# American Institute of Mathematical Sciences

May  2014, 8(2): 587-610. doi: 10.3934/ipi.2014.8.587

## A semi-blind source separation method for differential optical absorption spectroscopy of atmospheric gas mixtures

 1 Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, United States 2 Department of Chemistry/Mathematics, University of California at Irvine, Irvine, CA 92697, United States, United States 3 Department of Mathematics/Mathematics, University of California at Irvine, Irvine, CA 92697

Received  September 2012 Revised  December 2013 Published  May 2014

Differential optical absorption spectroscopy (DOAS) is a powerful tool for detecting and quantifying trace gases in atmospheric chemistry [22]. DOAS spectra consist of a linear combination of complex multi-peak multi-scale structures. Most DOAS analysis routines in use today are based on least squares techniques, for example, the approach developed in the 1970s [18,19,20,21] uses polynomial fits to remove a slowly varying background (broad spectral structures in the data), and known reference spectra to retrieve the identity and concentrations of reference gases [23]. An open problem [22] is that fitting residuals for complex atmospheric mixtures often still exhibit structure that indicates the presence of unknown absorbers.
In this work, we develop a novel three step semi-blind source separation method. The first step uses a multi-resolution analysis called empirical mode decomposition (EMD) to remove the slow-varying and fast-varying components in the DOAS spectral data matrix ${\bf X}$. This has the advantage of avoiding user bias in fitting the slow varying signal. The second step decomposes the preprocessed data $\hat{{\bf X}}$ in the first step into a linear combination of the reference spectra plus a remainder, or $\hat{{\bf X}} = {\bf A}\,{\bf S} + {\bf R}$, where columns of matrix ${\bf A}$ are known reference spectra, and the matrix ${\bf S}$ contains the unknown non-negative coefficients that are proportional to concentration. The second step is realized by a convex minimization problem ${\bf S} = \mathrm{arg} \min \mathrm{norm}\,(\hat{{\bf X}} - {\bf A}\,{\bf S})$, where the norm is a hybrid $\ell_1/\ell_2$ norm (Huber estimator) that helps to maintain the non-negativity of ${\bf S}$. Non-negative coefficients are necessary in order for the derived proportional concentrations to make physical sense. The third step performs a blind independent component analysis of the remainder matrix ${\bf R}$ to extract remnant gas components. This step demonstrates the ability of the new fitting method to extract orthogonal components without the use of reference spectra.
We illustrate utility of the proposed method in processing a set of DOAS experimental data by a satisfactory blind extraction of an a-priori unknown trace gas (ozone) from the remainder matrix. Numerical results also show that the method can identify trace gases from the residuals.
Citation: Yuanchang Sun, Lisa M. Wingen, Barbara J. Finlayson-Pitts, Jack Xin. A semi-blind source separation method for differential optical absorption spectroscopy of atmospheric gas mixtures. Inverse Problems & Imaging, 2014, 8 (2) : 587-610. doi: 10.3934/ipi.2014.8.587
##### References:
 [1] A. Bell and T. Sejnowski, An information-maximization approach to blind separation and blind deconvolution, Neural Computation, 7 (1995), 1129-1159. doi: 10.1162/neco.1995.7.6.1129.  Google Scholar [2] A. Bongartz, J. Kames, U. Schurath, CH. George, PH. Mirabel and J. L. Ponche, Experimental determination of hono mass accommodation coefficients using two different techniques, J. Atmos. Chem., 18 (1994), 149-169. doi: 10.1007/BF00696812.  Google Scholar [3] A. Bongartz, J. Kames, F. Welter and U. Schurath, Near-UV absorption cross sections and trans/cis equilibrium of nitrous acid, J. Phys. Chem., 95 (1991), 1076-1082. doi: 10.1021/j100156a012.  Google Scholar [4] J.-F. Cardoso, High-order contrasts for independent component analysis, Neural Computation, 11 (1999), 157-192. doi: 10.1162/089976699300016863.  Google Scholar [5] A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications, John Wiley and Sons, New York, 2005. Google Scholar [6] P. Comon and C. Jutten, Handbook of Blind Source Separation: Independent Component Analysis and Applications, Academic Press, 2010. Google Scholar [7] I. Daubechies, J. Lu and H.