Ex. PGS 2005
`
`EX. PGS 2005
`
`
`
`
`
`
`
`

`
`Prestack imaging with 3-D common-offset-vector gathers
`
`3-D common-offset-vector gathers
`
`Peter W. Cary
`
`ABSTRACT
`The natural extension of a common-offset gather (or section) from 2-D to 3-D is a
`common-offset-vector (COV) gather (or volume). A COV gather is a selection of
`traces with common inline-offset and common cross-line offset. It is similar, but not
`identical, to a common-offset-and-azimuth gather. For regular, wide-azimuth 3-D
`acquisition geometries, the prestack data can be subdivided naturally into N 1-fold
`COV gathers, where N is the nominal fold of the data. The resolution, as well as the
`data and operator aliasing issues, for a 3-D COV gather (or volume) can be analysed
`in the inline or crossline direction just like a 2-D common-offset gather (or section).
`
`INTRODUCTION
`Getting a reliable image from the prestack migration of 3-D land seismic surveys is
`difficult because of poor sampling of the prestack data (Canning and Gardner, 1998;
`Sun et al., 1997). 3-D Kirchhoff prestack migration involves the evaluation of a 5-
`dimensional integral of a highly oscillatory, irregularly sampled function that is
`usually aliased in at least 2 out of 4 spatial dimensions. It is difficult to accurately
`evaluate this integral with discrete summations.
`
`Kirchhoff algorithms are typically used in this situation because they can treat each
`trace separately, and thereby adapt the migration operator to the variable source-to-
`receiver azimuths and offsets of the data. However, the accuracy of the result is
`poorly controlled because the Kirchhoff integrals are essentially being estimated in a
`crude Monte Carlo fashion: each trace is migrated separately and added into the
`output volume. Coming out at the end is an image that we hope is reliable, but we
`have little real assurance that it is. Cary (1999) shows that the outcome is in some
`ways better, and in some ways worse, than doing migration after stack. In order to get
`a more uniformly accurate image, the true spatial distribution of traces has to be
`taken into account. A least-squares approach to migration (Nemeth et al., 1999) can,
`with the aid of proper model constraints, overcome these problems. However, this is a
`very expensive solution to the problem. Regularizing individual subsets of the data is
`a less expensive alternative (Chemengui and Biondi, 1999).
`
`With real 3-D data there is not only the issue of irregular acquisition geometry to
`contend with (crooked, missing and irregularly spaced receiver lines; missing,
`repeated or skidded shots), but even with perfectly regular sampling, it is not clear
`how to treat the data and operator aliasing issues with a typical land geometry of
`orthogonal shot and receiver lines.
`
`Vermeer and Grimbergen (1998) approached this problem by separately migrating all
`the cross-spreads in a 3-D prestack dataset, and “tiling” together the separate images,
`much like the cross-spread approach to 3-D DMO (Padhi and Holley, 1997).
`However, migrating cross-spreads in 3-D is much the same as migrating shot gathers
`in 2-D: the migrated image is dominated by edge effects (wavefronting). Obtaining
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 1
`
`

