Footprint Attenuation with 5D Interpolation - Arcis

Footprint Attenuation with 5D Interpolation 

Peter Cary and Mike Perz 

Arcis Seismic Solutions, Calgary, Alberta, Canada 

In recent years, 5D interpolation has become a very popular 

method for regularizing seismic data before 3D prestack 

migration. In addition to its main purpose of reducing the 

generation of migration artifacts due to irregular spatial 

sampling, 5D interpolation also has the very beneficial sideeffect 

of attenuating acquisition footprint during the interpolation 

process. An example is shown in Figure 1. The reason 

for why footprint attenuation occurs during 5D interpolation 

has been the subject of some debate. The footprint could 

either be attenuated because the noise or signal causing the 

footprint has been correctly interpolated and then regularly 

sampled within all CDP’s, or it could be attenuated because 

the signal or noise causing the footprint has not been interpolated 

correctly and therefore is reduced in the final image. 

In this article we examine the fidelity of the 5D interpolation 

of a real data example and show (with the aid of a new 5D 

interpolation quality control tool called 5D leakage) that footprint 

has been attenuated because the interpolation has been 

interpolating correctly in some parts of the 3D volume, and 

paradoxically enough, also because it has not been interpolating 

correctly in other parts. 

Acquisition footprint is a type of noise, so we are always 

pleased to see footprint being attenuated. However, we 

would prefer to see footprint attenuated because the source 

of the footprint has been regularized across all CDP’s, not 

because the interpolation has worked badly. The point is that 

interpolation ideally should not be used explicitly as a form 

of noise attenuation. A perfect interpolator, such as a sincfunction 

interpolator (Karl, 1989), is said to be perfect (zero 

error) because it is able to interpolate both signal and noise 

perfectly. In fact there is nothing in a perfect interpolation 

algorithm that distinguishes between signal and noise. As far 

as a perfect interpolation algorithm is concerned, the data is 

all just signal. 

All forms of 5D interpolators are capable of producing excellent 

results, but they are nevertheless imperfect in real 3D 

seismic data sampling scenarios. All forms of 5D interpolation 

(MWNI (Trad, 2009), ALFT (Xu et al., 2010), rank reduction 

(Trickett et al., 20) incorporate some sort of sparseness or 

simplicity constraint on the solution in order to interpolate a 

fully sampled 5D output wavefield from a relatively small 

number of input samples of the full wavefield. Whatever 

signal or noise is not sparse (i.e. sparse in the sense that the 

algorithm requires for correct interpolation) will be imperfectly 

interpolated. The danger, therefore, is that some 

complex parts of the signal may be attenuated during the 

interpolation process. 

So why is acquisition footprint actually attenuated during 5D 

interpolation Is it because the data is being correctly or 

incorrectly interpolated Without knowing whether the noise 

or signal that causes the footprint has been correctly or incorrectly 

interpolated, we cannot answer this question with any 

certainty. In order to answer questions such as these, we have 

come up with a method for measuring the error that occurs 

during 5D interpolation (Cary and Perz, 2012a,b). We call this 

measure of the error “5D leakage”. The 5D leakage is measured 

only at input trace locations because it is computed by 

using just the newly 5D interpolated traces to interpolate 

back into the original input traces locations. The difference 

between the true input traces and the interpolated estimates 

at those same input locations input is a simple measure of the 

5D interpolation error. Once these 5D leakage traces are 

computed, they can be stacked and viewed in various ways, 

including time slices of the CDP stack of the leakage traces. 

Acquisition footprint usually appears as a regular pattern of 

linear artifacts, or a mesh pattern, on time slices of 3D stacks, 

such as in Figure 1(a). Acquisition footprint is not an indication 

of the true subsurface geology. Instead it resembles some 

FOCUS ARTICLE 

Coordinated by Mauricio Sacchi* / Carson Yarham / Jason Noble 

1a) 1b) 

Figure 1. An example of 3D acquisition footprint attenuation as seen on the timeslice of a 3D orthogonal land dataset (a) before 5d interpolation (b) after 5d interpolation 

with the MWNI algorithm. 

Continued on Page 30 

November 2012 CSEG RECORDER 29

Focus Article Cont’d 


Continued from Page 29 

aspect of the acquisition geometry, such as the shot or receiver 

layout. The many causes of acquisition footprint have been investigated 

extensively (e.g. Hill et al., 1999). A likely reason for the 

strong footprint that appears in the shallow parts of many 3D 

images is the fact that the particular offsets and azimuths that are 

most contaminated with source-generated or backscattered noise 

at early times are sampled only within every n-th CDP in the inline 

or crossline directions, where n is the number of CDP’s between 

source or receiver lines. As a result, a regular pattern of noise 

appears at every n-th inline or crossline. 

