(1) |

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.

Figure 3

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:

(2) |

(3) |

(4) |

(5) |

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).

4/29/2001