# NMO stretching

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Figure 3.1-9b shows the CMP gather in Figure 3.1-7a after NMO correction. The rms velocity function shown in Figure 3.1-8 was used in equation (2b) for this correction. As a result of the NMO correction, a frequency distortion occurs, particularly for shallow events and at large offsets. This is called NMO stretching and is illustrated in Figure 3.1-10. The waveform with a dominant period T is stretched so that its period T0, after NMO correction, is greater than T. Stretching is a frequency distortion in which events are shifted to lower frequencies. Stretching is quantified by

 ${\displaystyle {\frac {\Delta f}{f}}={\frac {\Delta t_{NMO}}{t_{0}}},}$ (6)

where f is the dominant frequency, Δf is the change in frequency, and ΔtNMO is given by equation (2b). The derivation of equation (6) is given in Section C.1.

Table 3-2 lists the percent frequency changes caused by the NMO stretching associated with the velocity function in Table 3-1. Note that stretching is confined mainly to large offsets and shallow times. For example, a waveform with a 30-Hz dominant frequency at 2000-m offset and t0 = 0.25 s shifts to nearly 10 Hz after NMO correction.

 ΔtNMO, in s ΔtNMO, in s t0, s vNMO, m/s x = 1000 m x = 2000 m 0.25 2000 0.309 0.780 0.5 2500 0.140 0.443 1 3000 0.054 0.201 2 3500 0.020 0.080 4 4000 0.008 0.031
 %Δf/f for %Δf/f for t0, s vNMO, m/s x = 1000 m x = 2000 m 0.25 2000 123 312 0.5 2500 28 89 1 3000 5 20 2 3500 1 4 4 4000 0.2 0.8

Because of the stretched waveform at large offsets, stacking the NMO-corrected CMP gather (Figure 3.1-9b) will severely damage the shallow events. This problem can be circumvented by muting the stretched zones in the gather. Automatic muting is done by using the quantitative definition of stretching given by equation (6). Figures 3.1-9c and d show two versions of the CMP gather after NMO correction and muting; one version has a stretch limit of 50 percent, while the other has a stretch limit of 100 percent. The 50-percent stretch limit does not show significant frequency distortion. However, the stretch limit can be extended to 100 percent because we want to include as much of the CMP gather in the stack as possible without degradation.

A trade-off exists between the signal-to-noise ratio and mute. In particular, if the signal-to-noise ratio is good, then it may be preferable to mute more than stretch mute requirements to preserve signal bandwidth. On the other hand, if the signal-to-noise ratio is poor, it may be necessary to accept a large amount of stretch to get any events on the stack. A real data example is provided in Figure 3.1-11. Here, the stretched zone is seen as the low-frequency zone at the shallow part of the CMP gathers without mute applied.

Another method for optimum selection of the mute zone is to progressively stack the data. Figure 3.1-12a is an NMO-corrected CMP gather without mute applied. Figure 3.1-12b shows the stack traces derived from the CMP gather (Figure 3.1-12a). The far right trace is the same as the far right trace in the input CMP gather. The second trace from the right is the sum of the two near-offset traces, and so on, progressively increasing the stacking fold. The far left trace is the full-fold stack of the input CMP gather. By following the waveform along a certain event and observing where changes occur, the mute zone is derived as shown in Figure 3.1-12b. A similar procedure can be followed to determine an inside mute. This time, the stacking fold is progressively increased in the near-offset direction.

Aside from the signal-to-noise ratio, attenuation of multiples dictates the choice of a suitable mute pattern. Specifically, large offsets often are needed to attenuate multiple reflections based on moveout discrimination between primaries and multiples. An inside mute may be needed in addition to the application of a multiple attenuation technique to alleviate the small move-out discrimination between primaries and multiples at small offsets. An inside mute also may be applied to land records to suppress ground-roll energy and air waves associated with surface sources.

Muting also is dependent upon the ultimate use of the stacked data. If the stacked data are intended as input to amplitude inversion, you may want to apply a harsh mute to minimize the angle-dependency of reflection amplitudes inferred by the Zoeppritz equations (acoustic impedance estimation). If the stacked data are intended as input to poststack depth migration, then a limited-offset stack may be needed to minimize the amplitude and traveltime distortions caused by the nonhyperbolic moveout associated with reflections below complex overburden structures (introduction to earth imaging in depth).