# Fourth-order moveout

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

 ${\displaystyle t^{2}=C_{0}+C_{1}x^{2}+C_{2}x^{4}+C_{3}x^{6}+\cdots ,}$ (3)

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

Nevertheless, to compute a velocity spectrum using this equation requires scanning for two parameters — vrms and C2; thus, making equation (5a) cumbersome to use for velocity analysis. Below, a practical scheme to compute a velocity spectrum using equation (5a) is suggested:

1. Drop the fourth-order term to get the small-spread hyperbolic equation (4b). Compute the conventional velocity spectrum (Velocity analysis) by varying vrms in equation (4b), and pick an initial velocity function vrms(t0).
2. Use this picked velocity function in equation (5a) to compute a velocity spectrum by varying the parameter C2, and pick a function C2(t0).
3. Use the picked function C2(t0) in equation (5a) to recompute the velocity spectrum by varying vrms. Finally, pick an updated velocity function vrms(t0) from this velocity spectrum.

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

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

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

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.

1. Set S = 1 in equation (5b) to get equation (4b). Compute the velocity spectrum by varying vrms in equation (4b), and pick an initial velocity function vrms(t0).
2. Use this picked velocity function in equation (5b) and compute a velocity spectrum by varying the parameter S. Pick a function S(t0), and
3. use it in equation (5b) to recompute the velocity spectrum by varying vrms. Finally, pick an updated velocity function vrms(t0) from this velocity spectrum.

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

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

where t0 is the two-way zero-offset time, tp is related to the time at which the asymptotes of the hyperbolic traveltime trajectory converge (Section C.1), and vs is the reference velocity assigned to the layer below the recording surface (not the near-surface layer). When tp = t0, 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 vs. Then, for each output time t0 and for each offset x, apply time shift tp to traces in the CMP gather and compute the input time t for the offset under consideration. Compute a velocity spectrum for a range of tp values. Finally, pick a function tp(t0) from the velocity spectrum.

## References

1. Castle (1994), Castle, R. J., 1994, A theory of normal moveout: Geophysics, 59, 983–999.
2. De Bazelaire (1988), De Bazelaire, E., 1988, Normal moveout revisited: Inhomogeneous media and curved interfaces: Geophysics, 53, 143–157.
3. 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.