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`Ex. PGS 1043
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`EX. PGS 1043
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`Effects of irregular sampling on 3-D prestack migration
`
`Gerald H.F. Gardner* and Anat Canning, Houston Advanced Research Center
`
`SP4.7
`
`Summary
`
`In a direct application of Kirchhoff migration each trace
`is added to the migrated volume by spreading the data along
`impulse response curves with suitable change in the amplitude
`and shape of the wavelets. Overlapping impulse responses form
`the correct answer where they form an envelope and are
`For uniform data (constant
`supposed to cancel elsewhere.
`offset, constant azimuth, constant midpoint spacing and constant
`velocity) the cancellation is excellent and the reflectors have the
`correct amplitude. This paper shows how failure to meet any of
`these constancy criteria results in noise. Modified summation
`procedures can reduce the noise at the expense of a loss of
`resolution.
`
`within 9240 ft of a shot in the N-S direction, are active for each
`shot. Thus, at most 8 N-S lines are active at any one time, with
`84 receivers on each line. Because the distances are multiples of
`110, all the midpoints lie on a grid 110 ft in the N-S direction, by
`220 ft in the E-W direction. The fold varies from one at the
`edge to about 25 at the center.
`In the actual field implementation of this design, many
`vibrator points were displaced from the planned positions
`because of obstacles. The total number of traces was 250,000 in
`an area 19800 ft by 19800 ft (14 square miles). Figure 1 shows
`a plot of the actual VP stations and the actual receiver stations.
`As a result of this design, offset and azimuth vary from one
`midpoint to the next. The departures from the plan add more
`irregularities to the trace distribution.
`
`Comparison of post-stack and prestack migration
`
`To test the adequacy of the layout design, synthetic data
`were generated with this layout and processed in two ways.
`First, 3-D DMO was applied, followed by stack and migration;
`second, 3-D prestack Kirchhoff migration was applied. The
`synthetic data corresponded to a model with three horizontal
`interfaces at depths of 2000, 4000 and 6000 A, with equal
`reflection coefficients independent of the angle of incidence. The
`defined by
`wavelet
`complete
`cycle
`was
`one
` -0.5
` where f = 15 Hz and the beginning of
`the wavelet marks the reflection. The maximum frequency was
`about 40 Hz. The velocity was constant and equal to 11000
`ft/sec. For this velocity and maximum frequency the sampling
`
`interval for migration,
`, is about 66 ft.
` max
`Theoretically, because the velocity is constant, post-stack
`and prestack migration should yield identical results, whatever
`the data. However, the 3-D DMO applied in these tests has
`midpoint and offset filters embedded in the procedure, whereas
`the 3-D Kirchhoff migration does not.
`The input data were not muted, nor were any weighting
`factors applied to correct for irregularities in the spacing. Thus
`there is considerable stretching of the wavelet for the large
`offsets (~11,000 ft) and shallowest interface (2,000 A).
`Nevertheless, the post-stack and prestack migrations should be
`identical.
`Figures 2 and 3 show the results of post-stack and
`prestack migration for a N-S line as indicated in Figure 1.
`It is
`evident that the prestack migration is noisier. The post-stack
`migration is quite free of background noise, the wavelet is well-
`preserved,
`and the amplitude variation is smooth, increasing
`toward the center of the survey as the fold increases. In
`contrast, the prestack data show migration smiles, the wavelet is
`distorted and the amplitude is erratic.
`
` l
`
`Introduction
`
`Multi-fold 3-D surveys are often designed to facilitate
`stacking by arranging that many traces have midpoints close to a
`regular grid. The grid spacing is made as large as is reasonable
`to obtain a high fold. It is somewhat disconcerting to find that
`prestack Kirchhoff migration of such data may produce a noisier
`result than DMO, stack and migration. We illustrate this
`possibility using the coordinates from a land survey and synthetic
`data.
`
`The noise created by Kirchhoff summation can be
`coherent and have the appearance of reflections, or it may just
`obscure the correct images. One cause of the noise is too large a
`trace spacing, which leads to operator aliasing. Lumley et al.
`(1994) have shown that for regular trace spacing it is possible to
`design an operator filter that effectively reduces the noise, but
`their analysis does not apply to irregular trace spacing. We show
`in this paper that noise is also created by variations in offset or
`azimuth from one trace to the next, even when the midpoint
`spacing is regular and small. Because 3-D surveys contain
`irregular variations in midpoint, offset and azimuth we propose
`that the degree of filtering required can be determined
`empirically by applying a modified Kirchhoff surnmation using
`the actual coordinates of the survey and suitable synthetic data.
`
`The survey layout
`
`All the basic distances in the layout are multiples of 100
`A. There are 16 N-S receiver lines with a separation of 1320 ft
`and 8 E-W shot lines with a separation of 2640 A. Along the
`receiver lines the geophone station interval is 220 ft, and along
`the shot lines the vibrator station interval is 440 ft. There are
`three shots symmetrically placed between adjacent receiver lines.
`All receivers within 5280 ft of a shot in the E-W direction, and
`
`1553
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`Downloaded 03/12/14 to 173.226.64.254. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`Ex. PGS 1043
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`

