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In 3D seismic surveydesign, the parameterchoice has to satisfy awide variety of geo-physical, operational,and cost constraints.The designer of a sur-vey has to strike a bal-ance between thevarious requirementsand invariably willhave to settle for somekind of compromise.Hence, this designprocess could also be

viewed as an opti-mization process.

In an award-win-ning poster paperLiner et al. (1998) introduced the idea of survey design opti-mization. It was soon followed by another poster paper, butnow in TLE (Liner et al., 1999). The method described inLiner, Underwood, and Gobeli (1999, to be called the LUGmethod in this paper) honors geophysical requirementsonly. This method determines design parameters based onthe minimization of a cost function that is a weighted sumof deviations from the user-specified geophysical target val-ues.

Morrice, Kenyon, and Beckett (2001) introduced a dif-

ferent survey design optimization method (to be called theMKB method). The MKB method focuses on the minimiza-tion of the actual cost of a 3D survey, while satisfying a num-

ber of geophysical constraints. The geophysical constraintscan be formulated according to the same geophysicalrequirements as used in the LUG method. In the MKBmethod the cost function is minimized using the Solverfunction available in Microsoft Excel.

The MKB and LUG methods inspired this paper in whichI will present modifications and improvements to both. Thepublished methods deal with orthogonal geometry and sodoes this paper. To optimize the optimization method, a clearunderstanding of what constitutes a geophysically desirableconfiguration is essential. Therefore, this paper starts witha description of the main parameters of orthogonal geom-

etry and their interrelationships. Next, various geophysicalrequirements that may influence the choice of the geome-try parameters are reviewed. Then the LUG and the MKBmethods are reviewed and modifications are discussed. Themodified methods will be illustrated with some examplesof 3D survey design.

Description of orthogonal geometry.Figure 1 illustrates thedual description of regular orthogonal geometry. The leftside shows the way in which the data may be acquired inthe field; the right side shows that the acquisition geome-try may also be described in terms of cross-spreads. (Thegeometry might also be acquired with cross-spreads but

that would be highly inefficient.)The range of shots in the template defines the crossline

roll of the geometry. In Figure 1 the shot range equals onereceiver line interval leading to a one-line roll. This type ofrolling ensures that all cross-spreads are the same with thereceiver line cutting the spread of shots in the center (exceptalong the edges of the survey). If the range of shots equalsseveral receiver line intervals, say 3, then the crossline rollwill equal 3. In that case three different cross-spreads will

be acquired with differences between the positive and neg-

ative maximum cross-line offsets. Although those multilineroll acquisition geometries would produce constant fold atdeep levels (where no offset muting is applied), they wouldshow strong fold variations causing severe acquisition foot-prints at intermediate levels (see Vermeer, 2002, Section4.6.6). One-line roll geometries show much less variation infold at intermediate levels. Therefore, like in the MKB andLUG methods, I will only optimize for one-line roll.

The width of the midpoint area of the cross-spread equalshalf the length of the receiver spread. It is also equal to themaximum inline offset Xmax,i. Similarly, the height of the mid-point area of the cross-spread equals half the length of theshot spread, and this height is also equal to the maximumcrossline offset Xmax,x. The midpoint area of the next cross-spread in the inline direction is shifted over one shot line

interval in that direction. Therefore, to create constant inlinefoldMi, the shot line interval SLIshould fit an integer num-

ber of times on the width of the cross-spread. Similarly, con-stant crossline fold Mx is achieved by selecting a receiverline interval RLIthat fits an integer number of times on thecrossline dimension of the cross-spread.

Figure 1 also shows the unit cell of the geometry. Its areaequals RLI* SLI. This area fitsMi *Mx times on the area ofthe cross-spread, i.e., total foldM = Mi * Mx = (midpoint areaof cross-spread)/(area of unit cell). In Figure 1 total foldequals 4 * 4 = 16.

A one-line roll orthogonal geometry is defined by sixparameters only (Table 1); nS, nR, M i, andMx are integers.

934 T HE LEADING EDGE OCTOBER 2003 OCTOBER 2003 THE LEADING EDGE 00

3D seismic survey design optimization

Gijs J. O. Vermeer, 3DSymSamGeophysical Advice, The Netherlands

ACQUISITION/PROCESSING

Coordinated by William Dragoset

Figure 1. The same orthogonal geometry with template (left) and with cross-spread (right). Horizontal lines are receiver

lines; vertical lines are source lines. The template represents the way in which the data may be acquired in the field; inthis case, there are eight receiver lines with a number of shots in the center of the template. The cross-spread gathers alldata for receivers along the same receiver line that have listened to a range of shots along the same source line (afterVermeer, 2002).

