Born Modeling of a Supergather

In LSRTM, the Born modeling operator is used to fit the observed reflection data with a reflectivity model $m(\textbf{x})$ . Following Dai et al. (2012), the Born modeling of a supergather can be computed in the time domain as

$$(\bigtriangledown^{2}-\frac{1}{v^2}\frac{\partial^2}{\partial t^2... ...}(t,\textbf{x})=m(\textbf{x})\frac{\partial^2 {P}(t,\textbf{x})}{\partial t^2},$$ (39)

subsequent to equation 3.7. The frequency domain data $\tilde{d}(\omega_s,\textbf{g})$ can be extracted from ${d}(t,\textbf{g})$ according to

$$\tilde {d}(\omega_s,\textbf{g}) =\frac{1}{T} \int_{T}^{2T}{d}(t,\textbf{g})e^{-i\omega_s t}dt.$$ (310)

Digitizing $\tilde{d}(\omega_s,\textbf{g})$ yields a supergather $\tilde{\textbf{d}}_{i\omega_s,ig}$ containing the reflection waves. Equations 3.7, 3.9, and 3.10 represent the numerical computation of the Born modeling operator $\textbf{L}$ in equation 3.3.