|Waves in strata|
From the assumption that experimental data can be fit to hyperbolas (each with a different velocity and each with a different apex ) let us next see how we can fit an earth model of layers, each with a constant velocity. Consider the horizontal reflector overlain by a stratified interval velocity shown in Figure 3.10.
Figure 10. Raypath diagram for normal moveout in a stratified earth.
The separation between the source and geophone, also called the offset, is and the total travel time is . Travel times are not be precisely hyperbolic, but it is common practice to find the best fitting hyperbolas, thus finding the function .
An example of using equation (3.24) to stretch into is shown in Figure 3.11. (The programs that find the required and do the stretching are coming up in chapter .)
Figure 11. If you are lucky and get a good velocity, when you do NMO, everything turns out flat. Shown with and without mute.
Equation (3.21) shows that
the ``root-mean-square'' or
``RMS'' velocity defined by
an average of over the layers.
Expressing it for a small number of layers we get
Next we examine an important practical calculation,
getting interval velocities from measured RMS velocities:
in layer ,
the interval velocity
and the two-way vertical travel time .
Define the RMS velocity
of a reflection
from the bottom of the -th layer
to be .
Equation (3.25) tells us that for
reflections from the bottom of the first, second, and third layers we have
Normally it is easy to measure the times of the three hyperbola tops, , and . Using methods in chapter we can measure the RMS velocities and . With these we can solve for the interval velocity in the third layer. Rearrange (3.30) and (3.29) to get
|Waves in strata|