SEISMIC IMAGES are acquired by transmitting artificially

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1 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 9, NO. 2, APRIL A Genetic Algorithm for Automated Horizon Correlation Across Faults in Seismic Images Melanie Aurnhammer and Klaus D. Tönnies Abstract Finding corresponding seismic horizons which have been separated by a fault is still performed manually in geological interpretation of seismic images. The difficulties of automating this task are due to the small amount of local information typical for those images, resulting in a high degree of interpretation uncertainty. Our approach is based on a model consisting of geological and geometrical knowledge in order to support the lowlevel image information. Finding the geologically most probable matches of several horizons across a fault is a combinatorial optimization problem, which cannot be solved exhaustively since the number of combinations increases exponentially with the number of horizons. A genetic algorithm (GA) has been chosen as the most appropriate strategy to solve the optimization problem. Our implementation of a GA is adapted to this particular problem by introducing geological knowledge into the solution process. The results verify the suitability of the method and the appropriateness of the parameters chosen for the horizon correlation problem. Index Terms Combinatorial optimization, computer vision, correspondence analysis, genetic algorithms (GAs), image analysis. I. INTRODUCTION SEISMIC IMAGES are acquired by transmitting artificially generated sound waves through the Earth s interior. These waves are reflected at geological boundaries within the ground and their travel times are recorded. Several processing steps are necessary, until an image or data volume is obtained, which can be interpreted as a rough approximation of the underground structure. Geological interpretation of seismic images is a relatively new application field for computer vision and pattern recognition. In particular, structural interpretation has been a highly subjective process which involves a human interpreter extracting information by visually inspecting patterns on seismic sections. This approach is not only extremely time consuming but causes also nonrepetitive results. Structural interpretation comprises the localization and interpretation of faults, tracking of uninterrupted horizon segments and correlating, i.e., matching these segments across faults. Horizons indicate boundaries between rocks of different lithology. Faults are discrete fractures across which there is measurable displacement of strata (see Fig. 1). The amount of vertical displacement associated with a fault at any location is termed the throw of Manuscript received October 23, 2003; revised July 20, M. Aurnhammer is with the Computer Science Department, Queen Mary, University of London, London E1 4NS, U.K. ( Melanie@dcs.qmul. ac.uk). K. D. Tönnies is with the Computer Vision Group, Department of Simulation and Graphics, Otto-von-Guericke University, Magdeburg, Germany ( Klaus@isg.cs.uni-magdeburg.de). Digital Object Identifier /TEVC the fault. By plotting the amounts of vertical displacement of horizons against the horizon distances, a discrete function of fault throw is obtained. The typical characteristics of a fault throw function are an increase of fault throw from zero at the upper end of the fault, to a maximum in the central area of the fault, followed by a decrease to zero at the lower limit of the fault (see Fig. 2). While tracking of uninterrupted horizon segments has been automated satisfactorily by using low-level image processing methods, automatic interpretation of fault patterns on seismic sections poses many problems, especially those of correlating horizons across faults. Determining geologically valid horizon correlations is an important but difficult interpretation task, since it involves a high degree of uncertainty. Earlier approaches to solve this problem [1] [4] suffered from using exclusively local image information. In order to reduce interpretation uncertainties, we developed a fault model [5] using empirical knowledge from structural geology and geophysics, e.g., [6] [10]. In this paper, we describe the application of this model to automatically obtain geologically valid horizon correlations. Section II starts by describing the structure of the problem, before the geological model, which is adapted to this structure, is outlined. Thereafter, the reasons for selecting a genetic algorithm (GA) as optimization method are explained, and an overview of prior applications in geosciences is given. Our problem specific implementation is then shown in Section III, before experiments concerning parameter selection and GA performance are reported in Section IV. Finally, the results are discussed and conclusions are given. II. BACKGROUND A. Problem Structure The problem of correlating horizon segments across faults can be subdivided into two components. The first component comprises the formation of every possible horizon pair (HP), consisting of one horizon segment from the left side and one horizon segment from the right side of the fault. A measurement which assesses the similarity of those horizon segment pairs needs to be computed. The second component includes combining horizon segment pairs. A horizon pair combination consists of a set of horizon pairs.we seek to define a measure whose optimum corresponds to the geologically correct HPC. Calculating the global similarity by merely combining the similarity values of the horizon pairs in the combination is insufficient and should, therefore, be X/$ IEEE

