Next: Velocity increasing linearly with Up: CURVED WAVEFRONTS Previous: Nonhyperbolic curves

## Can I abandon the material in this section?

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 h4 and higher powers. Velocity then enters only as the coefficient of h2. 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 ith 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 p3 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(p3) terms when h is small. Substituting this value of p into equation () yields
 (41)
Squaring both sides and discarding terms of order h4 and p4 yields the advertised result, equation (24).

XXXXXXXX

Next: Velocity increasing linearly with Up: CURVED WAVEFRONTS Previous: Nonhyperbolic curves
Stanford Exploration Project
12/26/2000