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When a ray travels in a depth-stratified medium,
Snell's parameter is constant along the ray.
If the ray emerges at the surface,
we can measure the distance *x* that it has traveled,
the time *t* it took, and its apparent speed *dx*/*dt*=1/*p*.
A well-known estimate for the earth velocity contains this apparent speed.

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(18) |

To see where this velocity estimate comes from,
first notice that the stratified velocity *v*(*z*) can be parameterized
as a function of time and take-off angle of a ray from the surface.
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(19) |

The *x* coordinate of the tip of a ray with Snell parameter *p* is
the horizontal component of velocity integrated over time.
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(20) |

Inserting this into equation (18)
and canceling *p*=*dt*/*dx* we have
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(21) |

which shows that the observed velocity is the ``root-mean-square'' velocity.
When velocity varies with depth,
the traveltime curve is only roughly a hyperbola.
If we break the event into many short line segments where the
*i*-th segment has a slope *p*_{i} and a midpoint (*t*_{i},*x*_{i})
each segment gives a different and we have the unwelcome chore of assembling the best model.
Instead, we can fit the observational data to the best fitting hyperbola
using a different velocity hyperbola for each apex,
in other words,
find so this equation
will best flatten the data in -space.

| |
(22) |

Differentiate with respect to *x* at constant getting
| |
(23) |

which confirms that the observed velocity
in equation (18),
is the same as no matter where you measure
on a hyperbola.

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** Up:** CURVED WAVEFRONTS
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Stanford Exploration Project

12/26/2000