# Fourth-order moveout

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 |

A review of the moveout equation (**3**) to attain higher accuracy at far offsets is given in Section C.1. At first, it seems that including the terms up to the fourth-order in equation (**3**) should achieve this objective:

**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=C_0+C_1 x^2+C_2 x^4+C_3 x^6+\cdots,}****(**)

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

Nevertheless, to compute a velocity spectrum using this equation requires scanning for two parameters — *v _{rms}* and

*C*

_{2}; thus, making equation (

**5a**) cumbersome to use for velocity analysis. Below, a practical scheme to compute a velocity spectrum using equation (

**5a**) is suggested:

- Drop the fourth-order term to get the small-spread hyperbolic equation (
**4b**). Compute the conventional velocity spectrum (Velocity analysis) by varying*v*in equation (_{rms}**4b**), and pick an initial velocity function*v*(_{rms}*t*_{0}). - Use this picked velocity function in equation (
**5a**) to compute a velocity spectrum by varying the parameter*C*_{2}, and pick a function*C*_{2}(*t*_{0}). - Use the picked function
*C*_{2}(*t*_{0}) in equation (**5a**) to recompute the velocity spectrum by varying*v*. Finally, pick an updated velocity function_{rms}*v*(_{rms}*t*_{0}) from this velocity spectrum.

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

**Figure 3.1-3**NMO correction (equation**2a**) involves mapping nonzero-offset traveltime*t*onto zero-offset traveltime*t*_{0}. (a) Before and (b) after NMO correction.**Figure 3.1-4**Computational description of NMO correction. For a given integer value for*t*_{0}, and velocity*v*and offset*x*, compute*t*using equation (**1**). The amplitude at time*t*, denoted by*A*, does not necessarily fall onto an input integer sample location. By using two samples on each side of*t*(denoted by solid dots), we can interpolate between the four amplitude values to compute the amplitude value at*t*. This amplitude value then is mapped onto output integer sample*t*_{0}denoted by*A′*at the corresponding offset.**Figure 3.1-5**(a) CMP gather containing a single event with a moveout velocity of 2264 m/s, (b) NMO-corrected gather using the appropriate moveout velocity, (c) overcorrection because too low a velocity (2000 m/s) was used in equation (**2b**), and (d) undercorrection because too high a velocity (2500 m/s) was used in equation (**2b**).

Castle (1994)^{[1]} shows that a time-shifted hyperbola of the form

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

is an exact equivalent of the fourth-order moveout equation (**5a**). Here, *S* is a constant (Section C.1). For *S* = 1, equation (**5b**) reduces to the conventional small-spread moveout equation (**4b**).

As for the fourth-order moveout equation (**5a**), the time-shifted hyperbolic equation (**5b**) can, in principle, be used to conduct velocity analysis of CMP gathers.

**Figure 3.1-7**(a) A synthetic CMP gather derived from the velocity function depicted in Figure 3.1-8; (b), (c), and (d) are CMP gathers derived from the rms velocities (indicated at the top of each gather) associated with the second, third, and fourth reflectors from the top. The traveltimes in (a) were derived using the raypath integral equations for a horizontally layered earth model.

- Set
*S*= 1 in equation (**5b**) to get equation (**4b**). Compute the velocity spectrum by varying*v*in equation (_{rms}**4b**), and pick an initial velocity function*v*(_{rms}*t*_{0}). - Use this picked velocity function in equation (
**5b**) and compute a velocity spectrum by varying the parameter*S*. Pick a function*S*(*t*_{0}), and - use it in equation (
**5b**) to recompute the velocity spectrum by varying*v*. Finally, pick an updated velocity function_{rms}*v*(_{rms}*t*_{0}) from this velocity spectrum.

^{[2]} offers an alternative moveout equation to achieve higher-order accuracy at far offsets:

**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}},}****(**)

where *t*_{0} is the two-way zero-offset time, *t _{p}* is related to the time at which the asymptotes of the hyperbolic traveltime trajectory converge (Section C.1), and

*v*is the reference velocity assigned to the layer below the recording surface (not the near-surface layer). When

_{s}*t*=

_{p}*t*

_{0}, equation (

**5c**) reduces to the small-spread hyperbolic equation (

**4b**).

^{[3]} demonstrate the use of equation (**5c**) to obtain a stacked section with a higher stack power compared to the conventional stack derived from the small-spread moveout equation (**4b**). To use equation (**5c**) for velocity analysis, choose a fixed value of reference velocity *v _{s}*. Then, for each output time

*t*

_{0}and for each offset

*x*, apply time shift

*t*to traces in the CMP gather and compute the input time

_{p}*t*for the offset under consideration. Compute a velocity spectrum for a range of

*t*values. Finally, pick a function

_{p}*t*(

_{p}*t*

_{0}) from the velocity spectrum.

## See also

- main page Reflection_moveout.
- NMO for a flat reflector
- NMO in a horizontally stratified earth
- NMO stretching
- NMO for a dipping reflector
- NMO for several layers with arbitrary dips
- Moveout velocity versus stacking velocity
- Exercises
- Topics in moveout and statics corrections

## References

- ↑ Castle (1994), Castle, R. J., 1994, A theory of normal moveout: Geophysics, 59, 983–999.
- ↑ De Bazelaire (1988), De Bazelaire, E., 1988, Normal moveout revisited: Inhomogeneous media and curved interfaces: Geophysics, 53, 143–157.
- ↑ Thore and Kelly (1992), Thore, P. D. and Kelly, P., 1992, Stack enhancement using the three-term equation: Synthetic and real data examples: 62nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1550–1553.