-T. Wu, Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool, Applied and Computational Harmonic Analysis, 30 (2011), 243-261. doi: 10.1016/j.acha.2010.08.002.  Google Scholar [8] D. DeHaan, T. Brauers, K. Oum, J. Stutz, T. Nordmeyer and BJ. Finlayson-Pitts, Heterogeneous chemistry in the troposphere: Experimental approaches and applications to the chemistry of sea salt particles, Intern. Rev. Phys. Chem., 18 (1999), 343-385. Google Scholar [9] BJ. Finlayson-Pitts, LM. Wingen, AL. Sumner, D. Syomin and KA. Ramazan, The Heterogeneous Hydrolysis of $NO_2$ in Laboratory Systems and in Outdoor and Indoor Atmospheres: An Integrated Mechanism, Phys. Chem. Chem. Phys., 5 (2003), 223-242. Google Scholar [10] T. Gomer, T. Brauers, T. Heintz, J. Stutz and U. Platt, MFC Version 1.99,, 1995., ().   Google Scholar [11] N. Huang, Z. Shen, S. Long, M. Wu, H. Shih, Q. Zheng, NC. Yen, C. Tung and H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Lond. A., 454 (1998), 903-995. doi: 10.1098/rspa.1998.0193.  Google Scholar [12] J. Huber and E. Ronchetti, Robust Statistics, Wiley, 2009. Google Scholar [13] A. Hyvärinen, Fast and robust fixed-point algorithms for independent component analysis, IEEE Trans. on Neural Networks, 10 (1999), 626-634. Google Scholar [14] A. Juan and R. Tauler, Multivariate Curve Resolution (MCR) from 2000: Progress in Concepts and Applications, Critical Reviews in Analytical Chemistry, 36 (2006), 163-176. Google Scholar [15] E. J. Karjalainen, The Spectrum Reconstruction Problem: Use of Alternating Regression for Unexpected Spectral Components in two-dimensional Spectroscopies, Chemometrics and Intelligent Laboratory Systems, 7 (1989), 31-38. Google Scholar [16] L. Lin, Y. Wang and H. Zhou, Iterative filtering as an alternative algorithm for empirical mode decomposition, Adv. Adapt. Data Anal., 1 (2009), 543-560. doi: 10.1142/S179353690900028X.  Google Scholar [17] J. Liu, J. Xin and Y. Y. Qi, A Soft-Constrained Dynamic Iterative Method of Blind Source Separation, SIAM J. Multiscale Modeling Simulation, 7 (2009), 1795-1810. doi: 10.1137/080736168.  Google Scholar [18] J. Noxon, Nitrogen dioxide in the stratophere and troposphere measured by ground-based absorption spectroscopy, Science, 189 (1975), 547-549. doi: 10.1126/science.189.4202.547.  Google Scholar [19] J. Noxon, E. Whipple and R. Hyde, Stratospheric $NO_2$. 1. Observational method and behavior at midlatitudes, J. Geophys. Res., 84 (1979), 5047-5076. Google Scholar [20] D. Perner, D. Ehhalt, H. Patz, U. Platt, E. Roth and A. Volz, OH-radicals in the lower troposphere, Geophys. Res. Lett., 3 (1976), 466-468. doi: 10.1029/GL003i008p00466.  Google Scholar [21] U. Platt, D. Perner and H. Pätz, Simultaneous measurements of atmospheric $CH_2O$, $O_3$ and $NO_2$ by differential optical absorption, J. Geophys. Res., 84 (1979), 6329-6335. Google Scholar [22] U. Platt and J. Stutz, Differential Optical Absorption Spectroscopy: Principles and Applications, Springer, 2008. Google Scholar [23] J. Stutz and U. Platt, Numerical analysis and estimation of the statistical error of differential optical absorption spectroscopy measurements with least-squares methods, Appl. Optics., 35 (1996), 6041-6053. doi: 10.1364/AO.35.006041.  Google Scholar [24] G. Thakur, E. Brevdo, N. S. Fučkar and H.-T. Wu, The Synchrosqueezing algorithm for time-varying spectral analysis: Robustness properties and new paleoclimate applications, Signal Processing, 93 (2013), 1079-1094. doi: 10.1016/j.sigpro.2012.11.029.  Google Scholar [25] S. Voigt, J. Orphal, K. Bogumil and J. P. Burrows, The temperature dependence (203-293 K) of the absorption cross sections of $O_3$ in the 230 - 850 nm region measured by Fourier-transform spectroscopy, J. Photochem. Photobiol. A: Chemistry, 143 (2001), 1-9. Google Scholar [26] Y. Wang, G.-W. Wei and S. Yang, Iterative filtering decomposition based on local spectral evolution kernel, J. of Sci. Comp., 50 (2012), 629-664. doi: 10.1007/s10915-011-9496-0.  Google Scholar [27] J. White, Long Optical Paths of Large Aperture, J. Opt. Soc. Amer., 32 (1942), 285-288. doi: 10.1364/JOSA.32.000285.  Google Scholar [28] ZH. Wu and N. Huang, Ensemble empirical mode decomposition: A noise-assisted data analysis method, Advances in Adaptive Data Analysis, 1 (2009), 1-41. doi: 10.1142/S1793536909000047.  Google Scholar

