A Genetic Algorithm for Constrained Seismic Horizon Correlation Melanie Aurnhammer1 , Klaus T¨onnies1 and Rafael Mayoral2 1 Computer Vision Group, Department of Simulation and Graphics
Otto-von-Guericke University Magdeburg, Germany {aurnhamm, klaus}@isg.cs.uni-magdeburg.de 2 Departamento Autom´atica y Computaci´on
Universidad P´ublica de Navarra, Campus de Arrosad´ıa, Spain
[email protected]
Abstract. A new approach towards automating the interpretation of geological structures like horizons or faults in reflection seismic data images is presented. Although automatic horizon tracking across faults to thereby determine geologically valid correlations is an important and time consuming task, it has still not been solved satisfactorily. The reason for this is the difficulty involved in locating non-ambiguous local correlation features due to the small amount of local information contained in seismic images. The method described in this paper provides an enhancement against a solely local feature based analysis by imposing additional geological and geometrical constraints to find a geologically valid solution. We model this process as an activity of searching for an optimum combination of the available knowledge by introducing a genetic algorithm. The application of the method to typical seismic data images resulted in the successful matching of all major horizons across several normal faults.
discontinuities such as faults still remains problematical. The difficulties of this task are due to the images containing a small amount of local information, partially disturbed by vague or noisy signals. Previous attempts to solve the problem of correlating horizons across faults have been based on artificial neural networks [1, 2]; however, these solutions use only similarities of the seismic patterns. In this paper we describe an approach for automatically correlating several horizons simultaneously across discontinuities to find a global, geologically realistic solution. We formulate this task as an activity of searching for an optimum combination of all major horizons across a fault being constrained by geological criteria. In order to implement this activity, we introduce a genetic algorithm and apply it in a manner which is consistent with the characteristics of the problem solving process of a human interpreter.
2 Correlating Horizons across Faults The problem of correlating horizons across faults can be subdivided into two levels. At the first level, pairs consisting of one horizon from each side of the fault have to be formed. At the second level, these single pairs have to be combined to a global, geologically valid match for the complete area of interest. A-priori knowledge which is derived from the fault behaviour is introduced at each level in the following manner:
1 Introduction Reflection seismic data images consist of adjacent time series indicating the arrival of artificially created sound waves reflected from the interfaces between rock formations with differing physical properties. By analysing these traces, hypotheses about the underground structure can be developed which should merge into a consistent subsurface model. Strong reflection events are called horizons and indicate boundaries between rock formations or strata. Faults are discrete fractures across which there is measurable displacement of rock layering. An important task of the structural interpretation of seismic images is the location of faults and the tracking of horizons across these faults. While the tracking of uninterrupted horizons is well addressed by the standard repertoire of modern commercial interpretation software packages, the tracking of horizons across
First level: Horizon-pairs • Similarity of reflector sequences (constraint 1) • Consistent polarity (constraint 2) • Restricted fault throw (constraint 3) Second level: Combinations of horizon-pairs • Horizons must not cross (constraint 4) 1
• Expected sign of fault throw (constraint 5) • Behaviour of fault throw variation (constraint 6)
2 1 0
2.1 First Level: Single Horizon-Pairs In order to find characteristic horizon attributes, we consider the first constraint which is derived from the procedure a human interpreter usually applies: To find matches of horizons, the expert would look for characteristic patterns of reflector sequences which exist on either side of the fault. We model this process by using the cross-correlation coefficient (CC) as well as applying two constraints which a human expert would use to exclude certain matches of horizons. We calculate CC for each horizon-pair by using the average amplitude or grey value of three pixels in horizon direction over a neighbourhood of twenty pixels above and below the particular horizon. Since the strata of different sides of a fault may be unequally compressed, CC is also calculated for stepwise scaled functions of one side within a range of ±8 pixels. The maximum is then chosen among the diverse CC-values. We define the similarity Sk,l of two horizons k and l as their maximum CC. The second constraint concerns the polarity of horizons. Polarity can be illustrated by regarding the original seismic trace1 which shows positive or negative amplitudes representing boundaries of strata with different physical properties, depending on their sequence. Since generally the sequence of horizons remains constant on either side of a fault, the sign of the amplitude must be equal for corresponding horizon segments. The third constraint follows the investigated relationship between fault length L and the maximum vertical displacement of the horizons or the maximum fault throw D [3]: (1) D = C · Ln .