`
`Cary
`
`the full, correct image requires the summation of migrations from a regular, fine
`spacing of shots in 2-D or cross-spreads in 3-D. A completely artifact-free image
`could be obtained only with extremely expensive acquisition parameters.
`
`The accuracy of 2-D prestack migrations can be controlled with the use of common-
`offset gathers and sections since in principle each migrated common-offset section
`can stand alone as a complete image of the subsurface. Creating common-offset
`sections by binning and stacking together several true constant-offset gathers is a
`simple, approximate method of regularizing and dealiasing the input (and/or output)
`data for each common-offset migration. The advantages and disadvantages of
`migrating this approximate form of minimal dataset are examined elsewhere in this
`volume (Cary, 1999).
`
`Being able to separately migrate subsets of the full 3-D prestack volume in the same
`way that common-offset gathers are separately migrated for 2-D data would be
`extremely useful. Not only would it possibly allow each subset of data to be migrated
`with a non-Kirchhoff algorithm (e.g. F-K), but it would provide an ideal geometry to
`strive for during survey design or data regularization, at least as far as sampling signal
`is concerned. The requirements for sampling noise in order to minimize its effect on
`the final image is an important issue that is not examined here.
`
`WHAT IS A COMMON-OFFSET-VECTOR GATHER?
`So what is the subset of 3-D prestack data that is ideal for prestack imaging purposes?
`That is, what is the 3-D analog of 2-D common-offset gathers?
`
`The first, and simplest, guess at an answer to that question is a 3-D common-offset
`gather, where offset is understood to be the absolute value of the distance from the
`source to the receiver. 3-D common-offset gathers or stacks are commonly used in the
`processing of 3-D data because they can provide N volumes that cover the entire 3-D
`survey area, where N is the nominal fold of the data.
`
`However, in order for 3-D prestack migration or DMO to work properly, traces
`within the gather must have common azimuths as well as common offsets. This is
`because the migration operator varies with both offset and azimuth. Within a 3-D
`common-offset gather, traces in neighboring bins can have completely different
`azimuths, so migration will map the dipping reflections on those traces to separate
`places in the image space, so the migration wavefronts will not interfere correctly.
`We need a dataset that can be mapped from a continuous part of data space to a
`continuous part of image space.
`
`The usual second guess at an answer to the question is a common-offset-and-azimuth
`gather, since this would seem to overcome the shortcomings of common-offset
`gathers. In fact, common-offset-and-azimuth gathers would be an ideal type of gather
`for our needs, so attempts to use this type of gather have been made. For example,
`Chemingui and Biondi (1999) approach the problem of regularizing 3-D datasets that
`are acquired with irregular geometries with a method which they call ICO (inversion
`to common offset), which consists of regularizing the data into a number of common-
`offset-and-azimuth gathers with azimuth moveout.
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 2
`
`

`
`3-D common-offset-vector gathers
`
`The trouble with common-offset-and-azimuth gathers is obvious to anyone who has
`tried to form them from normal 3-D land data (or marine OBC data) that is acquired
`with orthogonal shot and receiver lines. Even when this type of acquisition geometry
`is perfectly regular, there is no way to choose the offsets and azimuths in a way that
`yields a trace at each CMP location. If the pie slices that define the shot-to-receiver
`azimuths are made too narrow, then there are a lot of empty CMP bins. If they are
`made wider, then the fold is very irregular from bin to bin. The width of the offset
`bins can be selected to try to overcome some of these difficulties, but there is no
`obvious way to do this.
`
`The problems with common-offset-and-azimuth gathers stem from the fact that polar
`coordinates are being used to gather the data, but Cartesian coordinates are used to
`acquire the data. The simple solution to the problem is to match the gathering
`geometry to the acquisition geometry. This naturally leads us to common-offset-
`vector gathers, which are simply the Cartesian coordinate version of common-offset-
`and-azimuth gathers.
`
`Instead of thinking of the source-to-receiver vector in terms of offset and azimuth
`(polar coordinates), it is fruitful to think in terms of inline-offset and crossline-offset
`(Cartesian coordinates), as illustrated in Figure 1. A common-offset-vector (COV)
`gather is simply a collection of traces that share the same inline-offset and the same
`crossline-offset. The inline-offset does not have to equal the crossline offset, but the
`inline-offsets of all traces must be the same and the crossline-offset of all traces must
`be the same. It turns out that COV gathers can be constructed and analysed in very
`much the same way as 2-D common-offset gathers.
`
`True COV gathers are normally sparsely sampled in the CMP domain, just like true
`common-offset gathers are sparsely sampled for a lot of land 2-D geometries. For 2-D
`data, the spacing between traces in true common-offset gathers is equal to the source
`interval, which is often several times larger than the receiver interval. For 3-D
`geometries the inline direction is normally along the receiver lines, and the crossline
`direction is normally along the source lines. This implies that the spacing between
`traces in true COV gathers is equal to the source-line spacing in the inline direction
`and the receiver-line spacing in the crossline direction. In order to obtain a 1-fold
`volume of traces that fills in every CMP location with similar values of inline-offset
`and crossline-offset, the inline-offsets and crossline-offsets need to be binned. It is
`easiest to use a real 3-D geometry to illustrate these points.
`
`HOW TO MAKE A COMMON-OFFSET-VECTOR GATHER
`In order to understand how COV gathers are constructed, an example that uses a real
`3-D geometry can be used. The field layout of shots and receivers from this 3-D
`survey are shown in Figure 2. As with most land 3-D surveys, the shot and receiver
`lines are orthogonal to each other. The acquisition geometry is approximately regular,
`although irregularities, as always, do exist. The rules for making COV gathers, which
`are based on the assumption of regular geometry, will work well where the geometry
`is regular, and will start to break down where the geometry is irregular. In other
`words, the guidelines for creating 3-D COV gathers will be as useful as the guidelines
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 3
`
`

