Theoretical Study in Estimating Mineral Compositions From Spectral Measurements With a Bayesian Approach
- Se Un Park (Schlumberger) | Michael D. Prange (Schlumberger-Doll Research (ret.))
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- February 2020
- Document Type
- Journal Paper
- 158 - 176
- 2020.Society of Petroleum Engineers
- machine learning, transmission Fourier-transform infrared spectroscopy (FTIR), Bayesian inversion, semi-blind deconvolution, mineralogy
- 16 in the last 30 days
- 73 since 2007
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|SPE Non-Member Price:||USD 35.00|
We propose a systematic Bayesian method for inferring the mineral composition of rock samples from transmission Fourier-transform infrared spectroscopy (FTIR) measurements. Currently available FTIR inversion methodologies depend on measuring pure minerals, using them within least-squares approaches with a hand-tuned noise model, and implementing ad hoc post-processing of the obtained solution. Within the proposed data-driven framework, we replaced these previous inversion approaches with the automatic training and estimation steps. In this approach, the linear operator, comparable with the FTIR spectral standards (FTIR spectra of end-members or pure minerals), is estimated or trained on a calibration set of rock samples for which the mineral composition has been accurately determined by other means in the laboratory. With this linear operator, we then quantify the spectral-noise covariance matrix from the calibration set, which forms the basis of our estimation of the Bayesian posterior uncertainty on a mineral-composition estimate. This quantification of uncertainty in mineral estimation is a novel feature that can be used as a reliability measure. The uncertainty also describes the correlation between estimates of mineral-weight fractions, indicating which pairs of minerals cannot be independently estimated. Our Bayesian model also addresses the uncertainty propagated from the estimated linear operator and thus captures a possible mismatch of the model parameter from the true operator (i.e., semiblindness of the model). We demonstrate the advantages of our approach by performing experiments with synthetic data.
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Bickel, P. J. and Levina, E. 2008. Regularized Estimation of Large Covariance Matrices. Ann. Statist. 36 (1): 199–227. https://doi.org/10.1214/009053607000000758.
Cai, T. T. and Yuan, M. 2012. Adaptive Covariance Matrix Estimation Through Block Thresholding. Ann. Statist. 40 (4): 2014–2042. https://doi.org/10.1214/12-AOS999.
Cai, T. T., Zhang, C.-H., and Zhou, H. H. 2010. Optimal Rates of Convergence for Covariance Matrix Estimation. Ann. Statist. 38 (4): 2118–2144. https://doi.org/10.1214/09-AOS752.
Charsky, A. and Herron, M. M. 2012. Quantitative Analysis of Kerogen Content and Mineralogy in Shale Cuttings by Diffuse Reflectance Infrared Fourier Transform Spectroscopy. Presented at the International Symposium of the Society of Core Analysts, Aberdeen, 27–30 August. SCA2012-27.
Craddock, P., Herron, M., and Herron, S. 2017a. Comparison of Quantitative Mineral Analysis by X-Ray Diffraction and Fourier Transform Infrared Spectroscopy. J Sediment Res 87 (6): 630–652. https://doi.org/10.2110/jsr.2017.34.
Craddock, P. R., Prange, M. D., and Pomerantz, A. E. 2017b. Kerogen Thermal Maturity and Content of Organic-Rich Mudrocks Determined Using Stochastic Linear Regression Models Applied to Diffuse Reflectance IR Fourier Transform Spectroscopy (DRIFTS). Org. Geochem. 110 (August): 122–133. https://doi.org/10.1016/j.orggeochem.2017.05.005.
Debba, P., Carranza, E. J. M., van der Meer, F. D. et al. 2006. Abundance Estimation of Spectrally Similar Minerals by Using Derivative Spectra in Simulated Annealing. IEEE Trans Geosci Remote Sens 44 (12): 3649–3658. https://doi.org/10.1109/TGRS.2006.881125.
Grundmann, A. and Mo¨ ller, H. M. 1978. Invariant Integration Formulas for the n-Simplex by Combinatorial Methods. SIAM J. Numer. Anal. 15 (2): 282–290. https://doi.org/10.1137/0715019.