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(a) Resulting possible matches: 1-0 and 0-0.
2 1 0
2 1 0
(b) Impossible matches: 1-2 and 0-2.
Figure 1. Resulting possible and impossible horizon-pairs for initial match of left horizon 2 and right horizon 1.
fault and then decrease to zero at the lower limit of the fault plane [4] as shown in Figure 2. The sign or direction of the fault throw can be calculated by considering the fault type: faults may be classified according to the direction of displacement of the blocks of strata on either side of the fault plane. The most common fault types are normal and reverse faults which have a displacement in a vertical sense. While the fault plane of a normal fault is vertical or dips towards the downthrown side of a fault, the fault plane of a reverse fault dips in the opposite sense, i. e. towards the upthrown side [3]. The occurrence of these fault classes is not arbitrary but can be ascribed to the forces which had influenced the area being studied. Therefore, certain assumptions about the fault class and therewith the expected sign can be made. In addition to this, changes of the sign within a combination indicate very unlikely solutions.
The examination of thousands of faults reported in [3] showed (2) D = 0.03 · L1.06 to be the best-fitting relationship, which we adopted for our model. Horizon-pairs showing displacement values greater than D are very likely to be wrongly matched.
2.2 Second Level: Combining Horizon-Pairs The combination of horizons to form a global horizon match across a fault requires additional knowledge. The fourth constraint we consider is a simple geometrical one: horizons within a scene must not cross (Figure 1). The fifth and the sixth constraint we use are derived from the characteristics of the fault throw function: The fault throw will increase from zero at the upper end of the fault plane to a maximum in the central portion of the 1A
Figure 2. Representation of a typical fault throw function in terms of arcs of circles of radius ±R joined at inflection points I1 and I2 [4].
Constraint six is employed to assess the behaviour of the fault throw within a global horizon combination. Only those combinations whose fault throw function shows not more than one maximum represent probable solutions.
seismic trace is represented by one column of the seismic im-
ages.
2
3 Genetic Algorithm to Correlate Horizons
3.3 Initial Population
The basic principles of Genetic Algorithms (GA) as a directed random search technique were invented by Holland [5]. Thereafter, a series of literature became available [7, 6, 8], followed by a large number of practical applications in different fields [9, 10, 11] as well as reports focussing the underlying theoretical principles [12, 13]. In GA, a population of individuals represents potential solutions to a problem. The solution is characterised by the chromosomes which form the individual. A fitness function decides on the development of the population. The main reasons for adopting GA are as follows. First, since the number of solutions increases nonpolynomially with the number of horizons, exhaustive search is not acceptable. Second, compared with other heuristic methods such as neural networks, it is more straightforward to precisely define the evaluation criteria. Finally, the style of search is similar to the one human experts would follow. That is, they iteratively try to build pairs of strong reflectors, combine them to a global match, check the consistency and replace pairs until a geometrically and geologically valid solution is found.
The initial population is created by randomly building combinations of horizon-pairs. However, we restrict the search space by applying constraints. First, the set of horizon-pairs is reduced by excluding those which do not follow constraint 2 and constraint 3. Second, we avoid the generation of combinations within which horizonpairs cross (constraint 4). This is achieved by restricting the random search in every step to the resulting possible horizon-pairs.
3.1 Solution Representation
Offspring strings are generated by choosing two parent strings, randomly selecting a single crossing location and exchanging the substrings bounded by that crossing location. Before evaluating the fitness of a new solution obtained by crossover, its geometrical validity is verified and, as the case may be, discarded.
3.4 Selection Operator As selection scheme, we adopted the usual roulette wheel procedure to pick r parents on the basis of their fitness [6]. Then (I − r) distinct individuals are taken to survive unchanged into the next generation. I denotes the number of individuals in a generation. The remaining r individuals which are not selected as survivors will be automatically replaced by the r offspring produced in the breeding phase.
3.5 Crossover Operator
An individual is represented by an integer string. While the index k of an integer within the string represents the left horizon number, its allocated value l(k) indicates the right horizon number. If a left horizon has no counterpart, the value −1 is assigned.
3.6 Mutation Operator
3.2 Fitness Function
A classical mutation strategy which changes randomly chromosomes would generate an unreasonably high rate of combinations which are invalid regarding the constraint of non-crossing horizons. Thus, we use a revised strategy which repeatedly produces mutations for a chromosome until constraint 4 is satisfied. A new horizon is generated by rounding a uniformly distributed random number in the range from −1 to the maximum number of horizons.
