3D seismic image processing for faults

3D seismic image processing for faultsXinming Wu & Dave Hale, Center for Wave Phenomena, Colorado School of Mines

SUMMARY

Fault surfaces are often represented by triangle or quadmeshes, which are more complex than the arrays used to rep-resent seismic images, and are more complex than necessaryfor subsequent processing tasks, such as that of automaticallyestimating fault slip vectors. To facilitate image processingfor faults, we propose a simpler linked data structure in whicheach sample of a fault corresponds to exactly one image sam-ple. We also propose a method, using this linked data struc-ture, to extract complete and intersecting fault surfaces with-out holes. We use the same structure in subsequent processingto estimate fault slip vectors, and to assess the accuracy of es-timated slips by unfaulting the seismic images.

INTRODUCTION

Automatic interpretation of faults from a seismic image of-ten includes three parts: (1) Fault images are first computedusing attributes such as semblance (Marfurt et al., 1998), co-herency (Marfurt et al., 1999), variance (Van Bemmel and Pep-per, 2000; Randen et al., 2001), and fault likelihood (Hale,2013b). (2) Then fault surfaces are extracted from these com-puted fault images using various methods (e.g., Pedersen et al.,2002, 2003; Gibson et al., 2005; Hale, 2013b). (3) From ex-tracted fault surfaces, fault slips are estimated by correlatingseismic horizons (Admasu, 2008) or reflectors (Hale, 2013b)on opposite sides of fault surfaces. Although various meth-ods have been proposed for the three parts, the problem of ex-tracting intersecting faults, like those shown in Figure 1, is notwell addressed. In addition, extracted fault surfaces are oftenrepresented by triangle or quad meshes, which are often morecomplex than necessary for subsequent processing.

In this paper, we first compute images of fault likelihood, strikeand dip, and then represent these three images, all at once, byfault samples as shown in Figure 1a. Each fault sample cor-responds to one and only one seismic image sample, and isdisplayed as a small square colored by fault likelihood and ori-ented by strike and dip. We then propose a method to linkthese oriented fault samples to construct complete fault sur-faces without holes, as shown in Figure 1b. These surfaces areactually linked lists of the fault samples in Figure 1a; they ap-pear as opaque surfaces because fault samples are representedwith larger and overlapping squares in Figure 1b. With com-plete surfaces without holes, we are able to accurately estimatefault slips, and then correctly undo faulting in a seismic imageto correlate seismic reflectors across faults.

FAULT IMAGES

To illustrate our 3D seismic image processing for (1) comput-ing fault samples, (2) linking fault samples to form fault sur-faces, and (3) estimating fault dip slip vectors for unfaulting,we created a synthetic 3D seismic image containing two inter-

b)a)

Figure 1: A 3D seismic image displayed with fault samples (a)and fault surfaces (b), all colored by fault likelihood.

b)a)

F-A

F-B

F-C F-D

b)

F-A

F-B

Figure 2: A 3D seismic image with manually interpreted faults(a) and computed fault likelihoods (b).

b)

F-A

F-B

a) b)

Figure 3: A thinned fault likelihood image (a), with mostly nullvalues, is represented by fault samples (b). Each sample corre-sponds to exactly one seismic image sample, and is displayedas a square that is colored by fault likelihood and oriented byfault strike and dip.

secting normal faults F-A and F-B, a reverse fault F-C, and asmaller normal fault F-D, as shown in Figure 2a.

From a 3D seismic image (Figures 2a), we use Hale’s (2013b)method to scan the image over a range of possible combina-tions of strike and dip to find the one orientation that maxi-mizes fault likelihood, for each image sample. The maximumfault likelihood for image sample is recorded in the fault like-lihood image as shown in Figure 2b, and the strike and dipangles that yield the maximum likelihood are also recorded infault strike and dip images, which are not shown in this paper.

