Perturbation field: Adjoint operator

In the adjoint operation, we begin by upward propagating the perturbation in wavefield at depth z:  

$$\Delta u^{z-1} = {T_0^{z}}' \Delta u^{z} + \Delta \i^{z-1}$$ (8)

where

• T₀^z' is the upward continuation operator at depth z.

We can then obtain the perturbation in slowness from the perturbation in wavefield by applying the adjoint of the scattering operator:  

$$\Delta s^{z} = {u_0^{z}}' {G_0^{z}}' \left[ {T_0^{z}}' \Delta u^{z+1}-\Delta u^{z} \right]$$ (9)

Equations (6) and (7) for the forward operator and equations (8) and (9) for the adjoint operator express the linear relation established between the perturbation in slowness ($\Delta S$) and the perturbation in image ($\Delta R$).