I define the transform so that, under some usual assumptions, the transformed data would provide an averaged, relative amplitude response of the subsurface region illuminated by a Snell beam. The assumptions are, that within the range of a CMP gather the subsurface geology can be approximated by a plane-layered, isotropic elastic medium; that a previous multiple-removal process had been applied; and that transmission losses are either negligible or previously corrected. However, only the first assumption is required for the validity of the transform, which properly compensates for the effects of spherical divergence, source and receiver radiation patterns and normal-moveout.
Figure shows two Snell beams illuminating a small subsurface region. Each beam is defined by the Snell parameters (horizontal slowness) of the boundary rays of the beam, and if we do not take into account transmission or absorption losses, the total energy in a wavefront inside the beam remains constant before and after the reflection. Therefore, the ratio between the energy in a wavefront recorded at the surface and the energy in a wavefront near the source will be affected only by the reflection losses.
A small region of a reflector is illuminated by two Snell beams.
The transformation is performed in the following way:
The analytic expression for the described transform is
I have tested two possibilities for the sampling rate of the horizontal slowness (i.e. the interval between adjacent output traces). In one case, the Snell parameter interval was kept constant, and in the other the interval between the square roots of two adjacent Snell parameters was fixed. I prefer this second option, because the transformed data shows a more uniform scanning of the x-t domain (Figure ).
Snell beams corresponding to (a) a constant interval between adjacent values of the square root of p, and (b) a constant interval between adjacent values of p.
For shallow reflectors and large values of horizontal slowness, the hyperbolic approximation for the reflected wavefront loses its validity and a ray-tracing routine (controlled by the background model) must be used to determine the stacking curve.
The main advantages of this transformation are that no further moveout correction is required; geometrical spreading correction is done in an appropriate way; the effects of source directivity are directly incorporated in the transform; the transformed dataset approximates a local plane-wave response at the reflector's position; the number of undesirable events that are summed to the signal is considerably smaller here than in the conventional slant-stack, since just a local sum is involved. Although the method described uses an apriori velocity model, a data-driven Snell ray-tracing can be implemented as described by Ottolini (1987) or using of a local dip estimation process (Claerbout, 1990).