SEG New Orleans Annual Meeting Page 1728

DOI http://dx.doi.org/10.1190/segam2015-5882154.1© 2015 SEG

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Image processing for faults

F-A

F-B

F-A F-B

F-Ab) c)a)

Figure 4: Close-up view (a) of a subset of fault samples from Figure 3b. Links built among nearby fault samples form three setsof linked samples (b) which represent three fault surfaces (or patches). Near the intersection of faults F-A and F-B, fault F-A isseparated into two independent patches, and fault F-B has a hole. New fault samples (colored by yellow and blue) are created tomerge the fault patches and fill the hole to construct more complete intersecting fault surfaces (c).

As discussed by Hale (2013b), one significant limitation of thisscanning method is in dealing with intersecting faults. Becauseonly a single fault likelihood value and its corresponding strikeand dip are recorded for each image sample, this method im-plicitly assumes that each image sample can be associated withonly one fault. This assumption is not valid for samples wheremultiple faults intersect. For example, in the intersection areahighlighted in the fault likelihood image before (Figure 2b)and after (Figure 3a) thinning, fault likelihoods for only faultF-B have been recorded. Fault likelihoods of fault F-A mightalso be high near this intersection, but have been discarded to-gether with corresponding strikes and dips, only because theywere smaller than the fault likelihoods computed for fault F-B.Therefore, fault surfaces directly extracted from such fault im-ages often have holes near intersections. We describe below amethod to fill holes when constructing fault surfaces.

FAULT SURFACES

The thinned fault likelihood image with mostly null values,as shown in Figure 3a, can be represented as fault samplesas shown in Figure 3b, and more clearly in Figure 4a. Eachfault sample corresponds to a seismic image sample, and isdisplayed as a square that is colored by fault likelihood andoriented by fault strike and dip. Therefore, each fault samplecontains the three attributes of fault likelihood, strike and dip.

Linking fault sample neighborsBeginning with a seed sample that has sufficiently high faultlikelihood, we grow a fault surface by linking nearby faultsamples with similar fault likelihoods, strikes and dips. Re-call that each fault sample corresponds to exactly one sampleof the seismic image. This means that we can use the imagesampling grid to efficiently search for neighbor samples thatshould be linked. In a 3D sampling grid, each fault samplehas 26 adjacent grid points in a 3⇥3⇥3 cube centered at thatsample, and from these adjacent grid points, we search for upto four neighbor fault samples, above and below (in fault dipdirections), left and right (in fault strike directions).

To find a neighbor above, we need only consider the upper 9adjacent points in the 3⇥ 3⇥ 3 cube of grid points. Amongthese 9 grid points, we search for a fault sample that lies near-est to the line defined by the center fault sample and its dipvector. Similarly, we search for a neighbor below among thelower 9 adjacent grid points. To find a neighbor right and left,we need only search in the 8 adjacent grid points with the samedepth as the center fault sample. The right neighbor is the onelocated in the strike direction and nearest to the line defined bythe center fault sample and its strike vector. The left neighboris the one in the opposite direction and closest to the same line.

This searching is repeated until no more neighbors can be foundto obtain a linked list of fault samples. Then a new seed withsufficiently high fault likelihood is chosen from unlinked sam-ples for growing a new fault surface. This process ends whenno remaining unlinked fault samples have sufficiently high faultlikelihood. We discard surfaces with small numbers of linkedsamples and keep only those with significant sizes. For exam-ple, in Figure 4b, we have kept only the three largest surfacesconstructed from the fault samples in Figure 4a.

As shown in Figure 4b, each sample in a fault surface is linkedto up to four neighbors. However, some neighbors may bemissing in areas where the seismic image is noisy or wherefaults intersect as shown in Figure 4a. These missing faultsamples can cause holes within a fault surface, like the faultF-B in Figure 4b, and can yield gaps which separate a faultsurface into independent patches, like those of fault F-A in Fig-ure 4b. To fill in these holes and gaps to construct more com-plete fault surfaces, we must interpolate missing fault samples.

Interpolating missing neighborsDuring the processing discussed above for linking neighborsto a fault sample, if any of the neighbors above, below, left orright are missing, we try to create them. We do not first con-struct fault surfaces or patches with holes (missing neighbors)as shown in Figure 4b, and then fill holes in each of the con-structed fault surfaces or patches, because in this way we can-not merge fault patches to form more complete fault surfaces.



