Frechet derivative for the elastic forward modeling operator

2 Frechet derivative for the elastic forward modeling operator Let us assume that the Lame velocities are known and fixed everywhere in space with the exception of some local domain D. We can find the equations for the Frechet derivative by applying the variational operator to both sides of the corresponding Lame equation (15.230) 

SCp (r) and Sc (r) are the square velocity variations, which are unequal to zero only within domain D, and is the corresponding elastic field variation. 

For the sake of simplicity, we assume that there are no density variations in equation (15.239), i.e. Sp = 0. We also take into account that the perturbation of an external source is equal to zero 5/ = 0. 

Applying integral representation (15.233) to the elastic field variation satisfying equation (15.239), we obtain 

Therefore, the Prbchet differential of the forward modeling elastic field operator is given by the following expression  

---