Model-based Approach to Automatic 3D Seismic Horizon Correlation across Faults

Transcription

1 Model-based Approach to Automatic 3D Seismic Horizon Correlation across Faults Fitsum Admasu and Klaus Toennies Computer Vision Group Department of Simulation and Graphics Otto-von-Guericke Universität, Magdeburg, Germany Abstract Seismic data provide detailed information about subsurface structures. Reflection events visible in the seismic data are known as horizons, and indicate boundaries between different rock layers. A fault is a surface along which one side of rock layers has moved relative to the other in a direction parallel to the surface. Faults are recognized in seismic data by discontinuities of horizons. Fault interpretation is an important but time-consuming task of seismic interpretations. It is least supported by automatic tools due to seismic data distortions near fault regions. In this paper, we present an automatic method for correlations of horizons across faults in 3d seismic data. As automating horizons correlations using only seismic data features is not feasible, we find optimal correlations by imposing geological constraints. We formulate the correlation task as a non-rigid continuous point matching between the two sides of the fault. A fault surface is identified and seismic features on both sides of the fault are gathered. One side of the fault serves as the floating image while the other side is the reference image. A fault displacement model is constructed by performing initial discrete match of some prominent regions. Then the simulated annealing optimization technique is used to perform continuous point matching between the two sides according to seismic feature similarity and the fault model. A possible multiresolution scheme for the simulated annealing optimization is discussed. The method was applied to real 3D seismic data, and has shown to produce geologically acceptable results. Key Words: automatic seismic interpretation, model-based correspondence analysis. 1 Introduction Faults are breaks or planar surfaces in rocks across which there are displacements of rock layers. They exert control over oil formation by acting as seals or conduits of hydrocarbons. As result accurate assessment of fault geometry and displacements are crucial for the petroleum industry. Since 1920s seismic data have been used extensively for hypothesizing subsurface structures. Three-dimensional seismic data consist of numerous closely-spaced seismic lines that provide measures of subsurface reflectivity. Different subsurface s rock layers have 1

2 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 [5]. Structural interpretation of seismic data attempts to create a 3D subsurface model. Some of the tasks involved in structural interpretation are tracking of uninterrupted horizon segments, fault surfaces identification and correlating horizon segments across faults [2]. These interpretation tasks are done manually for some parts of the seismic data shown on figure 1. The horizon segments which are offset by the fault line are matched by the arrows. Horizon segment Fault line Timeline Crossline Inline Figure 1: 3-D seismic data with some interpretations. Our work aims at developing a computer-based methodology for correlation of horizons across faults. Seismic interpreters find displaced seismic features and connect them to each other on the basis of reflection characters and geological reasoning. By automating this task, we can save much time and avoid the inconsistencies existing among human interpreters. However automating the correlation task is very challenging. Seismic data contain a little image information, which is further obscured by noise. Since the two sides of the fault may also undergo different geological processes, such as compression and erosion, there are scale differences and some regions on the one side may not have matches on the other side. 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 displacements computed at those 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 2