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Instead, we check for missing neighbors and create them as wegrow fault surfaces, and thereby directly obtain complete faultsurfaces without holes as shown in Figure 4c.

To create neighbors that are missing for a fault sample, wemust create adjacent fault samples in a 3⇥3⇥3 cube centeredat that sample, and then determine whether any of them couldbe the missing neighbors. To create fault samples within a 3⇥3⇥3 cube, we first construct images of fault likelihood, strikeand dip in a slightly larger cube; for example we use a 5⇥5⇥5cube in this paper. To construct a such a small fault likelihoodimage centered at the fault sample with missing neighbors, wefirst search nearby in a 31⇥ 31⇥ 31 cube to find N samplesthat have fault attributes similar to those for the center sample.

We then construct a 5⇥ 5⇥ 5 fault likelihood image by accu-mulating weighted anisotropic Gaussian functions generatedfrom the N samples:

f (xi) =XN

k=1f (xk)g(xk �xi), (1)

where f (xi) denotes a fault likelihood value computed for thei-th grid point in the 5⇥5⇥5 cube, and xi denotes the positionof that grid point. Here, f (xk) denotes the known fault likeli-hood of the k-th nearby fault sample, and g(x) is an anisotropicGaussian function: g(x) = exp(� 1

2 x>R>SRx), where

R =

2

4u>

kv>kw>

k

3

5 , and S =

2

64

1s 2

u0 0

0 1s 2

v0

0 0 1s 2

w

3

75 . (2)

Here, the unit column vectors uk and vk are the dip and strikevectors of the k-th nearby fault sample, respectively. The vec-tor wk = uk ⇥vk is normal to the plane of the k-th fault sample,and su, sv, and sw are specified half-widths of the Gaussianfunction in the dip (u), strike (v) and normal (w) directions,respectively. The matrix R rotates the anisotropic Gaussian tobe aligned with the vectors uk, vk and wk. We set sw = 1 andsu = sv = 15 samples, so that the Gaussian to be accumulatedextends primarily in the fault strike and dip directions.

When accumulating anisotropic Gaussian functions for the i-th sample in the 5⇥5⇥5 cube, we also accumulate weightedouter products of normal vectors for that sample:

D(xi) =XN

k=1f (xk)g(xk �xi)wkw>

k . (3)

We then apply eigen-decomposition to the 3⇥3 matrix D(xi),and choose the eigenvector corresponding to the largest eigen-value to be the normal vector wi for the i-th sample in the5⇥ 5⇥ 5 cube. From the normal vector wi, we then computestrike and dip angles for this i-th sample.

From the three constructed fault images centered at a faultsample with missing neighbors, we then create fault samplesadjacent to that fault sample, and search for missing neigh-bors among these new fault samples. Using both newly createdand original fault samples, we are able to construct intersectingfault surfaces without holes, as shown in Figure 4c. Figure 5ashows four fault surfaces extracted from the 3D seismic imageby using the method discussed above. These surfaces are re-ally just linked lists of fault samples shown in Figure 4c; theyappear as opaque surfaces, as shown in Figure 5b, because werepresent fault samples using larger and overlapping squares.

b)

F-AF-B

F-C

F-D

F-AF-B

F-C

F-D

a) b)

Figure 5: Links are built among consistent fault samples (Fig-ure 3b), and each set of linked fault samples in (a) representsa fault surface that appears opaque in (b), where fault samplesare displayed as larger overlapping squares.

F-AF-B

F-C

F-D

a) b)

Figure 6: Fault throws (a) for unfaulting (b) a seismic image.

FAULT DIP SLIPS

Fault dip slip is a vector representing displacement, in the dipdirection, of the hanging wall side of a fault surface relativeto the footwall side. To estimate dip slip, we first estimate itsvertical component called fault throw. Knowing the fault throwand fault surface with linked samples, as in Figure 4c, we canthen walk up or down the fault in the dip direction to computethe two corresponding horizontal components of dip slip.

