Next: Layered media Up: CURVED WAVEFRONTS Previous: CURVED WAVEFRONTS

## Root-mean-square velocity

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.
 (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.
 (19)
The x coordinate of the tip of a ray with Snell parameter p is the horizontal component of velocity integrated over time.
 (20)
Inserting this into equation (18) and canceling p=dt/dx we have
 (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 pi and a midpoint (ti,xi) 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.

Next: Layered media Up: CURVED WAVEFRONTS Previous: CURVED WAVEFRONTS
Stanford Exploration Project
12/26/2000