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Shot-record migration and velocity analysis

Wavefield reconstruction for multi-offset migration based on the one-way wave-equation under the shot-record framework is performed by separate recursive extrapolation of the source and receiver wavefields, ${{u}_s}$ and ${{u}_r}$. The wavefield extrapolation progresses forward in time (causal) for the source wavefield and backward in time (anti-causal) for the receiver wavefield:

$\displaystyle {{u}_s}_{z+\Delta z} \left ({\bf m}\right)$ $\textstyle =$ $\displaystyle e^{+ i {k_z}\Delta z}{{u}_s}_z \left ({\bf m}\right)$ (25)
$\displaystyle {{u}_r}_{z+\Delta z} \left ({\bf m}\right)$ $\textstyle =$ $\displaystyle e^{- i {k_z}\Delta z}{{u}_r}_z \left ({\bf m}\right)$ (26)

In equations 25-26, ${{u}_s}_z \left ({\bf m}\right)$ and ${{u}_r}_z \left ({\bf m}\right)$ represent the source and receiver acoustic wavefield for a given frequency $\omega $ at all positions in space ${\bf m}$ at depth $z$, and ${{u}_s}_{z+\Delta z} \left ({\bf m}\right)$ and ${{u}_r}_{z+\Delta z} \left ({\bf m}\right)$ represent the same wavefields extrapolated to depth $z+\Delta z$. The phase shift operation uses the depth wavenumber ${k_z}$ which is defined by the single square-root (SSR) equation
\begin{displaymath}
{k_z}= \sqrt{ \left [{ {\omega s} \left ({\bf m}\right)} \right]^2 - \left\vert {{{\bf k}_{\bf m}}} \right\vert^2}
\end{displaymath} (27)

The image is obtained from the extrapolated wavefields by selection of the zero cross-correlation lags in space of time between the source and receiver wavefields, an operation which is usually implemented as summation over frequencies:
\begin{displaymath}
{r}_z \left ({\bf m}\right)= \sum_\omega \overline{{{u}_s}_z...
..., \omega \right)} {{u}_r}_z \left ({\bf m}, \omega \right)\;.
\end{displaymath} (28)

An alternative imaging condition (Sava and Fomel, 2006) preserves the space and time cross-correlation lags in the image.

Linearizing the depth wavenumber given by the equation 27 relative to the background slowness $s_0 \left ({\bf m}\right)$ similarly to the case case of zero-offset migration, we can reconstruct the acoustic wavefields in the background model using a phase-shift operation

$\displaystyle {{u}_s}_{z+\Delta z} \left ({\bf m}\right)$ $\textstyle =$ $\displaystyle e^{+ i {k_z}_0 \Delta z}{{u}_s}_z \left ({\bf m}\right)\;,$ (29)
$\displaystyle {{u}_r}_{z+\Delta z} \left ({\bf m}\right)$ $\textstyle =$ $\displaystyle e^{- i {k_z}_0 \Delta z}{{u}_r}_z \left ({\bf m}\right)\;,$ (30)

which define the causal $ \mathcal{E}^{+}_{SRM}\left [{s_0}_z \left ({\bf m}\right),{u}_z \left ({\bf m}\right) \right]$ and the anti-causal $ \mathcal{E}^{-}_{SRM}\left [{s_0}_z \left ({\bf m}\right),{u}_z \left ({\bf m}\right) \right]$ wavefield extrapolation operators for shot-record migration constructed using the background slowness $s_0 \left ({\bf m}\right)$ and producing the wavefields ${{u}_s}_{z+\Delta z} \left ({\bf m}\right)$ and ${{u}_r}_{z+\Delta z} \left ({\bf m}\right)$ at depth $z+\Delta z$ from the wavefields ${{u}_s}_z \left ({\bf m}\right)$ and ${{u}_r}_z \left ({\bf m}\right)$ at depth $z$, respectively. A typical implementation of shot-record wave-equation migration follows the algorithm:


\begin{singlespace}
\hrule\vspace{0.1in}
{\sc shot-record migration algorithm}
\...
...\> \textcolor{black} {$\}$}
\end{tabbing}\hrule\vspace{0.1in}
\end{singlespace}
This algorithm is similar to the one used for zero-offset or survey sinking migration, except that the source and receiver wavefields are reconstructed separately using wavefield extrapolation. Unlike the zero-offset extrapolation operator, the shot-record extrapolation operator uses the background slowness $s_0$ since the operation involves sinking of the source and receiver wavefields from the surface toward the image positions. Wavefield extrapolation is usually implemented in a mixed domain (space-wavenumber), as briefly summarized in Appendix A.

Similarly to the derivation of the wavefield perturbation in the zero-offset migration case, we can write the linearized wavefield perturbation for shot-record migration as

$\displaystyle {\Delta {{u}_s}} \left ({\bf m}\right)$ $\textstyle \approx$ $\displaystyle +i \left. \frac{d {{k_z}}} {d s} \right\vert _{s_0} \Delta z\; {{u}_s} \left ({\bf m}\right)\Delta s \left ({\bf m}\right)$  
  $\textstyle \approx$ $\displaystyle +i\Delta z \frac{\omega {{u}_s} \left ({\bf m}\right)\Delta s \le...
...}_{\bf m}}} \right\vert}{ {\omega s} _0 \left ({\bf m}\right)} \right]^2} } \;,$ (31)

and
$\displaystyle {\Delta {{u}_r}} \left ({\bf m}\right)$ $\textstyle \approx$ $\displaystyle -i \left. \frac{d {{k_z}}} {d s} \right\vert _{s_0} \Delta z\; {{u}_r} \left ({\bf m}\right)\Delta s \left ({\bf m}\right)$  
  $\textstyle \approx$ $\displaystyle -i\Delta z \frac{\omega {{u}_r} \left ({\bf m}\right)\Delta s \le...
...}_{\bf m}}} \right\vert}{ {\omega s} _0 \left ({\bf m}\right)} \right]^2} } \;.$ (32)

