Automatic Method for Correlating Horizons across Faults in 3D Seismic Data

Transcription

1 Automatic Method for Correlating Horizons across Faults in 3D Seismic Data Fitsum Admasu Computer Vision Group University of Magdeburg, Germany Klaus Toennies Computer Vision Group University of Magdeburg, Germany Abstract Horizons are visible boundaries between certain sediment layers in seismic data, and a fault is a crack of horizons and it is recognized in seismic data by the discontinuities of horizons layers. Interpretation of seismic data is a timeconsuming manual task, which is only partially supported by computer methods. In this paper, we present an automatic method for horizon correlation across faults in 3d seismic data. As automating horizons correlations using only seismic data features is not feasible, we reformulated the correlation task as a non-rigid continuous point matching problem. Seismic features on both sides of the fault are gathered and an optimal match is found based on geological fault displacement model. One side of the fault is the floating image while the other side is the reference image. First, very prominent regions on both sides are automatically extracted and a match between them is found. Sparse fault displacements are then computed for these regions and they are used to calculate parameters for the fault displacement model. A multi-resolution simulated annealing optimization scheme is then used for the continuous point matching. The method was applied to real 3D seismic data, and has shown to produce geologically acceptable horizons correlations. Key Words: seismic image interpretation, model-based analysis, multi-resolution correspondence analysis. 1. Introduction Three-dimensional seismic data consist of numerous closely-spaced seismic lines that provide measures of subsurface reflectivity. Different subsurface s rock layers have different acoustic impedances. As result when seismic waves are sent to underground structures, the changes in the seismic wave velocities give strong reflections being visible in the seismic images. These strong reflection events are known as horizons. Faults rarely give reflection events rather they are recognized in seismic data by the discontinuities of horizons events [4]. Structural interpretation attempts to create 3D subsurfaces model and it consists of the following tasks: localization and interpretation of faults, tracking of uninterrupted horizon segments and correlating these segments across faults [2]. Figure 1 shows a seismic cube extracted from 3-d seismic data. The interpreted horizons and faults on this figure is done manually. The horizon segments which are offset by the fault line are matched by the arrows. Timeline Inline Horizon segment Fault line Crossline Figure 1: 3-D seismic data with some horizons interpretations. Seismic data interpreters perform manual horizon tracking mainly on 2-D projections of 3-D image or on 2-D slices of the 3-D data. It is a time-consuming task due to the large size of seismic data and has inconsistencies among interpreters. Auto-picking or auto-trackers (reviewed in [2], and [8]) have been commonly used to assist horizon tracking. Autopicking tools are aimed at extending manually selected seismic traces based on local similarity measures. They perform well if there are uninterrupted horizon features. But horizon interruptions are very common. Correlation of horizons across faults is one of the important task of structural interpretation. Interpreters find horizons and connect them to each other on the basis of reflection character and geological reasoning. The horizons offsets are used to calculate displacement on the fault surface. Fault displacements map allow a more objective assessment of subsurface interpretation. Our work is aimed at developing a computer-based methodology for correlation of horizons across faults in seismic data set. Besides reducing the time of interpreters, 1

