Fourth-order moveout

Seismic Data Analysis

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:

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

(3)

$t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{rms}^{2}}}+C_{2}x^{4}.$

(5a)

Nevertheless, to compute a velocity spectrum using this equation requires scanning for two parameters — v_rms 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 v_rms in equation (4b), and pick an initial velocity function v_rms(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 v_rms. Finally, pick an updated velocity function v_rms(t₀) from this velocity spectrum.

$t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{rms}^{2}}}.$

(4b)

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  Figure 3.1-2  Synthetic CMP gather associated with the geometry in Figure 3.1-1. Traveltime curve for a flat reflector is a hyperbola with its apex at zero-offset trace.

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  Figure 3.1-3  NMO correction (equation 2a) involves mapping nonzero-offset traveltime t onto zero-offset traveltime t₀. (a) Before and (b) after NMO correction.

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  Figure 3.1-4  Computational description of NMO correction. For a given integer value for t₀, 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 t₀ denoted by A′ at the corresponding offset.

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  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) shows that a time-shifted hyperbola of the form

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

(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.

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  Figure 3.1-6  A horizontally layered earth model geometry. The traveltime to the deepest reflector is described by equation (3). An approximate form of this equation used in practice is given by equation (4b).

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  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.

1. Set S = 1 in equation (5b) to get equation (4b). Compute the velocity spectrum by varying v_rms in equation (4b), and pick an initial velocity function v_rms(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 v_rms. Finally, pick an updated velocity function v_rms(t₀) from this velocity spectrum.

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

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

(5c)

where t₀ is the two-way zero-offset time, t_p is related to the time at which the asymptotes of the hyperbolic traveltime trajectory converge (Section C.1), and v_s is the reference velocity assigned to the layer below the recording surface (not the near-surface layer). When t_p = t₀, equation (5c) reduces to the small-spread hyperbolic equation (4b).

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 t₀ and for each offset x, apply time shift t_p 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 t_p values. Finally, pick a function t_p(t₀) 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

External links

find literature about Fourth-order moveout