We Prediction of Geological Characteristic Using Gaussian Mixture Model

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1 We Prediction of Geological Characteristic Using Gaussian Mixture Model L. Li* (BGP,CNPC), Z.H. Wan (BGP,CNPC), S.F. Zhan (BGP,CNPC), C.F. Tao (BGP,CNPC) & X.H. Ran (BGP,CNPC) SUMMARY The multi-attribute classification has been widely applied in recent years as one of the direct approachs of seismic reservoir prediction. However, the use of multi-attribute seismic volume classification is a difficult and time-consuming process. In this paper, we propose a simple effective unsupervised approach based on the Gaussian Mixture Model (GMM) to identify special geologic characteristics (geobodies) including fractures, submarine channels, and reefs. The method proposed is a nonlinear statistical process of parameter learning, in which each kind of geobody is represented with one or multiple Gaussian distributions and given a unique cluster label. This method enables different geobodies to be directly extracted from 3-D seismic data. Experimental results on the real field data from Western China show that our method is effective in automatically detecting special geobodies and reducing the uncertainties in reservoir characterization prediction.

2 Introduction The identification of 3-D geological characteristics is a fundamental challenge in reservoir analysis and prediction. Generally, it has been very difficult and infeasible to recognize geological characteristics with traditional interpretation operations. Seismic attributes from pre-stack or poststack seismic data, derived after mathematical change of the seismic wave geometry, kinematics, dynamics, and statistical information, contain a wealth of geological information that reflects the different stratigraphic, structure, and lithology characteristics. Over the last decades, seismic attribute analysis technology has been the main contents of the processing and interpretation of seismic characteristics (e.g., Chopra and Marfurt, 2008; Tomas et al., 2010). In general, seismic attribute is the reflection of a variety of complex geological factors, such that reservoir analysis and prediction using single seismic attribute is likely to yield multiple simultaneous solutions. Recently, multi-attribute analysis techniques have commonly been used in reservoir analysis, such as Kohonen self-organizing map methodology (Strecker and Richard, 2002), probabilistic neural networking (West et al., 2002), Multi-attribute seismic volume facies classification (Fernando and Harold, 2006), and Grey System Theory (AlMoqbel and Wang, 2010). The technique of seismic multi-attribute analysis is the comprehensive utilization of a variety of seismic attributes to predict 3-D geological characteristics, thereby enhancing the accuracy and reliability of the forecast. Traditional approaches such as supervised or unsupervised classification, however, may not perform well for the identification of special and complicated geological features, because of the lack of additional a priori knowledge such as well data, number of knots and hidden lays, initial parameter modeling, condition of convergence and initial number of classification. In this paper, we adopted an unsupervised probabilistic methodology based on the Gaussian Mixture Model (GMM; McLachlan, et al., 1988) to automatically estimate geological properties from seismic multi-attribute volume. The technique is an inherently nonlinear and completely data-driven method that requires no accurate initial model and no a priori operator linking the predicted reservoir properties with seismic inversion and seismic attributes. The proposed method assumes that multi-attribute data obtained from different geological properties have a certain probabilistic distribution function, which can be represented by the Gaussian Mixture Model. Therefore, the problem of identifying 2-D or 3-D geobodies can be considered to be a seismic multi-attribute data clustering process. Different geobodies will be expressed with one or more corresponding Gaussian component distributions in the GMM and will automatically be given a unique cluster label. By this method, different geologic bodies can be extracted directly from the seismic volume. In the experiment, two real 3-D seismic data sets are used to illustrate the performance of the proposed approach. In both cases, the algorithm is used to delineate geobodies, and has proven to be helpful for the submarine channel deposits, a reef carbonate reservoir, and a fracture reservoir. Gaussian mixture model A powerful statistical approach is the Gaussian Mixture Model, which has been widely studied and applied in the fields of image processing, signal analysis, pattern clustering, and speech recognition. As a parametric model of the probability distribution of continuous measurements, GMM can provide greater flexibility and precision in modeling the underlying statistics of sample data than traditional unsupervised clustering algorithms. Generally, all the GMM parameters are estimated from training data using the iterative Expectation Maximization (EM) algorithm (Dempster et al., 1977). In the field of multi-attribute volume clustering, the seismic multi-attributes data obtained from reservoir properties may not be a special or simple probability distribution function. Therefore, we try to simulate the real distribution function of the seismic multi-attribute data with the Gaussian mixture model, which is a linear combination of normal Gaussian functions. In the Gaussian mixture model, each kind of geobody is represented by a combination of one or more Gaussian components. In the classification framework, each multi-attribute seismic sample is represented as a d -dimensional random vector set, S {X u } N u 1 with X u u u u [x 1, x2,..., xd ] d, where N and d are the number of multi-