2 computer-assisted solutions, based on a quantitative model, provide repeatable or robust data analysis tool. Existing three-dimensional spatial relationships in the data (such as continuity) may be exploited directly whereas humans are only able to evaluate them from 2-d projections or 2-d slices of the data. However automating the horizon correlation task is very challenging. Seismic data contain only little image information, which is further obscured by noises. Since the two sides of the fault may also undergo different geological processes, such as compression and erosion, there are scale differences and some horizons on the one side side may not have matches on the other side Previous Work Alberts et.al. [1] explain a method for tracking horizons across discontinuities. They trained artificial neural networks to track similar seismic intensities. However, horizon tracking across faults using solely seismic patterns is infeasible due to large seismic data distortion near faults. To alleviate this matter, Aurnhammer [2] propose a model-based scheme for correlation of horizons at normal faults in 2D seismic images. The author extracts well-defined horizons segments on both sides of the fault and matchs the segments based on local correlation of seismic intensity and geological knowledge. Since exhaustive search for optimal solution of correlation is unfeasible, the author suggests genetic algorithm as optimization technique. However, a pure twodimensional approach lacks efficiency and is suitable only if the information of the 2D seismic slice is sufficient for evaluation of the geological constraints. We strive for exploiting existing three-dimensional spatial relationships in the data (such as continuity) directly for robust data analysis. The methodology that we envisage is fusing the seismic data with information from a geological model in an iterative fashion. The method first finds and matches point regions that contain sufficiently distinctive structures on the two sides of the fault regions. Then the parameters required for the geological model are estimated based on displacement computed for these regions. Finally simulated annealing global search technique is used to find the optimal correlation between the two sides of the fault, optimal in sense of maximizing seismic features similarities while maintaining geologically valid solutions. The optimization is done in a multi-resolution approach in order to take into account that horizons layers exist at different levels of resolution. The method has been applied to faults patches extracted from real 3D seismic data and has shown to be efficient in putting the displaced horizon into correspondence. 2. Correlating Horizons across Faults Horizons correlation is a task to gather sediment features on the two sides of the fault and to find an optimal match between them. We first identify a fault surface, approximated as a plane here, and its patch. The fault patch is then mapped onto two planes (left and right fault planes). Local features from the seismic data are mapped along the horizon direction onto two planes (see figure 2). The canny edge detector [5] is used for defining the direction along which values are integrated. The seismic features are averaged in 10 pixels size along the edge and mapped to the fault plane (see figure 2). Seismic information are distorted at locations close to the fault because of the geological process of fault creation. To correct for this fault distortion, averaging along the horizon starts at six pixel distance from the fault line. Two features values are computed for each mapped point on the planes: an averaged amplitude (gray-value) of the seismic data, and a reliability measure. The amplitude attributes (see Figure 3) are used to compute the local similarity between points on the two planes. However there is no guarantee that corresponding horizons on the two sides of a fault have equal feature values since sediments left and right of the fault may have suffered different fates during and after creation of the fault. Thus the reliability attribute is used to weigh the gray-value (amplitude) similarity computed at a local level. Reliability is computed as an average value of the coherence cube [3] attribute of the seismic data. Figure 4 shows seismic coherence attributes mapped onto the left and right fault planes; they correspond to the amplitude features of figure 3. The darker regions show lesser reliability values, which mean less weight are given for intensity correlation computed at those places. Horions Fault Patch Mapping from fault patch to fault planes Vertical displacment Displacment Sediment features at fault Left Right fault plane Horiontal displacment Figure 2: A fault patch is mapped onto left and right fault planes. Consequently, the correspondence analysis of the horizons is defined as finding displacement map for left fault image to overlap it with the right fault image so that corresponding positions in the two images are superposed. We define a match function, ξ : Z 2 R, which measures the degree of match between the left and right side of the fault planes as they are overlaid on top of each other. The two images (left and right side of the fault plane) are 2

3 Left plane Right plane C(x 0,y 0 )= n x,y= n (I lxy) I rxy ) n x,y= n (I lxy) 2 n x,y= n (I rxy) 2 (2) where I lxy = I l (x 0 x, y 0 y) I l (x 0,y 0 ) I rxy = I r (x 0 x, y 0 y) I r (x 0,y 0 ) Figure 3: Seismic amplitude attributes mapped onto the left and right fault planes. I l (x 0,y 0 ) and I r (x 0,y 0 ) are respectively the mean of I l and I r for n-neighborhood around point (x 0,y 0 ) and (x 0,y 0). The normalized local cross-correlation value at each point is weighted by the coherence cube feature value. Then the seismic similarity energy, E s, for a given transformation is computed as the sum of the values of the weighted normalized local cross-correlation over all points on the floating image. L. R Figure 4: Seismic coherence attributes mapped onto left (L) and right (R) fault planes. given as two dimensional arrays and denoted by I l and I r where each of I l (x, y) and I r (x, y) maps to its respective averaged seismic features. The right image, I r,servesas reference image while I l (x, y) is the floating image which is displaced with displacement field T (x, y). T (x, y) is a 2D spatial coordinate transformation, such that T (x, y) = (x,y ). The match function, ξ, is then given as ξ(t )=E s (T )+λ E g (T ) (1) where E s is the energy which is computed based on the seismic similarity, whereas E g is the energy that measures similarity of a given transformation and a geological model. They are described in details in section 2.1 and 2.2. λ is a negative scalar real value and used to balance E s and E g Computation of Seismic Similarity, E s Seismic similarity is defined as a mathematical measure of intensity similarity. In order to estimate the similarity, we choose the normalized cross-correlation technique. The main reasons for this choice are its potential accuracy and robustness with respect to noise as it is computed in some neighborhood of a point. For inputs I r, I l, and T, the normalized local crosscorrelation function at point (x 0,y 0 ) with T (x 0,y 0 ) = (x 0,y 0) is calculated as 2.2. Computation of Geometrical Energy, E g E g is computed by comparing similarity between the current observed displacement with the geological model. Our geological model is based mainly on pattern of displacement on fault surfaces Fault Displacement Model Layers of rock that have been moved by the action of faults show displacement on either side of the fault surface. The fault displacement is the offset of segments or points that were once continuous or adjacent. We deal only with normal faults. A normal fault is a type of fault in which the hanging wall moves down relative to the footwall (see figure 5). Thus normal fault displacements have only one direction, which means for any (x, y), T (x, y) =(x, y ). Normal faults are the most common ones and the displacement model which has been used here can be extended to other fault types. Footwall block Normal fault Thrust fault Strike-slip Fault Figure 5: Fault types. Hanging wall block 3

