Next: Nonhyperbolic curves Up: CURVED WAVEFRONTS Previous: Root-mean-square velocity

## Layered media

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 v(z) shown in Figure 10.

 stratrms Figure 10 Raypath diagram for normal moveout in a stratified earth.

The separation between the source and geophone, also called the offset, is 2h and the total travel time is t. Travel times are not be precisely hyperbolic, but it is common practice to find the best fitting hyperbolas, thus finding the function .

 (24)
where is the zero-offset two-way traveltime.

An example of using equation (24) to stretch t into is shown in Figure 11. (The programs that find the required and do the stretching are coming up in chapter .)

nmogath
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 (21) shows that is the root-mean-square'' or RMS'' velocity defined by an average of v2 over the layers. Expressing it for a small number of layers we get
 (25)
where the zero-offset traveltime is a sum over the layers:
 (26)
The two-way vertical travel time in the ith layer is related to the thickness and the velocity vi by
 (27)

Next we examine an important practical calculation, getting interval velocities from measured RMS velocities: Define in layer i, the interval velocity vi and the two-way vertical travel time .Define the RMS velocity of a reflection from the bottom of the i-th layer to be Vi. Equation (25) tells us that for reflections from the bottom of the first, second, and third layers we have
 (28) (29) (30)

Normally it is easy to measure the times of the three hyperbola tops, , and .Using methods in chapter we can measure the RMS velocities V2 and V3. With these we can solve for the interval velocity v3 in the third layer. Rearrange (30) and (29) to get

 (31) (32)
and subtract getting the squared interval velocity v32
 (33)
For any real earth model we would not like an imaginary velocity which is what could happen if the squared velocity in (33) happened to be negative. You see that this means that the RMS velocity we estimate for the third layer cannot be too much smaller than the one we estimate for the second layer.

Next: Nonhyperbolic curves Up: CURVED WAVEFRONTS Previous: Root-mean-square velocity
Stanford Exploration Project
12/26/2000