Unsupervised seismic facies from mixture models to highlight channel features Robert Hardisty* 1 and Bradley C. Wallet 1 1)

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1 Unsupervised seismic facies from mixture models to highlight channel features Robert Hardisty* 1 and Bradley C. Wallet 1 1) University of Olahoma, OK USA Summary Unsupervised seismic facies are a convenient and efficient tool for interpretation. Expanding upon Zhao et al. s (2016) study, Gaussian mixture models are used to show how features can automatically be generated using machine learning. The conventional expectation-maximization algorithm is compared to the neighborhood expectationmaximization algorithm to highlight the effects of spatial relations in the data in addition to the measurements of seismic attributes. The survey being used is a 3D seismic survey from the Canterbury basin, ew Zealand called Waa-3D Introduction Visual examination of seismic facies on large 3D seismic data sets where there is little a priori geologic information can be tedious and inaccurate. The process can be more automated and improved using machine learning. By teaching a computer how to recognize patterns, features can automatically be piced. This has the obvious benefit of quicer interpretations, but moreover it can highlight features that might otherwise go unnoticed. The Gaussian mixture model (GMM) provides a flexible framewor by which to accomplish this. Geologic setting The seismic survey is located on the Canterbury Basin, offshore ew Zealand (Figure 1). The area lies in the transition zone of the continental rise and continental slope. Figure 1: Aerial view of study area (Modified from Zhao et al., 2016) The data set contains many Cretaceous and Tertiary age paleocanyons and turbidite deposits. Sediments were deposited in a single transgressive-regressive cycle driven by tectonics (Zhao et al. 2016). A previously identified channel feature is analyzed using a Gaussian mixture model technique. Theory Gaussian mixture models for seismic attributes Gaussian mixture models (GMM) are a well-nown semiparametric density estimation technique using a weighted sum of normal, or Gaussian, distributions (Figure 2). An inherent assumption when using this technique is that the data comes from different Gaussian distributions. Figure 2: Example of a Gaussian mixture model with three mixture components. The overall density is estimated as the sum of the components. (Modified from Wallet et al., 2014) A multivariate Gaussian distribution can be defined as φ(x μ, C) = 1 e 1 (2π) d 2 C 1 2 (x μ) C 1 (x μ) 2 where µ is the mean, C is the covariance matrix, and d is the number of dimensions of x and µ. For seismic attributes, x is a voxel with dimensions equal to the number of attributes. A GMM can be expressed as p(x ψ) = π j φ(x μ j, C j ) j=1 where is the number of different Gaussian distributions or components, and πj is the weight of the j th component such that πj > 0 and π j = 1 j=1. Page 2289

Dynamic component allocation (DCA) as proposed by Vlassis and Lias (2002) is used to avoid user-defined initialization and to mae the process more unsupervised.

