# Difference between revisions of "NMO for several layers with arbitrary dips"

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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 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′''. | ||

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+ | [[file:ch03_fig1-15.png|thumb|{{figure number|3.1-15}} Geometry for the moveout for a dipping interface in an earth model with layers of arbitrary dips. Adapted from <ref name=ch03r18/>.)]] | ||

<ref name=ch03r18>Hubral and Krey (1980), Hubral, P. and Krey, T., 1980, Interval velocities from seismic reflection time measurements: Soc. Expl. Geophys.</ref> derived the expression for traveltime ''t'' along ''SDG'' as | <ref name=ch03r18>Hubral and Krey (1980), Hubral, P. and Krey, T., 1980, Interval velocities from seismic reflection time measurements: Soc. Expl. Geophys.</ref> derived the expression for traveltime ''t'' along ''SDG'' as | ||

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

− | + | {{NumBlk|:|<math>v^2_{rms}=\frac{1}{t_0}\sum _{i=1}^N v^2_i\Delta\tau_i,</math>|{{EquationRef|4a}}}} | |

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+ | {{NumBlk|:|<math>t^2=t^2_0+\frac{x^2}{v^{2}_{rms}}.</math>|{{EquationRef|4b}}}} | ||

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+ | {{NumBlk|:|<math>t^2=t^2_0+\frac{x^2}{v^{2}_{rms}}+C_2x^4.</math>|{{EquationRef|5a}}}} | ||

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+ | {{NumBlk|:|<math>t=t_0\left(1-\frac{1}{S}\right)+\sqrt{\left(\frac{t_0}{S}\right)^2+\frac{x^2}{Sv^{2}_{rms}}}</math>|{{EquationRef|5b}}}} | ||

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+ | {{NumBlk|:|<math>t=(t_0-t_p)+\sqrt{t^2_p+\frac{x^2}{v_s^2}},</math>|{{EquationRef|5c}}}} | ||

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+ | {{NumBlk|:|<math>v_{NMO}=\frac{v}{\sin \alpha}.</math>|{{EquationRef|8}}}} | ||

==See also== | ==See also== |

## Latest revision as of 12:08, 18 September 2014

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

**(**)

where the NMO velocity is given by

**(**)

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**).

**(**)

**(**)

**(**)

**(**)

**(**)

**(**)

## 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.