The fitness of a string is characterised by the combination of the sum of the local similarity of its chromosomes and its global consistency. Albeit a geologically valid solution may contain less horizon pairs than the maximal number of possible matches, it has to be considered that the reliability of the global match decreases with a decreasing number of horizon-pairs. This is reflected in our definition of the fitness function by favouring combinations consisting of a greater number of horizons. We thus calculate the fitness T Si of a string i consisting of a number n of chromosomes, i. e. horizon-pairs, j from T Si =
n j=1
2 Sk,l(k) − P1 − P2 .
4 Results We tested our method using horizons at several faults along 2D sections in a 3D seismic data set. The GA parameters were chosen in initial, non-exhaustive experiments. An appropriate mutation rate was observed to be 0.05. We increased the population size I proportional to the product of left and right horizons since the number of horizons influences the probability that the optimal solution is contained in the initial population. 80% of the population is replaced at each iteration step. Instead of using a fixed number of generations, we terminate the process if there are no further improvements over a specific number of generations.
(3)
P1 denotes a discount score with a fixed amount when constraint 5 is not fulfilled, and P 2 denotes a discount score with the same amount when an individual contradicts with constraint 6. Summing the squares of S k,l(k) instead of Sk,l(k) itself favours solutions which consist of horizon-pairs k with high single similarity values. 3
(a)
(b)
Figure 3. Examples of correctly matched horizons. Those horizons which have no counterpart are rightly unassigned.
Figure 3 shows results from two different examples of normal faults across which the displayed horizons have been correlated by our algorithm. The correctness of the correlations has been verified by comparing them to those chosen by geological experts.
[4] N. J. Price and J. W. Cosgrove, Analysis of Geological Structures (Cambridge University Press, 1994) 186–190. [5] J. H. Holland, Adaption in Natural and Artificial Systems (MIT Press, 1975). [6] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, 1989). [7] L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, 1991). [8] E. Falkenauer, Genetic Algorithms and Grouping Problems (John Wiley & Sons, 1998). [9] S. Bandyopadhyay, C. A. Murthy, and S. K. Pal, Pattern classification with genetic algorithms, Pattern Recognition Letters, 16, 1995, 801–808. [10] T. Sawaragi, J. Umemura, O. Katai, and S. Iwai, Fusing Multiple Data and Knowledge Sources for Signal Understanding by Genetic Algorithm, IEEE Transactions on Industrial Electronics, Vol. 43, No. 3, June 1996, 411–421. [11] J. Piper, Genetic algorithm for applying constraints in chromosome classification, Pattern Recognition Letters, 16, 1995, 857–864. [12] R. P. Srivastava, and D. E. Goldberg, Verification of the Theory of Genetic and Evolutionary Continuation, IlliGAL Report No. 2001007, Urbanda, IL: University of Illinois at Urbana-Champaign, Illinois Genetic Algoritms Laboratory. [13] F. Schmitt, and F. Rothlauf, On the Importance of the Second Largest Eigenvalue on the Convergence Rate of Genetic Algorithms, IlliGAL Report No. 2001021, Urbanda, IL: University of Illinois at Urbana-Champaign, Illinois Genetic Algoritms Laboratory.
5 Conclusions The Genetic Algorithm has proven to be an adequate method to correlate horizons across faults. Strategies were applied which follow analysis techniques commonly used by experts in seismic interpretation. Further developments will concern improvements of the geological constraints such as a strict curve fitting to the fault throw function as well as the investigation of additional constraints. The method will also be tested on other data sets and on different fault classes. We expect these improvements to lead to a much 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.
References [1] P. Alberts, M. Warner, and D. Lister, Artificial Neural Networks for simultaneous multi Horizon tracking across Discontinuities, 70th Annual International Meeting, SEG, Calgary, Canada, 2000. [2] L. F. Kemp, J. R. Threet, and J. Veezhinathan, A neural net branch and bound seismic horizon tracker, Expanded Abstracts, 62nd Annual International Meeting, SEG, Houston, USA, 1992. [3] B. A. van der Pluijm and M. Marshak, Earth structure. An introduction to structural geology and tectonics (McGrawHill, 1997) 145–177.
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