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The form of the shot continuation operator varies dramatically depending
on the distance the
shot is shifted. For illustration, Figure 5 depicts a shot gather
*s*_{1}, which is shifted various distances.
The original shot gather contains a single event at (*t*_{1} = 1.0, *h*_{1} = 0.25)
and it is shifted to *s*_{2} at the center of the geophone spread.
The distance shifted varies from frame to frame
as the shot location of *s*_{1} advances from the right to the left of the
geophone spread.

Each thin line in Figure 5 represents the kinematic solution
as it is derived and rendered by Mathematica. The background raster plot
is computed by the Ratfor routine `shct.rst` incorporating amplitude
variations along the operator.

**shct
**

Figure 5 Impulse Response.
The kinematic operator derived by the Mathematica script `smile.ma`
agrees with the raster-plot output of the corresponding Ratfor program
`shct.rst`. The operator varies strongly with the relative position
of the original and the extrapolated shot. The impulse response collapses
to a point when the original and extrapolated shot position is identical or
reciprocal.

The operator is a symmetric DMO-like smile when *s*_{2} coincides with
the midpoint associated with the input impulse at (*t*_{1}, *h*_{1}).
However, the operator skews sideways and finally collapses to a point
as the shot location *s*_{1} or the geophone *g*_{1} approaches *s*_{2}.
Beyond the identical and reciprocal shot position, the operator
turns into a frown. In addition to the variations induced by varying
shift distances, the operator also varies with different impulse event
times *t*_{1}.

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Stanford Exploration Project

11/17/1997