show all references

##### References:
 [1] A. Bell and T. Sejnowski, An information-maximization approach to blind separation and blind deconvolution, Neural Computation, 7 (1995), 1129-1159. doi: 10.1162/neco.1995.7.6.1129.  Google Scholar [2] A. Bongartz, J. Kames, U. Schurath, CH. George, PH. Mirabel and J. L. Ponche, Experimental determination of hono mass accommodation coefficients using two different techniques, J. Atmos. Chem., 18 (1994), 149-169. doi: 10.1007/BF00696812.  Google Scholar [3] A. Bongartz, J. Kames, F. Welter and U. Schurath, Near-UV absorption cross sections and trans/cis equilibrium of nitrous acid, J. Phys. Chem., 95 (1991), 1076-1082. doi: 10.1021/j100156a012.  Google Scholar [4] J.-F. Cardoso, High-order contrasts for independent component analysis, Neural Computation, 11 (1999), 157-192. doi: 10.1162/089976699300016863.  Google Scholar [5] A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications, John Wiley and Sons, New York, 2005. Google Scholar [6] P. Comon and C. Jutten, Handbook of Blind Source Separation: Independent Component Analysis and Applications, Academic Press, 2010. Google Scholar [7] I. Daubechies, J. Lu and H.-T. Wu, Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool, Applied and Computational Harmonic Analysis, 30 (2011), 243-261. doi: 10.1016/j.acha.2010.08.002.  Google Scholar [8] D. DeHaan, T. Brauers, K. Oum, J. Stutz, T. Nordmeyer and BJ. Finlayson-Pitts, Heterogeneous chemistry in the troposphere: Experimental approaches and applications to the chemistry of sea salt particles, Intern. Rev. Phys. Chem., 18 (1999), 343-385. Google Scholar [9] BJ. Finlayson-Pitts, LM. Wingen, AL. Sumner, D. Syomin and KA. Ramazan, The Heterogeneous Hydrolysis of $NO_2$ in Laboratory Systems and in Outdoor and Indoor Atmospheres: An Integrated Mechanism, Phys. Chem. Chem. Phys., 5 (2003), 223-242. Google Scholar [10] T. Gomer, T. Brauers, T. Heintz, J. Stutz and U. Platt, MFC Version 1.99,, 1995., ().   Google Scholar [11] N. Huang, Z. Shen, S. Long, M. Wu, H. Shih, Q. Zheng, NC. Yen, C. Tung and H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Lond. A., 454 (1998), 903-995. doi: 10.1098/rspa.1998.0193.  Google Scholar [12] J. Huber and E. Ronchetti, Robust Statistics, Wiley, 2009. Google Scholar [13] A. Hyvärinen, Fast and robust fixed-point algorithms for independent component analysis, IEEE Trans. on Neural Networks, 10 (1999), 626-634. Google Scholar [14] A. Juan and R. Tauler, Multivariate Curve Resolution (MCR) from 2000: Progress in Concepts and Applications, Critical Reviews in Analytical Chemistry, 36 (2006), 163-176. Google Scholar [15] E. J. Karjalainen, The Spectrum Reconstruction Problem: Use of Alternating Regression for Unexpected Spectral Components in two-dimensional Spectroscopies, Chemometrics and Intelligent Laboratory Systems, 7 (1989), 31-38. Google Scholar [16] L. Lin, Y. Wang and H. Zhou, Iterative filtering as an alternative algorithm for empirical mode decomposition, Adv. Adapt. Data Anal., 1 (2009), 543-560. doi: 10.1142/S179353690900028X.  Google Scholar [17] J. Liu, J. Xin and Y. Y. Qi, A Soft-Constrained Dynamic Iterative Method of Blind Source Separation, SIAM J. Multiscale Modeling Simulation, 7 (2009), 1795-1810. doi: 10.1137/080736168.  Google Scholar [18] J. Noxon, Nitrogen dioxide in the stratophere and troposphere measured by ground-based absorption spectroscopy, Science, 189 (1975), 547-549. doi: 10.1126/science.189.4202.547.  Google Scholar [19] J. Noxon, E. Whipple and R. Hyde, Stratospheric $NO_2$. 1. Observational method and behavior at midlatitudes, J. Geophys. Res., 84 (1979), 5047-5076. Google Scholar [20] D. Perner, D. Ehhalt, H. Patz, U. Platt, E. Roth and A. Volz, OH-radicals in the lower troposphere, Geophys. Res. Lett., 3 (1976), 466-468. doi: 10.1029/GL003i008p00466.  Google Scholar [21] U. Platt, D. Perner and H. Pätz, Simultaneous measurements of atmospheric $CH_2O$, $O_3$ and $NO_2$ by differential optical absorption, J. Geophys. Res., 84 (1979), 6329-6335. Google Scholar [22] U. Platt and J. Stutz, Differential Optical Absorption Spectroscopy: Principles and Applications, Springer, 2008. Google Scholar [23] J. Stutz and U. Platt, Numerical analysis and estimation of the statistical error of differential optical absorption spectroscopy measurements with least-squares methods, Appl. Optics., 35 (1996), 6041-6053. doi: 10.1364/AO.35.006041.  