Model-based analysis of geological structures in seismic images

Transcription

1 Model-based analysis of geological structures in seismic images Melanie Aurnhammer and Klaus Tönnies Computer Vision Group, Department of Simulation and Graphics Otto-von-Guericke University Magdeburg, Germany Abstract Subsurface models, obtained by structural interpretation of seismic images, underpin all decisions in hydrocarbon exploration and production. The main interest in this field is directed towards the interpretation of horizons and faults: Horizons are strong reflection events which indicate boundaries between rock layering while faults are discrete fractures along which appreciable displacement has taken place. Horizon tracking across faults and thereby determining geologically valid correlations is an important but time consuming task which has not been automated satisfactorily yet. The difficulties of matching horizon segments across faults are due to the fact, that those images contain only a small amount of local information, furthermore partially disturbed by vague or noisy signals. We describe an approach which reduces interpretation uncertainties by introducing geological constraints derived from a fault model. Two optimisation methods have been examined: an exhaustive search algorithm which reliably delivers the optimal solution presuming correctness of the model and a more viable strategy; namely, a genetic algorithm. Both methods successfully matched all selected horizons across normal faults in typical seismic data images. 1 Introduction Seismic data are aquired using the seismic reflection method which explores the subsurface by bouncing sound waves off the interfaces between rock layers with differing physical properties. 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 developed which should merge into a consistent subsurface model. All decisions in hydrocarbon exploration and production are underpinned by such models obtained by structural interpretation. Since drilling wells is very costly, as much information as possible should be derived from the seismic data to form an opinion about the probability of encountering petroleum in the structures [1]. Nevertheless, it is not possible to reliably determine whether an interpretation has been correct unless it has been verified by drilling. The test of a good interpretation is consistency rather than correctness [2]. Lawyer [3] defines minimum standards for a good interpretation: First, the interpretation has to be internally consistent with all of the data available; second, it has to be geologically reasonable; and third it has to be repeatable within the limits of the data. While repeatable results are an advantage of

2 automatically generated results, the main challenge is to establish a model which induces geologically reasonable solutions. Structural interpretation may be thought of as consisting of the following tasks: Localisation and interpretation of faults, tracking of uninterrupted horizon segments and correlating these segments across faults. Reflectors in seismic images usually correspond with horizons indicating boundaries between rocks of markedly different lithology. Faults are discrete fractures across which there is measurable displacement of rock layering. On seismic sections, 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 location is termed the throw of the fault. Previous attempts to solve the problem of correlating horizons across faults have been based on artificial neural networks [4, 5]; however, these solutions use only similarities of the seismic patterns. In this paper, we develop a structural model for correlating horizons across faults in order to achieve a geologically reasonable solution. A brief overview of the state-of-the-art in this field is given in chapter 2. Chapter 3 comprises a description of the process of horizon correlation as well as the definition of local horizon similarity and the derivation of the fault model. The practical implementation is described in chapter 4 where we present two different algorithms: an exhaustive search strategy and a genetic algorithm. Results are then shown in chapter 5 followed by our conclusions in chapter 6. faults horizon 1 horizon 2 Figure 1: Part of a seismic section showing typical normal faulting.

3 2 State of the Art Modern commercial interpretation software packages offer assistance for the interpretation of horizons and fault surfaces. The most commonly employed technique for horizon tracking is the so called autotracking or autopicking [6]. These algorithms require manually selected seed points as initial control for the autotracking operation. A similar feature is searched on a neighbouring trace; if it has been found within specified constraints, the tracker moves on to the next trace. Autotrackers are either feature based or correlation based. While the first class simply searches for a similar configuration of samples, the latter includes the neighbourhood of the trace and is therefore more robust and less sensitive to noise. The main disadvantage of autotracking algorithms is that they are unable to track horizons across discontinuities. Whenever any of the search criteria are not met, the autotracker stops at that trace. Computer-aided interpretation of fault surfaces is significantly less advanced than horizon interpretation [6]. Coherence measures are applied to seismic data for imaging geological discontinuities like faults or stratigraphic features. These coherence algorithms are based for example on cross correlation [7], semblance [8] or the eigenstructure of the data covariance matrix [9]. However, they produce only potential fault pixels, but do not generate the actual fault lines or surfaces. There exist methods for fault autotracking which use the same basic approach as horizon trackers, but with limited success [10]. The automatic methods described above have in common that they are based only on local features. On the other hand, the actual interpretation of fault surfaces and the correlation or tracking of horizons across faults are still done manually and are therefore highly subjective and time-consuming. The difficulties of automating those tasks are due to the seismic images which contain only a small amount of local information, furthermore partially disturbed by vague or noisy signals. Therefore, more sophisticated methods have to be developed which impose geological and geometrical knowledge in order to reduce interpretation uncertainties. 3 Correlating Horizon Segments across Faults The problem of correlating horizon segments across faults may be subdivided into two tasks. The first task consists of the calculation of a measurement which expresses the similarity between horizon segments from either side of the fault. After determining the similarity values for each possible horizon-pair, the second task is to find a global combination of horizon pairs over the complete area of interest. The global similarity may be thought of as a first approximation to the optimum solution since its geological plausibility is not ensured. An optimal solution has to show high similarity values as well as being geologically reasonable. In the following, we first describe the definition of local similarity before we specify our model of a geologically possible solution.