3 method has been applied to fault patches extracted from real 3D seismic data and produces acceptable correspondence results. This paper is organized as follows. We review some of the existing automated structural interpretation tools in chapter 2. We formulate the horizons correlations task as correspondence analysis between two sides of a fault in chapter 3. Seismic similarity measures and the fault model with indications of how we perform the parameter estimations are described in chapter 4. The optimization problem and the possible multi-resolution simulated annealing scheme are explained in chapter 5. Our experimental results appear in chapter 6, followed by concluding remarks in chapter 7. 2 State of the Art Due to the complexity and size of seismic data sets, several automated tools are developed for structural interpretation of seismic data. Auto-picking or auto-trackers (reviewed in [2], [9]) have been commonly used to assist horizon tracking. Auto-picking 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. Steen et. al. [15] describe a semiautomatic method for fault tracking. They extract directive attributes in user- or data-steered directions, then use trained neural networks to enhance fault features from the directive attributes. Further the outputs of the neural networks are passed to the Canny non-maximum suppression edge detection algorithm [6] to generate the potential fault pixels. Bahorich et.al. [4] introduce a technique for fault highlighting through local coherence analysis. The coherence analysis measures the cross correlation of neighboring traces. Then the fault features are extracted as poorly correlated adjacent traces. Another method, reported by Gibson et.al [10], uses a coherency measure to detect points of significant horizon discontinuities. Then a highest confidence first merging strategy with a flexible surface model is applied to estimate the 3D fault surfaces iteratively. Alberts et.al. [1] explain a method for tracking horizons across discontinuities. They trained artificial neural networks (ANN) to track similar seismic intensity. However, horizon tracking across faults using solely seismic patterns is infeasible due to large seismic data distortion near faults. To alleviate this matter, Aurnhammer et.al [3] propose a model-based scheme for correlation of horizons at normal faults in 2D seismic images. They extract well-defined horizons segments on both sides of the fault and match the segments based on local correlation of seismic intensity and geological knowledge. Since exhaustive search for optimal solution of correlation is unfeasible, they suggest genetic algorithm as optimization technique. However, a pure two-dimensional 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. 3

4 3 Problem Statement We formulate the horizons correlation task as a non-rigid continuous point matching between the two sides of the fault. Here continuous means each pixel on the left side of the fault is assigned a corresponding position on the right side. Our work concentrates on a single fault surface. Moreover, we assume that the fault surface has been already extracted as a plane with its associated patch. The seismic sediment features which are actually the grey (amplitude) values are averaged in five to ten pixels length along the horizons orientations starting at five pixels distance from the fault plane. The orientations of the horizons are defined by canny edge detector [6]. The average values are projected on the fault plane. These processes produce left and right fault feature 2D images (see Figure 2). The correspondence analysis problem 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. Fault Plane Left side of the fault mapped to the left plane Right side of the fault mapped to the right plane Horizons Fault patch Mapping from fault patch to fault plane Figure 2: A fault patch is mapped onto left and right fault planes. 4 Match Function We define a match function, ξ : Z 2 R, which measures the degree of match between the left and right side of the fault plane as they are overlaid on top of each other. The two images (left and right side of the fault plane) are 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, serves as 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 4

5 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 4.1 and 4.2. λ is a negative scalar real value and used to balance E s and E g. 4.1 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 cross-correlation function at point (x 0,y 0 ) with T (x 0,y 0 )=(x 0,y 0) is calculated as C(x 0,y 0 )= n x,y= n (I lxy) I rxy ) n x,y= n (I n lxy) x,y= n (I rxy) (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 ) 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 seismic similarity energy for given transformation is computed as the sum of the values of the normalized local cross-correlation at each point. 4.2 Computation of Geometrical Energy, E g The geometrical energy, E g, is computed as similarity between a given displacement map and a geological model. The geological model is based on pattern of displacement on a fault surface Fault Displacement Model Depending on the relative direction of displacement between the rocks on either side of the fault, faults are classified as normal, reverse or strike-slip [14](figure 3). The fault block above the fault surface is called the hanging wall, while the fault block below the fault is the footwall. We deal with normal faults. A normal fault is the most common fault type in which the hanging wall moves down relative to the the footwall. Thus normal fault displacements have only one direction, which means for any (x, y), T (x, y) =(x, y ). 5

6 Normal fault Footwall block Hanging wall block Thrust fault Strike-slip Fault Figure 3: Fault types. 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. [16] [17], 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 6

7 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) 5 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 nonlinearity 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) [11], a stochastic non-linear optimization technique. 5.1 Simulated Annealing Optimization We set up the metropolis algorithm [13] to perform the minimization form of equation (6), based on the assumption that though the match function, ξ, might be noisy, its optimal would be surrounded by points whose values are also good. The metropolis algorithm incorporates ξ with the regularizer term which imposes a priori smoothness constraint on the solution (after [12]). 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, [7]. 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. 7

