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Let *D*(*t*,*x*_{r}) be a common shot gather (CSG). Define to be a velocity-dependent operator that transforms the data *D*(*t*,*x*_{r})
into its image *I*(*t*_{0},*x*).

| |
(3) |

| (4) |

where *W*_{D} is a weighting function and *t* is defined in
equation (1). Notice that the operator is actually
a Kirchhoff migration operator of constant velocity *v*. Similarly, the
backward transformation from *I*(*t*_{0},*x*) to *D*(*t*,*x*_{r}) is
| |
(5) |

| (6) |

where *t*_{0} is defined in equation (2). may be
the inverse or transpose of depending on the choice of
weighting function *W*_{I}. The parameter *v* of the operators plays an
important role in both forward and backward transformations.
For the purpose of imaging, we
set *v* to be the velocity of the medium. As a result, *I*(*t*_{0},*x*)
will be the precise image of the earth. For other values of *v*, however,
*I*(*t*_{0},*x*) will be a distorted image. Our goal is to select a special
value of *v* such that after the forward transformation with operator
the images of the water bottom multiples are
as separable as possible from the images of other events.

** Next:** Choice of v
** Up:** PRINCIPLES OF THE ALGORITHM
** Previous:** Travel-time relations
Stanford Exploration Project

1/13/1998