Modeling regimes

Downward-continuation equations contain four main ingredients: the slowness of the medium at the geophone v(g)^-1; likewise at the shot v(s)^-1; the stepout in offset space $k_h / \omega$; and the dip in midpoint space $k_y / \omega$.These four ingredients all have the same physical dimensions, and modeling procedures can be categorized according to the numerical inequalities that are presumed to exist among the ingredients. One-dimensional work ignores three of the four--namely, dip, stepout, and the difference $v(g)^{-1} \,-\, v(s)^{-1}$.CMP slant stack includes the stepout $k_h / \omega$.Now we have a choice as to whether to include the dip or the lateral velocity variation. The lateral velocity variation is often severe near the earth's surface where the peglegs live. Recall the simple idea that typical rays in the deep subsurface emerge steeply at a low-velocity surface. When using continuation equations in the near surface, we are particularly justified in neglecting dip, that is $v^{-1} \gg k_y / \omega$.It is nice to find this excuse to neglect dip since our field experiments are so poorly controlled in dip out of the plane of the experiment. Offset stepout, on the other hand, is probably always much larger in the plane of the survey line than out of it.

Another important ingredient for modeling or processing multiple reflections is the coupling of upcoming and downgoing waves. This coupling introduces the reflectivity beneath the shot c(s) and the receiver c(g). An important possibility, to which we will return, is that c(s) may be different from c(g), even though all the angles may be neglected.