NMO for a dipping 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 3.1-13 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:

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}\sin ^{2}\alpha }{v^{2}}},}$ (7)

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 t0 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

 ${\displaystyle v_{NMO}={\frac {v}{\sin \alpha }}.}$ (8)

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

 ${\displaystyle \sin \alpha =\cos \phi ,}$ (9)

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

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}\cos ^{2}\phi }{v^{2}}}}$ (10)

and

 ${\displaystyle v_{NMO}={\frac {v}{\cos \phi }}.}$ (11)

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.

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

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 vNMO 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.