** Next:** APPENDIX C
** Up:** Jedlicka: Velocity analysis by
** Previous:** APPENDIX B

The aim of this section is to derive the expression for the operator
.We know that
| |
(36) |

for any from -space and from
(*t*,*x*)-space.
If we define
| |
(37) |

then
| |
(38) |

From this equation it follows
| |
(39) |

which is the expression for the operator .
Equation represents a
hyperbola in (*T*,*y*)-space passing through the point (*t*,*x*).
With changing *m* we obtain a set of hyperbolas passing through the
point (*t*,*x*) (Fig ). A weighting factor is assigned
to each hyperbola. The higher the slope of a hyperbola is, the
higher weighting factor is assigned.
The value of the operator at the point (*t*,*x*) is equal
to the weighted sum of all values lying on the hyperbolas passing
through the point (*t*,*x*).

This result may also be explained in the following way.
The sums along hyperbolas passing through the point (*t*,*x*)
form a parabola in -space (Jedlicka, 1989).
is obtained by weighted summation along this parabola.
Each point of the parabola represents a sum along a corresponding
hyperbola, from which the previous result follows immediately.

If *d*(*t*,*x*)=1 for all *t* and *x* then from equation (39)
it follows

| |
(40) |

This means that *count*(*t*,*x*) for velocity analysis is equal to the
sum of NMO counts over the range of velocities used in velocity analysis.
An expression for a count for NMO is derived in Appendix C.

** Next:** APPENDIX C
** Up:** Jedlicka: Velocity analysis by
** Previous:** APPENDIX B
Stanford Exploration Project

1/13/1998