Automatic Method for Correlating Horizons across Faults in 3D Seismic Data

Fitsum Admasu Klaus ToenniesComputer Vision Group Computer Vision Group

University of Magdeburg, Germany University of Magdeburg, Germanyadmasu@isg.cs.uni-magdeburg.de klaus@isg.cs.uni-magdeburg.de

Abstract

Horizons are visible boundaries between certain sedimentlayers in seismic data, and a fault is a crack of horizonsand it is recognized in seismic data by the discontinuitiesof horizons layers. Interpretation of seismic data is a time-consuming manual task, which is only partially supportedby computer methods. In this paper, we present an auto-matic method for horizon correlation across faults in 3dseismic data. As automating horizons correlations usingonly seismic data features is not feasible, we reformulatedthe correlation task as a non-rigid continuous point match-ing problem. Seismic features on both sides of the fault aregathered and an optimal match is found based on geologicalfault displacement model. One side of the fault is the float-ing image while the other side is the reference image. First,very prominent regions on both sides are automatically ex-tracted and a match between them is found. Sparse faultdisplacements are then computed for these regions and theyare used to calculate parameters for the fault displacementmodel. A multi-resolution simulated annealing optimizationscheme is then used for the continuous point matching. Themethod was applied to real 3D seismic data, and has shownto produce geologically acceptable horizons correlations.

Key Words: seismic image interpretation, model-basedanalysis, multi-resolution correspondence analysis.

1. IntroductionThree-dimensional seismic data consist of numerousclosely-spaced seismic lines that provide measures of sub-surface reflectivity. Different subsurface’s rock layers havedifferent acoustic impedances. As result when seismicwaves are sent to underground structures, the changes in theseismic wave velocities give strong reflections being visi-ble in the seismic images. These strong reflection eventsare known as horizons. Faults rarely give reflection eventsrather they are recognized in seismic data by the disconti-nuities of horizons events [4]. Structural interpretation at-tempts to create 3D subsurfaces model and it consists ofthe following tasks: localization and interpretation of faults,tracking of uninterrupted horizon segments and correlating

these segments across faults [2]. Figure 1 shows a seismiccube extracted from 3-d seismic data. The interpreted hori-zons and faults on this figure is done manually. The horizonsegments which are offset by the fault line are matched bythe arrows.

Timeline

Inline

Crossline

Fault lineHorizon segment

Figure 1: 3-D seismic data with some horizons interpreta-tions.

Seismic data interpreters perform manual horizon track-ing mainly on 2-D projections of 3-D image or on 2-D slicesof the 3-D data. It is a time-consuming task due to the largesize of seismic data and has inconsistencies among inter-preters.

Auto-picking or auto-trackers (reviewed in [2], and [8])have been commonly used to assist horizon tracking. Auto-picking tools are aimed at extending manually selected seis-mic traces based on local similarity measures. They per-form well if there are uninterrupted horizon features. Buthorizon interruptions are very common.

Correlation of horizons across faults is one of the impor-tant task of structural interpretation. Interpreters find hori-zons and connect them to each other on the basis of reflec-tion character and geological reasoning. The horizons off-sets are used to calculate displacement on the fault surface.Fault displacements map allow a more objective assessmentof subsurface interpretation.

Our work is aimed at developing a computer-basedmethodology for correlation of horizons across faults inseismic data set. Besides reducing the time of interpreters,

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computer-assisted solutions, based on a quantitative model,provide repeatable or robust data analysis tool. Existingthree-dimensional spatial relationships in the data (such ascontinuity) may be exploited directly whereas humans areonly able to evaluate them from 2-d projections or 2-d slicesof the data.

However automating the horizon correlation task is verychallenging. Seismic data contain only little image infor-mation, which is further obscured by noises. Since the twosides of the fault may also undergo different geological pro-cesses, such as compression and erosion, there are scale dif-ferences and some horizons on the one side side may nothave matches on the other side.

1.1. Previous Work

Alberts et.al. [1] explain a method for tracking horizonsacross discontinuities. They trained artificial neural net-works to track similar seismic intensities. However, horizontracking across faults using solely seismic patterns is infea-sible due to large seismic data distortion near faults. To al-leviate this matter, Aurnhammer [2] propose a model-basedscheme for correlation of horizons at normal faults in 2Dseismic images. The author extracts well-defined horizonssegments on both sides of the fault and matchs the segmentsbased on local correlation of seismic intensity and geolog-ical knowledge. Since exhaustive search for optimal solu-tion of correlation is unfeasible, the author suggests geneticalgorithm as optimization technique. However, a pure two-dimensional approach lacks efficiency and is suitable onlyif the information of the 2D seismic slice is sufficient forevaluation of the geological constraints. We strive for ex-ploiting existing three-dimensional spatial relationships inthe data (such as continuity) directly for robust data analy-sis.