2 202 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 9, NO. 2, APRIL 2005 Fig. 1. A 2-D cross section of a 3-D seismic volume. The horizontal axis represents the distance on the Earth s surface. The vertical axis corresponds to the two-way-travel-time for a sound wave to reach a reflector and return to the surface. Fig. 2. Representation of a typical fault throw function in terms of arcs of circles of radius 6R joined at inflection points I and I [12]. complemented by a global measurement [11]. The HPC which yields the highest total similarity may still be a geologically or geometrically impossible solution. Therefore, the total similarity should be thought of as a first approximation to the optimal solution. An optimal solution has to show high similarity values, as well as being geologically plausible. In order to obtain a geologically plausible solution, constraints derived from geological and geometrical knowledge were established. B. The Model Geological and geometrical constraints and measurements form our fault model. Within this model, measurements can be regarded as a first approximation to a solution, which are then complemented by various constraints in order to improve the probability of finding a geologically valid solution. The problem structure outlined above is reflected in the two components which constitute the model. First component: horizon pairs (HPs): local similarity: similarity of reflector sequences; polarity constraint: consistent polarity. Second component: horizon pair combinations (HPCs): global measurement: average displacement gradient deviation; intersection constraint: horizons must not intersect; sign constraint: sign of fault throw has to be consistent and correct; extremum constraint: throw function must not have more than one local extremum. The similarity of reflector sequences from both sides of a fault is measured by calculating the one-dimensional (1-D) cross correlation coefficients (CCs) of grey values. Estimating the CCs by considering grey value functions whose length is equal to the fault length, does not constitute a reasonable method. This is due to the behavior of a fault which is characterized by a change of throw along the fault. The throw or displacement of a fault typically increases from zero at the upper end to a maximum in the central portion, followed by a decrease to zero at the lower limit [12]. For this reason, it can only be assumed, that the throw remains approximately constant over a restricted neighborhood. Thus, the CCs for all possible HPs are calculated for a window of 30 pixels height at a small distance from the fault (five pixels), using grey values averaged over five pixels in horizon direction. To compensate for unequal compression of strata, the CCs are calculated for stepwise scaled functions of one side of the fault in the range. The local similarity of two horizon segments and is defined as their maximum normalized CC The calculation of the HP similarity is complemented by the polarity constraint. In case of differing polarities, the CC loses its significance and, hence, the HP similarity is set to zero. The polarity of a horizon is determined by the sign of its reflection coefficient, which depends on the order of two layers. Usually, dark values refer to positive and light values to negative polarity. The global measurement is derived from the theoretical throw function of a fault. Since there exists a relationship between fault length and maximum displacement, the global measurement employs expectations about the displacement gradient. The actual displacement gradient of belonging to, which consists of HPs is calculated from where denote the values of the horizon pixel which is the closest to the fault. The deviation from the estimated averaged displacement gradient is then calculated by (1) (2) (3)

3 AURNHAMMER AND TÖNNIES: GA FOR AUTOMATED HORIZON CORRELATION ACROSS FAULTS IN SEISMIC IMAGES 203 Finally, the mean displacement gradient deviation from where if if results Obviously, a constant displacement gradient does not correspond to the shape of the theoretical fault throw function (see Fig. 2). While near the inflection points the displacement gradient is approximately constant, it decreases toward zero at the fault tips an in the area of maximum displacement. Hence, it is more appropriate to use the data set dependent value as an upper boundary and consider only positive deviations in (4), while negative ones are neglected. The global similarity of a is defined as (4) where (5) The application of the three constraints intersection, sign, and extremum, yields simple binary decisions concerning the validity of a solution. The intersection constraint restricts valid HPCs to those within which horizon pairs do not intersect. The sign constraint takes into account the expected fault type and the behavior of the fault throw function. Presuming that the type and direction of the fault are known, the expected sign of the throw for HPCs is known as well. Furthermore, no changes of the displacement direction within a combination are allowed. The extremum constraint considers only those HPCs to be probable solutions, whose fault throw function does not show more than one local extremum. A detailed description of the geological measurements and constraints can be found in [5]. C. Optimization Problem Finding the combination of HPs which best fits our model is a combinatorial optimization problem. An appropriate optimization strategy should take