Lithofacies classification in Barnett Shale using proximal support ...

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Lithofacies classification in Barnett Shale using proximal support vector machines Tao Zhao*, Vikram Jayaram, and Kurt J. Marfurt, University of Oklahoma, Huailai Zhou, Chengdu University of Technology Summary Classification of different lithofacies and petrotypes is one of the main objectives of modern quantitative seismic interpretation. In this study, we present preliminary results of the application of a proximal support vector machine (PSVM) classification algorithm to seismic data. In this application we illustrate the PSVM method to differentiate limestone from shale in a Barnett Shale gas play. The PVSM’s low complexity feature compared to the standard vector machines could be well exploited in a data intensive computation such as the 3D seismic lithofacies classification. The paper reports two applications of this technique one for waveform classification and the other for the classification of well data. In both these applications PSVM classification results showed strong agreement with structural and stratigraphic interpretation results.

maximum likelihood classifier. In contrast, SVM and PSVM are supervised classifiers which assign classes with geological meanings before classification. Torres and Reveron (2013) tested binary PSVM classifiers on lithofacies classification between sand and shale. In this paper we use the PSVM classifier as a binary classifier, with the understanding that the workflow can be extended to multiple classes through an iterative eliminate workflow. In our study area, the Upper and Lower Barnett Shales are inter-bedded with Marble Falls and Forestburg Limestones. First we generated a nonlinear PSVM waveform classifier to delineate these two lithofacies in seismic data along a time window. Then we trained the classifier to recognize these two lithofacies on three wells using three different logs, and further performed a blind well test.

Introduction PSVM (Fung and Mangasarian, 2001, 2005) is a recent variant of support vector machine (SVM) (Cortes and Vapnik, 1995). The SVM’s are a powerful supervised machine learning technique widely used in text detection, image recognition and protein classification. In a binary (two cluster) classification problem, SVM defines a plane (called a decision boundary) that separates the two clusters with largest possible margin on both sides (Figure 1). In contrast, PSVM builds two parallel planes that approximate two data classes; the decision-boundary then falls between these two planes (Figure 2). One can also define nonlinear transformations or “kernel functions” to map data points into a higher dimensional feature space, providing a means to classify linearly inseparable problems (Figure 3). Researchers have also found that PSVM provides comparable classification correctness to standard SVM but at considerable computational savings (Fung and Mangasarian, 2001, 2005; Mangasarian and Wild, 2006) which is critical when dealing with large 3D seismic surveys. Self-organizing maps (SOM) is perhaps the most popular classifier of seismic data. Commonly called “waveform classification”, the waveform in SOM is not limited to seismic amplitude samples, but can be a suite of impedance samples, or a vector of appropriately scaled seismic attributes (Roy et al., 2013). In its simplest implementation, SOM is unsupervised, preventing interpreter control. Roy et al. (2013) introduced supervision by the addition of a

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Figure 1: (a) An illustration of a binary SVM in 2D space. Classes “+1” and “-1” can be separated by a decisionboundary (red line) with maximum margin, which is minimum of ‖𝝎‖. (b) SVM in 3D space. In this case the decision-boundary becomes a plane. d

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Lithofacies classification using proximal support vector machines which can be divided into two classes, 𝐴 + and 𝐴 −; 𝑫 ∈ 𝑅𝑚×𝑚 is a diagonal matrix of labels with a diagonal composed of “+1” for 𝐴 + and “−1” for 𝐴 −; 𝜈 is a nonnegative parameter; and 𝒆 ∈ 𝑅𝑚 is a column vector of ones. This optimization problem can be solved by using a Lagrangian multiplier 𝒖 ∈ 𝑅𝑚 : 1 1 𝐿(𝝎, 𝛾, 𝒚, 𝒖) = 𝜈 ‖𝒚‖2 + (𝝎′ 𝝎 + 𝛾 2 ) 2 2 − 𝒖′ (𝑫(𝑨𝝎 − 𝒆𝛾) + 𝒚 − 𝒆).

(4)

Figure 2: (a) An illustration of a Binary PSVM in 2D space. Classes “+1” and “-1” are approximated by two parallel lines that are pushed as far apart as possible. The decision-boundary then sits at the middle of these two lines. In this case, maximizing the margin is equivalent to minimizing (𝝎′ 𝝎 + 𝛾 2 )1⁄2 . (b) Two-class PSVM in 3D space. In this case the decision-boundary becomes a plane. Theory and Formulations Similarly to SVM, a PSVM decision is defined as > 0, 𝒙′ 𝝎 − 𝛾 { = 0, < 0,

𝑥 ∈ 𝐴+; 𝒙 ∈ 𝐴 + 𝑜𝑟 𝐴−; 𝑥 ∈ 𝐴−,

(1)

where 𝒙 ∈ 𝑅𝒏 is a 𝑛 dimensional vector data point to be classified, 𝝎 ∈ 𝑅𝒏 implicitly defines the normal of the decision-boundary, 𝛾 ∈ 𝑅 defines the location of the decision-boundary, and “𝐴 +” and “𝐴 −” are two classes of the binary classification. PSVM solves an easier optimization problem and takes the form of (Fung and Mangasarian, 2001): 1 1 min 𝜈 ‖𝒚‖2 + (𝝎′ 𝝎 + 𝛾 2 ), 2 2

𝝎,𝛾,𝒚

subject to

𝑫(𝑨𝝎 − 𝒆𝛾) + 𝒚 = 𝒆.

