Ex. PGS 1054
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`EX. PGS 1054
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`GEOPHYSICS. VOL. 49. NO. 8 (AUGUST
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`1984); P. 13861387. 3 FIGS
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`Short Note
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`The effect of binning on data from a feathered streamer
`
`Franklyn K. Levin*
`
`In a recent paper (Levin, 1983) I derived the common mid-
`point (CMP) time-distance relations for reflection from a plane
`of arbitrary orientation when the streamer carrying the hydro-
`phones was feathered by currents. The streamer was assumed
`to be straight but deviated at a constant angle y. This assump-
`tion corresponds well to what is observed during marine ex-
`ploration. I considered two cases: first, the CMP points chosen
`to be what they would have been in the absence of feathering
`and second, the CMP points selected to lie along a line perpen-
`dicular to the profile direction. The second choice is the one
`normally made and the only one I’ll consider in this short note.
`In the paper, I assumed the CMP points were allowed to
`extend from the origin, which was the point of coincident
`source and detector, to whatever distance from the line corre-
`sponded to the maximum source-to-detector separation. While
`this is a possible choice, it is not the choice of many of those
`who must process marine 3-D data. Instead they demand that
`all CMP points fall within a bin centered on the profile line. I
`shall call this choice “binning.”
`The algebra required to investigate binning is essentially the
`same as that laid out in the 1983 paper. There are minor
`differences, which are given in the Appendix. Here I shall
`simply show results. When the profile lines are shot in the dip
`direction, binning need not be considered, since, by my choice
`of points perpendicular to the profile direction, all the CMP
`points fall along strike and are the same distance from the plane
`reflector. Figure 1 is an example for rather large feathering (30
`degrees) and a 15 degree dip. Only for large values of source-to-
`detector separations X does the time-distance hyperbola differ
`appreciably from what it would be if there were no feathering.
`For profiles not shot in a dip direction, the effects of binning
`become apparent. The maximum distortion of time-distance
`plots due to feathering is seen for lines shot in the strike
`direction. Hence, they are data recorded for strike profiles we
`consider here and in the Appendix. When the 3-D lines are
`strike lines, binning tends to chop up the data. In order to have
`CMP points for all source-to-detector separations fall within a
`bin, traces must be pulled from different lines of the survey. For
`a survey shot in the strike direction, different lines of the survey
`are at different distances from the reflector: binning assembles
`data selected from time-distance hyperbolas having different t,
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`1 50
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`FIG. 1. Time-distance nlots for a dip line. The dip is 15 degrees;
`the feather angle, 30 degrees. The depth to the reflector is-3048
`m (10 000 ft) and the velocity is 3657.5 m/s (12 000 ftis).
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`values. The results for a specific choice of parameters are shown
`in Figures 2 and 3. Bins were selected to extend to half the
`separation of the survey lines. The data were broken into pieces
`by binning with those points of a given piece corresponding to
`CMP points that fell within the bin limits. No restriction was
`placed on the in-line extent of a bin, the requirement that bins
`extend to half the line separation being sufficient Depths to the
`reflector increased from 3048 m (10 000 ft) for a source-to-
`detector separation of zero to 3246 m (10 648 ft) for a source-
`to-detector separation of 3048 m for Figure 2 and decreased to
`2989 m (9806 ft) for Figure 3.
`Superimposed on the data of Figures 2 and 3 are the time-
`distance hyperbolas corresponding to no feathering and a
`depth to the reflector of 3048 m. In an average sense, the data
`fall around the hyperbolas. As shown in the Appendix, those
`points that lie at the center of the bin differ from the non-
`feathering values by amounts that are nearly proportional to
`the source-to-detector separation but are very small until that
`separation becomes large.
`A reader may remark on the smoothness of the points com-
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`Manuscript received by the Editor December 5, 1983; revised manuscript received February 3, 1984.
`*Exxon Production Research Company, P.O. Box 2189, Houston, TX 77001.
`(’ 1984 Society of Exploration Geophysicists. All rights reserved.
