Frechet derivative operators for wavefield problems

In Chapter 14 we developed expressions for the Frechet differentials of the forward modeling wavefield operators (14.29) and (14.84), which we reproduce here for convenience  

I For conciseness, the sccilar acoustic and vector wavefield equations have been combined into single equations in this summary, with alternative subscripts a and v to distinguish them it is understood, of course, that d , A , and Rv n (but not the adjoint operator F. ), would be printed in boldface in separate vector wavefield equations since they refer to 3-D vector elements in 

In these formulae, = Sj + As is the square slowness model for which we calculate the variation of the forward modeling operator 6s is the corresponding variation of the square slowness which is obviously equal to the variation of the anomalous square slowness, 6s — 6As expressions G (r r o ) and G (r r u ) stand for the Green s function and tensor defined for the given square slowness and the function p(r,u ) and vector u(r,cu) represent the total acoustic and vector wavefields for the given square slowness s.  

Note that in the RCG algorithm (15.171), the expression F v.n Ra,v,n) denotes the result of an application of the adjoint Frechet derivative operator to the corresponding acoustic or vector residual field = - a, (As ) — da on the n-th iteration. 

The expressions for the adjoint Pr6chet derivative operators are given by formulae (15.22) and (15.23). Based on these formulae, we can write 

---