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Once we have created an image perturbation , we can invert for the corresponding perturbation in slowness. Mathematically, this amounts to solving an optimization problem like Claerbout (1999)
| ![\begin{eqnarray}
\mathcal L\Delta \bf s&\approx& \Delta{\bf R}\\ \nonumber
\epsilon\mathcal A\Delta \bf s&\approx& 0,\end{eqnarray}](img1.gif) |
(1) |
| |
where
is a data-fitting operator, mainly composed of a scattering and a downward continuation operator, but which also incorporates the background wavefield Biondi and Sava (1999),
is a model-styling operator, either an isotropic Laplacian or an anisotropic steering-filter Clapp and Biondi (1998), which imposes smoothing on the model,
and
are respectively the slowness perturbation (the model) and the image perturbation (the data),
is a scalar parameter controlling the weight of each of the individual goals.
To speed-up the inversion procedure, we can precondition the model in Equation 1 and solve the system
| ![\begin{eqnarray}
\mathcal L\mathcal A^{-1}\Delta{\bf p}&\approx& \Delta{\bf R}\\ \nonumber
\epsilon\Delta{\bf p}&\approx& 0,\end{eqnarray}](img7.gif) |
(2) |
| |
where
is the preconditioned model variable.
Next: Slowness update and the
Up: Theory
Previous: Residual migration
Stanford Exploration Project
4/27/2000