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.
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 .
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 .)
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
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
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