Exploding-reflector modeling by upward continuation

The adjoint of exploding-reflector migration is exploding-reflector modeling: if we can migrate exploding-reflector data with

$$\begin{eqnarray} \hat{\bf m} & = & {\bf A}'_{\rm ER} \; \; {\bf d} \nonumber \\... ...\Sigma}_{\omega} \; {\bf D}^{-1} \; {\bf Z}_{N_\omega} \; {\bf d},\end{eqnarray}$$ (69)

then we should be able to forward model synthetic data with

$$\begin{eqnarray} \hat{\bf d} & = & {\bf A}_{\rm ER} \; \; {\bf m} \nonumber \\... ..._\omega} \; ({\bf D}')^{-1} \; {\bf \Sigma}_{\omega}' \; {\bf m}.\end{eqnarray}$$ (70)

A close look at the composite operator reveals that the first step is to apply ${\bf \Sigma}_{\omega}'$, which sprays the reflectivity model out to each frequency. The second step is to apply $({\bf D}')^{-1}$ which recursively marches the wavefield up through the earth. The final step is to truncate the data at the surface with the ${\bf Z}'_{N_\omega}$operator.