3 attribute seismic samples and the selected attributes, respectively. Thus, the Gaussian mixture model is a weighted sum of K -component Gaussian densities by the following equation: K K P(x K )= i (x i ), i 0, i 1 i 1, (1) i 1 where i is the set of parameters that defines the i th component, i is the mixing proportions, and the vector k =( 1,..., K, 1,..., K) is collectively represented for all parameters in the Gaussian mixture. In this case, K is the number of components, which defines the scale of the Gaussian mixture model. Each component density (x i )(i 1,..., K) is the Gaussian probability density with specified mean m j and covariance matrix j. Maximum Likelihood Parameter Estimation Given the multi-attributes data set S { x1, x2,... x N } and GMM configuration, we need to estimate the parameters of the GMM with the seismic attributes. There are several commonly used techniques applied to simulate this, including the maximum likelihood (ML) (Dempster et al., 1977), which is the most popular. The aim of ML estimation is to find the model parameters that maximize the likelihood of the GMM given the training data. Generally, we can construct the following maximum loglikelihood function. N N k log ps ( ) log ( x ) log ( x ) (2) k t k i t i t 1 t 1 i 1 Because ML is a non-linear function of the parameters, it can not typically be maximized directly. However, by applying the well-known Expectation-Maximization (EM) algorithm, we can easily estimate the parameters of the Gaussian mixture with the sample data such that the above ML function increases incrementally to a local maximum. Method Workflow The proposed method consists of the following steps: Load and pre-process seismic attributes to define the multi-attribute sample set, Perform the Principle Components Analysis (PCA) method, Estimate the parameters of GMM to identify different geobodies. In practice, each multi-attribute sample is often defined by a unique combination of physical and geometric attributes, which are treated as an n-dimensional vector. The PCA method (Hagen, 1982) was performed prior to running the classification to reduce the dimension of multi-attribute sample without losing any important information, corresponding to the 3-D seismic survey data volume. Following this procedure, all GMM parameters are updated automatically. Using this method, each type of geological characteristic can be related to one or more different Gaussian components in the GMM. Subsequently, the detailed description of geological characteristics can be obtained from the 3- D seismic volume. Figure 1 An example in 1D of the probability function for GMM (black). The joint probability can be represented by the linear combination of two or more Gaussian components (red, blue and green in the example). In the field of multi-attribute volume clustering analysis, each Gaussian component may identify one of the seismic facies.. Examples In the case study from Western China, the carbonate reservoir space is mainly composed of numerous fractures and channels. The GMM algorithm was used to automatically identify major/minor fractures