The methodology that we envisage is fusing the seismicdata with information from a geological model in an itera-tive fashion. The method first finds and matches point re-gions that contain sufficiently distinctive structures on thetwo sides of the fault regions. Then the parameters requiredfor the geological model are estimated based on displace-ment computed for these regions. Finally simulated anneal-ing global search technique is used to find the optimal corre-lation between the two sides of the fault, optimal in sense ofmaximizing seismic features similarities while maintaininggeologically valid solutions. The optimization is done in amulti-resolution approach in order to take into account thathorizons layers exist at different levels of resolution. Themethod has been applied to faults patches extracted fromreal 3D seismic data and has shown to be efficient in puttingthe displaced horizon into correspondence.

2. Correlating Horizons across FaultsHorizons correlation is a task to gather sediment features onthe two sides of the fault and to find an optimal match be-tween them. We first identify a fault surface, approximatedas a plane here, and its patch. The fault patch is then mappedonto two planes (left and right fault planes). Local featuresfrom the seismic data are mapped along the horizon direc-tion onto two planes (see figure 2). The canny edge detector[5] is used for defining the direction along which values areintegrated. The seismic features are averaged in 10 pixelssize along the edge and mapped to the fault plane (see fig-ure 2). Seismic information are distorted at locations closeto the fault because of the geological process of fault cre-ation. To correct for this fault distortion, averaging alongthe horizon starts at six pixel distance from the fault line.

Two features values are computed for each mapped pointon the planes: an averaged amplitude (gray-value) of theseismic data, and a reliability measure. The amplitude at-tributes (see Figure 3) are used to compute the local similar-ity between points on the two planes. However there is noguarantee that corresponding horizons on the two sides of afault have equal feature values since sediments left and rightof the fault may have suffered different fates during and af-ter creation of the fault. Thus the reliability attribute is usedto weigh the gray-value (amplitude) similarity computed ata local level. Reliability is computed as an average valueof the coherence cube [3] attribute of the seismic data. Fig-ure 4 shows seismic coherence attributes mapped onto theleft and right fault planes; they correspond to the amplitudefeatures of figure 3. The darker regions show lesser relia-bility values, which mean less weight are given for intensitycorrelation computed at those places.

Horions

Fault Patch

Mapping

from fault

patch to

fault planes

Vertical displacment

Displacment

Sediment

features at

fault

Left

Right

fault plane

Horiontal displacment

Figure 2: A fault patch is mapped onto left and right faultplanes.

Consequently, the correspondence analysis of the hori-zons is defined as finding displacement map for left faultimage to overlap it with the right fault image so that corre-sponding positions in the two images are superposed.

We define a match function, ξ : Z2 → �, which mea-sures the degree of match between the left and right side ofthe fault planes as they are overlaid on top of each other.The two images (left and right side of the fault plane) are

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Left planeRight plane

Figure 3: Seismic amplitude attributes mapped onto the leftand right fault planes.

L. R

Figure 4: Seismic coherence attributes mapped onto left (L)and right (R) fault planes.

given as two dimensional arrays and denoted by Il and Ir

where each of Il(x, y) and Ir(x, y) maps to its respectiveaveraged seismic features. The right image, Ir, serves asreference image while Il(x, y) is the floating image whichis displaced with displacement field T (x, y). T (x, y) is a2D spatial coordinate transformation, such that T (x, y) =(x′, y′).

The match function, ξ, is then given as

ξ(T ) = Es(T ) + λ ∗ Eg(T ) (1)

where Es is the energy which is computed based on theseismic similarity, whereas Eg is the energy that measuressimilarity of a given transformation and a geological model.They are described in details in section 2.1 and 2.2. λ is anegative scalar real value and used to balance Es and Eg .

2.1. Computation of Seismic Similarity, Es

Seismic similarity is defined as a mathematical measureof intensity similarity. In order to estimate the similarity,we choose the normalized cross-correlation technique. Themain reasons for this choice are its potential accuracy androbustness with respect to noise as it is computed in someneighborhood of a point.

For inputs Ir, Il, and T , the normalized local cross-correlation function at point (x0, y0) with T (x0, y0) =(x′

0, y′0) is calculated as

C(x0, y0) =

∑nx,y=−n(Ilxy) ∗ Irxy)√∑n

x,y=−n(Ilxy)2√∑n

x,y=−n(Irxy)2(2)

whereIlxy = Il(x0 − x, y0 − y) − Il(x0, y0)Irxy = Ir(x′

0 − x, y′0 − y) − Ir(x′

0, y′0)

Il(x0, y0) and Ir(x′0, y

′0) are respectively the mean of

Il and Ir for n-neighborhood around point (x0, y0) and(x′

0, y′0). The normalized local cross-correlation value at

each point is weighted by the coherence cube feature value.Then the seismic similarity energy, Es, for a given transfor-mation is computed as the sum of the values of the weightednormalized local cross-correlation over all points on thefloating image.