the specific characteristics of the problem into account. There are two important considerations arising from a geological point of view. A horizon segment may only be visible on one side of a fault. Thus assuming the existence of a counterpart for each horizon segment in the seismic data is inappropriate. Furthermore, increasing the number of input horizons reduces correlation uncertainties and, therefore, increases the reliability of the solutions. Additional requirements on an optimization method can be derived by considering the problem structure, as well as the type of constraints. First, the total quality of a solution or HPC cannot be calculated from its single components only, i.e., its horizon pairs. The quality also depends on a global measurement and global constraints. From this, it follows that the problem is not separable. Second, the search space is large and discontinuous. It was shown in [13] that the size of the search space grows exponentially with the number of input horizons. Third, no reasonable distance function or metric can be defined because no neighborhood relationship between solutions exists. After considering several optimization methods, we chose a GA [14], [15] as the most appropriate strategy to solve the problem at hand. GAs are a directed random search technique, which is able to optimize even in extremely complex search spaces, escape from local minima and simultaneously explore from a wide sampling of the search space. They are not fundamentally limited by restrictive assumptions about the search space, such as those concerning continuity, existence of derivatives or unimodality [15]. Falkenauer [16] points out that in practice GAs find solutions very near to the global optimum, in times which do not show the exponential growth of the enumerative methods. D. Application of Genetic Algorithms (GAs) in Geosciences GAs have been used for a broad range of applications, some of which concern the field of geophysics. The probably most prevalent type of application in this field is to the geophysical inverse problem. In geophysics, the inverse problem is to deduce information about the Earth s internal structure from geophysical measurements taken at, or near to, the Earth s surface [17]. Typical for inverse problems is the inherent ambiguity or nonuniqueness in the conclusions which can be drawn. Among the several geophysical inverse problems, GAs were in particular applied to the problem of seismic velocity analysis [18] [24]. Applications to other inverse geophysical problems include the residual statics problems [25], the location of earthquake hypocenters [26], [27], or the determination of the source of air pollutants [28]. GAs have as well been utilized as an optimization method in order to improve the performance of reservoir engineering systems and production operations in the oil and gas industry [29]. There also exist applications of GAs to problems in other fields which show similarities in their structure to the problem discussed in this work. They include the stereo matching of images [30] [32], image correlation [33], [34], chromosome classification [35], and grouping or partitioning problems [36] [38]. III. IMPLEMENTATION GAs were invented by Holland [14] as a robust and efficient directed random search technique for searching large spaces. They are based on drawing parallels between the mechanisms of biological evolution and mathematical modeling. GAs work with a population of individuals which are solutions to a problem, in order to enable a parallel search process. Each solution is represented by a string, called chromosome, composed of genes. A fitness function, as well as crossover, mutation, and selection operators determine the development of the population. In the following, a description of these GA components and their problem specific implementation is given. A. Solution Representation The original and most common encoding for candidate solutions are fixed-length, fixed-order binary strings. Thus, not only

4 204 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 9, NO. 2, APRIL 2005 For the purpose of comparing the measurements and, the values of are mapped to the same range as the values of the local measurement in every generation. In addition, conventional linear scaling [15] is performed, which helps to prevent premature convergence in earlier generations and to differentiate between very similar individuals in later generations. If the sign or the extremum constraint is violated, a penalty ( or ) is subtracted from the fitness value which depends on the average fitness at given generation. The value of is estimated only from the local and global measurements, without subtracting penalties. The amount of the penalties is calculated as Fig. 3. GA representation of candidate solutions. The indexes of a 1-D array correspond to the left horizons, the values to the right horizons. most applications, but also much of the existing GA theory is based on those representations. However, many authors emphasize that GAs will probably perform best when the encoding is as close as possible to the problem space. This implies for many problems in science and engineering the use of numbers of decimal form [39]. Likewise, an integer representation constitutes the most natural representation for the present case as HPCs may be represented as a mapping of left horizon to right horizon indices. In order to express this relation, we change our notation as follows. The left horizon becomes the domain variable and the right horizon variable onto which is mapped becomes. If