DIVIDING THE DATA INTO SNELL ZONES

Although the use of the critical horizontal slowness to perform the separation between P and S waves is effective for shallow reflectors, the finite size of the cable restricts its application to deep reflectors because a converted wave with horizontal slowness larger than the P wave critical slowness will be received at very large offsets.

A possible way to overcome the limitations associated with the finiteness of the field aperture is to use a criterion that somehow takes into account the depth of the reflector. Transforming the data into the $\tau$-p domain provides the necessary flexibility for use of a variable slowness cutoff. Tatham and Goolsbee (1984) showed also that better results can be achieved when a hyperbolic velocity filtering is used during the $\tau$-p transform to limit the range of reasonable stacking velocities.

As we will see, the separation of the data into ranges of Snell rays would seem to be another appropriate way to perform the slowness filtering with a variable-slowness cutoff.

A Snell ray (Ottolini, 1982) can be defined for a plane-layered earth as a ray that keeps a constant horizontal slowness (obeys the Snell law) while it propagates through the subsurface, as illustrated in Figure . The reflection points corresponding to a Snell ray with ray parameter (or horizontal slowness) p are defined by

$${\partial x \over \partial t} = p v^2(z) , \mbox{ \hspace{1... ...$v(z)$ \hspace{0.6cm} is the interval velocity at depth $z$. }$$

For the simple case of a plane-layered earth, the horizontal slowness of a P wave will always be lower than the horizontal slowness of a converted wave recorded at the same position in the x-t domain. It is possible then to choose a set of ray parameters and divide the data into regions whose P waves Snell rays are limited by two adjacent values (Figure ). Since the converted waves inside each region will have higher horizontal slownesses than the P waves, a different filter can be applied to each region; the cutoff value is given by the ray parameter of the Snell ray that defines the upper limit of that region.