2.2. Computation of Geometrical Energy, Eg

Eg is computed by comparing similarity between the cur-rent observed displacement with the geological model. Ourgeological model is based mainly on pattern of displace-ment on fault surfaces.

2.3. Fault Displacement ModelLayers of rock that have been moved by the action of faultsshow displacement on either side of the fault surface. Thefault displacement is the offset of segments or points thatwere once continuous or adjacent. We deal only with nor-mal faults. A normal fault is a type of fault in which thehanging wall moves down relative to the footwall (see figure5). Thus normal fault displacements have only one direc-tion, which means for any (x, y), T (x, y) = (x, y′). Nor-mal faults are the most common ones and the displacementmodel which has been used here can be extended to otherfault types.

Normal fault

Thrust fault

Strike-slip Fault

Hanging

wall blockFootwall

block

Figure 5: Fault types.

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Furthermore, it is known from structural geology thathorizons do not cross each other, that is for any T (x, a) =(x, a′) and T (x, b) = (x, b′), we have

a < b =⇒ a′ ≤ b′ (3)

According to heuristics of Walsh et.al. [12] [13], a nor-malized displacement, D, at a point on a fault surface isgiven by

D = 2(((1 + r)/2)2 − r2)2(1 − r) (4)

where r is normalized radial distance from the fault cen-ter. The normalized displacement is D = d

dmaxwhere d is

the fault displacement at a point and dmax is the maximumdisplacement on a fault surface.

2.3.1 Estimation of Fault Parameters

The fault model of equation (4) assumes that the center ofthe fault, the width of the fault and the maximum displace-ment on the fault are known. But faults are often not con-tained to their complete extents in seismic data set. Wehave estimated the extent of the fault by computing land-mark displacements. The simplest method for extraction oflandmark corresponding points is to manually specify them.However, it is very difficult to specify accurate correspond-ing points and time-consuming. Thus, landmark displace-ments are extracted automatically in the following fashion.

If the part of the fault ends in the seismic data, then thereare regions with zero displacement. We take partially over-lapping segments in those regions and propagate them to thenext slices. When any offset is found, the displacements arecalculated and serve as landmarks. But if neither end of thefault is included in the seismic data, we identify particularprominent linear structures from the two sides of the faultfeatures images. These prominent structures are extractedas segments by threshold of high contrasts. The segmentscan be matched with high confidence by maximizing the to-tal cross-correlation values of the intensities around smallneighborhood, as it is described in Aurnhammer [2]. Theresulting landmark displacements are plugged into equation(4). Then the Marquardt-Levenberg constrained nonlinearoptimization method, implemented in Matlab, is used to es-timate the parameters for the fault displacement model. Fi-nally, the theoretical transformation value for each point ofthe floating image is computed.

The geometrical energy, Eg , is computed as the leastsquare error between any given transformation, T , and thetheoretical transformation map, Tcomputed. i.e

Eg(T ) =∑

i

(Tcomputed,i − Ti)2 (5)

3. OptimizationAfter we have defined the transformation function and sim-ilarity measures, the next step is to find a suitable optimiza-tion procedure to generate the optimal transformation map,Tmax, which maximizes the value of the match function, ξ.

Tmax = argmaxT∈{T :Z2→Z2}(ξ(T)) (6)

The general difficulty of the optimization of the matchfunction in equation (6) is the non-linearity existing inthe search. It usually has many local maxima. Simplesearch strategies such as gradient decent are not appropri-ate. Therefore we use a simulated annealing (SA) [9], astochastic non-linear optimization technique.

3.1 Simulated Annealing Optimization

We set up the metropolis algorithm [11] to perform the min-imization form of equation (6). The metropolis algorithmincorporates ξ with the regularizer term which imposes apriori smoothness constraint on the solution (after [10]). Ithas to also make sure that all the current randomly gener-ated candidate solutions satisfy the geological constraint de-fined at equation (3).

We have used the geometric optimization schedule [6].The temperature is held fixed during each loop. At the endof each loop, k, the temperature, �, is dropped according tothe rule:

�k+1 = α ∗ �k (7)

The vales of the initial temperature (�0), α and the num-ber of iterations in each loop are to be determined experi-mentally.

