Adjoint Frechet derivative operator

From the last formula we conclude that the adjoint Frechet derivative operator Fg is given by the formula ... 

In a similar way we can obtain the formula for the adjoint Frechet derivative operator F ... 

In the section of this chapter describing the Kirchhoff inversion method and general non-linear inversion techniques, we have demonstrated that the calculation of the Kirchhoff adjoint operator (15.142) and the adjoint Frechet derivative operator... 

Adjoint Frichet derivative operator and back-propagating elastic field Now we can address the problem of calculating the adjoint Frechet derivative operator. Note that in the RCG algorithm (15.238) expression FJ, (Rl ) denotes the result of an application of the adjoint Frechet derivative operator to the corresponding residual field Rin = Ai,(m ) — di on the n-th iteration. 

Equation (5.7) can be simplified using the properties of the adjoint of the Frechet derivative operator, considered in Appendix D. 

The notion of the adjoint operator makes it possible to move a linear operator of a Frechet derivative, Fm, from the left to the right-hand side of the inner product in equation (5.7) ... 

We can also write the result of applying the adjoint Frechet operator to the residual field as a correlation of the time derivatives of the back-propagated residual field and the corresponding wavefield Pn, t) computed at the n-th iteration ... 

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