Next: Layered media
Up: CURVED WAVEFRONTS
Previous: CURVED WAVEFRONTS
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.
| ![\begin{displaymath}
\hat v \eq \sqrt{ {x\over t} \ {dx\over dt} }\end{displaymath}](img32.gif) |
(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.
| ![\begin{displaymath}
v(z) \eq v(x,z) \eq v'(p,t)\end{displaymath}](img33.gif) |
(19) |
The x coordinate of the tip of a ray with Snell parameter p is
the horizontal component of velocity integrated over time.
| ![\begin{displaymath}
x(p,t) \eq \int_0^t \ v'(p,t) \ \sin\theta(p,t)\ dt
\eq p\ \int_0^t v'(p,t)^2\ dt \ \end{displaymath}](img34.gif) |
(20) |
Inserting this into equation (18)
and canceling p=dt/dx we have
| ![\begin{displaymath}
\hat v \eq
v_{\rm RMS}\eq \sqrt{ {1\over t} \ \int_0^t v'(p,t)^2\ dt\ \ }\end{displaymath}](img35.gif) |
(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.
| ![\begin{displaymath}
t^2 \eq \tau^2 + x^2/V(\tau)^2\end{displaymath}](img38.gif) |
(22) |
Differentiate with respect to x at constant
getting
| ![\begin{displaymath}
2t\, dt/dx \eq 2x/V(\tau)^2\end{displaymath}](img39.gif) |
(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