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^{GC}3D Design Philosophy – Part 4: The EvenInteger Rule*
Bob Hardage^{1}
Search and Discovery Article #40664 (2010)
Posted December 17, 2010
*Adapted from the Geophysical Corner column, prepared by the author, in AAPG Explorer, December, 2010, and entitled “Last Call: The EvenInteger Rule”. Editor of Geophysical Corner is Bob A. Hardage ([email protected]). Managing Editor of AAPG Explorer is Vern Stefanic; Larry Nation is Communications Director. Click for remainder of series: Part 1 Part 2 Part 3
^{1}Bureau of Economic Geology, The University of Texas at Austin ([email protected])
This article is the fourth of a fourarticle series. The final guideline that should be used when designing a 3D survey is the use of the eveninteger rule for specifying the exact dimensions of a recording swath. This design principle can be stated as:
A recording swath should span an even number of receiver lines and an even number of sourceline spacings (Figure 1).
This rule defines how wide a 3D recording swath should be in the inline and crossline directions so stacking fold is a constant, nonoscillating value across 3D image space. This eveninteger rule does not replace the previously described concept of using the depth of the primary imaging target to define the size of the recording swath; the rule merely adjusts swath dimensions by small amounts to ensure a uniform stacking fold is achieved. For example: If the depth and size of the primary imaging target cause a designer to define the inline dimension of the recording swath to be 14,000 feet and the receiver station spacing to be 110 feet, the eveninteger rule might make a designer adjust the inline dimension to 13,200 feet (120 receiver stations) or to 14,080 feet (128 receiver stations), depending on how many receiver stations occur between adjacent source lines. When applied in the crossline direction, the eveninteger rule says the recording swath should span an even number of receiver lines. For example, a recording swath consisting of eight, 10 or 12 receiver lines is better than one consisting of nine, 11, or 13 lines. Note that the wording of the rule uses the phrase, “should span,” not the more restrictive condition, “must span.”
Eveninteger rule

The reason for this eveninteger guideline can be seen by referring to the equation for crossline stacking fold FXL described in Part 3 of this article series, which is:
FXL = (1/2) (Number of receiver lines in recording swath).
If the number of receiver lines used in that stackingfold calculation is an even integer – say eight – then the crossline fold FXL is a whole number: four. In contrast, if the number of receiver lines in the recording swath is an odd number – say nine – then the crossline stacking fold FXL is a fractional number: 9/2. Data processors can sum four seismic traces to create fourfold data or five traces to make fivefold data, but they cannot include onehalf of a trace in the summation process to create 4.5fold data. Instead, stacking fold in adjacent bins in the crossline direction oscillates between four and five so that, in an average sense, the crossline stacking fold is 4.5.
An oscillating stacking fold is not fundamentally wrong; it simply introduces dataprocessing challenges that if not properly addressed cause a 3D image to contain geometryinduced amplitude variations that have nothing to do with geology. When the eveninteger rule is applied in the inline direction, it requires the receiver lines to span an even number of sourceline spacings, which means an odd number of source lines should be included in the swath.
For example: A recording swath should span six, eight or 10 sourceline spacings (which would involve seven, nine or 11 source lines, respectively) rather than span five, seven or nine sourceline spacings (which would require six, eight or 10 source lines, respectively).
If for any reason  such as permitting constraints or lack of local surface access – a recording swath cannot span an even number of sourceline spacings, the eveninteger rule can be amended so the design requirement is:
Receiver lines in the recording swath should start and stop exactly on source lines.
The rationale for this rule is that to avoid oscillations in stacking fold in the inline direction, the stacking fold value FIL must be a whole number, not a fractional number. The only way to ensure FIL will be a whole number is to force the numerator in the FIL equation stated in last month’s article to be an even multiple of the denominator. Consequently, the dimension of a recording swath in the inline direction should be an even multiple of the sourceline spacing.
An example of the eveninteger rule in 3D design is illustrated as Figure 2 and Figure 3. The key geometrical parameters are:
● Sourceline spacing = 1,320 feet. ● Receiverline spacing = 880 feet. ● Sourcestation spacing = 220 feet. ● Receiverstation spacing = 110 feet.
Consequently, there are 12 receiver stations between adjacent source lines and four source stations between adjacent receiver lines. Two recording swaths, A and B, are shown overlaying the 3D grid on Figure 2. Swath A honors the eveninteger rule; swath B does not. In the crossline direction, swath A spans 10 receiver lines, which obeys the eveninteger requirement. Swath B violates the eveninteger rule in the crossline direction because it spans 11 receiver lines. In the inline direction, swath A spans 96 receiver stations, but swath B spans 84 receiver stations. For source stations a at the center of swath A, there are 48 receiver stations (that is, four sourceline spacings) north and south of the source position, causing swath A to span an even number (eight) of sourceline spacings.
For source stations b at the center of swath B, there are 42 receiver stations. Swath B thus spans an odd number (seven) of sourceline spacings and violates the eveninteger rule in the inline direction. Swath B is further undesirable because it does not start and stop on receiver lines. Because of these geometrical constraints, swath A creates whole number values of four and five for stacking fold parameters FIL and FXL, respectively, and a uniform stacking fold of 20 across the 3D grid. In contrast, swath B creates fractional (noninteger) values for inline, crossline and 3D stacking folds. Specifically for swath B:
� FIL = 3.5. ● FXL = 5.5. ● F = 19.25.
The 3D stacking fold patterns produced by swaths A and B are compared on Figure 3. Swath A, which honors the eveninteger design rule, creates a uniform stacking fold of 20 across the fullfold central portion of the 3D grid. Swath B, which violates the eveninteger rule, produces an oscillating stacking fold in both the inline and crossline directions, which results in a lessdesirable checkerboard pattern of variable fold across the grid. Because the 3D stacking fold is 19.25, the checkerboard pattern consists of abutted areas having stacking folds of 14, 18 and 22 that when averaged together give an average stacking fold of 19.25 over the fullfold portion of the image space. Although seismic data processors can usually adjust reflection amplitudes resulting from this type of irregular stacking fold so that the amplitudes are correctly balanced across the image space, it is prudent to use an acquisition geometry that does not create such dataprocessing problems.
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