Waves in strata |

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 , 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 . Since everything is symmetric in , we have only even powers. The practitioner's approach is to look at small offsets and thus ignore and higher powers. Velocity then enters only as the coefficient of . Let us why it is the RMS velocity, equation (3.25), that enters this coefficient.

The hyperbolic form of equation (3.24) will generally not be exact
when is very large.
For ``sufficiently'' small ,
the derivation of the hyperbolic shape follows
from application of Snell's law at each interface.
Snell's law implies that the Snell parameter , defined by

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

Solving equation () for gives , justifying the neglect of the terms when is small. Substituting this value of into equation () yields

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Waves in strata |

2009-03-16