Anisotropic 3d Seismic Features for RobustHorizons Correlation across Faults

Fitsum Admasu and Klaus ToenniesComputer Vision Group

Institut fuer Simulation und GraphikOtto-von-Guericke Universitaet Magdeburg

Abstract— While 3D seismic data become widespread and thedata-sets get larger, the demand for automation to speed upthe seismic interpretation process is increasing as well. However,the development of intelligent tools which can do more to assistinterpreters has been difficult due to low information contentin seismic data. In this paper, we present an image processingmethod in which a-priori geological knowledge is incorporatedto correlate horizons across faults. Our new method exploitsanisotropic spatial correlation of horizons for robustness andaims at developing an interpreter friendly interactive environ-ment. The results of this method are compared with previouslyproposed methods.

I. I NTRODUCTION

Seismic images are pictures of underground structures. Theyare obtained by sending seismic waves from a surface andrecording the reflected signals due to changes in acousticimpedance of underground layers. Three dimensional seismicdata are a sequence of slices, which image cross-sectionallayers at different depths [5].

Interpretation of faults and horizons are the backbone ofseismic data interpretation. Horizons are layered rocks whichare created through a long time sedimentation process. Afaulting process cuts and displaces horizons. Accordingly,faults are identified in seismic data as lateral discontinuitiesof horizons. Unless erosion occurred, the faulted horizonsusually have their corresponding part on the other side of thefault. The correspondence analysis between horizons across afault is important for describing a fault throw function, whichinfluences exploration decision. Thus, correlation of horizonsacross faults is an indispensable task of seismic interpretation.

Seismic data interpreters locate faults as lines from hori-zon discontinuities on seismic slices [3]. Horizon are thenconnected across faults on the basis of reflection characterand geological reasoning. Interpreters evaluate their correlationdecision on 2-d slices (see fig. 1). The 3-d nature of the dataset can be appreciated only if a sequence of slices is displayed.This is done frequently as the information from a single sliceis often inconclusive. Identification of the geological featuresin seismic sections by an interpreter is time consuming andsubjective.

The main focus of our research is developing a computer-based methodology for correlation of horizons across faults.Expected outcomes are to reduce the time-consuming manualtask, and to avoid the uncertainties associated with fault

Fig. 1. Seismic slice with manually interpreted faults and horizons.

interpretation by providing a repeatable and robust seismicdata analysis tool.

II. PREVIOUS WORKS

Partial automation success has been achieved in horizontracking by autotrackers, which track horizons between faultsby extending manually selected seismic traces based on localsimilarity measures [5].

Fault interpretation is more difficult as it involves detectionof faults and correlation of horizons across faults. Our mainconcern here is the horizons correlations. We are aware ofonly two works, which directly deal with the correlationacross a fault. The first is by Aurnhammer [2] who mentionsthat matching using solely seismic patterns is infeasible andproposes a model-based scheme for matching horizons atnormal faults in 2-d seismic images. Well-defined horizonssegments on both sides of the fault were extracted and matchedbased on local correlation of seismic intensity and geologicalknowledge. However, a pure 2-d approach lacks efficiencyand is suitable only if the information of the 2-d seismicslice is sufficient for evaluation of the geological constraints.The second is from Admasu et. al. [1] who introduce amulti-resolution continuous horizon correlation scheme wherethe correlation task is formulated as a non-rigid continuous

point matching between the two sides of the fault. It hasthe advantage that it does not require all horizons to bewell-defined. However, it is computational expensive and notsufficiently robust with respect to noise and artifacts in seismicdata. Besides, interactions from nearby faults distort the globalfault displacement model which was computed at the verycoarse level.

In this paper, we present a novel method, which strives forexploiting existing 3-d spatial relationships in the data directlyfor robust data analysis. We emphasis definitions of horizonsat different discrete levels so that strong horizons signals giveguidance for matching the weaker horizons signals.

