Waves in strata

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 that it has traveled, the time it took, and its apparent speed . 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 can be parameterized as a function of time and take-off angle of a ray from the surface.
 (19)

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

Inserting this into equation (3.18) and canceling 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 -th segment has a slope and a midpoint 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 at constant getting
 (23)

which confirms that the observed velocity in equation (3.18), is the same as no matter where you measure on a hyperbola.

 Waves in strata

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

2009-03-16