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Wavefield scattering

In migration by downward continuation, the wavefield at depth $z+\Delta z$ is obtained by phase-shift from the wavefield at depth z:  
\mathcal W\left({z+\Delta z} \right)= \mathcal W\left(z \right)e^{-i \k_z\Delta z}.\end{displaymath} (1)
Using a Taylor series expansion, the depth wavenumber ($\k_z$) depends linearly on its value in the reference medium (${\k_z}_o$) and the laterally varying slowness $s\left(x,y,z \right)$in the depth interval under consideration
\k_z\approx {\k_z}_o+ \left. \frac{d \k_z}
 {d s} \right\vert _{s=s_o}\left(s- s_o\right).\end{displaymath} (2)
so represents the constant slowness associated with the depth slab between the two depth intervals. $\left. \frac{d \k_z}
 {d s} \right\vert _{s=s_o}$ represents the derivative of the depth wavenumber with respect to the reference slowness and can be implemented in many different ways, e.g by the Fourier-domain method of Huang et al. (1999).

The wavefield downward continued through the background slowness $s_b\left(x,y,z \right)$ is
\mathcal W_b \left({z+\Delta z} \right)=\mathcal W\left(z \righ...
 {d s} \right\vert _{s=s_o}\left(s_b- s_o\right)\right]\Delta z},\end{eqnarray} (3)
and the full wavefield $\mathcal W\left({z+\Delta z} \right)$ depends on the background wavefield $\mathcal W_b \left({z+\Delta z} \right)$ by
\mathcal W\left({z+\Delta z} \right)= \mathcal W_b \left({z+...
 ...ft. \frac{d \k_z}
 {d s} \right\vert _{s=s_o}\Delta s\Delta z},\end{displaymath} (4)
where $\Delta s$ represents the difference between the true and background slownesses $\Delta s= s- s_b$.

Defining the wavefield perturbation $\Delta \mathcal W\left({z+\Delta z} \right)$ as the difference between the wavefield propagated through the medium with correct velocity, $\mathcal W\left({z+\Delta z} \right)$, and the wavefield propagated through the background medium, $\mathcal W_b \left({z+\Delta z} \right)$, we can write
\Delta \mathcal W\left({z+\Delta z} \right)&=& \mathcal W\left(...
 ...c{d \k_z}
 {d s} \right\vert _{s=s_o}\Delta s\Delta z} -1 \right].\end{eqnarray} (5)
Equation (5) represents the foundation of the wave-equation migration velocity analysis method. One major problem with Equation (5) is that the wavefield $\Delta \mathcal W$ and slowness perturbations $\Delta s$ are not linearly related. For inversion purposes, we need to find a linearization of this equation around the reference slowness, so. Biondi and Sava (1999) linearize Equation (5) using the Born approximation ($e^{i \phi} \approx 1 + i \phi$). With this choice, the WEMVA Equation (5) becomes  
\Delta \mathcal W\left({z+\Delta z} \right)= \mathcal W_b \l...
 ...rac{d \k_z}
 {d s} \right\vert _{s=s_o}\Delta s\Delta z\right].\end{displaymath} (7)
The wavefield perturbation $\Delta \mathcal W$ in Equation (7) turns into an image perturbation $\Delta \mathcal R$ after we apply an imaging condition, e.g. summation over frequencies. The WEMVA objective function is
\min_{\Delta s} \Vert \Delta \mathcal R- {\bf L} \Delta s\Vert\end{displaymath} (8)
where $\bf{L}$ is a linear operator defined recursively from Equation (7) at every depth level and frequency. We estimate the slowness update by minimizing this objective function through an iterative conjugate gradient optimization technique.

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Next: Differential image perturbation Up: Sava and Biondi: WEMVA Previous: Introduction
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