NMO in a horizontally stratified earth
We now consider a medium composed of horizontal isovelocity layers (Figure 3.1-6). Each layer has a certain thickness that can be defined in terms of twoway zero-offset time. The layers have interval velocities (v1, v2, …, vN), where N is the number of layers. Consider the raypath from source S to depth point D, back to receiver R, associated with offset x at midpoint location M.  derived the traveltime equation for this path as
where and C2, C3, … are complicated functions that depend on layer thicknesses and interval velocities (Section C.1). The rms velocity vrms down to the reflector on which depth point D is situated is defined as
where Δτi is the vertical two-way time through the ith layer and By making the small-spread approximation (offset small compared to depth), the series in equation (3) can be truncated to obtain the familiar hyperbolic form
When equations (1) and (4b) are compared, we see that the velocity required for NMO correction for a horizontally stratified medium is equal to the rms velocity, provided the small-spread approximation is made.
How much error is caused by dropping the higher order terms in equation (3)? Figure 3.1-7a shows a CMP gather based on the velocity model in Figure 3.1-8. Traveltimes to all four reflectors were computed by the raypath integral equations  that exactly describe wave propagation in a horizontally layered earth model with a given interval velocity function. We now replace the layers above the second shallow event at t0 = 0.8 s with a single layer with a velocity equal to the rms velocity down to this reflector — 2264 m/s. The resulting traveltime curve, computed using equation (4b), is shown in Figure 3.1-7b. This procedure is repeated for the deeper events at t0 = 1.2 and 1.6 s as shown in Figures 3.1-7c and d. Note that the traveltime curves in Figures 3.1-7b, c, and d are perfect hyperbolas. How different are the traveltime curves in Figure 3.1-7a from these hyperbolas? After careful examination, note that the traveltimes are slightly different for the shallow events at t0 = 0.8 and 1.2 s only at large offsets, particularly beyond 3 km. By dropping the higher order terms, we approximate the reflection times in a horizontally layered earth with a small-spread hyperbola.
Figure 3.1-7 (a) A synthetic CMP gather derived from the velocity function depicted in Figure 3.1-8; (b), (c), and (d) are CMP gathers derived from the rms velocities (indicated at the top of each gather) associated with the second, third, and fourth reflectors from the top. The traveltimes in (a) were derived using the raypath integral equations for a horizontally layered earth model.
- NMO for a flat reflector
- Fourth-order moveout
- NMO stretching
- NMO for a dipping reflector
- NMO for several layers with arbitrary dips
- Moveout velocity versus stacking velocity
- Topics in moveout and statics corrections
- Taner and Koehler, 1969, Taner, M. T. and Koehler, F., 1969, Velocity spectra — digital computer derivation and applications of velocity functions: Geophysics, 39, 859–881.
- Grant and West, 1965, Grant, F. S. and West, G. F., 1965, Interpretation theory in applied geophysics: McGraw-Hill Book Co.