2 The problem is to estimate (or learn) the parameters of the GMM, {π j, μ j, C j } for j= (1 ). Common practice is to learn the parameters of a Gaussian mixture through the expectation-maximization (EM) algorithm developed by Dempster et al. (1977). Dynamic component allocation (DCA) as proposed by Vlassis and Lias (2002) is used to avoid user-defined initialization and to mae the process more unsupervised. Dynamic component allocation (DCA) starts with a single component, and then alternates between optimization using the EM algorithm and allocation of a new component for the GMM. The first component is initialized using the population mean and covariance. Convergence of DCA occurs when the maximum number of components is reached. eighborhood expectation-maximization (EM) algorithm Learning of a GMM using the EM algorithm is a purely statistical construct and doesn t consider spatial correlations. In general, facies are expected to be at least laterally continuous to some extent. To account for spatial correlations of the latent space the eighborhood expectation-maximization (EM) algorithm is implemented and compared to the results of the conventional EM. The conventional EM algorithm can be viewed as a variant of coordinate descent on a certain objective function, D(W, ψ) = W ji [log{ W ji } log {π j φ(x μ j, C j )}] j=1 i=1 where Wji are the elements of the responsibility matrix, W (Hathaway, 1986). Ambroise et al. (1996) introduced a regularization term to tae into account the spatial information of the data, G(W) = 1 2 W ij W pj V ip j=1 i=1 p=1 where Vip are the elements of a neighborhood matrix, V. The new objective function then becomes U(W, ψ) = D(W, ψ) + β G(W) where β 0 and determines the weight of the spatial term, G(W). The neighborhood matrix, V, for this application has been chosen to be V ip = { 1 if x i and x p neighbors 0 else, and xi and xp are neighbors if they both lie within a userdefined window. The benefit of the EM algorithm is that the responsibilities of neighboring voxels are considered when deciding which mixture component a voxel belongs to. Methods Latent space modeling Lie all statistical classifiers, GMM s suffer from the curse of dimensionality. Latent space modeling is a powerful technique to project high dimensional data into a lower dimensional space. In this application, a two-dimensional latent space generated from Zhao et al. (2016) is considered. The latent space was generated using a distance-preserving SOM (DPSOM) technique with the attribute inputs being pea spectral frequency, pea spectral magnitude, coherent energy, and curvedness. The DPSOM algorithm mapped the 4D attribute input to a 2D SOM latent space resulting in 2 seismic attribute volumes, SOM latent axis 1 and SOM latent axis 2 (Figure 3). Using a GMM as a classifier on these two axes will produce a single partition volume and a number,, of mixture decomposition volumes for unsupervised seismic facies analysis. Figure 3: Horizon slice of (a) seismic amplitude, (b) SOM latent axis 1, (c) SOM latent axis 2. Purple lines and arrows show a feature previously interpreted as a muddy channel that cuts throug a sandy channel (orange arrow). Gaussian mixture models as a classifier Each component of a GMM attempts to model an underlying process that generated the data. A GMM is a model based clustering technique in that that each underlying process is assumed to be Gaussian in shape. The objective of a classifier is to find which component is responsible for producing each voxel. Page 2290

Usually finding the component responsible for each voxel is simply done by using the responsibility matrix, W, and assigning each voxel to the component with the highest responsibility.

3 Usually finding the component responsible for each voxel is simply done by using the responsibility matrix, W, and assigning each voxel to the component with the highest responsibility. However, due to the large size of seismic data a training set must be used due to memory and time constraints. The training set is used to learn the parameters of the GMM. The training set is constructed by uniformly sampling every 125 th voxel (one voxel for every 5 th inline, crossline, and time sample). Once the parameters of a GMM are learned using a training data set, the responsibility of each voxel can be calculated individually. For the conventional EM algorithm, this is simply done by implementing another E-step that includes the whole volume. The EM algorithm is done in a similar manner, but uses the training data set to approximate the total population when calculating the penalty term, G(W). Application The area of interest has been interpreted as a possible channel feature by Zhao et al. (2016). The area of interest consists of 456 crosslines x 576 inlines x 23 time samples. The SOM latent axis 1 and SOM latent axis 2 are used as inputs for two different GMM s; one GMM using the conventional EM algorithm and another using the EM algorithm. The number of components to be found is set to be four because four prototype vectors were used in the construction of the latent space axes. For the conventional EM case, DCA is used to find a GMM with four components. For the EM case, the parameters from the EM case are used for initialization and the spatial weight, β, is set to 0.1. Two cross sections are made, A-A and B-B, to show the channel feature in three dimensions. Previously this was interpreted by Zhao et al. (2016) as a possible muddy channel cutting through a sandy channel (Figure 3). In both the EM and EM case the sandy channel is dominated by the 4 th component of the mixture model and is colored tan. Liewise, the muddy channel is dominated by the 2 nd and 3 rd components of the mixture model, and are colored red and green respectively. The EM algorithm successfully segments the image into more spatially continuous facies. However, there are hard right angles similar to how acquisition footprint loos due to the uniform sampling of the training set of data. Cross section A-A shows the high amplitude channel being delineated by the tan colored facies and being surrounded by the blue colored facies. The EM algorithm improves the segmentation by removing the anomalous red facies above the high amplitude feature. In both EM and EM the red and green facies are not within the high amplitude feature. Figure 4: Horizon slice (a) seismic amplitude for reference (b) results from EM algorithm, (c) results frm EM algorithm. Blocy right angles can be noticed in (c) due to how the training data was sampled. A-A cuts across the flow direction of both channels. B- B goes along the flow direction of the tan-colored channel and cuts across the blue-green colored channel in the western end of the profile. Page 2291

The combination of red and green colored facies segment the channel well.

