Adjoint operator: Back-projection

Equation (B-12) enables us to compute the perturbation in image corresponding to a given perturbation in the velocity field, that is, to migrate the scattered field. The operation adjoint to migration, back-projection, can be derived from the same equation using the adjoint state system:  

$$\overline {\Delta \mathcal \S} = \left[\mathcal H(\mathcal I... ...T_0\mathcal G_0\widehat{\mathcal U_0}\right]' \Delta \mathcal R$$ (32)

where

• $\overline {\Delta \mathcal \S}$ is the back-projected perturbation in the velocity field obtained from the perturbation in the image.

Therefore, the back-projected perturbation in velocity ($\overline {\Delta \mathcal \S}$) is derived from the perturbation in the reflectivity image ($\Delta \mathcal R$) using the following expression:  

$$\fbox {$ \overline {\Delta \mathcal \S} = \widehat{\mathcal... ...-\mathcal T_0)^{-1} \right]' \mathcal H' \Delta \mathcal R $}$$ (33)

Equations (B-12) and (B-14) comprise a pair of adjoint operators that relate the perturbation in velocity to the perturbation in reflectivity image. We can use these two operators to invert for velocity from measurable perturbations in the image.