(2)

By setting the gradients of 𝐿 to zero, we obtain expressions for 𝝎, 𝛾 and 𝒚 explicitly in the knowns and 𝒖, where 𝒖 can further be represented by 𝑨, 𝑫 and 𝜈. Then by changing 𝝎 in equations 2 and 3 using its dual equivalent 𝝎 = 𝑨′𝑫𝒖, we can arrive at (Fung and Mangasarian, 2001): 1 1 min 𝜈 ‖𝒚‖2 + (𝒖′ 𝒖 + 𝛾 2 ), 2 2

𝝎,𝛾,𝒚

(3)

In this optimization problem, 𝒚 ∈ 𝑅𝑚 is the error variable; 𝑨 ∈ 𝑅𝑚×𝑛 is a sample matrix composed of 𝑚 samples,

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Figure 3: An illustration for separating linearly inseparable classes using kernel function mapping. SVM is shown for illustration purpose. (a) Binary SVM in 2D space for linearly inseparable samples. Classes “+1” and “-1” cannot be separated by a linear decision-boundary (line). (b) Mapping the same data in (a) into a higher dimensional feature space (3D). The nonlinear transformation allows the two classes to be separated by a plane.

subject to

𝑫(𝑨𝑨′𝑫𝒖 − 𝒆𝛾) + 𝒚 = 𝒆.

(5) (6)

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Lithofacies classification using proximal support vector machines

Equations 5 and 6 provide a more desirable version of the optimization problem since one can now insert kernel methods to solve nonlinear classification problems made possible by the term 𝑨𝑨′ in Equation 6. Utilizing the Lagrangian multiplier again (this time we denote the multiplier as 𝒗), we can minimize the new optimization problem against 𝒖, 𝛾, 𝒚 and 𝒗. By setting the gradients of these four variables to zero, we can express 𝒖, 𝛾 and 𝒚 explicitly by 𝒗 and other knowns, where 𝒗 is solely a dependent on the data matrices. Then for 𝒙 ∈ 𝑅1×𝑛 we write the decision conditions as > 0, 𝒙′ 𝑨′𝑫𝒖 − 𝛾 { = 0, < 0,

𝑥 ∈ 𝐴+; 𝒙 ∈ 𝐴 + 𝑜𝑟 𝐴−; 𝑥 ∈ 𝐴−,

algorithm, we selected training subsets of different size from the labeled set, and consequently used the corresponding leftover portion as testing subsets (Table 1). From Table 1 we can clearly see that even with as little as ten percent of labeled data as training group, we obtained more than 80 percent of correctness in the testing group. Also, when we used more than 50 percent of labeled data for training, the correctness is consistently over 90 percent, which is sufficient for a reliable classification.

(7)

with 𝑰

𝒖 = 𝑫𝑲′ 𝑫 ( + 𝑮𝑮′ )

−1

𝜈

𝑰

𝛾 = 𝑒 ′ 𝐷 ( + 𝑮𝑮′ ) 𝜈

−1

𝒆,

𝒆,

(8) (9)

and 𝑮 = 𝑫[𝑲 −𝒆].

(10)

Instead of 𝑨, we have 𝑲 in equations 8 and 10, which is a Gaussian kernel function of 𝑨 and 𝑨′ that has the form: 2

𝑲(𝑨, 𝑨′)𝑖𝑗 = exp (−𝜎‖𝑨′ 𝑖∙ − 𝑨′𝑗∙ ‖ ) , 𝑖, 𝑗 ∈ [1, 𝑚], (11) where 𝜎 is a scalar parameter. Finally, by replacing 𝒙′ 𝑨′ by its corresponding kernel expression, the decision condition can be written as: > 0, 𝑲(𝒙′, 𝑨′)𝑫𝒖 − 𝛾 { = 0, < 0,

𝑥 ∈ 𝐴+; 𝒙 ∈ 𝐴 + 𝑜𝑟 𝐴−; 𝑥 ∈ 𝐴−.

