NMO equations

The earth's velocity typically ranges over a factor of two or more within the depth range of a given data set. Thus the Pythagorean analysis needs reexamination. In practice, depth variable velocity is often handled by inserting a time variable velocity into the Pythagorean relation. (The classic reference, Taner and Koehler [1969], includes many helpful details). This approximation is much used, although it is not difficult to compute the correct nonhyperbolic moveout. Let us see how the velocity function v(z) is mathematically related to the NMO. Ignoring dip, NMO converts common-midpoint gathers, one of which, say, is denoted by P(h,t), to an earth model, say,  

$$Q(h,z)\ \eq \ \hbox{earth}(z) \ \times \ \hbox{const}(h)$$ (1)

Actually, Q(h,z) doesn't turn out to be a constant function of h, but that is the goal.

The NMO procedure can be regarded as a simple copying. Conceptually, it is easy to think of copying every point of the (h,t)-plane to its appropriate place in the (h,z)-plane. Such a copying process could be denoted as  

$$Q[h,z(h,t)]\ \eq \ P(h,t)$$ (2)

Care must be taken to avoid leaving holes in the (h,z)-plane. It is better to scan every point in the output (h,z)-plane and find its source in the (h,t)-plane. With a table t(h,z), data can be moveout corrected by the copying operation  

$$Q(h,z)\ \eq \ P[h,t(h,z)]$$ (3)

Using the terminology of this book, the input P(h,t) to the moveout correction is called a CMP gather, and the output Q is called a CDP gather.

In practice, the first step in generating the travel-time tables is to change the depth-variable z to a vertical travel-time-variable $\tau$.So the required table is $t(h, \tau$). To get the output data for location $(h, \tau$) you take the input data at location (h,t). The most straightforward and reliable way to produce this table seems to be to march down in steps of z, really $\tau$, and trace rays. That is, for various fixed values of Snell's parameter p, you compute $t(p, \tau )$ and $h(p, \tau )$ from $v( \tau )$ by integrating the following equations over $\tau$:

$$\begin{eqnarray} {dt \over d \tau}\ \ &=&\ \ {dz \over d \tau}\ {dt \over dz }\ ... ... \ {p \, v ( \tau )^2 \over \sqrt { 1\ -\ p^2 \, v ( \tau )^2 } }\end{eqnarray}$$ (4) (5)

(In equations (4) and (5) dt/dz and dh/dz are based on rays, not wavefronts). Given $t(p, \tau )$ and $h(p, \tau )$, iteration and interpolation are required to eliminate p and find $t(h, \tau )$.It sounds awkward--and it is--because at wide angles there usually are head waves arriving in the middle of the reflections. But once the job is done you can save the table and reuse it many times. The multibranching of the travel time curves at wide offset motivates a wave-equation based velocity analysis. The greatest velocity sensitivity occurs just where the classic hyperbolic assumption and the single-arrival assumption break down.