`
`Cary
`
`Crossline
`
`Crossline-
`Offset
`
`Offset
`
`Azimuth
`
`Source
`
`Inline-Offset
`
`Receiver
`
`Inline
`direction
`
`Fig. 1. A common-offset-vector gather is a selection of traces with common inline-offset and
`common crossline-offset (Cartesian coordinates), instead of common offset and azimuth
`(polar coordinates).
`
`88
`
`Inline
`
`1
`
`1
`
`135
`
`Crossline
`
`Fig. 2. The field layout for a real 3-D survey with shot-lines (vertical) spaced 400m apart and
`receiver-lines (horizontal) spaced 200m apart. Like most 3-D surveys, the geometry is mostly
`regular, but with some irregularities, so the rules for gathering COV traces are suitable over
`most of the survey.
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 4
`
`

`
`3-D common-offset-vector gathers
`
`for creating 2-D common-offset gathers, and the standard tricks for filling in gaps in
`the coverage (e.g. increasing the offset bin width) can be used in 3-D, just as in 2-D.
`
`The nominal acquisition geometry for the 3-D survey is as follows: the source-line
`interval is 400m, the receiver-line interval is 200m, the source interval is 64m, and
`the receiver interval is 30m. The first step in constructing a COV gather is to compute
`the inline-offset and the crossline-offset for every trace in the dataset. We can now
`consider the 3-D prestack spatial sampling function as a function of the inline-offset,
`crossline-offset, inline CMP number and crossline CMP number (Hampson, 1997).
`
`Figures 3 and 4 show two different 2-D projections of the true 4-D sampling function
`of the present dataset. These two projections illustrate the periodic nature of the
`inline-offsets and crossline-offsets in the prestack data. Figure 3 is the 2-D projection,
`or “stacking chart”, for traces selected only from inline 44, which runs through the
`middle of the 3-D. This plot of the wavefield sampling as a function of crossline-
`offset and crossline CMP number shows that the same crossline-offset is sampled by
`traces that are separated by the shot-line spacing, 400m. Notice that this is exactly the
`way that offsets repeat on stacking charts for 2-D data. Figure 4 is the “stacking
`chart” for crossline 67, which runs through the middle of the dataset in the orthogonal
`direction. This plot of the wavefield sampling as a function of the inline-offset and
`inline CMP number shows that the same inline-offset is sampled by traces that are
`separated by the receiver-line spacing, 200m. Notice that this is exactly the way that
`offsets would repeat on stacking charts for 2-D data if all receivers became shots and
`all shots became receivers.
`
`I have used the term “stacking chart” to describe Figures 3 and 4 because they both
`have the appearance of ordinary stacking charts that we are familiar with from 2-D
`data. These type of plots could be generated for every inline and crossline in the 3-D
`dataset. The four prestack coordinates for 3-D seismic data are orthogonal to each
`other, of course, so the stacking charts in the crossline direction (e.g. Fig. 3) are
`“decoupled” from the stacking charts in the inline direction (e.g. Fig. 4). This means
`that inline-offsets and crossline-offsets can be treated independently in the COV
`gathering process.
`
`We now see that a true COV gather should nominally have traces that are separated
`by the shot-line spacing along inlines and by the receiver-line spacing along
`crosslines. Offset bin widths will need to be defined for a “true” COV gather just as
`for 2-D common-offset gather in order to accommodate irregularities in the
`acquisition geometry and to accommodate a shot interval that is not an integer
`multiple of the receiver interval. The inline-offset and crossline-offset bin widths are
`given by the ratio of the shot-line spacing to the receiver interval (400/30) and the
`ratio of the receiver-line spacing to the shot interval (200/64), respectively.
`
`A true COV gather is not very useful because the sparse sampling will cause severe
`data aliased for most geometries. However, an approximate COV gather that has a
`trace at every CMP location can be formed by increasing the offset bin widths. These
`approximate COV gathers are extremely useful, for all the same reasons that
`approximate common-offset gathers are useful for 2-D data.
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 5
`
`