4 and channel deposit characteristics. Figure 2 shows the seismic section and the amplitude slice extracted from the seismic data volume. In this figure, a solid yellow line marks the target horizon. Along the seismic horizon, a 20 ms window (10 ms above and 10 ms below) is used to extract seismic attribute volumes. After numerous tests, six seismic attributes (most-positive curvature, most-negative curvature, coherence, dip-azimuth, peak amplitude and texture entropy) were calculated as input data to yield additional reservoir information. After PCA, three components were selected. Figure 3 depicts the results of the GMM algorithm applied to the 3-D real field. In Figure 3a, the characteristics of the major fractures (black) and channels (blue) are stressed strongly, and the characteristics of other facies (grey) are ignored. In Figure 3b, some of the subtle faults (green arrows) are clearly visible. These results show that the boundaries of each seismic facies are clear. Figure 2 2-D seismic section of part of the cross-line showing a 500 ms window of input seismic data (displayed with wiggles and variable density color). The yellow line is the chosen seismic horizon. Amplitude slice of the interest horizon. Color bar is amplitude. Figure 3 Clustering results of the major/minor fractures and channel deposit characteristics. The clustering results with major fractures and channel deposits. The sub- clustering results in which minor fractures are detected. For the second case study from western China, we focus on depicting the reef reservoirs. This type of geological body is characterized by large burial depths, compact lithological character and strong anisotropy. It is sometimes difficult to recognize the structure, reservoir, and the fluid characteristic in this area. In Figure 4a and 4b, the seismic section and the amplitude slice extracted from the seismic data volume are shown, respectively. The target horizon is shown by a purple line marked by a red arrow. Along the seismic horizon, a 20 ms window (10 ms above and 10 ms below) is used to extract the multi-attribute volumes. In order to identify different geologic characteristics, six conventional seismic attributes (most-positive curvature, most-negative curvature, coherence, energy ratio similarity, peak amplitude and phase) were used to construct the multi-attribute sample dataset. As shown by the results, the margin forms, sketch structures, and inner cavities of the reefs are clearly delineated (Figure 5). These areas may be significant for seismic exploration. Therefore, the experiment indicates that our method is efficient at identifying the characteristics of reef reservoirs. Conclusions In this paper, we applied the unsupervised statistical seismic facies method based on the Gaussian Mixture Model (GMM) to the identification of different geological features, which has advantages in

5 dealing with complex nonlinear problems. The proposed method combines seismic multi-attributes to enhance the predictive accuracy of the reservoir characteristics. Compared to the traditional clustering algorithms, the proposed method obtains more accurate and reasonable classification results. The proposed algorithm adaptively adjusts the number of the GMM Gaussian components with the training data, which not only reduces the influence of the inaccuracy in sampling points, but also increases the accuracy of the prediction. Finally, we applied the method to two real seismic data sets to illustrate its feasibility. Figure 4 2-D seismic section of part of inline showing a 700 ms window of input seismic data (displayed with wiggles and variable density color). The purple line is the chosen seismic horizon. Amplitude slice of the horizon of interest. Color bar is amplitude. Figure 5 Classification results from 3-D cluster label cube using the GMM algorithm. As shown by the results, the margin forms, sketch structures and inner cavities of the reefs are delineated clearly, respectively. References AlMoqbel, A.M. and Wang, Y. [2010] Carbonate reservoir characterization based on grey prediction theory. EAGE Expanded Abstract. Chopra, S. and Marfurt, K.J. [2008] Emerging and future trends in seismic attributes. The Leading Edge, 27, Dempster, A.P., Laird, N.M. and Rubin, D.B. [1977] Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society, 39(1), Fernando, A.N. and Harold, T. [2006] Multi-attribute seismic volume facies classification for predicting fractures in carbonate reservoirs. The Leading Edge, 25(6), Hagen, D.C. [1982] The application of principal component analysis to seismic data set. Geoexploration, 20, McLachlan, G., ed. [1988] Mixture Models. Marcel Dekker, New York, NY. West, B.P., May, S.R., Eastwood, J.E. and Rossen, C. [2002] Interactive seismic facies classification using textural attributes and neural networks. The Leading Edge, 21(10), Strecker, U. and Richard, U. [2002] Data mining of 3D poststack seismic attribute volumes using Kohonen self-organizing maps. The Leading Edge, 21, Tomas, V.H., Stephane, G. and Jim, P. [2010] Geometric attributes for seismic stratigraphic interpretation. The Leading Edge, 29,