(B) Iterative linearized migration/inversion

We can define a minimizing problem that aims at finding $\delta m^{*}$ by minimizing the following cost function:  

$$f\left(\delta \textbf{m} \right) =\Vert A \delta \textbf{m}-\delta \textbf{d} \Vert^{2}_{2} .$$ (47)

The Newton iterative algorithms can be used for solving the minimizing problem. The standard Newton iterative algorithm is

$$\delta \textbf{m}^{\left( k+1\right) }=\delta \textbf{m}^{\l... ...ight)\right]^{-1} \nabla f\left(\delta \textbf{m}^{(k)}\right).$$ (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:  

$$H^{DFP}_{k+1}=H_{k}+\frac{ \textbf{p}^{(k)}\left(\textbf{p}^... ...ht)^{T}}{\left(\textbf{q}^{k} \right)^{T}H_{k}\textbf{q}^{(k)}}$$ (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  

$$\delta \textbf{m}^{\left( k+1\right) }=\delta \textbf{m}^{\l... ...right) } -H_{k+1} \nabla f\left(\delta \textbf{m}^{(k)}\right).$$ (50)