8 In Simulation und Visualisierung 2004, Magdeburg, March Multi-resolution Optimization Though simulated annealing (SA) algorithms could find the global optimal results, it has high computational complexity [7]. We propose to take advantage of a multi-resolution analysis to increase the convergence rate of SA. The multi-resolution 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 4 shows the result of the continuous 1-D wavelet coefficients using Daubechies wavelet [8]. The correspondence analysis between (a) and (b) on this figure is faster if we perform the analysis starting from the coarser-level and go to the finer-level. (a) (b) (c) (d) (e) (f) Figure 4: Wavelet decomposition: (a) and (b) show the left and the right-side fault feature images. (c) and (d) are respectively decompositions of (a) and (b) at coarse level. (e) and (f ) are respectively decompositions of (a) and (b) at coarser level. In this multi-resolution scenario, the simulated annealing optimization starts from lower 8

9 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, lower search temperature (that is, less number of iterations) can be used for searching at higher resolutions. But the multi-resolution analysis of the fault displacement constraint requires further investigations. This issues will be addressed in a paper to follow this. The results described in following chapter are done by single resolution optimization scheme. 6 Results and Discussion Our experimental data consist of 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 5. 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 6. These slices are taken from the restored 3D fault patch at locations I and II of figure 5 (a) and (b). The correlations are done according to the displacement map of figure 5 (d). The correlations have been verified by comparing them with manual interpretation. Our method fails in two test cases. The reason for failure appears to be insufficient information in seismic data to make correct initial discrete segments match. Interactions from nearby faults distort the fault displacement model and lead to incorrect global constraint (figure 7). However an overall analysis of the test cases reveals that the resulting correspondence analysis of our method is satisfactory with very good performance. 7 Summary and Conclusions We have presented an automatic method, which aligns locations of the left-side of the fault onto those of the right-side of the fault. The 3D spatial information of the seismic data is exploited in designing the fault displacement model and the smoothness constraints. Optimal displacement vector fields for correlating the displaced horizons are searched for maximizing the similarity of seismic image information in both sides while maintaining the fault model constraints. Since the search space is large and has many local maxima, we have utilized the stochastic stimulated annealing global search procedure. To speed up the convergence rate of the simulated annealing, the possible multi-resolution framework 9

10 In Simulation und Visualisierung 2004, Magdeburg, March I II (a) (c) I II (b) (d) Figure 5: (a) and (b) show respectively left and right fault images. (c) is the deformed image of (b) after aligning it with (a). (d) is displacement map of the alignment. is described. While the results are somewhat preliminary, they clearly demonstrate the applicability of our approach to real seismic data. Besides reducing the seismic interpreters time, the presented method, based on a quantitative model, provides repeatable data analysis tool. Moreover, the method can be used not only as an automatic verification tool for manually interpreted faults but also as a visualization tool for studying the full extent of the fault when the fault is not fully included in the seismic data set. However, the employed fault model needs to be modified to include fault interactions and lateral displacements. 8 Acknowledgements 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/

11 (a) (b) Figure 6: (a) and (b) seismic slices extracted at positions I and II of figure 5. The black arrows for some horizons indicate optimal correlation results of SA. The white curves show the fault lines. (a) (b) Figure 7: 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. References [1] P. Alberts, M. Warner and D. Lister, Artificial Neural Networks for Simultaneous Multi Horizon Tracking across Discontinuities, 70th Annual International Meeting, Expanded Abstracts, Society of Exploration Geophysicists, Calgary, Canada, [2] M. Aurnhammer, Model-based Image Analysis for Automated Horizon Correlation across Faults in Seismic Data, PhD Thesis, University of Magdeburg, [3] M. Aurnhammer and K. Tönnies, Model-based analysis of geological structures in seismic images, In Simulation und Visualisierung, pp , Magdeburg, March, [4] M. Bahorich and S. Farmer, 3-D seismic Discontinuity for faults and stratigraphic features, The leading edge, Vol. 14, No. 10, pp , October,

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