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S and R have to fit an integer number nS

and nR

on thereceiver line interval and the shot-line interval, respectively,to ensure that inline roll and crossline roll distances involvemoving an integer number of station spacings. So, any opti-mization process has to find four integers and two stationspacings to define a geometry that will satisfy all geophys-ical requirements.

Pairwise, the six parameters in Table 1 define the threecharacteristic aspect ratios of the geometry: the aspect ratioof the binsAb = S/R, the aspect ratio of the unit cellAu =RLI/SLI, and the aspect ratio of the cross-spread Ac =Xmax,x/Xmax,i.

Geophysical requirements. Figure 1 describes a fully sym-metric geometry with all aspect ratios equal to 1. From a

physical point of view, this constitutes the ideal situation,because physically, there is no difference between proper-ties in the inline direction and in the crossline direction. Inparticular the receiver sampling interval R along thereceiver line and the shot sampling interval S along theshot line should be the same and small enough for propersampling of the cross-spread. Proper sampling means thatthe underlying continuous wavefield can be fully recon-structed from the sampled wavefield. Only in case of over-sampling will the sampling intervals of inline and crosslinedirection be different. Similarly to Morrice et al. (2001), I willuse the requirement of equal station spacings in the opti-mization procedure. Liner et al. (1999) allow different shotand receiver station spacings.

Ideally, the aspect ratios of the unit cell and of the cross-spread should be equal to one as well. A square unit cell is

better than a rectangular unit cell for a number of reasons.One reason is that the unit cell acts as the basis of offset-vector tile (OVT) gathers (see Vermeer, 2002). In Figure 1the cross-spread consists of 16 different unit cells. Takingthe same cell (tile) from all cross-spreads of the geometryproduces a single-fold OVT-gather. Such a gather is thenearest one can get in orthogonal geometry to a common-offset vector (COV) gather. The inevitable variation in off-set in an OVT-gather is smallest for square OVTs. Gesbert(2002) and Vermeer (2002) describe the use of those gathersin prestack migration.

Vermeer (2002) also describes that offset-vector orientedmuting can achieve constant fold at each time level in the

3D survey, thus minimizing acquisition footprint. This mut-ing procedure is based on splitting each cross-spread in 4Mquarter-unit-cell-sized rectangles. The procedure is mostefficient for square unit cells as Figure 2 illustrates. This fig-ure compares geometries 1 and 2 that have the same totalfoldM, and the same square cross-spread as the basic sub-set. Geometry 1 hasMi =Mx = 6, and geometry 2 hasMi =4 and Mx = 9. In case no offset-vector-oriented muting isapplied, fold at each time level will vary, and the variationin fold will be largest for the geometry with rectangular unitcells. A rule of thumb would be to choose at most a factortwo difference between shot line interval and receiver lineinterval.

The desirability ofAc = 1 is quite obvious. All traces in

a cross-spread with the same absolute offset are lying on acircle with its midpoint at the cross-spread center. Hence,the maximum useful offset is also defined by a circle, andthis leads to Xmax,x = Xmax,i. Assuming equal shot and receiverstation spacings, a square cross-spread produces commonreceiver gathers that are common shot gather look-alikeswith the same number of traces. 3D velocity filtering(Galibert et al., 2002; Girard et al., 2002; Karagul andCrawford, 2003) stands the best chance of success in squarecross-spreads. Another benefit ofAc = 1 is that it maximizesthe useful extent of the cross-spread in both inline andcrossline directions, thus minimizing the number of cross-spread edges for a given fold. The fewer the edges, the

smaller the impact of migration artifacts caused by spatialdiscontinuities. In practice, other considerationsmainlycostmay lead toAc 1. Again, a good rule of thumb might

be 0.5

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In general, the fold at shallow targets may be smaller than atdeep targets, and the designer may specify average fold as

needed at various target levels. The average fold at a shallowtarget Fsh for a given geometry and given mute offset Xsh atthat level may be computed using simple geometrical rela-tions. Total foldM = Mi * Mx = Xmax,i/SLI * Xmax,x/RLI= (areaof cross-spread)/(area of unit cell). The right side of Figure 1illustrates this formula for a 16-fold geometry. If mutingrestricts the range of offsets that contributes, the average fold

Mav = (area of cross-spread not muted)/(area of unit cell), seeFigure 3.

