Angle-domain illumination gathers by wave-equation-based methods

Angle-domain image and illumination

Because the subsurface offset is linked to the scattering angle and dip angle through local ray parameters, the sensitivity kernel (and consequently the reflectivity image and the Hessian) can be transformed from the subsurface-offset domain into the angle domain through a simple Fourier domain mapping, or equivalently a space-domain slant stack (Sava and Fomel, 2003). In this section, we first demonstrate that a Fourier-domain mapping using the depth and horizontal-subsurface-offset wavenumbers produces scattering angles that are implicitly averaged over illuminated dips. It is useful for point scatterers. Dip-dependent scattering-angle illumination, however, is required for accurately predicting illumination strength for planar reflectors. We restrict our discussion only in 2-D for simplicity, where ${\bf x}=(x,z)$ , ${\bf h}=(h_x,h_z)$ , ${\bf x}_s=(x_s,z_s=0)$ and ${\bf x}_r=(x_r,z_r=0)$ . But the extension to 3-D should be straight forward and would be discussed in further publications.