4 Furthermore, it is known from structural geology that horizons do not cross each other, that is for any T (x, a) = (x, a ) and T (x, b) =(x, b ), wehave a<b= a b (3) According to heuristics of Walsh et.al. [12] [13], a normalized displacement, D, at a point on a fault surface is given by D = 2(((1 + r)/2) 2 r 2 ) 2 (1 r) (4) where r is normalized radial distance from the fault center. The normalized displacement is D = d d max where d is the fault displacement at a point and d max is the maximum displacement on a fault surface Estimation of Fault Parameters The fault model of equation (4) assumes that the center of the fault, the width of the fault and the maximum displacement on the fault are known. But faults are often not contained to their complete extents in seismic data set. We have estimated the extent of the fault by computing landmark displacements. The simplest method for extraction of landmark corresponding points is to manually specify them. However, it is very difficult to specify accurate corresponding points and time-consuming. Thus, landmark displacements are extracted automatically in the following fashion. If the part of the fault ends in the seismic data, then there are regions with zero displacement. We take partially overlapping segments in those regions and propagate them to the next slices. When any offset is found, the displacements are calculated and serve as landmarks. But if neither end of the fault is included in the seismic data, we identify particular prominent linear structures from the two sides of the fault features images. These prominent structures are extracted as segments by threshold of high contrasts. The segments can be matched with high confidence by maximizing the total cross-correlation values of the intensities around small neighborhood, as it is described in Aurnhammer [2]. The resulting landmark displacements are plugged into equation (4). Then the Marquardt-Levenberg constrained nonlinear optimization method, implemented in Matlab, is used to estimate the parameters for the fault displacement model. Finally, the theoretical transformation value for each point of the floating image is computed. The geometrical energy, E g, is computed as the least square error between any given transformation, T, and the theoretical transformation map, T computed. i.e E g (T )= i (T computed,i T i ) 2 (5) 3. Optimization After we have defined the transformation function and similarity measures, the next step is to find a suitable optimization procedure to generate the optimal transformation map, T max, which maximizes the value of the match function, ξ. T max = argmax T {T :Z2 Z 2 }(ξ(t)) (6) The general difficulty of the optimization of the match function in equation (6) is the non-linearity existing in the search. It usually has many local maxima. Simple search strategies such as gradient decent are not appropriate. Therefore we use a simulated annealing (SA) [9], a stochastic non-linear optimization technique. 3.1 Simulated Annealing Optimization We set up the metropolis algorithm [11] to perform the minimization form of equation (6). The metropolis algorithm incorporates ξ with the regularizer term which imposes a priori smoothness constraint on the solution (after [10]). It has to also make sure that all the current randomly generated candidate solutions satisfy the geological constraint defined at equation (3). We have used the geometric optimization schedule [6]. The temperature is held fixed during each loop. At the end of each loop, k, the temperature,, is dropped according to the rule: k+1 = α k (7) The vales of the initial temperature ( 0 ), α and the number of iterations in each loop are to be determined experimentally. 3.2 Multi-resolution Optimization Though simulated annealing (SA) algorithms could find the global optimal results, it has high computational complexity [6]. We propose to take advantage of a multi-resolution analysis to increase the convergence rate of SA. The multiresolution analysis is obtained by wavelet decomposition of the left and right side fault images. The decomposition is done by calculating the coefficient of a one dimensional continuous wavelet transform. Each column of the two dimensional image is fed to the one-dimensional continuous wavelet transform which computes the continuous wavelet coefficients of the input column vector at real, positive scales using a given type of wavelet. Figure 6 shows the result of the continuous 1-D wavelet coefficients using Daubechies wavelet [7]. The correspondence analysis between figure 6 and figure 6 is faster if we perform the analysis starting from the coarser-level and go to the finerlevel. 4