`
`Cary
`
`Crossline
`Offset
`(m)
`
`∆x
`
`Crossline CMP Number
`
`Fig.3. Stacking diagram for inline 44. The interval, ∆x, indicates that the periodicity of the
`crossline-offset is the source-line spacing.
`
`Inline
`Offset
`(m)
`
`∆y
`
`Inline CMP Number
`
`Fig.4. Stacking diagram for crossline 67. The interval, ∆y, indicates that the periodicity of the
`inline-offset is the receiver-line spacing.
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 6
`
`

`
`3-D common-offset-vector gathers
`
`The recipe for making a 2-D common-offset gather with a trace at every CMP is to
`make the offset bin width equal to twice the shot interval (which follows from the
`periodicity of offsets in 2-D stacking charts). The periodicities of inline and crossline
`offsets illustrated in the “stacking charts” in Figures 3 and 4 indicate that the
`corresponding recipe for making a COV gather with a trace at each CMP location is
`to make the inline-offset bin width equal to twice the source interval and to make the
`crossline-offset bin width equal to twice the receiver interval.
`
`There are “positive” and “negative” offset vectors for 3-D data, just like there are
`positive and negative offsets for 2-D data. If we assume reciprocity of shots and
`receivers, then “positive” and “negative” offset vectors can be gathered together,
`since the migration operator for P-P data is insensitive to the exchange of shots and
`receivers.
`
`Figure 5 shows the CMP coverage of the “positive” COV gather counterpart (inline
`offset = 1200 ± 400m; crossline offset = 200 ± 100m) together with its “negative”
`COV counterpart (inline offset = -1200 ± 400m; crossline offset = -200 ± 100m).
`Grouping together positive and negative COV gathers can increase the efficiency of
`the migration algorithm, as well as help to regularize the geometry by filling in gaps
`in CMP coverage. Fig. 5 shows that the COV gathering recipe works well where the
`geometry is regular, and starts to break down where the shot lines are irregular.
`
`Several additional characteristics of 2-D common-offset gathers also apply to 3-D
`COV gathers. For example, the total number of 3-D COV gathers that can be formed
`with reasonably large coverage of the survey area is equal to the nominal fold of the
`data, just like 2-D common-offset gathers. Figure 6 shows the selection of COV
`gathers that are available for the present 3-D survey. The number of offset bins in the
`inline and crossline directions is given by the rule for calculating the inline fold and
`the crossline fold (Stone, 1994, p.136), and the total number of bins equals the
`product of the inline fold and the crossline fold.
`
`Finally, there are parallels between the way that the coverage of 2-D common-offset
`gathers and 3-D COV gathers vary with offset. As offset increases, the edges of 2-D
`common-offset gathers, within which live data is recorded, generally move down in
`time and inward from the survey edges since the mute time increases with offset, and
`the large offsets are generally not sampled at the ends of the 2-D line. The same kind
`of thing happens with 3-D COV gathers. Knowledge of how the coverage of COV
`gathers decreases with inline-offset and crossline-offset is important information for
`reducing edge effects in applications such as 3-D prestack migration.
`
`A SIMPLE EXAMPLE
`Perhaps the simplest process that illustrates how COV gathers can be used for
`imaging wide-azimuth 3-D surveys is common-conversion-point (CCP) binning of 3-
`D converted-wave (P-S) data. CCP binning maps P-S data in a depth-variant manner
`to their true reflection points under the assumption that all reflectors are flat. In most
`cases, the P-S reflection point is located between the midpoint and the receiver
`location.
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 7
`
`

