Adjoint Frechet derivative operator and back-propagating elastic field

3 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. 

The explicit expression for the adjoint Frechet derivative operator F can be found according to the definition  

Using the definitions (15.236) and (15.237) of inner products and the expression (15.241) for the Frechet derivative operator, we can rewrite formula (15.244) 

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

The surface integral term in the last formula can be treated as the c omplcx conjugate elastic field, (r,a ), 

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