III. D ATA REPRESENTATION

Depth-converted seismic data can be represented as 3-dsceneS = (V, F ), where V is the set of all voxels,c, inthe data.c = (x, y, z) denotes the coordinates of any voxel.x andy span the cross sectional area covered by the seismicsurvey whilez spans the depth.F is a function that assignsto every voxelcεV seismic attributes vectorF (c).

Horizons are represented as topological surfaces in the 3dseismic data [8]. A fault is a 3d damage zone on the layersof horizons. A fault plane is a regression surface that fits thedamage zone.

IV. PROBLEM REPRESENTATION

We restrict ourself to correspondence analysis across a sin-gle normal fault surface. Normal faults are the most commonfault types where one side of the fault block (hangingwall)moves down relative to the other side (footwall). Furthermore,we assume that the fault plane,τ , is already given and definedasτ(x, y, z) = 1, where(x, y, z)εV .

We define a mapping process which transforms the seismicdata into a horizon and fault patch dependent coordinatesystem (see fig. 2).

Consider(hx, hy) as a point on the fault plane coordinatesystem, and let(xh, yh, zh) represents such point in respect toS with τ(xh, yh, zh) = 1. SupposeHf = (Hf1,Hf2...,Hfm)and Hh = (Hh1,Hh2, ...,Hhn) represent the projected hori-zons onto the fault plane from the foot- and the hanging-wallrespectively. Then for1 ≤ i ≤ m

Hfi(hx, hy) ={

Σjai,jFij , if gfi(xh, yh) = zhφ, otherwise

whereFij = F (xh+ j, yh+ j, gfi(xh+ j, yj + j)) is a vectorof seismic complex attributes which are computed from theHilbert transform of the data.gfi defines the trend surface fora footwall horizon at depth leveli, and0 < gf1 < gf2 < ... <gfm. ai,j are weights for the integration.φ means null vector.Similar definition follows for the hangingwall by replacingthe subscriptf by h. Then, the horizons correlations consistof finding optimal matches betweenHf andHh.

Fig. 2. A fault patch is mapped onto two 2-d planes.

V. HORIZON CORRELATION AS A LABELLING

The horizon correlation problem can be seen as a registra-tion or stereo correspondence problem between the two featureplane images mapped from a fault patch. However, the appli-cation of classical stereo correspondence [9] or registrationalgorithms [4] to perform correspondence between the featureimages is not feasible due to little intensity information toguide the utilization of optical flows and presence of localdistortions.

We define the horizon correlation as a labelling problemin which Hf serves as sites whileHh serves as labels. Thecorrelation problem may then be solved within a MAP-MRFframework advocated by Geman and Geman [7]. The labellingfunction, ζ, is defined as

ζ : Hf → Hh ∪ {Φ} (1)

where Φ represents not-horizons regions and is assigned tosites where there are no corresponding labels fromHh. Thisis where some horizons on the footwall do not have matcheson the hangingwall due to lacking data.

Each site is considered as a random variable, and the la-belling as events. When the sites have some labelling assignedto them, we have a configurationT such thatT = {(i, j) :iεHf ∧ jεHh ∪ {Φ}}. However, the admissible labels maynot be common to all sites due to the geological constraintsthat horizons must not cross each other. Furthermore, offsetshave only one direction as we deal with normal faults. Theseimpose constraints on the search for wanted configurations.

As it was pointed out before in [2] [1], we can not relyonly on the seismic information to solve the correlation task.It needs to be guided by a-priori knowledge of displacementpatterns on the fault surface. Therefore, we need an objectivefunction which maps a candidate configuration solution to areal number measuring the quality of the solution in termsof seismic similarity as well as geological knowledge. Suchobjective functionξ is defined as follows.

ξ(T ) = α ∗ Es(T ) + β ∗ Eg(T ) + γ ∗ Ec(T ) (2)

whereα, β, andγ are weights for balancing, and

Es(T ) =∑

(i,j)εT

Sij (3)

Sij = cross corr cof(Hfi, Hhj)

where cross corr cof is cross correlation coefficientbetween horizonsHfi and Hhj . The cross-correlationtechnique has been already successfully used before by [2][1] to measure the similarity between seismic signals. Itsstrength comes from its ability to measure linear relationshipsof the seismic features.