4 3.1 Similarity of Horizon Segments Since any individual seismic reflection is defined only by its amplitude, polarity and wavelength, it is insufficiently distinct to be correlatable on its own. In order to find attributes which are able to express the similarity between horizon segments on either side of the fault, we compare sequences of reflectors. Reflector sequences are distinguished by characteristic patterns which can usually be found on either side of the fault. We use the cross correlation coefficient (CC) to compare those sequences. 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 S l,r of two horizons l and r as their maximum CC. The estimation of the horizon similarity is complemented by a local constraint which concerns the polarity of horizons. Polarity can be illustrated by regarding the original seismic trace 1 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 should be equal for corresponding horizon segments. This additional condition (hereafter, constraint 1) causes that, in case of differing polarities, the cross correlation coefficient loses its significance. 3.2 Fault Model The combination of horizons which leads to the highest similarity may be a geologically or geometrically impossible solution. Interpretation of seismic data requires a conceptual model of the portion of the earth involved in seismic measurements to counteract the lack of local information. The model is a simplification of the actual earth which comprises only those 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], mathematical expressions have to be found to underpin the automatic interpretation task. The model we introduce consists of constraints which are deduced from geometrical and geological knowledge about faults. Faults are usually classified according to the direction of displacement of the blocks of strata on either side of the fault plane following Anderson [12]: normal, thrust (reverse) or strike-slip. Figure 2 shows diagrammatically the types of displacement involved. Normal and reverse faults have a displacement in a vertical sense whereas the displacement of a strike-slip fault is quoted as a horizontal displacement. While the 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 [13]. The occurrence of these fault classes is not arbitrary but can be ascribed to the forces which had influenced 1 A seismic trace is represented by one column in the seismic images.

5 the area being studied. Normal faults are generally associated with tensional stress, reverse faults 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 zero at the upper end of the fault plane to a maximum in the central portion of the fault and then decreases 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 but could 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

6 These constraints are described in detail in the following. R R Pre-fault R x x I 1 I 2 Post-fault configuration R Figure 3: Representation of a typical fault throw function in terms of arcs of circles of radius ±R joined at inflection points I 1 and I 2 [14] Horizons must not cross The second constraint we consider is a simple geometrical one: horizons within a scene must not cross (Figure 4) (a) Resulting possible matches: 1-0 and 0-0. (b) Impossible matches: 1-2 and 0-2. Figure 4: Resulting possible and impossible horizon-pairs for initial match of left horizon 2 and right horizon Sign of fault throw Since we we restrict our model to normal faults, the expected sign of the throw for horizon segments combined across faults can be deduced from the fault direction. In addition to this, changes of the sign within a combination indicate very unlikely solutions.

