# NMO for several layers with arbitrary dips

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

Figure 3.1-15 shows a 2-D subsurface geometry that is composed of a number of layers, each with an arbitrary dip. We want to compute the traveltime from source location S to depth point D, then back to receiver location G, which is associated with midpoint M. Note that the CMP ray from midpoint M hits the dipping interface at normal incidence at D′, which is not the same as D. The zero-offset time is the two-way time along the raypath from M to D′.

[1] derived the expression for traveltime t along SDG as

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{NMO}^{2}}}+higher\ order\ terms,}$ (12)

where the NMO velocity is given by

 ${\displaystyle v_{NMO}^{2}={\frac {1}{t_{0}\cos ^{2}\beta _{0}}}\sum \limits _{i=1}^{N}v_{i}^{2}\Delta t_{i}\prod \limits _{k=1}^{i-1}\left({\frac {\cos ^{2}\alpha _{k}}{\cos ^{2}\beta _{k}}}\right).}$ (13)

The angles α and β are defined in Figure 3.1-15. For a single dipping layer, equation (13) reduces to equation (8). Moreover, for a horizontally stratified earth, equation (13) reduces to equation (4). As long as the dips are gentle and the spread is small, the traveltime equation is approximately represented by a hyperbola (equation 5), and the velocity required for NMO correction is approximately the rms velocity function (equation 4).

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).