Back projection

Converting the $\gamma$ value into the $\bf \Delta t$ term needed for tomography is troublesome. Etgen (1990) showed that residual migration R_mig is the vector sum of residual normal moveout R_nmo, residual dip moveout R_dmo and residual zero offset migration R_zoff components. What we want to back project in image domain tomography is the R_nmo and R_dmo components. We can use simple trigonometry to convert a $\gamma$ term to an approximate R_nmo term through

$$\Delta t ({\bf x},\alpha)=( \gamma -1)* \left(\frac{1}{\cos(\alpha)}- 1\right),$$ (9)

where $t({\bf x}, \alpha)$ is the traveltime to the surface of ray pair starting from $\bf x$ at the opening angle $\alpha$.This approximation is not accurate, but has approximately the correct behavior.