[5] The imaging gather of migration/inversion

Equation (84) or (85) gives the imaging matrix:  

$$\left[ \begin{array} {llcl} r_{11} & r_{12} & \cdots & r_{1Q... ...} & r_{P2} & \cdots & r_{PQ} \\ \end{array}\right]_{P\times Q},$$ (89)

where P is the number of scatterers and Q is the number of incident angles for a scatterer. A row of the reflectivity matrix is an angle gather. However, the angle is not evenly sampled in a constant angle interval in complex medium. Physically, a reflection point or scatterer is not illuminated with a constant incident angle interval; mathematically, an angle gather evenly sampled with a constant angle interval for the point can be yielded with Fourier transform approaches. Weglein and Stolt (1999) and Sava and Fomel (2003) gave an approach for creating the angle gathers from the imaged data set. However, in our mind, we should know the difference. The uneven-incident-angle illumination will cause imaging noises. And this will also cause amplitude distortions in angle gathers. Fig.5 shows that a reflection point or scatterer is not illuminated with a constant incident angle interval.

 

even_incident_angle
Figure 5 the illumination with the uneven incident angles