B i n S i z e a n d L i n e a r v ( z )
`Christopher L. Liner *, University of Tulsa; Ralph Gobeli, Mercury International Technology
`
`ACQ 2.5
`
`S u m m a r y
`
`The choice of bin size is an important issue in 3-D
`seismic survey design. It effects acquisition cost, as
`well as quality of processing and interpretation. The
`standard formula for calculating bin size is based, ul-
`timately, on constant velocity assumptions. Here we
`demonstrate that a useful formulation of the prob-
`lem can be developed based on a linear V(Z) velocity
`model. The bin size calculated by the V(X) method
`can differ by 40% or more from the constant velocity
`estimate.
`
`I n t r o d u c t i o n
`
`Bin size for a 3-D seismic survey is designed to elim-
`inate, or minimize, spatial aliasing of the recorded
`wavefield. Data aliasing is detrimental to many seis-
`mic processing steps, and ultimately degrades the
`final image quality. But while data quality argues
`for a small bin, economics drive the system toward
`larger bins which lower acquisition and processing
`costs. For a given acquisition area, the costs increase
`dramatically with decreasing bin size. Clearly, bin
`size determination is an important technical and eco-
`nomic issue.
`
`In this paper we present a method for estimating bin
`size based on a linear
` velocity model. This rep
`resents a move toward reality from the standard prac-
`tice of constant velocity calculations (Yilmaz, 1987).
`To be fair, we expect that many major companies and
`operators are currently using some form of
` for
`bin calculations. However, the only mention of this
`practice we find in the literature is Bee et al. (1994),
`where linear
` seems to be applied to migration
`distance estimates not bin size calculations.
`
`Spatial Aliasing Review
`
`Spatial aliasing occurs when wavelet time delay be-
`tween adjacent traces is greater than half the wavelet
`period (Yilmaz, 1987). This is termed “data alias-
`ing” to distinguish it from “process aliasing” which
`can (but should not) be produced by migration, dip-
`moveout (DMO), radon transforms, and other seismic
`processing steps. For a band-limited wavelet, data
`aliasing has the effect of aligning sidelobes along a
`slope antithetic to the true slope. Performance of
`multichannel programs like migration and DMO are
`degraded by these false slopes. There has been work
`done on “de-aliasing” 3-D seismic data, but the pre-
`ferred path is to record unaliased data in the field.
`
`47
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`The conventional condition for
`aliasing for a reflected arrival is
`
`avoidance of data
`
`
`
` Period/Slope ,
`
`where the wavelet period is given by
`
`is the dominant frequency. The slope used
`and
`in this calculation is the constant velocity, zero offset
`slope (Yilmaz, 1987) given by
`
`where y is the midpoint coordinate,
` is velocity, and
` is the geological dip. With these definitions, the
`bin estimate becomes
`
`
`
`
`
` 2
`
`
`
` l
`
` This
` =
`The safest bin estimate is based on
`allows for fault diffraction limbs which have a slope
`equivalent to a 90° dipping bed.
`
`Construction For Linear v(z)
`
`The theory of seismic raypaths in linear v(z) media
`is well-known (Slotnick, 1959; Telford et al., 1976).
`Figure 1 shows the geometry for our problem. The
`source and receiver are coincident on the earth sur-
`face, and velocity in the earth increases linearly with
`depth,
` =
` +
` where k is the velocity gradi-
` The constant velocity raypath is
`ent in
`straight and its reflection time is t,, while the v(z)
`ray is a circular arc with reflection time t,. Both
`the source point and the curved ray reflection point
`are along radii from the center of a circle above the
`acquisition surface (Slot nick, 1959).
