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Layered media

From the assumption that experimental data can be fit to hyperbolas (each with a different velocity and each with a different apex $ \tau $) 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 3.10.

Figure 10.
Raypath diagram for normal moveout in a stratified earth.
[pdf] [png] [scons]

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 $V^2(\tau)$.

t^2 \eq \tau^2 + \frac{4h^2}{V^2(\tau)}
\end{displaymath} (24)

where $ \tau $ is the zero-offset two-way traveltime.

An example of using equation (3.24) to stretch $t$ into $ \tau $ is shown in Figure 3.11. (The programs that find the required $V(\tau )$ 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.
[pdf] [png] [scons]

Equation (3.21) shows that $V(\tau )$ is the ``root-mean-square'' or ``RMS'' velocity defined by an average of $v^2$ over the layers. Expressing it for a small number of layers we get

V^2(\tau) \eq \frac{1}{\tau} \sum_i v^2_i \Delta\tau_i
\end{displaymath} (25)

where the zero-offset traveltime $ \tau $ is a sum over the layers:
\tau \eq \sum_i  \Delta\tau_i
\end{displaymath} (26)

The two-way vertical travel time $\tau_i$ in the $i$th layer is related to the thickness $\Delta z_i$ and the velocity $v_i$ by
\Delta\tau_i \eq \frac{2 \Delta z_i}{v_i}   .
\end{displaymath} (27)

Next we examine an important practical calculation, getting interval velocities from measured RMS velocities: Define in layer $i$, the interval velocity $v_i$ and the two-way vertical travel time $\Delta\tau_i$. Define the RMS velocity of a reflection from the bottom of the $i$-th layer to be $V_i$. Equation (3.25) tells us that for reflections from the bottom of the first, second, and third layers we have

$\displaystyle V_1^2$ $\textstyle =$ $\displaystyle {v_1^2\Delta\tau_1
\over \Delta\tau_1 }$ (28)
$\displaystyle V_2^2$ $\textstyle =$ $\displaystyle {v_1^2\Delta\tau_1+ v_2^2\Delta\tau_2
\over \Delta\tau_1 + \Delta\tau_2 }$ (29)
$\displaystyle V_3^2$ $\textstyle =$ $\displaystyle {v_1^2\Delta\tau_1+ v_2^2\Delta\tau_2 +v_3^2\Delta\tau_3
\over \Delta\tau_1 + \Delta\tau_2 +\Delta\tau_3}$ (30)

Normally it is easy to measure the times of the three hyperbola tops, $\Delta\tau_1$, $\Delta\tau_1 + \Delta\tau_2$ and $\Delta\tau_1 + \Delta\tau_2 +\Delta\tau_3$. Using methods in chapter [*] we can measure the RMS velocities $V_2$ and $V_3$. With these we can solve for the interval velocity $v_3$ in the third layer. Rearrange (3.30) and (3.29) to get

$\displaystyle (\Delta\tau_1 + \Delta\tau_2 +\Delta\tau_3)
V_3^2$ $\textstyle =$ $\displaystyle v_1^2\Delta\tau_1+ v_2^2\Delta\tau_2 +v_3^2\Delta\tau_3$ (31)
$\displaystyle (\Delta\tau_1 + \Delta\tau_2)
V_2^2$ $\textstyle =$ $\displaystyle v_1^2\Delta\tau_1+ v_2^2\Delta\tau_2$ (32)

and subtract getting the squared interval velocity $v_3^2$
v_3^2 \eq {
(\Delta\tau_1 + \Delta\tau_2 +\Delta\tau_3) V_3^2 -
(\Delta\tau_1 + \Delta\tau_2 ) V_2^2
\end{displaymath} (33)

For any real earth model we would not like an imaginary velocity which is what could happen if the squared velocity in (3.33) 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.

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