Similar to Hale (2013b), we use the dynamic image warpingmethod (Hale, 2013a) to estimate fault throws by correlatingseismic reflectors on opposite sides of a fault surface. Differ-ent from Hale’s (2013b) method that represents a fault surfaceas a quad mesh, we represent a fault surface using the simplerlinked data structure as shown in Figure 4c; this structure facil-itates gathering seismic amplitudes on opposite sides of a faultfor dynamic warping.

It is advantageous that the fault surfaces represented in Fig-ure 4c do not have holes. Holes in fault surfaces like thoseshown in Figure 4b make it difficult to access seismic ampli-tudes alongside a fault for dynamic warping. Holes also makeit difficult for the dynamic warping method to enforce the con-straints that fault throws vary smoothly. For these reasons,fault throws estimated using fault surfaces, like those shown inFigure 4c, are more accurate than those for fault surfaces withholes, like those shown in Figure 4b. As shown in Figure 6a,estimated fault throws for fault F-C are negative because thisfault is a reverse fault. Estimated fault throws for faults F-A, F-B, and F-C generally increase in magnitude with depth, whilethrows for fault F-D first increase, then decrease with depth.



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a) b)

c)

large throw

d)

Figure 7: Fault samples colored by fault likelihood (a) are computed, and linked to form fault surfaces (b), which are used toestimate fault throws (c) for computing an unfaulted image (d).

With fault throw, the vertical component of dip slip estimatedfor each sample on a fault surface, we can use the links withinthe fault surface to walk upward or downward in fault dip di-rections, to determine the horizontal inline and crossline com-ponents of slip for that sample. The dip slip vectors estimatedin this way are used in Wu et al. (2015) to obtain an unfaultedimage shown in Figure 6b. This unfaulted image illustrates thatestimated fault slips are accurate because seismic reflectors arealigned across all the faults.

A REAL IMAGE EXAMPLE

From a real seismic image shown in Figure 7a (and in thesmaller subset shown in Figure 1), (1) we first compute faultsamples, which are displayed as squares oriented by strikesand dips, and colored by fault likelihood in the right-upperpanel of Figure 7a. As each fault sample corresponds to a seis-mic image sample, the same fault samples can be displayedas a fault likelihood image overlaid with the seismic imageslices shown in Figure 7a. (2) We then link the oriented faultsamples to form fault surfaces, displayed in Figure 7b. Com-pared to the three image slices in Figure 7a, some fault samplesare removed when constructing surfaces, because they cannotbe linked to form surfaces with significant sizes. Also, new

fault samples are created to fill holes that occur where faultsintersect. (3) We further use these fault surfaces to estimatefault dip slips, and display fault throws (vertical component ofslips) in Figure 7. After estimating fault slips, the number offault surfaces is reduced, because we keep only fault surfacesfor which dip slips are significant. The estimated fault dip slipvectors are then used to undo faulting in the seismic image, sothat reflectors are aligned across faults, as shown in Figure 7d.

CONCLUSION

We propose to represent fault surfaces by linked lists of faultsamples, each of which corresponds to one and only one seis-mic image sample. Therefore, the processing for faults dis-cussed in this paper is mostly just image or array processing.We also propose a method that uses this simple data structureto construct complete and intersecting fault surfaces withoutholes. With complete fault surfaces, we can accurately esti-mate fault dip slips, and then undo faulting in a seismic image.

ACKNOWLEDGMENTS

The real 3D seismic image used in this paper was graciouslyprovided by Kees Rutten and Bob Howard, via TNO (Nether-lands Organisation for Applied Scientific Research).



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REFERENCES

Admasu, F., 2008, A stochastic method for automated match-ing of horizons across a fault in 3D seismic data: PhDthesis, Otto-von-Guericke-Universitat Magdeburg, Univer-sitatsbibliothek.

Gibson, D., M. Spann, J. Turner, and T. Wright, 2005, Faultsurface detection in 3-D seismic data: Geoscience and Re-mote Sensing, IEEE Transactions on, 43, 2094–2102.

Hale, D., 2013a, Dynamic warping of seismic images: Geo-physics, 78, S105–S115.

——–, 2013b, Methods to compute fault images, extract faultsurfaces, and estimate fault throws from 3D seismic images:Geophysics, 78, O33–O43.