Equations 31-32 define the forward scattering operators $ \mathcal{F}^{\pm}_{SRM}\left [{u} \left ({\bf m}\right),s_0 \left ({\bf m}\right),\Delta s \left ({\bf m}\right) \right]$ producing the scattered wavefields ${\Delta {u}} \left ({\bf m}\right)$ from the slowness perturbation $\Delta s \left ({\bf m}\right)$, based on the background slowness $s_0 \left ({\bf m}\right)$ and background wavefield ${u} \left ({\bf m}\right)$. In this case, the symbol ${u}$ stands for either ${{u}_s}$ or ${{u}_r}$, given the appropriate choice of sign in the forward scattering operator. The image perturbation at depth $z$ is obtained from the source and receiver scattered wavefields using the relation
\begin{displaymath}
\Delta {r} \left ({\bf m}\right)= \sum_\omega \left (\overli...
...ega \right)} {{u}_r} \left ({\bf m}, \omega \right)\right)\;,
\end{displaymath} (33)

which corresponds to the frequency-domain zero-lag cross-correlation of the source and receiver wavefields required by the imaging condition.

Given an image perturbation $\Delta {r}$, we can construct the scattered source and receiver wavefields by the adjoint of the imaging condition

$\displaystyle {\Delta {{u}_s}} \left ({\bf m}\right)= {{u}_r} \left ({\bf m}\right)\overline{\Delta {r} \left ({\bf m}\right)}\;,$     (34)
$\displaystyle {\Delta {{u}_r}} \left ({\bf m}\right)= {{u}_s} \left ({\bf m}\right)\Delta {r} \left ({\bf m}\right)\;,$     (35)

for every frequency $\omega $. Then, the slowness perturbations due to the source and receiver wavefields at depth $z$ under the influence of the background source and receiver wavefields at the same depth $z$ can be written as
$\displaystyle \Delta s_s \left ({\bf m}\right)$ $\textstyle \approx$ $\displaystyle -i \left. \frac{d {{k_z}}} {d s} \right\vert _{s_0} \Delta z\; \overline{{{u}_s} \left ({\bf m}\right)} {\Delta {{u}_s}} \left ({\bf m}\right)$  
  $\textstyle \approx$ $\displaystyle -i\Delta z \frac{\omega \overline{{{u}_s} \left ({\bf m}\right)} ...
...}_{\bf m}}} \right\vert}{ {\omega s} _0 \left ({\bf m}\right)} \right]^2} } \;,$ (36)

and
$\displaystyle \Delta s_r \left ({\bf m}\right)$ $\textstyle \approx$ $\displaystyle -i \left. \frac{d {{k_z}}} {d s} \right\vert _{s_0} \Delta z\; \overline{{{u}_r} \left ({\bf m}\right)} {\Delta {{u}_r}} \left ({\bf m}\right)$  
  $\textstyle \approx$ $\displaystyle -i\Delta z \frac{\omega \overline{{{u}_r} \left ({\bf m}\right)} ...
...}_{\bf m}}} \right\vert}{ {\omega s} _0 \left ({\bf m}\right)} \right]^2} } \;.$ (37)

Equations 36-37 define the adjoint scattering operators $ \mathcal{A}^{\pm}_{SRM}\left [{u} \left ({\bf m}\right),s_0 \left ({\bf m}\right),{\Delta {u}} \left ({\bf m}\right) \right]$, producing the slowness perturbation $\Delta s \left ({\bf m}\right)$ from the scattered wavefield ${\Delta {u}} \left ({\bf m}\right)$, based on the background slowness $s_0 \left ({\bf m}\right)$ and background wavefield ${u} \left ({\bf m}\right)$. In this case, ${u}$ stands for either ${{u}_s}$ or ${{u}_r}$, given the appropriate choice of sign in the adjoint scattering operator. A typical implementation of shot-record forward and adjoint scattering follows the algorithms:


\begin{singlespace}
\hrule\vspace{0.1in}
{\sc shot-record forward scattering alg...
...\> \textcolor{black} {$\}$}
\end{tabbing}\hrule\vspace{0.1in}
\end{singlespace}



\begin{singlespace}
\hrule\vspace{0.1in}
{\sc shot-record adjoint scattering alg...
...\> \textcolor{black} {$\}$}
\end{tabbing}\hrule\vspace{0.1in}
\end{singlespace}
These algorithms are similar to the one used for zero-offset or survey sinking migration, except that the source and receiver wavefields are reconstructed separately using wavefield extrapolation. Unlike the zero-offset scattering operators, the shot-record scattering operators use the background slowness $s_0$ since the operation involves sinking of the source and receiver wavefields from the surface toward the image positions. Both forward and adjoint scattering algorithms require the background wavefields, ${{u}_s} \left ({\bf m}\right)$ and ${{u}_r} \left ({\bf m}\right)$, to be precomputed at all depth levels. Scattering and wavefield extrapolation are implemented in the mixed space-wavenumber domain, as briefly explained in Appendix A.


next up previous [pdf]

Next: Summary of operators Up: Wave-equation migration and velocity Previous: Survey-sinking migration and velocity

2013-08-29