Model-based analysis of geological structures inseismic images

Melanie Aurnhammer and Klaus TonniesComputer Vision Group, Department of Simulation and Graphics

Otto-von-Guericke University Magdeburg, Germany

Abstract

Subsurface models, obtained by structural interpretation of seismic images, under-pin all decisions in hydrocarbon exploration and production. The main interest in thisfield is directed towards the interpretation of horizons and faults: Horizons are strongreflection events which indicate boundaries between rock layering while faults are dis-crete fractures along which appreciable displacement has taken place. Horizon trackingacross faults and thereby determining geologically valid correlations is an important buttime consuming task which has not been automated satisfactorily yet. The difficultiesof matching horizon segments across faults are due to the fact, that those images con-tain only a small amount of local information, furthermore partially disturbed by vagueor noisy signals.

We describe an approach which reduces interpretation uncertainties by introducinggeological constraints derived from a fault model. Two optimisation methods have beenexamined: an exhaustive search algorithm which reliably delivers the optimal solutionpresuming correctness of the model and a more viable strategy; namely, a genetic algo-rithm. Both methods successfully matched all selected horizons across normal faults intypical seismic data images.

1 Introduction

Seismic data are aquired using the seismic reflection method which explores the subsurfaceby bouncing sound waves off the interfaces between rock layers with differing physicalproperties. Figure 1 shows an example of a vertical 2-D section of a seismic data volume.By analysing the recorded signals, hypotheses about the underground structure can be de-veloped which should merge into a consistent subsurface model. All decisions in hydro-carbon exploration and production are underpinned by such models obtained by structuralinterpretation. Since drilling wells is very costly, as much information as possible shouldbe derived from the seismic data to form an opinion about the probability of encounteringpetroleum in the structures [1]. Nevertheless, it is not possible to reliably determine whetheran interpretation has been correct unless it has been verified by drilling. The test of a goodinterpretation is consistency rather than correctness [2]. Lawyer [3] defines minimum stan-dards for a good interpretation: First, the interpretation has to be internally consistent withall of the data available; second, it has to be geologically reasonable; and third it has tobe repeatable within the limits of the data. While repeatable results are an advantage of

automatically generated results, the main challenge is to establish a model which inducesgeologically reasonable solutions.Structural interpretation may be thought of as consisting of the following tasks: Localisa-tion and interpretation of faults, tracking of uninterrupted horizon segments and correlatingthese segments across faults. Reflectors in seismic images usually correspond with horizonsindicating boundaries between rocks of markedly different lithology. Faults are discretefractures across which there is measurable displacement of rock layering. On seismic sec-tions, faults are usually identified where reflectors can be seen to be displaced vertically(see Figure 1). The amount of vertical displacement associated with a fault at any locationis termed the throw of the fault.Previous attempts to solve the problem of correlating horizons across faults have beenbased on artificial neural networks [4, 5]; however, these solutions use only similarities ofthe seismic patterns.In this paper, we develop a structural model for correlating horizons across faults in orderto achieve a geologically reasonable solution. A brief overview of the state-of-the-art inthis field is given in chapter 2. Chapter 3 comprises a description of the process of horizoncorrelation as well as the definition of local horizon similarity and the derivation of thefault model. The practical implementation is described in chapter 4 where we present twodifferent algorithms: an exhaustive search strategy and a genetic algorithm. Results are thenshown in chapter 5 followed by our conclusions in chapter 6.

horizon 1

horizon 2

faults

Figure 1: Part of a seismic section showing typical normal faulting.

2 State of the Art

Modern commercial interpretation software packages offer assistance for the interpreta-tion of horizons and fault surfaces. The most commonly employed technique for horizontracking is the so called autotracking or autopicking [6]. These algorithms require manu-ally selected seed points as initial control for the autotracking operation. A similar featureis searched on a neighbouring trace; if it has been found within specified constraints, thetracker moves on to the next trace. Autotrackers are either feature based or correlationbased. While the first class simply searches for a similar configuration of samples, the lat-ter includes the neighbourhood of the trace and is therefore more robust and less sensitiveto noise. The main disadvantage of autotracking algorithms is that they are unable to trackhorizons across discontinuities. Whenever any of the search criteria are not met, the auto-tracker stops at that trace.Computer-aided interpretation of fault surfaces is significantly less advanced than horizoninterpretation [6]. Coherence measures are applied to seismic data for imaging geologicaldiscontinuities like faults or stratigraphic features. These coherence algorithms are basedfor example on cross correlation [7], semblance [8] or the eigenstructure of the data covari-ance matrix [9]. However, they produce only potential fault pixels, but do not generate theactual fault lines or surfaces. There exist methods for fault autotracking which use the samebasic approach as horizon trackers, but with limited success [10].The automatic methods described above have in common that they are based only on localfeatures. On the other hand, the actual interpretation of fault surfaces and the correlation ortracking of horizons across faults are still done manually and are therefore highly subjec-tive and time-consuming. The difficulties of automating those tasks are due to the seismicimages which contain only a small amount of local information, furthermore partially dis-turbed by vague or noisy signals. Therefore, more sophisticated methods have to be devel-oped which impose geological and geometrical knowledge in order to reduce interpretationuncertainties.

