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Let us denote as the operator of summation along hyperbolas.
The operator transforms (*t*,*x*)-space into -space:
| |
(2) |

An adjoint operator transforms -space into
(*t*,*x*)-space:
| |
(3) |

We can also define a stacking operator in -space. Because
maps a point onto a parabola (Jedlicka, 1989b), it is natural
to define an operator as an operator of summation along parabolas:
| |
(4) |

The adjoint operator ,
| |
(5) |

may serve as an alternative operator for velocity analysis.
Here each point in (*t*,*x*)-space is spread onto a parabola
in -space.
Both operators and are linear.
The relationship between them is shown in Appendix A.

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Stanford Exploration Project

1/13/1998