`
`Cary
`
`Inline
`
`Crossline
`Fig. 5. CMP coverage for positive and negative COV gathers with inline-offset = 1200 ± 200m
`and crossline-offset = 200 ± 100m. The coverage is uniform except where the source lines
`are irregularly spaced.
`
`1800
`
`Inline
`Offset
`(m)
`
`0
`
`-1800
`
`-1400
`
`0
`Crossline Offset (m)
`
`1400
`
`Fig. 6. The distributions of COV gathers for the 3-D survey. The total number of gathers is the
`nominal fold of the data. The arrows indicate the two “positive” and “negative” COV gathers
`whose coverage is shown in Fig. 5.
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 8
`
`

`
`3-D common-offset-vector gathers
`
`If we look at the region of coverage of a COV volume extracted from a real 3-C, 3-D
`dataset, which on input is binned according to midpoint, as in Figure 7(a), then CCP
`binning maps the data at 600ms to a large continuous portion of the total output grid,
`Figure 7(b), which is what is required of an ideal minimal 3-D dataset. In contrast, a
`cross-spread that is binned by midpoint, as in Figure 8(a), is mapped to a relatively
`small continuous portion of the total output grid, Figure 8(b). In both Figures 7(b) and
`8(b), gaps appear within the “continuous” portion of each output grid because the
`binning has been done with nearest neighbor interpolation. If sinc-function
`interpolation had been used, which takes the bandlimited nature of the wavefield into
`account, then the empty bins would be filled with data from neighboring bins. An
`analysis of stacking charts indicates that P-S fold would be more uniform if the data
`were binned with a “natural” CCP bin size (Eaton and Lawton, 1992) that is larger
`than the CMP bin size. However, this analysis ignores the bandlimited nature of the
`data. The average subsurface separation of P-S reflection points for each pseudo-
`common-offset gather is the CMP interval (Cary, 1999), so the correct bin size for
`imaging P-S data is still the CMP bin size.
`
`REMARKS AND CONCLUSIONS
`A common-offset-vector gather is an ideal subset of a 3-D seismic dataset since it can
`provide a complete image of most of the output 3-D volume. For regular acquisition
`geometry, each COV gather can be separately imaged. Ideally, stacking several
`independently imaged COV volumes should improve the signal-to-noise ratio of the
`final image, but not its fidelity.
`
`Since true COV gathers are rarely sampled properly, simple guidelines for making
`COV gathers that will populate all CMP bins, if geometry is regular, have been
`described that are based on nearest-neighbor interpolation of inline-offsets and
`crossline-offsets. The interleaved CMP sampling between true 3-D COV gathers
`leads to approximately dealiased pseudo-COV gathers, with a trace at each CMP
`location, in the same way that interleaved CMP sampling between true 2-D common-
`offset gathers leads to approximately dealiased pseudo-common-offset gathers (Cary,
`1999). Therefore, COV gathers are the natural choice of geometry to use for 3-D data
`regularization schemes instead of common-offset-and-azimuth gathers.
`
`Since COV gathers can be analysed independently in the inline and crossline
`directions, we can take the results from the imaging tests on 2-D common-offset
`gathers performed by Cary (1999) and apply them directly to 3-D COV gathers. By
`partial stacking into well-sampled COV gathers before migration, the high
`frequencies of dipping events in the inline and crossline directions will be smeared in
`the same way as for 2-D common-offset gathers. However, the events with small dip
`will be imaged coherently and with reliable waveforms.
`
`On the other hand, the 2-D results (which ignore the impact of noise) suggest that 3-D
`surveys that are acquired with widely-spaced source and receiver lines should be
`capable of generating coherent images of the signal by stacking into COV gathers
`after migration, but the waveforms will be unreliable because of imprinting of the
`acquisition geometry. To obtain coherent images and reliable waveforms in
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 9
`
`