Eg(T ) =∑

(i,j)εT

∑

dεDi,j

(d−DT (r))2 (4)

where Di,j , is the set of offsets of hanging wall horizonj from footwall horizon i for a given configurationT , andrε{(hx, hy) : Hfi(hx, hy) 6= φ}. DT is an ideal fault throwfunction proposed by Walsh et.al. [10]:

DT (r) = (2(((1 + nr)/2)2 − n2r)

2(1− nr))/dTmax (5)

wherenr = (R− r)/W such thatR is the fault center,Wthe width of the fault, anddTmax is a maximum displacementon the fault surface.

Ec measures a priori smoothness.

Ec(T ) =∑

iεHf

(ζ(i)− ζ(i− 1))2 (6)

The solution for the horizon correlation is a configurationTmax, which maximizes the value of the objective function ofequation 2. Geman and Geman [7] provide a proof for suchclaim, assuming MRF distribution. Since searching forTmax

is not trivial due to the non-linearity and many local maxima,we use a simulated annealing search optimization technique.

VI. EXPERIMENTS AND RESULTS

We have conducted experiments to test our hypothesesthat the lateral horizon’s homogeneity leads to more robustcorrelation, and we compare the results with the techniquesintroduced before.

Fault patches were isolated from seismic data, which consistof large numbers of faults and fault systems. Each isolatedfault patch contains a fault surface with seismic sectionson the two sides of the surface. The seismic sections wereconsidered to be displaced only under the influence of thissingle fault surface. The identification of the fault surface wasdone semiautomatically as follows.

Fault features were highlighted on seismic slices by apply-ing a set of log-Gabor filters [6] at different orientation anddifferent scales. A fault line was drawn on a slice to depict thefault to be tracked. Then starting from this line, a fault planewas extracted automatically by propagating and identifyingthe fault lines in the successive inline slices using orientationconstraints and linear regression of the filter responses of log-Gabor filter. The fault lines generation steps are shown onfigure 3.

From each side of the fault plane, seismic features wereprojected along the horizons’ orientations onto the fault planeas described in section IV. The trend surfaces were determined

Fig. 3. (a) Seismic slice. (b). Fault enhancing by log-Gabor filters. (c).Automatically detected fault line on the post-processed image of (b). (d)Automatically detected fault line on the original seismic slice.

by gradient structure tensor (GST), a tool to analyze localstructure in images [3]. GST (5S 5 St) models the seismicsection at sides of the fault as being locally planer and yieldsan estimate of the orientation of the structure. Then, theorientation estimation was used to steer the integration of theseismic features. The coefficients were from a gaussian kernel.As the fault plane was only an approximation of the 3d faultregion, seismic information is distorted at locations close to thefault plane. To correct for this, integration along the horizonwas started at a five pixel distance from the fault plane.

The simulated annealing procedure was set up for maxi-mizing the objective function at equation 2. The parameterswere fine tuned for converging to optimal correlation solutions.Then, tests were conducted by varying the lateral extents ofthe fault plane. The increments in lateral extent yield morecorrect correlation. This was expected as more information isavailable to make the correlation decision. We have isolated15fault patches. Within the context of the individual fault patch,geologically acceptable automatic correlations are obtained on12 out of the15 fault patches. The rejections of the3 faultpatches are mainly due to mismatched horizons at deeperlocation where the seismic horizon definitions deteriorate andso affect the similarity measures. We compare our method withpreviously proposed methods.

Aurnhammer [2] introduces a method, which extracts dis-crete prominent horizon segments on 2d slices and findmatches between them. The method works only if there isenough information on 2d seismic section and also tends toleft a large number of horizons unmatched. Fig. 4 comparesthis method’s result with ours. For comparison, we selectedthe same slice and fault used in [2]. Our method outperformsdue to the smoothness constraints, which imposes a continuityon the correlation as well as the integrations of seismic datawhich serves more information and as noise repression.