`
`From the geometry, we derive a relationship for the
`reflection time
` The event slope can then be
` and the bin size,
` is found
`derived from
`by applying Equation (1). The result is
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`Downloaded 04/18/14 to 173.226.64.254. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`1
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`ION 1044
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`

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`Bin Size and Linear v(z)
`
`Conclusions
`
`Linear
` can be incorporated into bin size calcu-
` bin, bin,, tends to be less than
`lations. The
`the constant velocity bin,
` but this is not always
`the case. If
` is used in 3-D survey design and
` bin, then steep dips will tend to be aliased.
`bin,
` bin, then extra costs are incurred for a bin
`If bin,
`smaller than required. Fine tuning the bin size may
`allow cost savings and data quality improvements,
`particularly with respect to amplitude information
`which is easily degraded by data aliasing.
`
`References
`
`Bee, M. F., Bearden, J. M., Herkenhoff, E. F.,
`Supiyanto, H., and Koestoer, B., 1994, Efficient
`3-D seismic surveys in a jungle environment,
`First Break, vol. 12, no.5 , p.253-259.
`
`Slotnick, M. M., 1959, Lessons in Seismic Comput-
`ing, Society of Exploration Geophysicists, Tulsa,
`OK. (reprint 1989)
`
`Telford, W. M., Geldart, L. P., Sheriff, R. E., and
`Keys, D. A., 1976, Applied Geophysics, Cam-
`bridge University Press, New York, NY. (reprint
`1981)
`
`Yilmaz, O., 1987, Seismic Data Processing, Society
`of Exploration Geophysicists, Tulsa, OK.
`
` is the
`In these formulae, p is the ray parameter,
`take-off angle, and w is a temporary variable to sim-
`plify the slope expression.
`
`E x a m p l e s
`
`Consider a Gulf Coast 3-D survey being planned over
`a target horizon at a depth of 5000 feet. We use the
`standard velocity function of
` = 6000 + 0.6
`ft/sec (Slotnick, 1959). For the constant velocity cal-
`culation we use the average velocity of 7500 ft/sec.
`The dominant frequency is set to 40 Hz. The bin size
`calculation is done for
` = 45 and 85 degrees. The
`results are
`
`
`
` = 85 :
`
`b i n ,
`.
`
` 1 3 2 . 6 f t
`= 1 3 5 . 2 f t
` = 9 4 . 5 f t
`= 7 5 . 6 f t
`
`.
`
`If the 85 degree bin, estimate is used in an area
`where this velocity gradient is present, dips beyond
`about 60 - 65 degrees will be spatially aliased (see
`Figure ??). Repeating the same calculation for a
`stronger gradient, k = .9, yields
`
` = 45 :
`
`b i n , = 1 4 5 . 8 f t
`.
`= 1 5 1 . 2 f t
` = 1 0 3 . 5 f t
`v = 75.9 ft .
`
`.
`
`These examples do not suggest a simple rule such as
`
` Figure 2 shows a plot of bin size versus
`
`dip. Parameters are those of the first example above.
`It can be seen that the constant velocity calculation
`recommends too-small a bin for
`
` 45, and too-large
`a bin for
`
` 45. This is also seen in Figure 3, where
`the effect of variable velocity gradient is added. The
`percent difference between bin, and
` is largest for
`steep dips and strong velocity gradients.
`
`48
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`
`2
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`Bin Size and Linear v(z)
`
`Figure 1: Geometry for the linear
`arc. See text for details.
`
` dipping reflector problem. As is well known, the
`
` ray is a circular
`
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`3
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`Bin Size and Linear
`
` Parameters
`Figure 2: Comparison of bin size recommendation based on constant velocity and linear
`for this test are given in the text. By using a constant velocity bin, we either use too small a bin (dip<45)
`and waste money, or use too large a bin (dip>45) and risk spatial aliasing.
`
`Figure 3: Percent difference between bin, and bin,, as a function of dip and velocity gradient, k. Negative
`values indicate bin, > bin,. For steep dips and strong velocity gradients, the constant velocity bin can easily
`be 30% too large. This could lead to significant spatial aliasing.
`
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