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Joint-inversion

Instead of solving the two equations in equation (6) independently, we combine them to form a joint system of equations

$\displaystyle \left [ \begin{array}{cc} {\bf\widetilde L}_{0} & {\bf0 } \\ {\bf...
...array}{cc} {\bf\widetilde d}_{0} \\ {\bf\widetilde d}_{1} \end{array} \right ],$ (11)

for which a solution is obtained by minimizing the objective function

\begin{displaymath}\begin{array}{ccc} S({\bf m_0}, {\bf m_1})= \left \vert\left\...
...end{array} \right ] \right \vert \right \vert ^2.\\ \end{array}\end{displaymath} (12)

Neglecting numerical stability issues, the computational cost of minimizing equations 12 is the same as the cost of minimizing the two objective functions in equation 9. Because several shots are encoded and directly imaged, the computational cost of this approach is considerably reduced relative to non-encoded (or single source) data sets. Equivalent formulations for conventional time-lapse seismic data sets have been shown by previous authors (Ajo-Franklin et al., 2005; Ayeni and Biondi, 2008).


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Next: Regularization and Preconditioning Up: Linear least-squares migration/inversion Previous: Linear least-squares migration/inversion

2009-09-25