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We could work out the mathematical problem
of finding an analytic solution for
the travel time
as a function of distance in an earth with stratified *v*(*z*),
but the more difficult problem is
the practical one which is the reverse,
finding from the travel time curves.
Mathematically we can
express the travel time (squared)
as a power series in distance *h*.
Since everything is symmetric in *h*,
we have only even powers.
The practitioner's approach is to look at small offsets
and thus ignore *h*^{4} and higher powers.
Velocity then enters only as the coefficient of *h*^{2}.
Let us why it is the RMS velocity,
equation (25),
that enters this coefficient.

The hyperbolic form of equation (24) will generally not be exact
when *h* is very large.
For ``sufficiently'' small *h*,
the derivation of the hyperbolic shape follows
from application of Snell's law at each interface.
Snell's law implies that the Snell parameter *p*, defined by

| |
(36) |

is a constant along both raypaths in Figure 10.
Inspection of Figure 10 shows that
in the *i*th layer
the raypath horizontal distance and travel time are given on the left below by
| |
(37) |

| (38) |

The center terms above arise by using equation ()
to represent and as a function of hence *p*,
and the right sides above come from expanding in powers of *p*.
Any terms of order *p*^{3} or higher will be discarded,
since these become important only at large values of *h*.
Summing equation () and () over all layers
yields the half-offset *h* separating the midpoint
from the geophone location and the total travel time *t*.
| |
(39) |

| (40) |

Solving equation () for *p* gives ,justifying the neglect of the *O*(*p*^{3}) terms when *h* is small.
Substituting this value of *p* into equation () yields
| |
(41) |

Squaring both sides and discarding terms of order *h*^{4} and *p*^{4}
yields the advertised result, equation (24).
XXXXXXXX

** Next:** Velocity increasing linearly with
** Up:** CURVED WAVEFRONTS
** Previous:** Nonhyperbolic curves
Stanford Exploration Project

12/26/2000