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A velocity analysis panel is obtained by transformation of a CMP gather. In this
paper an operator of summation will be used. This operator transforms
(*t*,*x*)-space into -space; i.e., it is an operator of transformation
from the time-offset domain into the time-sloth domain.
We are interested in the image in -space of a point
in (*t*,*x*)-space.
Let a gather *G*_{tx} contain only one non-zero amplitude
that is located at time *t* of a trace with the offset *x* (Fig. a).
Let us apply the operator on the gather *G*_{tx}.
The operator gives non-zero results only for hyperbolas passing through
the point (*T*,*X*) (Fig. ). Each hyperbola is determined by
the time and the sloth *m*, which satisfy the equation

| |
(1) |

which can be written
| |
(2) |

This is the equation of a parabola, so we have
| |
(3) |

A parabola corresponding to the point (*T*,*X*) is shown in Figure b.

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Stanford Exploration Project

1/13/1998