3 Correlating Horizon Segments across Faults

The problem of correlating horizon segments across faults may be subdivided into twotasks. The first task consists of the calculation of a measurement which expresses the sim-ilarity between horizon segments from either side of the fault. After determining the simi-larity values for each possible horizon-pair, the second task is to find a global combinationof horizon pairs over the complete area of interest. The global similarity may be thought ofas a first approximation to the optimum solution since its geological plausibility is not en-sured. An optimal solution has to show high similarity values as well as being geologicallyreasonable. In the following, we first describe the definition of local similarity before wespecify our model of a geologically possible solution.

3.1 Similarity of Horizon Segments

Since any individual seismic reflection is defined only by its amplitude, polarity and wave-length, it is insufficiently distinct to be correlatable on its own. In order to find attributeswhich are able to express the similarity between horizon segments on either side of thefault, we compare sequences of reflectors. Reflector sequences are distinguished by char-acteristic patterns which can usually be found on either side of the fault. We use the crosscorrelation coefficient (CC) to compare those sequences.We calculateCC for each horizon-pair by using the average amplitude or grey value ofthree pixels in horizon direction over a neighbourhood of twenty pixels above and belowthe particular horizon. Since the strata of different sides of a fault may be unequally com-pressed,CC is also calculated for stepwise scaled functions of one side within a rangeof ±8 pixels. The maximum is then chosen among the diverseCC-values. We define thesimilarity Sl,r of two horizonsl andr as their maximumCC.The estimation of the horizon similarity is complemented by a local constraint which con-cerns the polarity of horizons. Polarity can be illustrated by regarding the original seismictrace1 which shows positive or negative amplitudes representing boundaries of strata withdifferent physical properties, depending on their sequence. Since generally the sequenceof horizons remains constant on either side of a fault, the sign of the amplitude should beequal for corresponding horizon segments. This additional condition (hereafter, constraint1) causes that, in case of differing polarities, the cross correlation coefficient loses its sig-nificance.

3.2 Fault Model

The combination of horizons which leads to the highest similarity may be a geologicallyor geometrically impossible solution. Interpretation of seismic data requires a conceptualmodel of the portion of the earth involved in seismic measurements to counteract the lack oflocal information. The model is a simplification of the actual earth which comprises onlythose elements which are expected to be most important in affecting the measurements.While the model of a human interpreter is a rather vague mental picture [11], mathematicalexpressions have to be found to underpin the automatic interpretation task. The modelwe introduce consists of constraints which are deduced from geometrical and geologicalknowledge about faults.Faults are usually classified according to the direction of displacement of the blocks ofstrata on either side of the fault plane following Anderson [12]: normal, thrust (reverse) orstrike-slip. Figure 2 shows diagrammatically the types of displacement involved. Normaland reverse faults have a displacement in a vertical sense whereas the displacement ofa strike-slip fault is quoted as a horizontal displacement. While the the fault plane of anormal fault is vertical or dips towards the downthrown side of a fault, the fault plane of areverse fault dips in the opposite sense, i. e. towards the upthrown side [13]. The occurrenceof these fault classes is not arbitrary but can be ascribed to the forces which had influenced

1A seismic trace is represented by one column in the seismic images.

the area being studied. Normal faults are generally associated with tensional stress, reversefaults with compressional stress and strike-slip faults with shear stress.

(a) Normal fault

(b) Thrust (reverse) fault (c) Strike-slip fault

Figure 2: The geometric features of the three main types of faults

The throw, i. e. the vertical displacement of a fault is not constant but increases from zeroat the upper end of the fault plane to a maximum in the central portion of the fault and thendecreases to zero at the lower limit of the fault plane [14] as shown in Figure 3.The model we develop in the following is applicable to simple single normal faults butcould be modified for thrust or strike-slip faults.The constraints we deduce from the geological knowledge described above are:

• Constraint 2: Horizons must not cross

• Constraint 3: Sign of fault throw has to be consistent and correct

• Constraint 4: Fault throw function must not have not more than one local maximum

• Constraint 5: Displacement gradient is restricted

These constraints are described in detail in the following.

