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Next: (C) Non-linear waveform inversion Up: comparison among migration/inversion methods Previous: (3) Least-squares migration/inversion

(B) Iterative linearized migration/inversion

We can define a minimizing problem that aims at finding $\delta m^{*}$ 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} (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} (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} (49)
where $\textbf{p}^{(k)}=\delta \textbf{m}^{k}-\delta \textbf{m}^{(k+1)}$, and $\textbf{q}^{(k)}=\nabla f\left(\delta \textbf{m}^{(k+1)}\right) - \nabla f\left(\delta \textbf{m}^{(k)}\right) $.

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} (50)


next up previous print clean
Next: (C) Non-linear waveform inversion Up: comparison among migration/inversion methods Previous: (3) Least-squares migration/inversion
Stanford Exploration Project
11/1/2005