Another method, which treats the horizon matching asa non-rigid continuous image registration, is proposed byAdmasu and Toennies [1]. This method was able to findsatisfactory correlations only for8 fault patches out of the15fault patches that are used for our test here. Our new method

Fig. 4. (a) Original seismic slice. The broken white line shows the faultline. (b) Automatic correlation results of [2].(c) Correct automatic correlationresults of our method.

is more robust and faster because it matches segments whichhave more discriminating features than points (see Fig. 4). Fur-thermore, feature integration and smoothness constraint enableguiding the correspondence analysis in low SNR regions byresults from high SNR regions.

Fig. 5. (a) Original slice, (b) results from [1], (c) our new method.Improvements of(c) over (b) are shown in white.

Our algorithm was not successful for faults shown on fig.6. These failures are mainly due to local features disturbanceswhich are also partially resulted from incorrect definitions ofthe fault surface.

Fig. 6. Black arrows show the automatic correlation results.(a) and (b)show the original seismic slice and the correlation results for fault 1.(c) and(d) do the same for fault 2. For some incorrect results, the white arrows showthe manual correlations.

VII. C ONCLUSIONS

We have shown that the consideration of the anisotropicspatial correlation of the seismic data provides more robust

results. With the change from isotropic features of the realpart of the signal to anisotropic features of the complex signal,we increased the discriminative power for the local matchingattribute. However, the usefulness for providing a reliabilitymeasure has to be determined. At very noisy regions of theseismic data, the cross-correlation coefficient is not reliableenough to estimate the seismic similarity.

Our current implementation of the method using Matlabtakes from 15 to 20 minutes from the fault plane extractiontill the correlations result. Further improvement could beachieved, if the simulated annealing procedure is embeddedin a multi-resolution framework.

Since the ultimate goal is production, seismic interpretationis done iteratively to get a good reliable and consistentmeasures of the interpretation. To this end, our new horizoncorrelation method can take also as input manually pickedhorizons to make the interpreters interactivity more effective.

VIII. A CKNOWLEDGEMENTS

We would like to acknowledge Shell for the seismic dataand stimulating discussions. We also thank Dr. Stefan Backat RWTH Aachen University of Technology for his expertiseadvices regarding geology. This research was supported byDFG Grant TO-166/8-1.

REFERENCES

[1] F. Admasu and K. Toennies, “Automatic method for correlating horizonsacross faults in 3d seismic data.”In Pro. the IEEE CVPR,WashingtonDC, June, 2004.

[2] M. Aurnhammer,Model-based Image Analysis for Automated HorizonCorrelation across Faults in Seismic Data,PhD Thesis, University ofMagdeburg, 2003.

[3] P. Bakker,Image structure analysis for seismic interpretation,PhD Thesis,Technical University of Delft, 2002.

[4] L. Brown, “A Survey of Image Registration Techniques”,ACM Comput-ing Surveys, Vol. 24, No. 4, pp. 325-376, 1992.

[5] G. Dorn, “Modern 3-D Seismic Interpretation,”The Leading Edge,Vol.17, No. 9, pp. 1262-1273, 1998.

[6] D. Field, “Relations Between the Statistics of Natural Images and theResponse Properties of Cortical Cells”,Journal of The Optical Society ofAmerica A, Vol. 4, No. 12, pp.2379-2394, December 1987.

[7] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distribution andthe Bayesian restoration of images.”IEEE Trans. on PAMI,Vol. 6, No.6, pp. 721-741, 1984.

[8] B. Nielsen,P. Mostad,J. Gjerde “Stochastic structural modelling,”Mathe-matical Geology, Vol. 35, No. 8, 2003.

[9] D. Scharstein and R. Szeliski and R. Zabih, “A taxonomy and evaluationof dense two-frame stereo correspondence algorithms.”In Proceedings ofthe IEEE Workshop on Stereo and Multi-Baseline Vision,Kauai, HI, Dec.2001.

[10] J. Walsh and J. Watterson, “Distributions of cumulative displacement andseismic slip on a single normal fault surface”,J. of Structural Geology,Vol. 9, No. 8, pp. 1039-1046, 1987.