RR

R R

Pre-fault

Post-fault configuration

x xI1 I2

RR

R R

Pre-fault

Post-fault configuration

x xI1 I2

RR

R R

Pre-fault

Post-fault configuration

x xI1 I2

Figure 3: Representation of a typical fault throw function in terms of arcs of circles ofradius±R joined at inflection pointsI1 andI2 [14].

3.2.1 Horizons must not cross

The second constraint we consider is a simple geometrical one: horizons within a scenemust not cross (Figure 4).

2

1

00

1

22

1

00

1

22

1

00

1

2

(a) Resulting possible matches: 1-0 and0-0.

2

1

0

2

1

0

2

1

0

2

1

0

2

1

0

2

1

0

(b) Impossible matches: 1-2 and 0-2.

Figure 4: Resulting possible and impossible horizon-pairs for initial match of left hori-zon 2 and right horizon 1.

3.2.2 Sign of fault throw

Since we we restrict our model to normal faults, the expected sign of the throw for horizonsegments combined across faults can be deduced from the fault direction. In addition tothis, changes of the sign within a combination indicate very unlikely solutions.

3.2.3 Behaviour of fault throw

This constraint is employed to assess the behaviour of the fault throw within a global hori-zon combination. Only those combinations whose fault throw function shows not morethan one maximum represent probable solutions. We determine the number of zero cross-ings of the first derivative of the fault throw functions which arise from the combinations ofhorizon-pairs. Acceptable are only those combinations whose throw function shows eitherone ore no zero crossings since functions with a higher number of zero crossings indicateeither a mismatch of horizons or converging faults.

3.2.4 Restricted displacement gradient

The fifth constraint follows the investigated relationship between maximum displacementon a fault and the dimensions of the fault surface [15, 16]. The displacement gradient ona fault is a measure of the rate at which displacement changes along the fault plane in aspecified direction. It is given by

Gvm =D

R(1)

whereD is the maximum displacement along the measured section andR the radius (halfthe length) of this section [17]. Since these values are less than 1 it is often more convenientto refer to the reciprocal of the displacement gradient. Elliot [18] suggested that a linear re-lationship betweenR andD is likely and that the characteristic value of theR/D ratio isapprox. 7. Walsh and Watterson [19] showed that for a variety of faults, the relationshipbetweenD andR is non-linear and suggestedR/D values from 5 to 1000, depending onscale and material properties. For a known fault-length, the displacement gradient can beestimated by following the investigated relationship between fault lengthL and the maxi-mum vertical displacement of the horizons or the maximum fault throwD = C ·Ln. Typicalvalues for a variety of faults have been found to beC = 0.03 andn = 1.06 [13]. However,faults are often not contained to their complete extent in seismic dataset and hence, it is notpossible to determine the length of the fault. Therefore we use the value suggested by Elliotas a rough estimate which is compared to the actual displacement gradient. The variaton ofthe displacement gradient in relation to the distance to the fault center is not considered inthis calculation but nevertheless, this approximation is sufficient for our purposes.

4 Implementation

We examined two optimisation methods in order to find a geologically valid solution. First,we implemented an exhaustive search algorithm which reliably delivers the optimal solu-tion presuming correctness of the model. Therefore, this approach is suitable to serve as avalidation method. Since for an increasing number of horizons the exhaustive search ap-proach is not viable, we examined stochastic methods to solve the optimisation problem.We found a genetic algorithm to be an appropriate method to represent our problem.

4.1 Input Data

The horizon segments which we use as input for our algorithms are skeletons of strongreflections. The skeletal pixels can be considered to be the medial axis of the reflections.We use a classical thinning algorithm for bi-level images. The seismic image is convertedinto two binary images by a using a threshold to obtain the strong positive respectively thestrong negative amplitudes.Since these operations occasionally converge horizon segments across faults which do notbelong together (Figure 5(a)), we use the output of a fault highlighter (Figure 5(b)) toseparate them again. Fault lines are generated by defining manually the region of interestin the discontinuity image and generating a fault line by interpolation of the presumed faultpixels (Figure 5(c)).Horizon segments are then assigned either to the class ”left” or the class ”right” segmentsand cut at the same distance to the interpolated fault line in order to objectify the fault throwcalculation. The user has the possibility to decide which of the generated horizon segmentsare to be used for correlation.An advantage of this method is, that no seed points are required as initial step for thehorizon tracking.

(a) Skeletons of reflectors (b) 2-D section of an outputvolume of a fault highlighter

(c) Horizon segments andinterpolated fault line

Figure 5: Creation of input horizon segments

4.2 Exhaustive Search Algorithm

The basic steps of our exhaustive search method are as follows:

1. estimating the single similarity of all possible horizon-pairs;

2. calculating the total similarity of each global correlation; and

3. the application of geological constraints to find the optimum solution.