4 Figure 5: Profile A-A of (a) seismic amplitude, (b) EM algorithm, and (c) EM algorithm. Cuts perpindicular to the flow direction of tan colored channel. The blac arrow indicates a high amplitude feature Cross section B-B goes more or less along the flow direction of the tan colored channel (Figure 6). The combination of red and green colored facies segment the channel well. The EM algorithm removes many of the red colored facies in in the high amplitude areas and replaces them with tan colored facies. Figure 6: Profile B-B of (a) seismic amplitude, (b) EM algorithm, and (c) EM algorithm. The channel outlined in purple is composed of all the facies. In the EM algorithm, C, constrains the red facies to mostly the channel fill unlie the EM algorithm. Conclusions Gaussian mixture models are a convenient way to characterize seismic attributes and generate unsupervised seismic facies to let the data spea for itself. Results may not correlate to all the geology, but can highlight features that may be of geological interest. The EM algorithm can act lie a smoothing operator in the spatial domain to ensure that facies have some spatial continuity. Different ways of defining the neighborhood matrix, along with different values of the spatial weight, β, should be investigated further. The unsupervised seismic facies in this paper are using GMM s as a partitioning method lie -means; future wor using GMM s as a fuzzy clustering method may more reveal more complexity in the data. Acnowledgements We would lie to than the ew Zealand Petroleum and Minerals for maing the Waa-3D seismic data public. We would also lie to than the sponsors of the Attribute- Assisted Seismic Processing and Interpretation (AASPI) Consortium at the University of Olahoma. Horizon slices were generated using Petrel licenses courtesy of Schlumberger. A special thans to Tao Zhao for use of his latent space axes. And thans to all our colleagues for their valuable insights. Page 2292

5 EDITED REFERECES ote: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2017 SEG Technical Program Expanded Abstracts have been copyedited so that references provided with the online metadata for each paper will achieve a high degree of lining to cited sources that appear on the Web. REFERECES Ambroise, C., M. Dang, and G. Govaert, 1997, Clustering of spatial data by the EM algorithm, in geoev I Geostatistics for Environmental Applications, 9, , Dempster, A.P.,. M. Laird, and D. B. Rubin, 1977, Maximum lielihood from incomplete data via the EM algorithm: Journal of Royal Statistical Society, B, 39, Hathaway, R. J., 1986, Another interpretation of the EM algorithm for mixture distributions: Statistics & Probability Letters, 4, 53 56, Vlassis,., and A. Lias, 2002, A greedy EM for Gaussian mixture learning: eural Processing Letters, 15, 77 87, Wallet, B. C., R. M. Slatt, and R. P. Altimar, 2014, Unsupervised classification of λρ and µρ attributes derived from well log data in the Barnett Shale: 84th Annual International Meeting, SEG, Expanded Abstracts, , Zhao, T., J. Zhang, F. Li, and K. J. Marfurt, 2016, Characterizing a turbidite system in Canterbury Basin, ew Zealand using seismic attributes and distance-preserving self-organizing maps: Interpretation, 4, SB79 SB89, Page 2293

SUMMARY INTRODUCTION METHOD. Review of VMD theory

SUMMARY INTRODUCTION METHOD. Review of VMD theory Bin Lyu*, The University of Olahoma; Fangyu Li, The University of Georgia; Jie Qi, Tao Zhao, and Kurt J. Marfurt, The University of Olahoma SUMMARY The coherence attribute is a powerful tool to delineate