Note that in this computation use is made of continuousgeometry; i.e., areas are measured rather than number of tracesin bins. If the sampling intervals are not too large, there willnot be much difference between the two computational tech-niques, but using continuous geometry (infinitely small sam-pling intervals) facilitates the computations considerably.

The mute offset at various target levels is determined fromthe mute function. This function is often computed from thevelocity function and the maximum allowable NMO stretchfactor. In the optimization procedure discussed later, it isassumed that mute offsets at target levels have been deter-mined beforehand.

The parameters maximum offset Xmax, smallest maximumoffset Xmaxmin, and largest minimum offset Xmin can also bedetermined in a simple way using continuous geometry. Thisis illustrated in the cross-spread of Figure 1. If we take all quar-ter-unit-cell-sized areas in the corners of all cross-spreads ofthe geometry (black squares in Figure 1), single-fold coverageis constructed containing the largest available offsets in thegeometry. The smallest of these largest offsets Xmaxmin can be

found from the corners closest to the center of the cross-spread.Xmaxmin may play a role in AVO analysis. It could be chosenas the smallest offset needed in all CMPs to cover the requiredrange of angles of incidence. Xmin determines the shallowestlevel where at least complete single-fold coverage is reached.

The LUG optimization method. Having established therequirements of all geophysical parameters as discussed in theprevious section, the designer has to determine an acquisitiongeometry that satisfies all requirements in an optimal way. Thiscan be quite a challenge, and therefore Liner et al. devised anoptimization technique for survey parameter design (the LUGmethod).

Liner et al. also used the fact that regular orthogonal

geometry can be described by six parameters. Next to shotand receiver intervals in x andy (corresponding to the firstfour parameters in Table 1), they used number of livereceivers in x- andy-direction rather than maximum inlineoffset and maximum crossline offset.

In the LUG method a cost function is minimized byusing the six parameters as decision variables. The costfunction describes deviations from desired values in thefollowing way:

pi is a geophysical parameter, Dpi its target (desired) value,and wi a weight attached to parameterpi. tx and ty describethe x- andy-dimension of the template, respectively. Variousgeophysical parameters used in the LUG method are bin-size bx in x and by in y, total fold M, number of availablechannels, and maximum offset Xmax. The template term in(1) describes the desirability of a square template. Liner etal. computed tx and ty as distances between first and lastsample rather than as effective length, i.e., (n 1) ratherthan n. As a consequence, maximum offset expressed asXmax = 0.5(tx

2+ ty2) did not get the correct value.

Liner et al. first tried a minimization technique, avail-

able in Mathematica, but to overcome various problemswith that technique a special program was written in C toestablish from all possible solutions (84 million cases weretested) the one that minimizes the geophysical cost func-tion. However, the number of possible solutions can bereduced significantly by selecting R, S, nS, nR,Mi andMx(Table 1) as the six parameters to be established. The sim-plified problem formulation can be solved easily using theSolver function available in MS Office program Excel in asimilar way as in Morrice et al.

The MKB optimization method. The LUG method onlyoptimizes for geophysical parameter requirements. The costof the acquisition is another important decision factor, andMorrice et al. formulated the survey design optimization

problem as a minimization problem for the cost subject togeophysical and operational constraints (the MKB method).

The cost function includes such things as permittingcost, surveying cost, cutting and clearing cost, drilling cost,daily cost of the crew, and more. The number of decisionvariables in the MKB method is no less than 13. Two of thoseare number of receiver channels moved per day andnumber of shots per day. Obviously, these two variablesare part of the cost computation. However, for any choiceof geometry parameters, these two parameters depend oneach other. Also, the other decision parameters appear inter-dependent and it turns out that a modified MKB methodcan be formulated with only five independent decision vari-ables (assuming fixed station spacings for shots and

receivers).The constraints in the MKB method are formulated asLeft Hand Side (LHS), Right Hand Side (RHS) or as LHS RHS. For instance one constraint is Xmax DXmax. Otherconstraints specify that certain variables can only be inte-gers. All this is fed to the Excel Solver, which finds a solu-tion for the optimization problem.

Modified MKB method. The MKB method and the spread-sheet given in Morrice et al. are used as a starting point. Thecost function and cost elements were not changed, but I intro-duced some significant improvements to the rest of theproblem description and solution. First, the number of deci-sion variables is reduced from 13 to five: nS, nR,Mi,Mx, (Table

938 T HE LEADING EDGE OCTOBER 2003 OCTOBER 2003 THE LEADING EDGE 00

Figure 3.Average fold computed from mute offset and midpoint area ofcross-spread. The mute offset Xsh cuts a circular area with radius Xsh/2

from the midpoint area of a cross-spread. The (truncated) circular areacovers all traces that contribute to average fold at the level of the shallowtarget. In case of offset-vector-oriented muting (Figure 2), the circulararea is approximated by small squares, and fold is constant.

