Spherical divergence

As suggested by the preceding result, spherical divergence should be included in the DMO process. Indeed, for a given shot-geophone traveltime, the zero-offset ray path when the reflector is dipping is shorter than when the reflector is horizontal. Since the decrease in amplitude of a spherical wave is inversely proportional to the distance traveled, the spherical spreading is clearly related to the dip of the structures and, thus, must be included in the DMO process.

Gardner 1990 expressed the spherical spreading factor as a function of k ($\sqrt{h^2-x^2}$) and t₂ (t₀). As a function of h, t_n, and x, it becomes  

$$F_{ss} = h^2 \left( 1 + \frac{v^2t_n^2}{4h^2} \right) \left( 1 - \frac{x^2}{h^2} \right).$$ (6)

The two first terms of expression (6) account for the offset and the depth of the reflector, whereas the third term clearly accounts for its dip.