The primary attribute of an integral method is to respect the kinematic
component of the process. However, in order to yield a *consistent* stack
of the operators illuminating a given location, the integration should be
a weighted sum. In other words, an amplitude function should be applied
along the operator. The integral DMO process will be *consistent*
if it obeys the following rules.

**Rule 1.** According to Hale 1991,
``The impulse responses [obtained by Fourier Transform DMO]
may be used as a standard by which to judge integral DMO methods''.
Because (*f*,*k*) DMO methods have a perfect
behavior with respect to amplitude, the integral DMO operator should
be as close as possible to the (*f*,*k*) DMO operator in amplitude and
phase. Thus, we expect the integral impulse response to have a low
amplitude and a high-frequency content near *x*=0 and a high amplitude
and a low-frequency content when the slope of the operator becomes steeper.

**Rule 2.** Flat events must not be affected in amplitude and phase
by the DMO process. This rule, clearly stated by Hale
1991, is perfectly respected by any (*f*,*k*)
DMO process Hale (1983); Liner (1990),
but it represents a challenging test for integral DMO processes.

**Rule 3.** Events of a given reflectivity must show balanced
amplitude after the DMO process, whatever their dip. This rule
is essential in order not to spoil the data for a possible AVO study.

The first section of this paper explains how to avoid the aliasing of the operator. In the next section, the three rules stated above help us choose the most convenient weighting among three amplitude schemes selected from the literature. Finally, a brief section discusses how to apply the operator on a 3-D grid.

11/17/1997