a left horizon does not have a counterpart, is set to. A simple example is given in Fig. 3. The two main advantages of this representation are a straightforward solution interpretation, as well as simplifications regarding tests for geological validity (see Section III-F). B. Fitness Function Decisions on the performance of candidate solutions require the definition of a fitness function, which is composed of local and global measurements as well as global constraints. The fitness of a solution consisting of a number of genes, i.e., horizon pairs is calculated from where the local measurement and is calculated by (1). Since is defined only for for. The global measurement is calculated according to (5). and denote penalty terms accounting for the sign and extremum constraint. The summation of correlation values in (7) encourages combinations consisting of a larger number of horizon-pairs. This can be considered as a reliability factor since the reliability of a global match decreases with decreasing number of horizonpairs; although a geologically valid solution may contain less horizon pairs than the maximum number of possible matches. (6) (7) The reason for this is to give those individuals, which violate both constraints but whose local and global fitness values are above average, a chance of being selected for reproduction. These individuals probably possess parts which may contribute to geologically possible offsprings with high fitness values, if chosen for crossover. C. Objective Function In most GA implementations, an additional measure is required in order to ensure a generation independent quality assessment. This measure is commonly referred to as objective function. The raw, unscaled values of the objective function are used to measure the development of the total quality over several generations and for the decision on the best solution (see Section III-E), whereas selection is performed on the scaled fitness values. In our implementation, the objective values are calculated from where denotes the maximum theoretically possible number of HPs within a HPC, the local measurement [see (7)], and is the average displacement gradient deviation calculated from (4). Substituting the generation dependent by in the objective function ensures quality comparison of candidate solutions in all generations. D. Initial Population To start the optimization procedure, a set of initial candidate solutions or initial population is required. The initial population can either be created randomly by a random number generator, or by employing a priori knowledge about the given problem. The advantage of the second approach is that the algorithm converges to a local or global optimum in less time since the GA starts with a set of approximately known solutions [40]. The present implementation uses a priori knowledge. Solutions are generated by randomly building HPCs. However, the search space is restricted by applying constraints. First, the set of HPs is reduced by excluding those which do not follow the polarity constraint. Second, the generation of HPCs which violate the intersection constraint is avoided. In order to generate an HPC, (8) (9)

5 AURNHAMMER AND TÖNNIES: GA FOR AUTOMATED HORIZON CORRELATION ACROSS FAULTS IN SEISMIC IMAGES 205 HPs are selected in turn, while the random search is stepwise restricted to the respective resulting possible HPs. The population size is set proportional to the product of left and right horizons in order to allow for suitable exploring capabilities of the solution space for an increasing number of possible HPCs. E. Selection Operator Most parent selection methods are based on the assumption that a fitter individual produces a higher number of offspring and, thus, has a higher chance that its characteristics survive into the subsequent generation. Common selection schemes include proportionate selection [15], ranking methods, and tournament selection. Roulette wheel selection [15], which is commonly used to implement the proportionate selection scheme, is applied in this work. The two basic strategies for replacement of the old generation are generational replacement and steady-state reproduction. While in generational replacement the entire population is replaced by an equal number of offspring, in steady-state reproduction only a few individuals are replaced once in the population. Generational replacement is usually combined with an elitist strategy, where a few of the best chromosomes are copied into the next generation. The elitist strategy appears to improve the performance; however, it may increase the speed of domination of a population by a super individual. In order to find a balance between avoiding to lose the best individuals while reducing the probability of premature convergence, the following variant of elitist strategy is applied [41]. The procedure starts by picking parents on the basis of their fitness from a population containing individuals to produce offspring (roulette wheel selection). Then, disparate individuals are selected according to the same procedure to survive unchanged into the next generation. The remaining individuals which are not selected as survivors will be automatically replaced by the offspring produced in the breeding phase. The relation determines the crossover rate. Instead of using a fixed number of generations, the process is terminated when the number of disparate individuals is less than. However, this strategy does not guarantee that the very best individual will be carried over into the next generation. This also means that the possibility