# 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′.

Figure 3.1-15  Geometry for the moveout for a dipping interface in an earth model with layers of arbitrary dips. Adapted from [1].)

[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).

 ${\displaystyle v_{rms}^{2}={\frac {1}{t_{0}}}\sum _{i=1}^{N}v_{i}^{2}\Delta \tau _{i},}$ (4a)

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{rms}^{2}}}.}$ (4b)

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{rms}^{2}}}+C_{2}x^{4}.}$ (5a)

 ${\displaystyle t=t_{0}\left(1-{\frac {1}{S}}\right)+{\sqrt {\left({\frac {t_{0}}{S}}\right)^{2}+{\frac {x^{2}}{Sv_{rms}^{2}}}}}}$ (5b)

 ${\displaystyle t=(t_{0}-t_{p})+{\sqrt {t_{p}^{2}+{\frac {x^{2}}{v_{s}^{2}}}}},}$ (5c)

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