D

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1) and number of shots per day. The first four decisionvariables are constrained to be integer; implicitly this makestotal foldM integer as well, which was not so in the origi-nal MKB method. The number of shots per day is nowallowed to be noninteger, as it concerns a variable that mayvary from day to day.

The number of constraints following the LHS RHS rulehas been reduced from 16 to nine. Currently, there are con-straints for Xmin, Xmax, Fsh (fold at shallow target), Fdp (foldat deep target), aspect ratiosAc andAu, number of receiver

channels, number of shots per day, and number ofreceiver channels moved per day.

Xmin is constrained as (Xmin DXmin)/DXmin

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The cost structure for different contractors will be differentand is difficult to capture in a simple problem descriptionas discussed here. Moreover, the data corresponding to theone-line crossline roll assumed in this paper may also beacquired in more cost-effective ways using different acqui-sition techniques (Vermeer, 2002). Yet, the MKB method canprovide a quick alternative solution for situations whereshots are known to be more expensive than receivers.

Conclusions. In this paper the survey design optimizationmethods proposed by Liner et al. (1999) and by Morrice etal. (2001) have been improved and combined into a singlespreadsheet method. The concise description of the one-lineroll orthogonal geometry with two equal station spacingsand four integers (corresponding to shot line interval,receiver line interval, maximum inline offset, and maxi-mum crossline offset) reduces the number of possible para-meter choices considerably and allows the Excel Solver toproduce fully satisfactory solutions at one click of the mouse.The combined use of the two methods allows a quick com-parison between alternative solutions, also in relation tosurvey cost.

The Excel spreadsheet discussed in this paper is includedon the CD-Rom enclosed in Vermeer (2002). Afree copy mayalso be obtained by sending an e-mail to the author.

Suggested reading. Practical aspects of 3D coherent noise fil-

tering using (f-kx-ky) or wavelet transform filters by Galibertet al. (SEG 2002 Expanded Abstracts). From acquisition footprintsto true amplitude by Gesbert (GEOPHYSICS, 2002). Contributionof new technologies and methods for 3D land seismic acquisi-tion and processing applied to reservoir structure enhance-mentBlock 10, Yemen by Girard et al. (SEG 2002 Expanded

Abstracts). 3D seismic survey design by Hornman et al. (FirstBreak, 2000). Use of recent advances in 3D land processing: acase history from the Pakistan Badin area by Karagul et al.(EAGE 2003 Extended Abstracts). 3-D seismic survey design asan optimization problem by Liner et al. (SEG 1998 Expanded

Abstracts). 3-D seismic survey design as an optimization prob-

lem by Liner et al. (TLE, 1999). Optimizing operations in 3-D land seismic surveys by Morrice et al. (GEOPHYSICS, 2001).3-D Seismic Survey Designby Vermeer (SEG, 2002). TLE

Corresponding author: gijs@3dsymsam.nl

0000 T HE LEADING EDGE OCTOBER 2003 OCTOBER 2003 THE LEADING EDGE 94

Table 2. Comparison of LUG and MKB solutions to two different problems

Station spacingsMute offset shallowMute offset deepDXminDF

shDFdp

SLI

RLIMiMxMXminXmax,iXmax,xFshFdpLUG cost

MKB cost

MKB-b40 m

2303 m3492 m600 m

4060

400 m280 m

61060

488 m2400 m2800 m

37.259.5

0.251

28039

MKB-a40 m

2303 m3492 m600 m

4060

400 m320 m

8756

512 m3200 m2240 m

32.454.50.312

27131

LUG-b40 m

2303 m3492 m600 m

4060

360 m320 m

7963

482 m2520 m2880 m

36.262.0

0.085

29255

LUG-a40 m

2303 m3492 m600 m

4060

320 m320 m

88

64453 m2560 m2560 m

40.763.8

0.065

31170

MKB50 m

1250 m2800 m600 m

1030

500 m300 m

4728

583 m2000 m2100 m

8.227.90.488

18301

LUG-250 m

1250 m2800 m600 m

1030

400 m350 m

56

30532 m2000 m2100 m

8.829.9

0.040

20071

LUG-150 m

1250 m2800 m600 m

1030

350 m400 m

6530

532 m2100 m2000 m

8.829.90.040

21232

Example 1 (Morrice) Example 2 (EAGE)