of loosing the best solution generated in the whole GA run exists. We, thus, improve the performance of the GA by storing the fittest individual in every generation. When a run of the GA has terminated, the set of fittest individuals is reduced by those which violate one or both constraints (sign and extremum). Among the remaining individuals, the solution with the highest objective function value is chosen as the best solution of this particular run of the GA. F. Crossover Operator The simplest recombination operator is single-point or onepoint crossover: two individuals are randomly selected as parents, a single crossing location is selected randomly, and the substrings bounded by that crossing location are exchanged. Although single-point crossover is able to recombine short, loworder, schemata [14] in an advantageous manner, it has been criticized because of several reasons. First, it cannot combine all possible schemata. 1 Second, schemata with long defining lengths are likely to be destroyed under single-point crossover. Third, some loci are treated preferentially: the segments exchanged between the two parents always contain the endpoints of the strings. In order to avoid the shortcomings of single-point crossover, many practical GA implementations use two-point crossover, in which two positions are chosen randomly and the segments in between are exchanged. Two-point crossover is able to combine more schemata and is less prone to disruption of schemata with large defining lengths than single-point crossover. Besides, the segments which are exchanged do not necessarily contain the endpoints of the parents. However, just as with single-point crossover, there are schemata which two-point crossover cannot generate. Parameterized uniform crossover [42], in which an exchange is performed at each position with probability, can potentially recombine any schemata contained at different positions in the parents. However, the disadvantage of this operator is that it can be highly disruptive of any schemata. In the present case, the recombination of two individuals might lead to an offspring which violates the intersection constraint, i.e., it represents a HPC within which horizons cross. Therefore, the geometrical validity of a new solution candidate is verified before its fitness is evaluated. In case of a valid offspring, the fitness function is evaluated and the solution is carried over into the new generation; otherwise, the offspring is discarded. Two-point crossover is applied in this implementation in order to reduce the likelihood of schemata disruption on the one hand, and on the other hand, ensure that the number of geometrically invalid offsprings remains small. G. Mutation Operator Mutation in a simple binary representation is straightforward: with every new generation all individuals in the population are visited bit-by-bit and the values are reversed according to a specified rate, usually in the order However, like other components of GAs, many versions of the mutation operator have been developed in the literature, depending on the problem and its representation. A mutation strategy which randomly changes integer values would in the present case generate an unreasonably high rate of HPCs which are invalid regarding the intersection constraint. Thus, a revised strategy has been developed where values from a restricted set are chosen randomly. Eligible values are those which represent a valid HP for the selected bit plus the value no combination (coded as ). A valid HP means in this case, that the polarity constraint is not violated. Mutations for the selected gene are produced by repeatedly extracting candidates from the eligible values until the intersection constraint is fulfilled. IV. EXPERIMENTS AND RESULTS A. Solution Definition In order to evaluate the experimental results, it is necessary to define what constitutes a solution which is considered to solve 1 Single crossover cannot combine for example, instances of and to

6 206 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 9, NO. 2, APRIL 2005 the problem. In the present case, solutions can belong to one of the three classes: optimal solution; acceptable solution; unacceptable solution. The optimal solution is defined as the result, which a human interpreter would consider the geologically most plausible one. This is the solution which is supposed to be the result of the exhaustive search algorithm or the global optimum of the GA. However, there exists a range of solutions, which deviate slightly from the optimal solution but are almost as good in a geological sense. These solutions are referred to as acceptable solutions and comprise all those solutions which include at most 10% less correct assignments of horizon segments than the theoretically possible number of HPs but no wrong HP compared with the optimal solution. The theoretically possible number is used in order to consider also horizon segments which are correctly not assigned because of a lacking counterpart. A missing correlation is often merely due to inaccuracies of input horizons caused by the simple horizon generation procedure and does not lead to a major deterioration of the solution quality. Does a solution contain any wrong HP, it is considered geologically implausible and constitutes, therefore, an unacceptable solution. Unacceptable solutions are also those with more than 10% missing HPs. B. Test Data Four subsets of three-dimensional (3-D) seismic data sets from different geographic regions were used for the experiments. Altogether, 14 test faults were selected