`
`Cary
`
`960m
`
`720 m
`
`Crossline
`
`Inline
`
`Fig. 7(a). CMP coverage of a common-offset-vector gather before CCP binning. The main arrows
`indicate the average inline-offset and crossline-offset of traces within the gather, and the box
`indicates the offset bin widths used to define the gather.
`
`Crossline
`
`Fig. 7(b). The COV gather after CCP binning with nearest-neighbor interpolation. The traces
`have been shifted downward and to the left (toward the receiver) by CCP binning.
`
`Inline
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 10
`
`

`
`3-D common-offset-vector gathers
`
`Crossline
`
`Inline
`
`Receiver line
`
`Shot line
`
`Fig. 8(a). The CMP coverage of a single cross-spread before CCP binning.
`
`Fig. 8(b). The CCP coverage of the cross-spread in Fig. 8(a). The traces have moved toward
`the receivers after CCP binning.
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 11
`
`

`
`Cary
`
`structurally complex areas, small shot-line and receiver-line spacings appear to be
`required, unless some method such as least-squares migration, or least-squares DMO
`is capable of removing the footprint effect.
`
`It is common to design wide-azimuth 3-D surveys with the aim of making the offset
`and azimuth distributions uniform within CMP bins. If prestack imaging is going to
`be performed with COV gathers instead of with common-offset-and-azimuth gathers,
`a more appropriate design criterion would be to make the inline-offset and crossline-
`offset distributions uniform within bins. This design criterion would ensure that COV
`gathers are well-sampled. Since COV gathers emerge naturally from regular
`orthogonal acquisition geometries, the COV design criteria should be easy to satisfy,
`as long as the geometry can be kept fairly regular.
`
`As far as amplitude-versus-offset (AVO) and velocity analysis are concerned,
`imaging with COV gathers would have an impact. If dips are small, then stacking
`into COV gathers before prestack migration would yield reliable amplitudes for AVO
`and velocity analysis, but there would be a relatively small number of absolute offsets
`and azimuths sampled. If dips are large, and shot-line and receiver-line spacings are
`large, stacking into COV gathers after prestack migration is to be preferred, but it is
`doubtful that AVO analysis would be reliable because of the acquisition footprint that
`originates from the prestack migration. If azimuth-dependent velocity analysis before
`or after migration is required, this would be workable with COV gathers since
`azimuths are still being sampled in a somewhat regular fashion.
`
`REFERENCES
`
`Canning, A., and G.H.F. Gardner, 1998, Reducing 3-D acquisition footprint for 3-D DMO and 3-D
`prestack migration: Geophysics, 63, 1177-1183.
`Cary, P.W., 1999, Generalized sampling and “beyond Nyquist” imaging: 11th Annual CREWES
`Report (this volume).
`Chemingui, N., and Biondi, B., 1999, Data regularization by inversion to common offset (ICO):
`Extended Abstracts, 69th Annual Internat. Mtg., Soc. Expl. Geophys.,1398-1401.
`Eaton, D.W.S., and D.C.Lawton, 1992, P-SV stacking charts and binning periodicity: Geophysics, 57,
`745-748.
`Hampson, G., 1997, 3-D wavefield sampling in the CMP method: Extended Abstracts, 67th Annual
`Internat. Mtg., Soc. Expl. Geophys., 55-58.
`Nemeth, T., Wu, C. and Schuster, G.T., 1999, Least-squares migration of incomplete reflection data:
`Geophysics, 64, 208-221.
`Padhi, T. and Holley, T.K., 1997, Wide azimuths—why not?: The Leading Edge, 16, 175-177.
`Sun, Y., G.T.Schuster, and K.Sikorski, 1997, A quasi-Monte Carlo approach to 3-D migration: theory:
`Geophysics, 62, 918-928.
`Stone, D.G., 1994, Designing Seismic Surveys in Two and Three Dimensions:Society of Exploration
`Geophysicists, 244pp.
`Vermeer, G.J.O, and J.L.T.Grimbergen, 1998, 3D prestack migration with cross-spreads: Extended
`Abstracts, 60th EAGE Conf. & Tech. Exhib., 1-51.
`
`CREWES Research Report — Volume 11 (1999)
`
`Ex. PGS 2005
`Page 12