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Sava and Fomel (2003) define an image space transformation from subsurface offset to reflection and azimuth angle as:
| ![\begin{eqnarray}
{\bf m}({\bf x},{\Theta}) &=& {\bf T'} (\Theta,{\bf h}){\bf m}({\bf x},{\bf h}),
\end{eqnarray}](img37.gif) |
(12) |
where
are the reflection and the azimuth angles, and
is the adjoint of the angle-to-offset transformation operator (slant stack).
Substituting the prestack migration image (subsurface offset domain) in equation 7 into equation 12 we obtain the expression for the prestack migration image in the angle-domain that follows:
| ![\begin{eqnarray}
{\bf m}({\bf x},{\Theta})&=& {\bf T'} (\Theta,{\bf h}) {\bf L}'...
...{'}\sum_{{\bf x}}{\bf d}({\bf x}_s,{\bf x}_r;\omega). \nonumber
\end{eqnarray}](img39.gif) |
(13) |
| |
The synthetic data can be modeled (as the adjoint of equation 14) by the chain of linear operator
and the angle-to-offset transformation operator acting on the model space,
| ![\begin{eqnarray}
{\bf d}({\bf x}_s,{\bf x}_r;\omega)&=&{\bf L}{\bf T} (\Theta,{\...
...m_{\omega}
{\bf T} (\Theta,{\bf h}){\bf m}({\bf x},{\Theta}),
\end{eqnarray}](img40.gif) |
|
| (14) |
The quadratic cost function is
| ![\begin{eqnarray}
S({\bf m}) &=& \frac{1}{2} \sum_{\omega}\sum_{{\bf x}_s}\sum_{{...
...\left[ {\bf d}({\bf x}_s,{\bf x}_r;\omega)-{\bf d}_{obs} \right],
\end{eqnarray}](img41.gif) |
|
| (15) |
while its first derivative with respect to the model parameters
is
| ![\begin{eqnarray}
\frac{\partial{S({\bf m})}}{\partial{{\bf m}({\bf x},\Theta)}}=...
...) {\bf G}({\bf x+h},{\bf x}_s;\omega){\bf T} (\Theta,{\bf h})
\}
\end{eqnarray}](img43.gif) |
|
| (16) |
and its second derivative with respect to the model parameters
and
is the angle-domain Hessian
| ![\begin{eqnarray}
{\bf H}({\bf x,\Theta};{\bf x',\Theta'})&=&
\frac{\partial^2{S...
...bf h}){\bf H}({\bf x,h};{\bf x',h'}) {\bf T}(\Theta',{\bf h'}).
\end{eqnarray}](img45.gif) |
|
| |
| |
| |
| |
| (17) |
Next: Explicit vs. implicit Hessian
Up: Expanding Hessian dimensionality
Previous: Subsurface-offset Hessian
Stanford Exploration Project
10/31/2005