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`1388
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`Ex. PGS 1054
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`FIG. 2. Time-distance plots for a strike line. The dip is 15
`degrees; the feather angle, 30 degrees (updip). The depth to the
`reflector is 3048 m (10 000 ft) and the velocity is 3658 m/s
`(12 000 ft/s).
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`FIG. 3. Time-distance plots for a strike line. The dip is 15
`degrees; the feather angle, - 30 degrees (downdip). The depth
`to the reflector is 3048 m (10 000 ft) and the velocity is 3658 m/s
`(12 000 ft/s).
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`prising each piece of Figures 2 and 3. That smoothness stems
`from the assumption that any source-to-detector separation
`needed to let a CMP point fall precisely on the line perpendicu-
`lar to the profile is available. In practice, only by chance would
`these assumed values be those allowed by a fixed arrangement
`of hydrophone stations and preassigned shot positions. We
`would expect deviations from the perfection of each small piece.
`All the limitations laid out in earlier papers (Levin, 1971,
`1983) are applicable to the results discussed here. The system
`being considered is unrealistically simple but is the one often
`assumed for velocity determination. To the extent that the
`behavior illustrated by Figures 1 to 3 corresponds to that found
`in the field, the moral is clear: if currents are expected to be
`strong, lay out your lines in a dip direction or be prepared to
`correct the recorded data.
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`REFERENCES
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`Levin, F. K., 1971, Apparent velocity from dipping interface reflec-
`tions: Geophysics, 36,51@516.
`~~ 1983, The effects of streamer feathering on stacking: Geophys-
`ics.48, 1165-l 171.
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`LO, (X/2) sin Y, 01.
`If we bin our data, points within a bin have coordinates
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`(A-3)
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`[O, Y + (X/2) sin y, 01
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`and we select data from different parallel lines such that
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`I Y + (X/2) sin y I
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`is smaller than the half-width of the bin chosen. The center of
`the bin is at (0, 0, 0). This implies that at the center of the bin
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`Y + (X/2) sin y = 0.
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`Substituting from equation (A-4) into (A-2) gives
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`D’ = D + (X/2) sin y sin 4
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`(A-4)
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`(A-5)
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`for the center of the bin.
`We want to know by what amount the time for data at the
`center of the bin differs from the time we’d compute for the
`same SGD if there were no feathering. Write
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`&V2t2) = 4(D’)‘(l - sin* 4 sin2 u)
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`APPENDIX
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`+ (X - 20’ sin $ sin r)* - [4D* + X2].
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`(A-6)
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`The time-distance relation for reflection data collected along
`a strike line is
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`Vzfz = 4D2(1 - sin’ 4 sin* y)
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`+ (X - 20 sin I$ sin r)*,
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`(A-l)
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`where 4 is the dip angle and y is the feathering angle. There is a
`misprint in equation (8) of Levin (I 983). D is the perpendicular
`distance from the origin (CMP point) to the reflecting plane. X
`is the source-to-detector distance SGD and I/, the average
`velocity to the reflector. If instead of collecting data along the
`x-axis we collect data along a line parallel to the x-axis but
`displaced a distance Y, we replace D with
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`Substituting from equation (A-5) into equation (A-6) gives
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`@V2t2) = 4(D + X/2 sin + sin r)*(l
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`- sin* 4 sin* y)
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`+ (X - 20 sin 4 sin y - X sin* 4 sin* r)2
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`- [4D2 + X2],
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`s(V2r2) = -(X sin + sin r)*,
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`or
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`6r2 = -[(X/V)
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`sin + sin r]‘.
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`If we assume fit* is a continuous function, we can write
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`(A-6a)
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`(A-7)
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`D’ = D - Y sin 4.
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`(A-2)
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`6t = &2/(2t).
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`(A-8)
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`For CMP points chosen in a direction perpendicular to the
`profile line (x-axis), the CMP points have coordinates
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`Except for sign, equations (A-7) and (A-8) also hold for the
`differences between time squared and time for dip lines.
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`Ex. PGS 1054