# NMO for several layers with arbitrary dips

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 3.1-15 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

**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}{v^2_{NMO}}+higher\ order\ terms,}****(**)

where the NMO velocity is 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^2_{NMO}=\frac{1}{t_0\cos^2\beta_0}\sum\limits^{N}_{i=1}v^2_i\Delta t_i \prod\limits^{i-1}_{k=1}\left(\frac{\cos^2\alpha_k}{\cos^2\beta_k}\right).}****(**)

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**).

**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^2_{rms}=\frac{1}{t_0}\sum _{i=1}^N v^2_i\Delta\tau_i,}****(**)

**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}{v^{2}_{rms}}.}****(**)

**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}{v^{2}_{rms}}+C_2x^4.}****(**)

**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=t_0\left(1-\frac{1}{S}\right)+\sqrt{\left(\frac{t_0}{S}\right)^2+\frac{x^2}{Sv^{2}_{rms}}}}****(**)

**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=(t_0-t_p)+\sqrt{t^2_p+\frac{x^2}{v_s^2}},}****(**)

**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}.}****(**)

## See also

- NMO for a flat reflector
- NMO in a horizontally stratified earth
- Fourth-order moveout
- NMO stretching
- NMO for a dipping reflector
- Moveout velocity versus stacking velocity
- Exercises
- Topics in moveout and statics corrections

## References

- ↑
^{1.0}^{1.1}Hubral and Krey (1980), Hubral, P. and Krey, T., 1980, Interval velocities from seismic reflection time measurements: Soc. Expl. Geophys.