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Forward Modeling of a Supergather

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

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

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



Wei Dai 2013-07-10