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We can define a minimizing problem that aims at finding
by minimizing the following cost function:
| ![\begin{displaymath}
f\left(\delta \textbf{m} \right) =\Vert A \delta \textbf{m}-\delta \textbf{d} \Vert^{2}_{2} .\end{displaymath}](img51.gif) |
(47) |
The Newton iterative algorithms can be used for solving the minimizing problem. The standard Newton iterative algorithm is
| ![\begin{displaymath}
\delta \textbf{m}^{\left( k+1\right) }=\delta \textbf{m}^{\l...
...ight)\right]^{-1} \nabla f\left(\delta \textbf{m}^{(k)}\right).\end{displaymath}](img58.gif) |
(48) |
However, the inverse of the Hessian is difficult to calculate. The Quasi-Newton algorithms are used commonly. The inverse of the Hessian matrix can be calculated with the DFP formula:
| ![\begin{displaymath}
H^{DFP}_{k+1}=H_{k}+\frac{ \textbf{p}^{(k)}\left(\textbf{p}^...
...ht)^{T}}{\left(\textbf{q}^{k} \right)^{T}H_{k}\textbf{q}^{(k)}}\end{displaymath}](img118.gif) |
(49) |
where
, and
.
The Quasi-Newton iterative algorithm is
| ![\begin{displaymath}
\delta \textbf{m}^{\left( k+1\right) }=\delta \textbf{m}^{\l...
...right) } -H_{k+1} \nabla f\left(\delta \textbf{m}^{(k)}\right).\end{displaymath}](img119.gif) |
(50) |
Next: (C) Non-linear waveform inversion
Up: comparison among migration/inversion methods
Previous: (3) Least-squares migration/inversion
Stanford Exploration Project
11/1/2005