Deconvolution in the Fourier domain

Jacobs (1982) compares various imaging methods for shot-profile migration. He shows that the deconvolution imaging condition  

$${\bf r(\omega)}=\frac{ {\bf u(\omega)\overline{d}(\omega)}}{ {\bf d(\omega)\overline{d}(\omega)}+\varepsilon^2}$$ (6)

is stable. The same imaging condition was used by Lee et al. (1991) in split-step migration.

Since we are only interested in the zero-time lag, the reflection strength can be computed as  

$${\bf r}(x,z,\tau =0)=\sum_{\omega}^{\omega_{Nyq}} {\bf r}(x,z,\omega).$$ (7)

where $\omega_{Nyq}$ is the Nyquist frequency.

An advantage of working in the Fourier domain is that the problem does not need to be stated as an inversion problem.