Various aspects of reflected signal, such as residual normal 

moveout, can also cause acquisition footprint. For example, a 

multiple reflection, or a primary reflection that is not flat across 

all offsets due to an NMO velocity error may generate a regular 

footprint pattern due to the regular wavelet sampling at particular 

offsets within a pattern of CDP’s. Even amplitude variations 

with offset (AVO) on a perfectly flat reflector will generate some 

degree of amplitude footprint on the final stacked image due to 

the pattern of offset sampling within the CDP grid. 

After 5D interpolation, the offset and azimuth sampling of traces 

within neighboring CDP’s is regularized. Regardless of whether 

it was some aspect of signal or noise that was the original cause 

of the footprint, the fact that the signal and noise are sampled 

identically at all offsets and azimuths within every CDP after 5D 

interpolation will make the regular footprint pattern disappear. 

At least that is the argument, if you believe that 5D interpolation 

is interpolating the full wavefield (i.e. signal plus noise) perfectly. 

The trouble is that, even though the 5D interpolated data may 

look excellent, we have already established that 5D interpolation 

is not perfect, so there is at least some possibility that footprint is 

being attenuated because the aspect of the noise or signal that 

caused the footprint has not been correctly interpolated. To 

generate footprint, you require two things: (1) a variation of the 

wavefield with offset (and/or azimuth), and (2) a somewhat 

regular, but sparse, sampling of offsets and azimuths within 

CDPs. To attenuate the footprint with an interpolation method, 

you need to do one (or both) of two things: either remove 

requirement (1) with an imperfect interpolation, or remove 

requirement (2) with a good interpolation/regularization 

method. How is one to know which of these two situations is 

causing the footprint attenuation in any particular case 

An examination of the accuracy of the interpolation, as measured 

by the 5D leakage, can answer this question. If the aspect of the 

wavefield that is causing the footprint is being correctly interpolated, 

then the stack of the 5D leakage traces will not contain any 

sign of the footprint. This would be the case if the source-generated 

noise, or the multiple, or whatever signal or noise was the 

origin of the footprint in the first place was being correctly interpolated 

into all new trace positions. The source of the footprint 

would then also disappear from the 5D leakage. On the other 

hand, if the footprint does appear in the 5D leakage, then we 

know that the aspect of the signal or noise that was the origin of 

footprint was not being interpolated correctly. Footprint in the 

5D leakage implies a poor interpolation. Absence of footprint in 

the 5D leakage implies a good interpolation. 

Figure 2 is taken from the shallow section (300ms) of the 3D 

volume from an orthogonal land 3D dataset. The original stack 

2a) 

2b) 

2c) 

Figure 2. An example of footprint attenuation at the shallow time level (300ms) 

of a 3D dataset (a) before 5d interpolation (b) after 5d interpolation, and (c) the 

5d leakage (a measure of the error in the interpolation). The fact that the 5d 

leakage contains much of the footprint that has been removed between (a) and (b) 

implies that the aspect of the wavefield that is causing the footprint (probably 

some type of noise) has been interpolated imperfectly at this time level. 

Continued on Page 32 

30 CSEG RECORDER November 2012




shows a fairly large amount of footprint which is probably 

caused by some remnant of source-generated noise that has not 

been completed removed during processing. The attenuation of 

the footprint by 5D interpolation is both dramatic and beneficial 

to the final image. The time slice of the 5D leakage stack shows a 

large amount of footprint, which indicates that the noise has not 

been perfectly interpolated. 

Figure 3 is taken from a deeper section (1500ms) of the same 3D 

volume as in Figure 2. The footprint in this case is less strong 

than in Figure 2, and its true cause (whether due to signal or 

noise) is not obvious from examining the data but it is likely due 

3a) 

to some variation with offset in the moveout or amplitude on a 

primary or multiple reflection. The footprint attenuation by 5D 

attenuation is more subtle in this case than in Figure 2, but the 

attenuation is no less important at this time level because of the 

similarity of some of the real geologic features to the linear footprint. 

At this time level, the time slice of the 5D leakage shows 

only slight hints of footprint, so in this case, whatever signal or 

noise was the original cause of the footprint is being very well 

interpolated. 

The examination of the 5D leakage in Figures 2 and 3 has enabled 

us to know what is really happening when footprint is attenuated 

by 5D interpolation. The answer in the case of this 3D 

volume is that footprint attenuation is occurring for two different 

reasons. In the shallow section, footprint was attenuated as a 

result of imperfect interpolation, and in the deeper section, footprint 

was attenuated as a result of a good interpolation. 

