Migration imaging in the time domain

We develop now a migration imaging method based on an approximate regularized solution of the time domain inverse problem by introducing the parametric functional 

Following the basic principles of the Newton method, we try to find the minimum of the parametric functional in one iteration. To do so, we perturb the iteration step, Act, and find the corresponding variation of the parametric functional (11.89). According to (11.83) and (11.87), it is equal to 

As in the frequency domain case, considered above, we use linearization of the forward modeling operator for the predicted electric field Ep ctb + Act)  

Taking into account expansion (11.91), we can evaluate the first variation of the energy flow functional at the point (at -f Aa) as 

Observing that according to Theorem 81 of Appendix D, the first variation of the parametric functional at the minimum must be equal to zero for any -variation 6Aa, we find at once the regularized normal equation for the optimum step, Act  

---