Clearly the selection of the stacking velocities must be done to correct for the moveout of the primary reflections and not for the multiples. At a given zero-offset arrival time the velocity of a primary reflection is greater than that of a multiple, which according to Equation (1) implies a smaller moveout. This difference in moveout makes it possible to flatten the primary reflections while leaving the multiples under-corrected with a moveout approximately parabolic Hampson (1986). The Parabolic Radon Transform exploits this difference by summing trace amplitudes along parabolas of different zero-offset time and curvature. Hence, the transform can be considered a mathematical operator that maps parabolas in the t-x domain to small regions of the parabolic moveout (p) and zero-offset time () domain.
This is schematically shown in Figure 3 which shows that the horizontal events in t-x domain map to a vertical strip in the -p domain at p=0. The multiple reflections, on the other hand, are mapped in the -p domain to a region away from the p=0 vertical line. This separation allows for the suppression of the multiple energy by zeroing out the -p region to the right of the dashed line in Figure 3. The inverse -p transform would then return the primaries to the t-x domain.
In practice the process is applied a little differently: it is the energy of the primaries that is suppressed (energy to the left of the dashed line in Figure 3) and inversely transformed to the t-x domain. The primaries are computed by subtracting the multiples from the original data in this domain. This method was first introduced with the name ``inverse velocity stacking'' Hampson (1986).
The mathematical equivalent of the qualitative description given before for the Parabolic Radon Transform is a set of two equations:
The need to work with discrete equations may give raise to aliasing problems Yilmaz (1987) as well as to some stability problems related to the selection of the number of parabolas used in the transform. In commercial software packages the transform is normally implemented in the f-x domain because of issues related to the amplitude of the inverse transform which is computed via a numerical optimization process. The discussion of these details, which are very important for the successful application of the method, are out of the scope of this paper. See for example Anderson (1993) and Alvarez (1995).