from these subsets, four each from data set A, B, C, and two from the smaller data set D. C. Parameter Selection Important control parameters of a GA include population size, as well as crossover and mutation rates. Appropriate parameter values for the problem at hand are estimated in the following. The effect of these parameters on the performance of a GA has been studied by several researchers, e.g., [15] and [43] [45], whose findings influenced this parameter selection. Estimating an appropriate setting for the parameters is not straightforward because of their nonlinear interaction. In order to avoid testing all possible parameter settings, experiments were restricted to varying only one parameter at a time. This, however, requires starting with already near optimal parameter settings, as can be found in a previous work [13]. Five test faults were chosen for the parameter selection tests. Following [45], ten runs were performed for each parameter setting and test fault. Since this implementation focuses on the reliability of the GA, the evaluation criterion was the number of times, or frequency, of finding the global optimum, while neglecting computational cost. The maximum number of generations was set to 150 for each run which was found to be sufficient for each considered crossover rate and test fault. 1) Population Size: A large population size increases the computation time per iteration compared to a small population Fig. 4. Values of f as a function of crossover rate averaged over five test faults. size, but increases the probability of convergence to a global optimum since more samples from the search space are used. In determining the population size for the present implementation, the exponentially large search space of the problem has to be taken into account. Furthermore, due to the varying number of input horizon segments, a fixed number is inappropriate. Thus, the population size was estimated by, where and denote the number of left and right horizons, respectively. In [13], values larger than were reported to give adequate results concerning the number of acceptable solutions. For the experiments, was chosen, since a further increase of the population size did not yield considerable improvements of the solution quality. This value was found appropriate for all datasets considered in this work. 2) Crossover Rate: A large crossover rate may lead to a dominating individual, or at most a few, superindividuals. As a result, new areas of the problem space become unreachable. A low crossover rate may decrease the speed of convergence to a promising region of the search space. Some common settings which have been previously reported for the crossover rate are [43], [44], and [45]. Within this implementation, as it is commonly done, recombination is applied deterministically to % of the population, thus neglecting the probabilistic character of the crossover rate as originally proposed by Holland [14]. The experiments began with varying the crossover rate while the mutation rate was fixed at, according to [13]. Fig. 4 shows the frequency, or the number of times the global optimum was reached as a function of crossover rate. This frequency was approximately constant in the broad range from to. Within this range, the crossover rate to perform the following experiments was estimated using the weighted average, which resulted in. 3) Mutation Rate: A high mutation rate introduces high diversity in the population but might also cause instability, whereas a too low mutation rate might cause difficulties to find the global optimum. Common previously proposed settings for the mutation rate are [43], [44], and [45]. Four mutation rates were tested in this work:. The maximum frequency in finding

7 AURNHAMMER AND TÖNNIES: GA FOR AUTOMATED HORIZON CORRELATION ACROSS FAULTS IN SEISMIC IMAGES 207 Fig. 5. Values of f as a function of mutation rate averaged over five test faults. the global optimum was obtained for in Fig. 5., as illustrated D. Discussion of Parameter Selection The experiments reported in Section IV-C showed the crossover rate to be a robust parameter yielding good results for, which was also found in previous investigations, e.g., [45]. The deterioration of the results for is due to the selection procedure used in this work (see Section III-E). Decreasing means increasing the number of disparate individuals carried over intact into the next generation, which in case of low crossover rates may only be feasible for a small number of generations. A mutation rate of did actually yield the best results for the tested combination of and. Furthermore, the results indicate that the population size is sufficiently large for this problem. Reducing the population size in order to reduce computation time is not a priority of this work and was, thus, not tested. E. Experiments After having selected appropriate parameter settings, experiments were performed in order to assess the behavior of the GA. In particular, the following questions were examined. 1) Does the global optimum of the GA correspond to an acceptable solution? 2) How repeatable are the results of the GA? The first criterion which was assessed was the appropriateness of the fitness function, i.e., the correspondence to the manually selected solution. Furthermore, the performance of the GA was examined, which concerns mainly the robustness, i.e., the independence of the results from starting conditions. 