next up previous print clean
Next: Differential image perturbation Up: Sava and Biondi: WEMVA Previous: Introduction

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:  
 \begin{displaymath}
\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
\begin{displaymath}
\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
\begin{eqnarray}
\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
\begin{displaymath}
\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
   \begin{eqnarray}
\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)
(6)
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  
 \begin{displaymath}
\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
\begin{displaymath}
\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.


next up previous print clean
Next: Differential image perturbation Up: Sava and Biondi: WEMVA Previous: Introduction
Stanford Exploration Project
7/8/2003