# NMO for a dipping reflector

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

Figure 3.1-13 depicts a medium with a single dipping reflector. We want to compute the traveltime from source location *S* to the reflector at depth point *D*, then back to receiver location *G*. For the dipping reflector, midpoint *M* is no longer a vertical projection of the depth point to the surface. The terms *CDP gather* and *CMP gather* are equivalent only when the earth is horizontally stratified. When there is subsurface dip or lateral velocity variation, the two gathers are different. Midpoint *M* and the normal-incidence reflection point *D′* remain common to all of the source-receiver pairs within the gather, regardless of dip. Depth point *D*, however, is different for each source-receiver pair in a CMP gather recorded over a dipping reflector.

^{[1]}, using the geometry of Figure 3.1-13, derived the following two-dimensional (2-D) traveltime equation for a dipping reflector:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t^2=t_0^2+\frac{x^2\sin^2\alpha}{v^2},}****(**)

where the two-way traveltime *t* is associated with the nonzero-offset raypath *SDG* from source *S* to reflection point *D* to receiver *G*, the two-way zero-offset time *t*_{0} is associated with the normal-incidence raypath *MD′* at midpoint *M*, and *α* is the angle between the normal to the dipping reflector and the direction of the line of recording (Figure 3.1-13). The moveout velocity is then given by

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_{NMO}=\frac{v}{\sin \alpha}.}****(**)

For the 2-D geometry of the dipping reflector shown in Figure 3.1-13, note that

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sin\alpha=\cos\phi,}****(**)

where *ϕ* is the dip angle of the reflector. Hence, equations (**7**) and (**8**) are written in terms of the reflector dip *ϕ*

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t^2=t^2_0+\frac{x^2\cos^2\phi}{v^2}}****(**)

and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_{NMO}=\frac{v}{\cos \phi}.}****(**)

The traveltime equation (**10**) for a dipping reflector represents a hyperbola as for the flat reflector (equation **1**). However, the NMO velocity now is given by the medium velocity divided by the cosine of the dip angle as defined by equation (**11**). This equation indicates that proper stacking of a dipping event requires a velocity that is greater than the velocity of the medium above the reflector.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t^2=t_0^2+\frac{x^2}{v^2},}****(**)

In conclusion, the NMO velocity for a dipping reflector depends on the dip angle. The larger the dip angle, the higher the moveout velocity, hence the smaller the moveout. There is a 4 percent difference between moveout velocity *v _{NMO}* and medium velocity

*v*for a 15-degree dip. The difference is 50 percent at a 30-degree dip and rapidly increases at steep dips. An accompanying observation is that a horizontal layer with a high velocity can yield the same moveout as a dipping layer with a low velocity, as illustrated in Figure 3.1-14.

**Figure 3.1-14**Moveout for low-velocity event (a) is larger than for high-velocity event (b). Moveout for low-velocity dipping event (c) may not be distinguishable from high-velocity horizontal event (b). These observations are direct consequences of equation (**7**).

## See also

- NMO for a flat reflector
- NMO in a horizontally stratified earth
- Fourth-order moveout
- NMO stretching
- NMO for several layers with arbitrary dips
- Moveout velocity versus stacking velocity
- Exercises
- Topics in moveout and statics corrections

## References

- ↑ Levin (1971), Levin, F. K., 1971, Apparent velocity from dipping interface reflections: Geophysics, 36, 510–516.