5 We have tested the method on seven fault patches taken from 3D seismic data which were surveyed from two different geographical locations. We have extracted the fault plane by linearly interpolating between manually provided seed points. The optimal solutions are defined as the would be solutions if experts performed the correlation. For the match function, we observed that the appropriate values of λ which compensates between the local seismic features and the global geological constraint range from 0.5 to 0.3. The initial temperature for the simulated annealing (SA) is set to values between 1000 and 1200 but still needs further experiments. We found that it is favorable for the optimization structure to cool down quickly and to stay more at lower temperatures. Thus the temperature is decremented at each step by 93% as cooling proceeds. The results of SA for a sample fault patch are shown on figure 7. To verify the results, we have restored the feature images to the 3D fault patch by inverting the feature mapping process (see figure 2). Some optimally correlated horizons are shown on two seismic slices on figure 8. These slices are taken from the restored 3D fault patch at locations I and II of figure 7 and. The correlations are done according to the displacement map of figure 7 (d). The correlations have been verified by comparing them with manual interpretation. I II I II (c) (d) (e) (f) Figure 6: Wavelet decomposition: and show the left and the right-side fault feature images. (c) and (d) are respectively decompositions of and at coarse level. (e) and (f) are respectively decompositions of and at coarser level. (c) (d) In this multi-resolution scenario, the simulated annealing optimization starts from lower resolution level, then at higher resolution it searches with the convergence solutions reached at the lower resolution. Since the convergence solutions at lower resolution are close to the higher resolution optimum solutions, less number of iterations can be used for searching at higher resolutions. 4. Results and Discussion Figure 7: and show respectively left and right fault images. (c) is the deformed image of after aligning it with. (d) is displacement map of the alignment. Figure 8: and seismic slices extracted at positions I and II of figure 7. The black arrows for some horizons indicate optimal correlation results of SA. The white curves show the fault lines. Our method fails in two test cases. Interactions from nearby faults distort the fault displacement model and lead to incorrect global constraint (figure 9). The method also need proper initial discrete segments match to give correct alignment. But in some cases it was not easy to detect and match those segments automatically. Thus we need to switch in such cases to semi-automatic version of the method where human interpreters find the horizons segments and calculate the landmark displacements. However an overall analysis of the test cases reveals that the resulting correspondence analysis of our method is satisfactory with 5

6 [1] P. Alberts, M. Warner, and D. Lister, Artificial Neural Networks for Simultaneous Multi Horizon Tracking across Discontinuities, 70th Annual International Meeting, SEG, Houston, USA, [2] M. Aurnhammer, Model-based Image Analysis for Automated Horizon Correlation across Faults in Seismic Data, PhD Thesis, University of Magdeburg, [3] M. Bahorich,and S.Farmer, The coherence cube, The Leading Edge, Vol.14, pp ,1995. Figure 9: Seismic slices. The black arrows indicate correlation for horizons found by SA algorithm, while the white arrows show experts manual correlations for incorrect correlation of the black arrows. The white curves show fault lines. very good performance. 5. Summary and Conclusions We have presented an automatic method, which aligns locations of the left fault plane onto those right fault plane. An optimal displacement vector field for the alignment was found based on the combination of seismic image information and a fault displacement model. The fault displacement model is constructed by performing initial discreet match of some prominent regions. Multi-resolution based simulated annealing optimization strategy is adopted to find the optimal solution. While the results are somewhat preliminary, they clearly demonstrate the applicability of our approach to real seismic data. Besides its main application, automatic fault displacement calculation, our method can be also used as an automatic verification tool for manually interpreted faults. And also as a visualization tool for studying the full extent of the fault when the fault part is not fully included in the seismic data set. Further we consider additional geological constraints and seismic attributes which can improve our method. The method will be extend to include lateral displacement. 6 Acknowledgements [4] A. Brown, Interpretation of Three-Dimensional Seismic Data, American Association of Petroleum Geologists, 5th edition, December, [5] J. Canny, A computational approach to edge detection, IEEE Trans. patt. anal. mach. intell., Vol. 8, No. 6, pp , [6] H. Cohn and M. Fielding, Simulated annealing: searching for an optimal temperature schedule, Society for Industrial and Applied Mathematics, Journal of Optimization, Vol.9, No. 3, pp , [7] I. Daubechies, Ten lectures on wavelets, Society for Industrial and Applied Mathematics, Philadelphia, [8] G. Dorn, Modern 3-D Seismic Interpretation, The Leading Edge, Vol. 17, No. 9, pp , [9] S. Kirkpatrick, J.C.D. Gelatt, M.P. Vecchi, Optimization by Simulated Annealing, Science, Vol. 220(4598), pp ,1983. [10] S. Li, Markov Random Field Modeling in Computer Vision, Springer-Verlag, [11] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, Equations of state calculations by fast computing machines, J. Chemical Physics, Vol. 21, No. 6, pp , [12] 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, pp , [13] J. Walsh and J. Watterson, Analysis of the relationship between displacements and dimensions of faults, Journal of Structural Geology, Vol. 10, No. 3, pp , We would like to acknowledge Shell for the seismic data and stimulating discussions. We also thank Stefan Back and Janos Urai for their expertise advices regarding geology. The work presented here was supported by DFG Grant TO-166/8-1. References 6