(12)

and 𝑲(𝒙′, 𝑨′)𝑖𝑗 = exp(−𝜎‖𝒙 − 𝑨′ 𝑖∙ ‖2 ) , 𝑖 ∈ [1, 𝑚]. (13) The formulations above represent a non-linear PSVM classifier. Validation and Classification A. Seismic Waveform Classification To validate the usefulness of our PSVM algorithm, we first propose a binary classification between the Forestburg Limestone and Upper/ Lower Barnett Shales on a 3D seismic data. The average time thickness of the Forestburg in the study area is 15 ms, such that we extract waveforms within a time window between 1370 – 1384 ms which brackets the average depth of Forestburg. The time structure map of Forestburg is shown in Figure 4. In this validation study, 161 traces across the whole survey were randomly selected and labeled based on a traditional structural interpretation of reflector surfaces. The training data were assigned a value “+1” if it fell in the limestone and “-1” for shale. To further test the robustness of the

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Figure 4: Time structure map of Forestburg. The formation is dipping to the east. % of Traces # of Training # of Testing Correctness (%) Used in Training Traces Traces 10 16 145 83.45 20 32 129 87.6 30 48 113 84.1 40 64 97 80.41 50 80 81 90.12 50 81 80 93.75 60 97 64 93.75 70 113 48 93.75 80 129 32 90.63 90 145 16 93.75

Table 1: Testing correctness for different training subsets. We note that even using only 10 percent of “truth” data as training group we still achieved correctness over 80%. When using over 50% of traces for training, the testing result became stable. After the tests, we used all labeled traces for training, which resulted in a 94.41% training correctness. We utilized the trained model to build a classifier and perform classification for the whole study area. The classification map is given in Figure 5. The blue curves in Figure 5 are top and base Forestburg formation horizons from traditional structural interpretation as it crosses the center of the time window, which will serve as validation of our classifier. In Figure 5, PSVM classifies the gray areas as limestone and black areas as shale. Note the excellent match between the blue curves and the grey/ black boundaries, which means the derived PSVM classification

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Lithofacies classification using proximal support vector machines

result shares a strong agreement with the structural interpretation result. One can identify some misclassified traces throughout the survey, most of which are in the Lower Barnett Shale area. One explanation is that Lower Barnett Shale is thick and contains many reflection events with great variation, therefore produces greater chances of misclassification. B. Classification on Wells The second application is still for lithologic classification (limestone versus shale), but from well logs instead of seismic waveforms. Among the four wells available to us from the Barnett Shale, three were used for training, with one as a blind test well. The three training wells are widely separated across the survey, giving sufficient representation of the lithology distribution. The testing well is located in the center, approximately equally far from each training well. Density, gamma ray and sonic logs are used as inputs, with outputs still being “limestone” or “shale”. Blind well test result is shown in Figure 6. The training correctness is around 89%, and blind well test can give 88% correctness. One may notice the high misclassification rate in the Upper Barnett Shale segment, where the log responses are very similar to those of limestone segments. This is because we have limited dimensions of inputs, the classifier is not robust enough to distinguish such shale from limestone. Adding more independent but lithologic responsive inputs may give better classification capability.

Also thanks to the National Natural Science Foundation of China (seismic multi-wave fields characteristics analysis of the thin interbedded reservoirs, Grant No. 41204091).

Figure 5: Bottom: Lithofacies classification result centered at 𝑡 = 1377ms. Blue curves are the boundary of Forestburg Limestone from structural interpretation. Grey and black represent limestone and shale from PSVM classification, respectively. Strong agreement can be identified between the PSVM limestone class and the conventional structural interpretation. The eastmost gray area is the Marble Falls Limestone which overlies on the Upper Barnett. Top: Cartoon illustration of lithology distribution in a cross section at the white line. Names are listed for each formation as depicted in the figure. Formations are dipping to the east.

Conclusions The PSVM lithofacies classification showed promising results in both seismic and well log data. This demonstrated the validity of PSVM classifier in binary classification between shale and limestone. Multiclass PSVM classifiers work in a sequential manner, starting with “shale” and “not shale”, followed subdividing the “not shale” into “limestone” and “not limestone” as the second step. A simple extension for this study is to apply PSVM to a vector of attributes (such as lambda-rho, mu-rho, density, peak frequency) rather than a vector of amplitude samples providing a 3D multi-attribute rather than a 2D amplitude waveform classification. We also anticipate comparisons between low complexity PSVM and other supervised (e.g. artificial neural networks or ANN) and unsupervised (e.g. SOM, generative topographic mapping or GTM) classification algorithms. Acknowledgement Thanks to Devon Energy for providing the data, all sponsors of Attribute Assisted Seismic Processing and Interpretation (AASPI) consortium group for their generous sponsorship, and colleagues for their valuable suggestions.

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Figure 6: PSVM classification in blind well. Left: well curves showing sonic, gamma ray and density logs; Middle: lithology interpreted from well; Right: lithology predicted by PSVM.

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http://dx.doi.org/10.1190/segam2014-1210.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2014 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

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