Adjoint operator: Modeling

Equation (A-8) enables us to compute the image corresponding to a given velocity field from data measured at the surface of the earth, that is, to migrate the data. The operation adjoint to migration, modeling, can be derived from the same equation using the adjoint state system  

$$\overline \mathcal D= [\mathcal H(\mathcal I- \mathcal T_0)^{-1}]' \mathcal R_0$$ (18)

where

• $\overline \mathcal D$ is the modeled data, computed for a given velocity field.

Therefore, we can obtain the modeled data ($\overline \mathcal D$) from the reflectivity image ($\mathcal R_0$) by writing  

$$\fbox {$ \overline \mathcal D= \left[ (\mathcal I- \mathcal T_0)^{-1} \right]' \mathcal H' \mathcal R_0 $}$$ (19)