7 3.2.3 Behaviour of fault throw This constraint 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. We determine the number of zero crossings of the first derivative of the fault throw functions which arise from the combinations of horizon-pairs. Acceptable are only those combinations whose throw function shows either one ore no zero crossings since functions with a higher number of zero crossings indicate either a mismatch of horizons or converging faults Restricted displacement gradient The fifth constraint follows the investigated relationship between maximum displacement on a fault and the dimensions of the fault surface [15, 16]. The displacement gradient on a fault is a measure of the rate at which displacement changes along the fault plane in a specified direction. It is given by G vm = D (1) R where D is the maximum displacement along the measured section and R the radius (half the length) of this section [17]. Since these values are less than 1 it is often more convenient to refer to the reciprocal of the displacement gradient. Elliot [18] suggested that a linear relationship between R and D is likely and that the characteristic value of the R/D ratio is approx. 7. Walsh and Watterson [19] showed that for a variety of faults, the relationship between D and R is non-linear and suggested R/D values from 5 to 1000, depending on scale and material properties. For a known fault-length, the displacement gradient can be estimated by following the investigated relationship between fault length L and the maximum vertical displacement of the horizons or the maximum fault throw D = C L n. Typical values for a variety of faults have been found to be C = 0.03 and n = 1.06 [13]. However, faults are often not contained to their complete extent in seismic dataset and hence, it is not possible to determine the length of the fault. Therefore we use the value suggested by Elliot as a rough estimate which is compared to the actual displacement gradient. The variaton of the displacement gradient in relation to the distance to the fault center is not considered in this 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 solution presuming correctness of the model. Therefore, this approach is suitable to serve as a validation method. Since for an increasing number of horizons the exhaustive search approach 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.

8 4.1 Input Data The horizon segments which we use as input for our algorithms are skeletons of strong reflections. 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 converted into two binary images by a using a threshold to obtain the strong positive respectively the strong negative amplitudes. Since these operations occasionally converge horizon segments across faults which do not belong together (Figure 5(a)), we use the output of a fault highlighter (Figure 5(b)) to separate them again. Fault lines are generated by defining manually the region of interest in the discontinuity image and generating a fault line by interpolation of the presumed fault pixels (Figure 5(c)). Horizon segments are then assigned either to the class left or the class right segments and cut at the same distance to the interpolated fault line in order to objectify the fault throw calculation. The user has the possibility to decide which of the generated horizon segments are to be used for correlation. An advantage of this method is, that no seed points are required as initial step for the horizon tracking. (a) Skeletons of reflectors (b) 2-D section of an output volume of a fault highlighter (c) Horizon segments and interpolated 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.

9 The single similarity of all horizon-pairs is determined in step (1) by calculating their crosscorrelation coefficient combined with constraint 1 as described in 3.1. These pairs are connected in step (2) by building a solution-tree wherein each possible horizon-pair combination 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 of the single horizon-pairs. The results are then used in step (3) in an evaluation cycle, which applies constraints 1, 3, 4 and 5 to find the optimum combination of horizon-pairs. According to our model, this means the solution with the highest total similarity which fulfills all geological 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 inadmissibly large since the number of combinations increases exponentially with the number of horizons. Hence, we examined stochastic methods to find the optimum horizon combination. 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 to precisely define the evaluation criteria. Another advantage is, that the search space does neither have to be connected nor compact. In genetic algorithms [21], a population of individuals represents potential solutions to a problem. 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 the development of the population. In our implementation [22], we use an integer string to represent an individual. While the index l of an integer within the string represents the left horizon number, its allocated value r(l) indicates the right horizon number. If a left horizon has no counterpart, the value 1 is assigned. 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 1. Second, we avoid the generation of combinations within which horizon-pairs cross (constraint 2). This is achieved by restricting 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 composed of the cross-correlation coefficients of the chromosomes, and its global consistency. To provide global consistency, fixed amounts are subtracted from the similarity value if constraint 3, 4 or 5 is violated. 5 Results We tested both methods presented using horizons at several faults along 2-D sections in a 3-D seismic data set.

10 Figure 6 and 7 show results from three different examples of normal faults across which the displayed horizons have been correlated by our exhaustive search algorithm. The correctness of the correlations has been verified by comparing them to those chosen by geological experts. In no case, the correct solution has been found by maximising the local similarity only. The algorithm has been successful in each of the cases which have been tested but this is also due to the fact that the geological structure in the data set is relatively simple but nonetheless common. In more complicated structures it is expected that the consistency check does not use sufficient knowledge for a correct selection. The application of the genetic algorithm has led to the same solutions as the exhaustive search algorithm in 2 of the 3 test cases shown above. Figure 7(b) shows the third case where the genetic algorithm has found only a near optimum solution which contains one geologically incorrect correlation. The reason for this may be an insufficient consideration of fault throw behaviour. (a) (b) Figure 6: Examples of correctly matched horizons. Those horizons which have no counterpart are rightly unassigned. 6 Conclusions The exhaustive search strategy has proven to be an adequate method to correlate horizons across faults. Because of the small amount of local image information, horizon tracking across discontinuities requires geological constraints to be successful. The results indicate the suitability of the underlying fault model. Strategies have been applied which follow analysis techniques commonly used by experts in seismic interpretation. However, since the number of combinations increases exponentially with the number of horizons, the exhaustive search strategy is not viable for a higher number of horizon segments. This has led us to the examination of stochastic methods among which we have found genetic algorithms to be an appropriate search strategy.

