Snell-wave stacks and CMP slant stacks

Setting the takeoff angle S to zero also reduces the double-square-root equation to a single-square-root equation. The meaning of S = 0 is that k_s = 0 or equivalently that the data should undergo a summation (without time shifting) over shot s. Such a summation simulates a downgoing plane wave. The imaging principle behind the summation would be to look at the upcoming wave at the arrival time of the downgoing wave. S could also be set equal a constant, to simulate a downgoing Snell wave.

A Snell wave is a generalization of a downgoing plane wave at nonvertical incidence. The shots are not fired simultaneously, but sequentially at an inverse rate of $dt/ds\,=\,S/v$.This could be simulated with field data by summing across the (t,s)-plane along a line of slope dt/ds. Setting S to be some constant, say $S = v\ dt/ds$, also reduces the double-square-root equation to a paraxial wave equation, just the equation needed to downward continue the downgoing Snell wave experiment. Snell waves could be constructed for various $p\,=\,dt/ds$ values. Each could be migrated and imaged, and the images stacked over p. These ideas have been around longer than the DSR equation, yet they have gained no popularity. What could be the reason?

A problem with Snell wave simulation is that the wavefield is usually sampled at coarse intervals along a geophone cable, which itself never seems to extend as far as the waves propagate. Crafty techniques to interpolate and extrapolate the data are frustrated by the fact that on a common-geophone gather, the top of the hyperbola need not be at zero offset. For dipping beds the earliest arrival is often off the end of the cable. So the data processing depends strongly on the missing data.

These difficulties provide an ecological niche for the common-midpoint slant stack, namely, H = p v. (A fuller explanation of slant stack comes later.) At common midpoint the hyperbolas go through zero offset with zero slope. The data are thus more amenable to the interpolation and extrapolation required for integration over a slanted line. Setting H = p v yields  

$$k_z \eq -\ { \omega \over v }\ \left[ \ \sqrt{ 1 \ -\ (Y\,+\,pv)^2 } \ +\ \sqrt{ 1 \ -\ (Y\,-\,pv)^2 } \ \right]$$ (73)

This has not reduced the DSR equation to a paraxial wave equation, but it has reduced the problem to a form manageable with the available techniques, such as the Stolt or phase-shift methods. Details of this approach can be found in the dissertation of Richard Ottolini [1982].