1) Assessment of Fitness Function: The results from the 14 test cases were obtained by using the parameter settings, as explained in Section III-E. Ten cases yielded the optimal solution, and in another two cases, an acceptable solution was obtained. However, for two of the tested faults, the solutions were geologically unacceptable. One result image from the first two solution classes is shown exemplarily in Fig. 6. The displayed horizons were correlated by a typical run of the GA. One correlation is missing in the acceptable solution shown in Fig. 6(b): horizon Fig. 6. Examples of two different faults where (a) the optimal and (b) an acceptable solution were found, respectively. 15 left and 14 right (numbering from bottom to top) should have been matched as well. In both acceptable cases, i.e., where the global optimum deviates slightly from the geologically most probable solution, the missing correlations of matching horizon segments do not result from a misconception of the geological model but are due to inaccuracies of the input horizons. Those two cases, where the global optimum corresponds to an unacceptable solution are shown in Fig. 7. The geologically most probable solution (right image) and the unacceptable result obtained by the GA (middle image) are compared. The test fault shown in Fig. 7(a) constitutes a more difficult problem for the algorithm compared to the preceding cases. The difficulty in a geological sense arises from a higher throw gradient than the average value estimated for this data set. The considerable change of fault throw, which the optimal solution shows, yielded a low global measurement. Moreover, the local measurement did not give reliable values: in only one case, the CC of the correct assignment of right to left horizon segments corresponded to the maximum value among all possible assignments of right segments to the particular left one. The reason for the failure of the local measurement can easily be seen in the raw seismic data. In the middle part, reflections are hardly visible,

8 208 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 9, NO. 2, APRIL 2005 Fig. 7. Two different faults for which unacceptable solutions were found by the GA. (a) One correlation in lower part of middle image is missing, two are correct in the middle part, and the correlations on top are incorrect. (b) Correlations in lower part of middle image are incorrect. while in the upper part some reflection patterns are quite similar for several horizons and, thus misleading correlation factors. Besides, the selection of input horizons in the upper part of the fault was quite unfortunate, since the most clearly attributable horizon segments are not represented by input horizons on the right side. Correlating the horizon segments across the test fault shown in Fig. 7(b) is, especially in the lower part, not a straightforward task. The upper ten assignments of horizon segments are correct, whereas the lower four assignments are incorrect. The problem here is, that the reflections in the middle and deeper part are very weak and, hence, the local measurement is unreliable. For the same reason, input horizons were difficult to generate, which caused missing counterparts of the few strong reflections in the middle part. This led to further uncertainties of the correlation task. 2) Performance of the GA: The repeatability of the GA was examined for those 12 test faults for which either the optimal solution or an acceptable one corresponds to the global optimum. For each fault, 30 runs were performed and the results classified in optimal, acceptable, and unacceptable solutions. If the global optimum of the fitness function corresponds to a geologically acceptable solutions, other acceptable solutions are determined with reference to the geologically optimal solution. TABLE I REPEATABILITY OF SOLUTIONS Table I shows how often the respective solution classes were obtained. For nine of the test faults, an acceptable solution was found with a high frequency, ranging from 93% to 100%. In contrast, an acceptable solution was obtained only in 60% to 80% of the runs for test faults B2, D1, and C3. However, most of the unacceptable solutions were due to a higher number of missing HPs

9 AURNHAMMER AND TÖNNIES: GA FOR AUTOMATED HORIZON CORRELATION ACROSS FAULTS IN SEISMIC IMAGES 209 than admissible, but far less frequently caused by wrong HPs. This results from the conception of the geological model, where solutions with missing correlations are favored above solutions with wrong HPs. While the local measurement supports solutions consisting of many HPs, the global measurement uses an average value, which improves by eliminating bad HPs. Even if correct HPs are added to an HPC, the total contribution to the objective function might be very small, due to a deterioration of the global measurement. The procedure employed to scale the fitness function might then lead to a better assessment of the HPC without the additional HPs. However, the HPs not contained in the solution, although correct are generally the less reliable ones and, thus, this behavior of the algorithm is generally desirable. The computation time for the test case with the highest number of input horizons (26 26) was approximately 300 s on a PC with Pentium II, 266 MHz processor, assuming a fixed number of generations of 150. However, in the present prototypical implementation with the Interactive Data Language (IDL), there is still room for great improvement of the computational efficiency. V. DISCUSSION AND CONCLUSION Finding the most geologically probable combination of HPs constitutes a combinatorial optimization problem. A solution was sought which takes into account all the problem s characteristics, such