3.2 Multi-resolution Optimization

Though simulated annealing (SA) algorithms could find theglobal optimal results, it has high computational complex-ity [6]. We propose to take advantage of a multi-resolutionanalysis to increase the convergence rate of SA. The multi-resolution analysis is obtained by wavelet decompositionof the left and right side fault images. The decomposi-tion is done by calculating the coefficient of a one dimen-sional continuous wavelet transform. Each column of thetwo dimensional image is fed to the one-dimensional con-tinuous wavelet transform which computes the continuouswavelet coefficients of the input column vector at real, pos-itive scales using a given type of wavelet. Figure 6 showsthe result of the continuous 1-D wavelet coefficients usingDaubechies wavelet [7]. The correspondence analysis be-tween figure 6(a) and figure 6(b) is faster if we perform theanalysis starting from the coarser-level and go to the finer-level.

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(a) (b)

(c) (d)

(e) (f)

Figure 6: Wavelet decomposition: (a) and (b) show the leftand the right-side fault feature images. (c) and (d) are re-spectively decompositions of (a) and (b) at coarse level. (e)and (f) are respectively decompositions of (a) and (b) atcoarser level.

In this multi-resolution scenario, the simulated anneal-ing optimization starts from lower resolution level, then athigher resolution it searches with the convergence solutionsreached at the lower resolution. Since the convergence so-lutions at lower resolution are close to the higher resolutionoptimum solutions, less number of iterations can be usedfor searching at higher resolutions.

4. Results and Discussion

We have tested the method on seven fault patches takenfrom 3D seismic data which were surveyed from two dif-ferent geographical locations. We have extracted the faultplane by linearly interpolating between manually providedseed points. The optimal solutions are defined as the wouldbe solutions if experts performed the correlation.

For the match function, we observed that the appropri-ate values of λ which compensates between the local seis-mic features and the global geological constraint range from−0.5 to −0.3. The initial temperature for the simulated an-nealing (SA) is set to values between 1000 and 1200 butstill needs further experiments. We found that it is favorablefor the optimization structure to cool down quickly and tostay more at lower temperatures. Thus the temperature isdecremented at each step by 93% as cooling proceeds. Theresults of SA for a sample fault patch are shown on figure7. To verify the results, we have restored the feature imagesto the 3D fault patch by inverting the feature mapping pro-

cess (see figure 2). Some optimally correlated horizons areshown on two seismic slices on figure 8. These slices aretaken from the restored 3D fault patch at locations I and IIof figure 7 (a) and (b). The correlations are done accordingto the displacement map of figure 7 (d). The correlationshave been verified by comparing them with manual inter-pretation.

(b)(a)

(d)(c)

II I III

Figure 7: (a) and (b) show respectively left and right faultimages. (c) is the deformed image of (b) after aligning itwith (a). (d) is displacement map of the alignment.

(a) (b)

Figure 8: (a) and (b) seismic slices extracted at positionsI and II of figure 7. The black arrows for some horizonsindicate optimal correlation results of SA. The white curvesshow the fault lines.

Our method fails in two test cases. Interactions fromnearby faults distort the fault displacement model and leadto incorrect global constraint (figure 9). The method alsoneed proper initial discrete segments match to give correctalignment. But in some cases it was not easy to detectand match those segments automatically. Thus we needto switch in such cases to semi-automatic version of themethod where human interpreters find the horizons seg-ments and calculate the landmark displacements. Howeveran overall analysis of the test cases reveals that the resultingcorrespondence analysis of our method is satisfactory with

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(b)(a)

Figure 9: Seismic slices. The black arrows indicate corre-lation for horizons found by SA algorithm, while the whitearrows show experts’ manual correlations for incorrect cor-relation of the black arrows. The white curves show faultlines.

very good performance.

5. Summary and ConclusionsWe have presented an automatic method, which aligns lo-cations of the left fault plane onto those right fault plane.An optimal displacement vector field for the alignment wasfound based on the combination of seismic image informa-tion and a fault displacement model. The fault displacementmodel is constructed by performing initial discreet match ofsome prominent regions. Multi-resolution based simulatedannealing optimization strategy is adopted to find the opti-mal solution. While the results are somewhat preliminary,they clearly demonstrate the applicability of our approachto real seismic data.

Besides its main application, automatic fault displace-ment calculation, our method can be also used as an auto-matic verification tool for manually interpreted faults. Andalso as a visualization tool for studying the full extent of thefault when the fault part is not fully included in the seismicdata set.

Further we consider additional geological constraintsand seismic attributes which can improve our method. Themethod will be extend to include lateral displacement.

6 Acknowledgements

We would like to acknowledge Shell for the seismic dataand stimulating discussions. We also thank Stefan Backand Janos Urai for their expertise advices regarding geol-ogy. The work presented here was supported by DFG GrantTO-166/8-1.

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