These findings should not surprise us if we reflect on the data 

requirements for a good 5D interpolation. The smooth spatial variations 

of moveout and AVO on primary and multiple reflections 

can be described by relatively simple wavenumber spectra, and 

such simple variations in the data are well reconstructed by 5D 

interpolation given its sparseness assumption. By contrast, the 

rapid offset variations of steeply-dipping source-generated noise 

are less well-suited to the assumptions of 5D interpolation so we 

should not be surprised that this type of noise is not being perfectly 

reconstructed. Since this 3D dataset is typical of most land orthogonal 

3D datasets in terms of its signal and noise sampling issues, 

we would expect to see that footprint is attenuated by 5D attenuation 

in many other 3D datasets for both of these reasons. 

In summary, this article has examined the issues surrounding the 

question of why 5D interpolation attenuates 3D acquisition footprint. 

We have found that attenuation occurs because 1) signal 

in the deeper section has been interpolated and then sampled 

3b) 

3c) 

Figure 3. An example of footprint attenuation at a deeper time level (1500ms) of a 3D dataset (a) before 5d interpolation (b) after 5d interpolation, and (c) the 5d leakage. 

The fact that the 5d leakage contains very little footprint implies that the aspect of the wavefield that is causing the footprint (probably some aspect of the signal) has been 

interpolated correctly at this time level. 

32 CSEG RECORDER November 2012 

Continued on Page 33



regularly in the output CDP’s and 

2) noise in the shallow section has 

not been properly interpolated so 

its effect has been reduced in the 

final image. We have come to 

these conclusions based on the 

recently devised method of measuring 

5D leakage (Cary and Perz, 

2012a,b). 5D leakage has proven 

to be very useful not only for 

enquiring into this footprint 

attenuation issue, but also for 

enquiring into other important 

issues such as the preservation of 

resolution. R 

References 

Cary, P. and M. Perz, 2012a, “High Resolution 

5D Interpolation: Measuring and 

Compensating for 5D Leakage”, CSEG 

Conference Abstracts. 

Cary, P. and M. Perz, 2012b, “5D leakage: 

measuring what 5D interpolation misses”, 

82nd Annual International Meeting, SEG, 

Expanded Abstracts. 

Hill, S., M. Shultz and J. Brewer, 1999, 

“Acquisition footprint and fold-of-stack plots”, 

The Leading Edge, 18, 686-695. 

Karl, J. H., 1989, An Introduction to Digital 

Signal Processing, Academic Press, 341pp. 

Trad, D., 2009, Five-dimensional interpolation: 

Recovering from acquisition constraints: 

Geophysics, 74,, no. 6, V123-V132. 

Trickett, S., L. Burroughs, A. Milton, L. 

Walton, and R. Dack, 2010, Rank-reductionbased 

trace interpolation: 81st Annual 

International Meeting, SEG, Expanded 

Abstracts, 1989-1992. 

Xu, S., Z. Yu, and G. Lambare, 2010, 

“Antileakage Fourier transform for seismic data 

regularization in higher dimensions”, 75, no. 6, 

WB113-WB120. 


Peter Cary has B.Sc. and M.Sc. degrees in physics, and a B.A.degree in 

philosophy from the University of Toronto, and a Ph.D. in geophysics (1987) 

from Cambridge University, England. He worked for Chevron both in 

Calgary and in La Habra, California from 1982 to 1984 and was Manager of 

Geophysical Research with Pulsonic Geophysical Ltd. from 1988 to 1996 and 

Chief Geophysicist with Sensor Geophysical Ltd. 1996 to 2011. He is 

presently Chief Geophysicist, Processing with Arcis Seismic Solutions. He 

has presented and published many papers on seismic processing, and 

served as technical program chairman of the SEG 2000 Annual Meeting and 

of the 1993 CSEG Annual Meeting. He served as CSEG president in 2004-05 

and was 2nd V.P. of the CSEG in 1996-97. He was an associate editor (seismic 

processing) of Geophysics from 1998-2001. One of his specialities is 

processing and writing software for multicomponent seismic data. 

Mike Perz received his BSc in Physics from the University of Toronto in 

1990 and his MSc in Geophysics from the University of British Columbia 

in 1993. Shortly thereafter, he joined Pulsonic Geophysical where his main 

focus was writing prestack time migration software. Four years later he 

moved to Geo-X Systems Ltd, later part of Divestco Inc., where he spent 

thirteen years, first writing geophysical applications software then managing the R and D 

group. In 2010, Mike moved over to Arcis Corporation where he was employed as Vice 

President of Technology and Integration at the time the organization was purchased by 

TGS. His current title is Manager Technology, Arcis Processing and Reservoir Services, and 

his role is two-tiered, focusing on both geophysical software and business development. His 

research interests span all elements of land processing including data regularization, 

prestack migration, fracture detection, deconvolution, tomostatics and multiple attenuation. 

November 2012 CSEG RECORDER 33

Footprint Attenuation with 5D Interpolation - Arcis

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