The single similarity of all horizon-pairs is determined in step (1) by calculating their cross-correlation coefficient combined with constraint 1 as described in 3.1. These pairs are con-nected in step (2) by building a solution-tree wherein each possible horizon-pair combina-tion is represented. However, the number of solutions is reduced by following constraint 2.The total similarity for each combination is evaluated by combining the similarity values ofthe single horizon-pairs. The results are then used in step (3) in an evaluation cycle, whichapplies constraints 1, 3, 4 and 5 to find the optimum combination of horizon-pairs. Accord-ing to our model, this means the solution with the highest total similarity which fulfills allgeological constraints.A detailed description of the implementation can be found in [20].

4.3 Genetic Algorithm to Correlate Horizons

We showed in [20] that the computational cost of an exhaustive search strategy is inad-missibly large since the number of combinations increases exponentially with the numberof horizons. Hence, we examined stochastic methods to find the optimum horizon combi-nation. We found a genetic algorithm to be an appropriate strategy for our problem since,compared with other heuristic methods like neural networks, it is more straightforward toprecisely define the evaluation criteria. Another advantage is, that the search space doesneither have to be connected nor compact.In genetic algorithms [21], a population of individuals represents potential solutions to aproblem. The solution is characterised by the chromosomes which form the individual.A fitness function as well as the genetic operators mutation and crossover decide on thedevelopment of the population.In our implementation [22], we use an integer string to represent an individual. While theindexl of an integer within the string represents the left horizon number, its allocated valuer(l) indicates the right horizon number. If a left horizon has no counterpart, the value−1is assigned.The initial population is created by randomly building combinations of horizon-pairs. How-ever, we restrict the search space by applying constraints. First, the set of horizon-pairs isreduced by excluding those which do not follow constraint 1. Second, we avoid the gener-ation of combinations within which horizon-pairs cross (constraint 2). This is achieved byrestricting the random search in every step to the resulting possible horizon-pairs.The fitness of a string is characterised by combining its local similarity, which is composedof the cross-correlation coefficients of the chromosomes, and its global consistency. To pro-vide global consistency, fixed amounts are subtracted from the similarity value if constraint3, 4 or 5 is violated.

5 Results

We tested both methods presented using horizons at several faults along 2-D sections in a3-D seismic data set.

Figure 6 and 7 show results from three different examples of normal faults across which thedisplayed horizons have been correlated by our exhaustive search algorithm. The correct-ness of the correlations has been verified by comparing them to those chosen by geologicalexperts. In no case, the correct solution has been found by maximising the local similarityonly. The algorithm has been successful in each of the cases which have been tested butthis is also due to the fact that the geological structure in the data set is relatively simplebut nonetheless common. In more complicated structures it is expected that the consistencycheck does not use sufficient knowledge for a correct selection.The application of the genetic algorithm has led to the same solutions as the exhaustivesearch algorithm in 2 of the 3 test cases shown above. Figure 7(b) shows the third casewhere the genetic algorithm has found only a near optimum solution which contains onegeologically incorrect correlation. The reason for this may be an insufficient considerationof fault throw behaviour.

(a) (b)

Figure 6: Examples of correctly matched horizons. Those horizons which have nocounterpart are rightly unassigned.

6 Conclusions

The exhaustive search strategy has proven to be an adequate method to correlate horizonsacross faults. Because of the small amount of local image information, horizon trackingacross discontinuities requires geological constraints to be successful. The results indicatethe suitability of the underlying fault model. Strategies have been applied which followanalysis techniques commonly used by experts in seismic interpretation.However, since the number of combinations increases exponentially with the number ofhorizons, the exhaustive search strategy is not viable for a higher number of horizon seg-ments. This has led us to the examination of stochastic methods among which we havefound genetic algorithms to be an appropriate search strategy.

(a) Correct solution,found by exhaustivesearch.

(b) Non-optimal so-lution found by thegenetic algorithm.

(c) Subset of theseismic line with-out correlation

Figure 7: Comparison of results from both search strategies

The results presented above confirm that, in principle, genetic algorithms may be applicableto our problem. Nevertheless, the parameterisation as well as the solution representationand the fitness function have to be further examined to enhance the reliability of the geneticalgorithm.Further developments will be concerned with improvements of the geological constraintsas well as the investigation of additional constraints. The method will also be tested onother data sets and on different fault classes. We expect these improvements to lead to amuch broader application and extend its use to the analysis of quite disparate data sets.

Acknowledgements

We would like to acknowledge Shell for the seismic data and stimulating discussions.

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