as nonseparability of the problem, or the exponential increase of the size of the search space with the number of input horizon segments. A GA was found to be a method which fulfills the requirements defined. Experiments concerning the selection of appropriate GA-parameter settings have been reported, followed by assessments of fitness function and repeatability of the GA. The high rate of test faults for which the global optimum or an acceptable solution was obtained, shows the appropriateness of the fitness function for the cases tested. This rate, as well as the high repeatability of the GA in most cases, indicate the suitability of the GA parameters selected in Section IV-C. Moreover, the amount subtracted from the fitness of an individual as penalty for violated constraints (see Section III-B) was found to be appropriate. Even if the global optimum of the fitness function did not in every test case correspond to the optimal solution, only two of the solutions, i.e., 14% were geologically unacceptable. It can be concluded that the obtained results proved the GA to be an appropriate method to solve the problem of correlating horizons across faults. ACKNOWLEDGMENT The authors would like to thank Shell for the seismic data and stimulating discussions. Furthermore, they wish to thank the anonymous reviewers for their constructive and detailed comments. 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10 210 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 9, NO. 2, APRIL 2005 [30] H. Saito and M. Mori, Application of genetic algorithms to stereo matching of images, Pattern Recogn. Lett., vol. 18, no. 8, pp , [31] K. P. Han, K. W. Song, E. Y. Chung, S. J. Cho, and Y. H. Ha, Stereo matching using genetic algorithm with adaptive chromosomes, Pattern Recogn., vol. 34, no. 9, pp , [32] J. Y. Goulermas and P. Liatsis, Hybrid symbiotic genetic optimization for robust edge-based stereo correspondence, Pattern Recogn., vol. 34, no. 12, pp , [33] R. R. Brooks, S. S. Lyengar, and J. Chen, Automatic correlation and calibration of noisy sensor readings using elite genetic algorithms, Artif. Intell., vol. 84, no. 1 2, pp , [34] E. Piazza, Surface movement radar image correlation using genetic algorithm, in Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 2001, vol. 2037, Proc. Appl. Evol. Comput., pp [35] J. Piper, Genetic algorithm for applying constraints in chromosome classification, Pattern Recogn. Lett., vol. 16, pp , [36] D. Smith, Bin packing with adaptive search, in Proc. 1st Int. Conf. Genetic Algorithms, J. J. Grefenstette, Ed., 1985, pp [37] D. R. Jones and M. A. Beltramo, Solving partitioning problems with genetic algorithms, in Proc. 4th Int. Conf. Genetic Algorithms, R.K. Belew and L. B. Booker, Eds., 1991, pp [38] R. van Driessche and R. Piessens, Load balancing with genetic algorithms, in Proc. 2nd Conf. Parallel Prob. Solving from Nature, R. Männer and B. Manderick, Eds., 1992, pp [39] D. A. Coley, An Introduction to Genetic Algorithms for Scientists and Engineers. Singapore: World Scientific, [40] D. T. Pham and D. Karaboga, Intelligent Optimization Techniques: Genetic Algorithms, Tabu Search, Simulated Annealing, and Neural Networks. Berlin, Germany: Springer-Verlag, [41] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs. Berlin, Germany: Springer-Verlag, [42] W. M. Spears and K. A. De Jong, On the virtues of parameterized uniform crossover, in Proc. 4th Int. Conf. Genetic Algorithms, 1991, pp [43] K. A. De Jong, An analysis of the behavior of a class of genetic adaptive systems, Ph.D. dissertation, Univ. Michigan, Ann Arbor, MI, [44] J. J. Grefenstette, Optimization of control parameters for genetic algorithms, IEEE Trans. Syst., Man, Cybern., vol. 16, no. 1, pp , [45] J. D. Schaffer, R. A. Caruana, L. J. Eshelman, and R. Das, A study of control parameters affecting online performance of genetic algorithms for function optimization, in Proc. 3rd Int. Conf. Genetic Algorithms, J. Schaffer, Ed., 1989, pp Melanie Aurnhammer received the B.Sc. degree in business engineering from the University of Applied Science, Offenburg, Germany, in 1998, and the M.Sc. and Ph.D. degrees in computational visualistics from the Otto-von-Guericke-University, Magdeburg, Germany, in 2000 and 2003, respectively. In 2003, she was a Visiting Fellow at the Public University of Navarre, Pamplona, Spain. Since January 2004, she has been with the Vision Group, Queen Mary, University of London, London, U.K., as a Postdoctoral Research Assistant. Her current research interests are in the field of computer vision and pattern recognition, focussing on seismic image analysis, evolutionary optimization strategies, machine learning, and semantic video analysis. Klaus D. Tönnies received the Diploma (M.Sc.) and Ph.D. degrees in computer science from the Technische Universität, Berlin, Germany, in 1983 and 1987, respectively. From 1987 to 1989, he was a Research Assistant Professor at the University of Pennsylvania, Philadelphia. From 1989 to 1995, he was an Assistant Professor at the Technische Universität. From 1996 to 1998, he headed the Image Processing Group, Department of Radiology, Inselspital, Universität Bern, Bern, Switzerland. Since 1998, he has been a Professor of Image Processing and Image Understanding at the Otto-von- Guericke-Universität Magdeburg, Magdeburg, Germany. His research focuses on investigating geometric models for image segmentation, as well as on using three-dimensional visualization techniques to interact with model knowledge in image analysis with applications in seismic and medical imaging.