`

`2
`
`3-D DMO
`
`Effects of irregular sampling
`
`Examples of operator aliasing
`
`The result of the 3-D DMO followed by stack is shown
`in Figure 4. This shows that the smoothing of the data occurs at
`this step. The velocity-independent DMO was done by applying
`a log-stretch to the input traces, taking an FFT of each input
`trace, multiplying each frequency component by a filter factor for
`each replacement point between the shot and receiver, adding
`into the four grid points nearest to the midpoint and into the two
`offsets nearest to the new offset. Since no NMO was applied,
`the filtering effect of this summation procedure increases as
`offset increases. An inverse FFT and log-stretch restores the
`data to the time domain.
`
`3-D Kirchhoff Migration
`
`In the Kirchhoff migration each trace was added to the
`answer using an obliquity factor based on the offset of the trace,
`i.e., as if the trace were part of a constant-offset, constant
`azimuth, survey. The formula for the summation is
`
`
`
`
`
`where
`
`= output time at output depth point Q ,
`
`
`
`
`
`=
`= travel time from receiver to Q ,
`= travel time from shot to Q ,
`
`
`Velocity Analysis and AVO Analysis
`
`Another comparison of Kirchhoff summation and DMO-
`PSI can be made by looking at CMP gathers after imaging but
`Figure 5a shows a gather using Kirchhoff
`before stack.
`summation to accumulate migrated traces by the offset of the
`original trace (common image gather); Figure 5b shows the
`gather after DMO + PSI + NMO; and Figure 5c shows an ideal
`constant azimuth traces with
`result using constant-offset,
`midpoints on a regular grid. It is clear that the irregularity of the
`coordinates makes Kirchhoff summation much noisier than the
`This example shows that the effect of
`DMO-PSI process.
`irregularities is more severe in the prestack domain and may
`cause problems with velocity analysis and with AVO analysis.
`
`Figures 6a - 6d illustrate some operator aliasing
`problems. First, for a single-fold, constant offset (2,000 R),
`constant azimuth, small regular spacing (55 R), Kirchhoff
`migration gives an almost noise-free result as shown in Figure
`6a. The wavelet and model are the same as for the previous
`examples; the output trace spacing is 55 ft in all the examples.
`Aliasing is introduced if the offset is changed from trace
`to trace. In this case, the coverage was obtained by having 16
`receivers in a square array with spacing 110 ft record a shot at a
`distance from the array of 2000 ft. This template was moved
`over shot positions on a square grid with spacing 220 ft (Figure
`7). The result is single-fold with midpoints at 55 ft and offsets
`between 2000 ft and 2400 ft from one midpoint to the other
`(Figure 6b). Even this small a variation in offset produces a
`visible background noise.
`As another example, the same template was used, but
`with a receiver spacing of 990 ft and a shot offset of 1000 ft.
`Again, the midpoint spacing is 55 ft but the jumps in offset are
`large. The noise level is increased by the increase in offset
`variability, as show in Figure 6c. A fine midpoint spacing by
`itself does not guarantee a noise-free image.
`On the other hand, the noise resulting from a coarse grid
`spacing (Figure 6e) is diminished by a high fold.
`If the receiver
`grid spacing is changed to 880 ft the midpoint spacing becomes
`220 ft and there are 16 traces at each midpoint. Figure 6d shows
`the result:
`the noise level is about the same as in Figure 6c.
`Finally, Figure 6e shows one-fold data on a 220 x 220 ft
`grid with a constant offset of 1000 ft. Here offset does not vary,
`but midpoint spacing is much too large and therefore aliasing
`It creates events that could be mistaken for
`noise is large.
`horizontal reflectors.
`
`Conclusions
`
`Examples of Kirchhoff prestack migration show that
`operator aliasing creates noise whenever the spacing is too large.
`When the offset is not constant, or the azimuth varies (for
`dipping reflectors) aliasing noise increases. The design of an
`operator filter to attenuate these effects will be discussed in the
`oral presentation.
`
`References
`
`Lumley, D. E., Claerbout, J. F. and Beve, D., 1994, Anti-aliased
`Kirchhoff 3-D migration, 64th Ann. Intemat. Mtg., Soc. Expl.
`Geophys., Los-Angeles.
`
`Acknowledgment
`
`This research was carried out as part of the 3-D
`consortium project at the Houston Advanced Research Center
`(HARC). We gratefully acknowledge the support provided by
`the sponsors of this project. We would especially like to thank
`Mitchell Energy & Development Corp. who provided the
`coordinates of a land 3-D survey.
`
`1554
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`Downloaded 03/12/14 to 173.226.64.254. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`Ex. PGS 1043
`
`