11 (a) Correct solution, found by exhaustive search. (b) Non-optimal solution found by the genetic algorithm. (c) Subset of the seismic line without correlation Figure 7: Comparison of results from both search strategies The results presented above confirm that, in principle, genetic algorithms may be applicable to our problem. Nevertheless, the parameterisation as well as the solution representation and the fitness function have to be further examined to enhance the reliability of the genetic algorithm. Further developments will be concerned with improvements of the geological constraints 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] W. M. Telford, L. P. Geldart, and R. E. Sheriff, Applied Geophysics, Cambridge University Press, 1990, pp [2] N. A. Anstey, How do we know we are right? Geophysical Prospecting, Vol. 21, 1974, pp [3] L. C. Lawyer, From the Other Side, The Leading Edge, Vol. 17, No. 9, 1998, pp

12 [4] P. Alberts, M. Warner, and D. Lister, Artificial Neural Networks for Simultaneous Multi Horizon Tracking across Discontinuities, 70th Annual International Meeting, SEG, Calgary, Canada, [5] 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, [6] G. A. Dorn, Modern 3-D Seismic Interpretation, The Leading Edge, Vol. 17, No. 9, [7] M. S. Bahorich, and S. L. Farmer, 3-D Seismic Discontinuity for Faults and Stratigraphic Features, The Leading Edge, Vol 14, No. 10, 1995, pp [8] K. J. Marfurt, R. L. Kirlin, S. L. Farmer, and M. S. Bahorich, 3-D Seismic Attributes Using a Semblance-Based Coherency Algorithm, Geophysics, Vol. 63, No. 4, 1998, pp [9] A. Gersztenkorn, and K. J. Marfurt, Eigenstructure-Based Coherence Computations as an Aid to 3-D Strucural and Stratigraphic Mapping, Geophysics, Vol. 64, No. 5, 1999, pp [10] G. Fehmers, Shell Research, Netherlands, personal communications. [11] R. E. Sheriff and L. P. Geldart, Exploration Seismology, 2nd ed., Cambridge University Press, 1995, pp [12] E. M. Anderson, The Dynamics of Faulting and Dyke Formation, with Applications to Britain, Edinburgh, Oliver and Boyd, [13] B. A. van der Pluijm and M. Marshak, Earth Structure. An Introduction to Structural Geology and Tectonics McGraw-Hill, 1997, pp [14] N. J. Price and J. W. Cosgrove, Analysis of Geological Structures, Cambridge University Press, 1994, pp [15] J. J. Walsh and J. Watterson, Distributions of Cumulative Displacement and Seismic Slip on a Single Normal Fault Surface, Journal of Structural Geology, Vol. 9, No. 8, 1987, pp [16] J. J. Walsh and J. Watterson, Analysis of the Relationship between Displacements and Dimensions of Faults, Journal of Structural Geology, Vol. 10, No. 3, 1988, pp [17] D. Meier, Abschiebungen: Geometrie und Entwicklung von Störungen im Extentionsregime, Enke: Stuttgart, [18] D. Elliott, Energy Balance and Deformation Mechanisms of Thrust Sheets, Philosophical Transactions of the Royal Society of London, A283, 1976, pp

13 [19] J. J. Walsh and J. Watterson, Displacement Gradients on Fault Surfaces, Journal of Structural Geology, Vol. 11, No. 3, 1989, pp [20] M. Aurnhammer and K. Tönnies, Horizon Correlation across Faults Guided by Geological Constraints, Proceedings of SPIE, Vol. #4667, Electronic Imaging 2002, January, San Jose, California USA. In press. [21] J. H. Holland, Adaption in Natural and Artificial Systems, MIT Press, [22] M. Aurnhammer and K. Tönnies, A Genetic Algorithm for Constrained Seismic Horizon Correlation, Proceedings of the International Conference on Computer Vision Pattern Recognition and Image Processing (CVPRIP 2002), March, Durham, North Carolina USA. In press.