Marfurt, K. J., R. L. Kirlin, S. L. Farmer, and M. S. Bahorich,1998, 3-D seismic attributes using a semblance-based co-herency algorithm: Geophysics, 63, 1150–1165.

Marfurt, K. J., V. Sudhaker, A. Gersztenkorn, K. D. Crawford,and S. E. Nissen, 1999, Coherency calculations in the pres-ence of structural dip: Geophysics, 64, 104–111.

Pedersen, S. I., T. Randen, L. Sonneland, and Ø. Steen,2002, Automatic fault extraction using artificial ants: 72ndAnnual International Meeting, SEG, Expanded Abstracts,512–515.

Pedersen, S. I., T. Skov, A. Hetlelid, P. Fayemendy, T. Randen,and L. Sønneland, 2003, New paradigm of fault interpreta-tion: 73rd Annual International Meeting, SEG, ExpandedAbstracts, 350–353.

Randen, T., S. I. Pedersen, L. Sønneland, et al., 2001, Auto-matic extraction of fault surfaces from three-dimensionalseismic data: 81st Annual International Meeting, SEG, Ex-panded Abstracts, 551–554.

Van Bemmel, P. P., and R. E. Pepper, 2000, Seismic signalprocessing method and apparatus for generating a cube ofvariance values. (US Patent 6,151,555).

Wu, X., S. Luo, and D. Hale, 2015, Moving faults while un-faulting 3D seismic images: CWP Report 838.



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EDITED REFERENCES Note: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2015 SEG Technical Program Expanded Abstracts have been copyedited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

Admasu, F., 2008, A stochastic method for automated matching of horizons across a fault in 3D seismic data: Ph.D. dissertation, University of Magdeburg.

Gibson, D., M. Spann, J. Turner, and T. Wright, 2005, Fault surface detection in 3-D seismic data: IEEE Transactions on Geoscience and Remote Sensing, 43, no. 9, 2094–2102.

Hale, D., 2013a, Dynamic warping of seismic images: Geophysics, 78, no. 2, S105–S115, http://dx.doi.org/10.1190/geo2012-0327.1.

Hale, D., 2013b, Methods to compute fault images, extract fault surfaces, and estimate fault throws from 3D seismic images: Geophysics, 78, no. 2, O33–O43, http://dx.doi.org/10.1190/geo2012-0331.1.

Marfurt, K. J., R. L. Kirlin, S. L. Farmer, and M. S. Bahorich, 1998, 3-D seismic attributes using a semblance-based coherency algorithm: Geophysics, 63, 1150–1165, http://dx.doi.org/10.1190/1.1444415.

Marfurt, K. J., V. Sudhaker, A. Gersztenkorn, K. D. Crawford, and S. E. Nissen, 1999, Coherency calculations in the presence of structural dip: Geophysics, 64, 104–111, http://dx.doi.org/10.1190/1.1444508.

Pedersen, S. I., T. Randen, L. Sonneland, and Ø. Steen, 2002, Automatic fault extraction using artificial ants: 72nd Annual International Meeting, SEG, Expanded Abstracts, 512–515.

Pedersen, S. I., T. Skov, A. Hetlelid, P. Fayemendy, T. Randen, and L. Sønneland, 2003, New paradigm of fault interpretation: 73rd Annual International Meeting, SEG, Expanded Abstracts, 350–353.

Randen, T., S. I. Pedersen, and L. Sønneland, 2001, Automatic extraction of fault surfaces from three-dimensional seismic data: 81st Annual International Meeting, SEG, Expanded Abstracts, 551–554.

Van Bemmel, P. P., and R. E. Pepper, 2000, Seismic signal processing method and apparatus for generating a cube of variance values: U. S. Patent 6151555.

Wu, X., S. Luo, and D. Hale, 2015, Moving faults while unfaulting 3D seismic images: Colorado School of Mines Center for Wave Phenomena (CWP) Report 838.



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http://dx.doi.org/10.1190/geo2012-0327.1

http://dx.doi.org/10.1190/geo2012-0331.1

http://dx.doi.org/10.1190/1.1444415

http://dx.doi.org/10.1190/1.1444508

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3D seismic image processing for faults