Selecting an optimal aperture in Kirchhoff migration using dip-angle images

Point diffractor

The hyperbolic reflector in equation A-1 creates a point diffractor at coordinates $\{x_0,z_0\}$ when $\beta=\pi/2$. In this case, equation A-5 simplifies to (Klokov and Fomel, 2012)

$$t_m(x_m) = \frac{2 \cos{\alpha}}{v} \frac{\gamma (x_m-x_0... ...amma^2 \sin^2{\alpha}\right)}}{1-\gamma^2 \sin^2{\alpha}}\;.$$ (18)

At a correct velocity ($\gamma=1$),

$$t_m(x_m) = \frac{2}{v} \frac{(x_m-x_0) \sin{\alpha} + \sqrt{(x_m-x_0)^2+z_0^2 \cos^2{\alpha}}}{\cos{\alpha}}\;,$$ (19)

which is equivalent to equation 2 in the main text. The dip of the image is

$$\tan{\alpha_m} = \frac{v}{2} t_m'(x_m) = \tan{\alpha} + ... ..._0}{\cos{\alpha} \sqrt{(x_m-x_0)^2+z_0^2 \cos^2{\alpha}}}\;.$$ (20)

It is easy to verify that, above the diffraction point ($x_m=x_0$), $\alpha_m=\alpha$.