Han, Y. and Misra, S. 2018. Joint Petrophysical Inversion of Multifrequency Conductivity and Permittivity Logs Derived From Subsurface Galvanic, Induction, Propagation, and Dielectric Dispersion Measurements. Geophysics 83 (3): D97–D112. https://doi.org/10.1190/geo2017-0285.1.
Herron, M., Matteson, A., and Gustavson, G. 1997. Dual-Range FT-IR Mineralogy and the Analysis of Sedimentary Formations. Presented at the International Symposium of the Society of Core Analysts, Calgary, 7–10 September. SCA-9729.
Herron, S., Herron, M., Pirie, I. et al. 2014. Application and Quality Control of Core Data for the Development and Validation of Elemental Spectroscopy Log Interpretation. Presented at the SPWLA 55th Annual Logging Symposium, Abu Dhabi, 18–22 May. SPWLA-2014-LLL.
Hillier, S. 2000. Accurate Quantitative Analysis of Clay and Other Minerals in Sandstones by XRD: Comparison of a Rietveld and a Reference Intensity Ratio (RIR) Method and the Importance of Sample Preparation. Clay Miner 35 (1): 291–302. https://doi.org/10.1180/000985500546666.
Makni, S., Ciuciu, P., Idier, J. et al. 2004. Semi-Blind Deconvolution of Neural Impulse Response in fMRI Using a Gibbs Sampling Method. Proc., International Conference on Acoustics, Speech, and Signal Processing, Montreal, Quebec, Canada, 17–21 May, Vol. 5, 601–604. https://doi.org/10.1109/ICASSP.2004.1327182.
MathWorks. 2012. MATLAB, Version 7.14.0 (R2012a). Natick, Massachusetts: MathWorks.
Matteson, A. and Herron, M. M. 1993. Quantitative Mineral Analysis by Fourier Transform Infrared Spectroscopy. Presented at the Society of Core Analysts Annual Symposium, Houston, 9–11 August. SCA-9308.
Onn, S. and Weissman, I. 2011. Generating Uniform Random Vectors Over a Simplex With Implications to the Volume of a Certain Polytope and to Multivariate Extremes. Ann Oper Res 189 (1): 331–342. https://doi.org/10.1007/s10479-009-0567-7.
Park, S. U., Dobigeon, N., and Hero, A. O. 2012. Semi-Blind Sparse Image Reconstruction With Application to MRFM. IEEE Trans Image Process 21 (9): 3838 –3849. https://doi.org/10.1109/TIP.2012.2199505.
Park, S. U., Dobigeon, N., and Hero, A. O. 2014. Variational Semi-Blind Sparse Deconvolution With Orthogonal Kernel Bases and Its Application to MRFM. Signal Process 94 (January): 386–400. https://doi.org/10.1016/j.sigpro.2013.06.013.
Petersen, K. B. and Pedersen, M. S. 2012. The Matrix Cookbook, version 20121115. Copenhagen, Denmark: Technical University of Denmark.
Pillonetto, G. and Cobelli, C. 2007. Identifiability of the Stochastic Semi-Blind Deconvolution Problem for a Class of Time-Invariant Linear Systems. Automatica 43 (4): 647–654. https://doi.org/10.1016/j.automatica.2006.10.009.
Robert, C. P. and Casella, G. 2004. Monte Carlo Statistical Methods, second edition. New York City: Springer Science+Business Media.
Rodriguez-Yam, G., Davis, R. A., and Scharf, L. L. 2004. Efficient Gibbs Sampling of Truncated Multivariate Normal With Application to Constrained Linear Regression. Technical report, Colorado State University, Fort Collins, Colorado, March 2004.
Silvia, D. S. and Skilling, J. 2006. Data Analysis: A Bayesian Tutorial, second edition. New York City: Oxford University Press.
Wolfram. 2012. Mathematica, Version 9.0. Champaign, Illinois: Wolfram.