The PSPM operator defined in equations
(1)
can be written as function of
the half-offset *h*, half-velocity *V*, and source-receiver
travel time *t*_{h}:

(2) |

Defining the variables ,we can rewrite equations (2) as

(3) |

Equations (3) are parametric
representations of the PSPM
operator, function of two new variables:
and .The variable is the
NMO corrected constant-offset travel
time, while the variable is the ** p_{x}=dt / dx** parameter.

In Figure 1 the PSPM operator is mapped using equations (3). The elliptic curves represent curves of constant parameter, while the vertical curves are curves of constant parameter. The PSPM operator is represented by and coordinates, which is a velocity independent characterization. The velocity cutoff is contained in the parameter , which is increasing toward the edges of the plot.

The cutoff is determined by the parameter . Higher velocities
introduce the restriction in the parameter which cannot be
higher than 1 / *V*, where *V* is half-velocity.
A medium with velocity *V _{1}* will have a
tighter cutoff than a medium with lower velocity

Equations (3) map the PSPM operator as a
function of the
variables and .
In a constant velocity medium, for each point in the
(, )space there corresponds a pair of values
(, ) and vice-versa.
For a medium with variable velocity with depth ** V(z)**, it was
shown
that for certain velocity models the DMO impulse response
presents triplications
(Popovici and Biondi, 1989).
Supposing the PSPM operator is represented in a
system
of coordinates
(, ),
for certain velocity models
the transformation
is not unique, thus generating the triplications in the PSPM curve.
In a medium with variable velocity, this can be determined
by following the variation of the parameter along
the equal travel time reflector. When the value of
is repeated along the same reflector with a given ,the PSPM operator will have multiple values of the
pair of parameters (

1/13/1998