Migration imaging in the frequency domain

We have learned in the previous chapters of this book, that an electromagnetic inverse problem is ill-posed. Therefore, direct minimization of the energy flow functional (11.13) could lead to an unstable solution. In order to generate a regularized solution, we have to minimize the parametric functional  

The Newton method is based on the idea that one can find the minimum of the parametric functional in one iteration. We now perturb the iteration step, Act, and find the corresponding variation of the parametric functional (11.37). According to (11.31) and (11.35), it is equal to 

The first variation of the energy flow functional at the point (ct(, + Act) can be evaluated using linearization of the forward modeling operator for the predicted electric field Ey r c b + Act)  

Substituting expression (11.39) back into (11.38), and taking into account that, according to Theorem 81 of Appendix D, the first variation of the parametric functional at the minimum must be equal to zero, we have 

Note that equation (11.42) must hold for any variation SAa. Therefore, from equation (11.42) we find at once the regularized normal equation for the optimum 

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