# NMO for a flat reflector

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

Figure 3.1-1 shows the simple case of a single horizontal layer. At a given midpoint location M, we want to compute the reflection traveltime t along the raypath from shot position S to depth point D then back to receiver position G. Using the Pythagorean theorem, the traveltime equation as a function of offset is

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}}{v^{2}}},}$ (1)

where x is the distance (offset) between the source and receiver positions, v is the velocity of the medium above the reflecting interface, and t0 is twice the traveltime along the vertical path MD. Note that vertical projection of depth point D to the surface, along the normal to the reflector, coincides with midpoint M. This occurs only when the reflector is horizontal.

Equation (1) describes a hyperbola in the plane of two-way time versus offset. Figure 3.1-2 is an example of traces in a common-midpoint (CMP) gather. The figure also represents a common-depth-point (CDP) gather, since all the raypaths associated with each source-receiver pair reflect from the same subsurface depth point D. The offset range in Figure 3.1-2 is 0 to 3150 m, with a 50-m trace separation. The medium velocity above the reflector is 2264 m/s. All of the traces in this CMP gather contain a reflection from the same depth point.

From equation (1), we see that velocity can be computed when offset x and two-way times t and t0 are known. Once the NMO velocity is estimated, the traveltimes can be corrected to remove the effect of offset as shown in Figure 3.1-3. Traces in the NMO-corrected gather then are summed to obtain a stack trace at the particular CMP location.

The numerical procedure involved in hyperbolic moveout correction is illustrated in Figure 3.1-4. The idea is to find the amplitude value at A′ on the NMO-corrected gather from the amplitude value at A on the original CMP gather. Given quantities t0, x, and vNMO, compute t from equation (1). Assume that this is 1003 ms. If the sampling interval were 4 ms, then this time is equivalent to the 250.25 sample index. The amplitude value at this time can be computed using the amplitudes at the neighboring integer sample values, two on each side — at 248, 249, 251, and 252 sample indexes. This is done by an interpolation scheme that involves the four samples.

An alternative numerical method for mapping trace amplitudes from a nonzero-offset to zero offset involves a nearest-neighbor sample as the output value. Accurate implementation of this method requires, first, oversampling the traces in a CMP gather along the time axis. Specifically, for each trace in the CMP gather, perform 1-D Fourier transform and pad the frequency axis with zeroes, usually by a factor of eight. Then, inverse transform back to the time domain and obtain a trace which has eight times as many samples at a sampling interval that is one-eighth of the original sampling rate. Now, given quantities t0, x, and vNMO, again, compute t from equation (1). Assume that this is t = 1003.4 ms. If the original sampling interval were 4 ms, after oversampling, the new sampling interval is 0.5 ms. Then the amplitude at t = 1003.4 ms can be borrowed from the nearest-neighbor sample with an index of 2006 without much sacrifice in accuracy.

The NMO correction is given by the difference between t and t0:

 ${\displaystyle \Delta t_{NMO}=t-t_{0},}$ (2a)

or, by way of equation (1),

 ${\displaystyle \Delta t_{NMO}=t_{0}\left[{\sqrt {1+\left({\frac {x}{v_{NMO}t_{0}}}\right)^{2}}}-1\right].}$ (2b)

Table 3-1 shows the moveout corrections for two different offset values using a realistic velocity function that increases with reflector depth.

From Table 3-1, note that the NMO increases with offset and decreases with zero-offset time, hence, with depth. The NMO also is smaller for higher velocities, and the combined effect of higher velocities at larger depths makes it much smaller.

For a flat reflector with an overlying homogeneous medium, the reflection hyperbola can be corrected for offset if the correct medium velocity is used in the NMO equation. From Figure 3.1-5, if a velocity higher than the actual medium velocity (2264 m/s) is used, then the hyperbola is not flattened completely. This is called undercorrection. On the other hand, if a lower velocity is used, then overcorrection results.

 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

Figure 3.1-5 also illustrates the basis of conventional velocity analysis. NMO correction is applied to the input CMP gather using a number of trial constant velocity values in equation (2b). The velocity that best flattens the reflection hyperbola is the velocity that best corrects for NMO before stacking the traces in the gather. Furthermore, for a simple case of a single horizontal reflector, this velocity also is equal to the velocity of the medium above the reflector.