delta \mathcal U= {\bf L}\left(\mathcal U_o\right)\left[\Delta s\right]\;.$$ (8)

 

• Linearization
• Born image perturbation
• Rytov image perturbation
• Differential image perturbation