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Optimization of the objective function in Equation (19) requires computation of its gradient with respect to slowness. The objective function J can be rewritten using the inner product as:
 (20)
A perturbation of the function J is related to a perturbation of the wavefield by the relation:
 (21)
If we replace from Equation (11) we obtain:
 (22)
therefore the gradient of the objective function can be written as
 (23)

Following the definition of the operator , we can write
 (24)

Finally, the expression for the gradient of the objective function with respect to slowness becomes
 (25)
which takes special forms depending on our choice of the operators and :

 WEMVA by TIF WEMVA by DSO

The gradient in Equation (25) is computed using the adjoint state method, which can be summarized by the following steps:

1.
Compute by downward continuation the wavefield
 (26)
2.
Compute by upward continuation the adjoint state wavefield
 (27)
i.e. solve the adjoint state system
 (28)
3.