`

`Effects of irregular sampling
`
`3
`
`Fig. 1. Layout of the 3-D survey. Shot locations are marked
`with
` receiver locations are marked with
` line A
`mark the position of the section that is presented in the
`following Figures.
`
`Fig. 3. Result of 3-D DMO and migration
`of the real 3-D survey (@ line A).
`
`using the coordinates
`
`Fig. 2. Result of 3-D prestack migration using the coordinates of
`the real 3-D survey (@ line A).
`
`Fig. 4. Result of 3-D DMO and stack using the coordinates of
`the real 3-D survey (@ line A).
`
`1555
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`Downloaded 03/12/14 to 173.226.64.254. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`Ex. PGS 1043
`
`

`

`4
`
`Effects of irregular sampling
`
`Fig. 6. (c) 3-D prestack Kirchhoff migration using a single fold,
`small regular spacing (55 ft) but large variation in
`offset (between 1990 ft. and 4300 ft).
`
`Fig. 5. (a) One common image gather after 3-D prestack
`Kirchhoff migration using the real survey coordinates.
`(b) One CMP gather after 3-D DMO, PSI and NMO
`using the real survey coordinates. (c) One common
`image gather after 3-D prestack Kirchhoff migration of
`a regular survey.
`
`Fig. 6. (d) 3-D prestack Kirchhoff migration using a single
`large regular spacing (220 ft) but 16 fold.
`
`-4.0
`
`l 6.0
`
`8.0
`fold,
`
`0.0
`
`6.0
`
`Fig. 6. (a) 3-D prestack Kirchhoff migration using a single fold,
`constant offset (2000 ft), constant azimuth, small
`regular spacing (55 ft).
`
`Fi
`
`8.0
`g. 6. (e) 3-D prestack Kirchhoff migration using a single fold,
`constant offset (2000 ft), constant azimuth, large
`regular spacing (220 R).
`
`s h o t
`
`2000ft
`
`110ft
`
`0
`
`0
`
`0
`
`0
`
`110ft
`
`0
`
`0
`
`0
`
`0
`
`0
`
`
`
`
`
`0
`
`0
`
`0
`
`0
`
`receiver array
`
`Fig. 6. (b) 3-D prestack Kirchhoff migration using a single fold,
`small regular spacing (55 ft) but small variation in
`offset (between 2000 A and 2400 ft).
`
`1556
`
`Fig.7: Template used to build a 3-D survey for Figure 6b.
`
`Downloaded 03/12/14 to 173.226.64.254. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`Ex. PGS 1043
`
`