A review of the moveout equation (3) to attain higher accuracy at far offsets is given in Section C.1. At first, it seems that including the terms up to the fourth-order in equation (3) should achieve this objective:
Nevertheless, to compute a velocity spectrum using this equation requires scanning for two parameters — vrms and C2; thus, making equation (5a) cumbersome to use for velocity analysis. Below, a practical scheme to compute a velocity spectrum using equation (5a) is suggested:
- Drop the fourth-order term to get the small-spread hyperbolic equation (4b). Compute the conventional velocity spectrum (Velocity analysis) by varying vrms in equation (4b), and pick an initial velocity function vrms(t0).
- Use this picked velocity function in equation (5a) to compute a velocity spectrum by varying the parameter C2, and pick a function C2(t0).
- Use the picked function C2(t0) in equation (5a) to recompute the velocity spectrum by varying vrms. Finally, pick an updated velocity function vrms(t0) from this velocity spectrum.
Figure 3.1-3 NMO correction (equation 2a) involves mapping nonzero-offset traveltime t onto zero-offset traveltime t0. (a) Before and (b) after NMO correction.
Figure 3.1-4 Computational description of NMO correction. For a given integer value for t0, and velocity v and offset x, compute t using equation (1). The amplitude at time t, denoted by A, does not necessarily fall onto an input integer sample location. By using two samples on each side of t (denoted by solid dots), we can interpolate between the four amplitude values to compute the amplitude value at t. This amplitude value then is mapped onto output integer sample t0 denoted by A′ at the corresponding offset.
Figure 3.1-5 (a) CMP gather containing a single event with a moveout velocity of 2264 m/s, (b) NMO-corrected gather using the appropriate moveout velocity, (c) overcorrection because too low a velocity (2000 m/s) was used in equation (2b), and (d) undercorrection because too high a velocity (2500 m/s) was used in equation (2b).
Castle (1994) shows that a time-shifted hyperbola of the form
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.
- Set S = 1 in equation (5b) to get equation (4b). Compute the velocity spectrum by varying vrms in equation (4b), and pick an initial velocity function vrms(t0).
- Use this picked velocity function in equation (5b) and compute a velocity spectrum by varying the parameter S. Pick a function S(t0), and
- use it in equation (5b) to recompute the velocity spectrum by varying vrms. Finally, pick an updated velocity function vrms(t0) from this velocity spectrum.
 offers an alternative moveout equation to achieve higher-order accuracy at far offsets:
where t0 is the two-way zero-offset time, tp is related to the time at which the asymptotes of the hyperbolic traveltime trajectory converge (Section C.1), and vs is the reference velocity assigned to the layer below the recording surface (not the near-surface layer). When tp = t0, equation (5c) reduces to the small-spread hyperbolic equation (4b).
 demonstrate the use of equation (5c) to obtain a stacked section with a higher stack power compared to the conventional stack derived from the small-spread moveout equation (4b). To use equation (5c) for velocity analysis, choose a fixed value of reference velocity vs. Then, for each output time t0 and for each offset x, apply time shift tp to traces in the CMP gather and compute the input time t for the offset under consideration. Compute a velocity spectrum for a range of tp values. Finally, pick a function tp(t0) from the velocity spectrum.
- NMO for a flat reflector
- NMO in a horizontally stratified earth
- 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
- Castle (1994), Castle, R. J., 1994, A theory of normal moveout: Geophysics, 59, 983–999.
- De Bazelaire (1988), De Bazelaire, E., 1988, Normal moveout revisited: Inhomogeneous media and curved interfaces: Geophysics, 53, 143–157.
- Thore and Kelly (1992), Thore, P. D. and Kelly, P., 1992, Stack enhancement using the three-term equation: Synthetic and real data examples: 62nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1550–1553.