Forward Modeling of a Supergather

All the shots in a supergather can propagate at the same time in the time domain,

$$(\bigtriangledown^{2}-\frac{1}{v^2}\frac{\partial^2}{\partial t^2... ...bf{x})=-\sum_{s}Re[W(\omega_{s})e^{i\omega_{s}t}]\delta(\textbf{x}-\textbf{s}).$$ (37)

At the receiver locations, the observed time domain data ${P}(t,\textbf{g})$ need to be transformed into the frequency domain and each shot selects the component according to its frequency encoding

$$\tilde {P}(\omega_s,\textbf{g}) = \frac{1}{T} \int_{T}^{2T}{P}(t,\textbf{g})e^{-i\omega_s t}dt.$$ (38)

In this case, the above integration should be carried out from $T$ to $2T$ where $T=\frac{2\pi}{d\omega}$ (Nihei and Li, 2006). Digitizing $\tilde {P}(\omega_s,\textbf{g})$ yields a supergather containing the blended full wavefields.