Google Scholar [24] G. Thakur, E. Brevdo, N. S. Fučkar and H.-T. Wu, The Synchrosqueezing algorithm for time-varying spectral analysis: Robustness properties and new paleoclimate applications, Signal Processing, 93 (2013), 1079-1094. doi: 10.1016/j.sigpro.2012.11.029.  Google Scholar [25] S. Voigt, J. Orphal, K. Bogumil and J. P. Burrows, The temperature dependence (203-293 K) of the absorption cross sections of $O_3$ in the 230 - 850 nm region measured by Fourier-transform spectroscopy, J. Photochem. Photobiol. A: Chemistry, 143 (2001), 1-9. Google Scholar [26] Y. Wang, G.-W. Wei and S. Yang, Iterative filtering decomposition based on local spectral evolution kernel, J. of Sci. Comp., 50 (2012), 629-664. doi: 10.1007/s10915-011-9496-0.  Google Scholar [27] J. White, Long Optical Paths of Large Aperture, J. Opt. Soc. Amer., 32 (1942), 285-288. doi: 10.1364/JOSA.32.000285.  Google Scholar [28] ZH. Wu and N. Huang, Ensemble empirical mode decomposition: A noise-assisted data analysis method, Advances in Adaptive Data Analysis, 1 (2009), 1-41. doi: 10.1142/S1793536909000047.  Google Scholar
 [1] Marko Filipović, Ivica Kopriva. A comparison of dictionary based approaches to inpainting and denoising with an emphasis to independent component analysis learned dictionaries. Inverse Problems & Imaging, 2011, 5 (4) : 815-841. doi: 10.3934/ipi.2011.5.815 [2] Meng Yu, Jack Xin. Stochastic approximation and a nonlocally weighted soft-constrained recursive algorithm for blind separation of reverberant speech mixtures. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1753-1767. doi: 10.3934/dcds.2010.28.1753 [3] Yitong Guo, Bingo Wing-Kuen Ling. Principal component analysis with drop rank covariance matrix. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2345-2366. doi: 10.3934/jimo.2020072 [4] Zheng Dai, I.G. Rosen, Chuming Wang, Nancy Barnett, Susan E. Luczak. Using drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution based data analysis software for transdermal alcohol biosensors. Mathematical Biosciences & Engineering, 2016, 13 (5) : 911-934. doi: 10.3934/mbe.2016023 [5] Imre Csiszar and Paul C. Shields. Consistency of the BIC order estimator. Electronic Research Announcements, 1999, 5: 123-127. [6] Samitha Samaranayake, Axel Parmentier, Ethan Xuan, Alexandre Bayen. A mathematical framework for delay analysis in single source networks. Networks & Heterogeneous Media, 2017, 12 (1) : 113-145. doi: 10.3934/nhm.2017005 [7] Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems & Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067 [8] Dominique Duncan, Thomas Strohmer. Classification of Alzheimer's disease using unsupervised diffusion component analysis. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1119-1130. doi: 10.3934/mbe.2016033 [9] Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357 [10] Jinting Wang, Linfei Zhao, Feng Zhang. Analysis of the finite source retrial queues with server breakdowns and repairs. Journal of Industrial & Management Optimization, 2011, 7 (3) : 655-676. doi: 10.3934/jimo.2011.7.655 [11] Bilal Saad, Mazen Saad. Numerical analysis of a non equilibrium two-component two-compressible flow in porous media. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 317-346. doi: 10.3934/dcdss.2014.7.317 [12] Dominique Duncan, Paul Vespa, Arthur W. Toga. Detecting features of epileptogenesis in EEG after TBI using unsupervised diffusion component analysis. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 161-172. doi: 10.3934/dcdsb.2018010 [13] Qingshan You, Qun Wan, Yipeng Liu. A short note on strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2013, 7 (1) : 305-306. doi: 10.3934/ipi.2013.7.305 [14] Shouming Zhou, Shanshan Zheng. Qualitative analysis for a new generalized 2-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4659-4675. doi: 10.3934/dcdss.2021132 [15] Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic & Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317 [16] Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Solidification and separation in saline water. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 139-155. doi: 10.3934/dcdss.2016.9.139 [17] Pavel Krejčí, Elisabetta Rocca, Jürgen Sprekels. Phase separation in a gravity field. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 391-407. doi: 10.3934/dcdss.2011.4.391 [18] Boris Andreianov, Nicolas Seguin. Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 1939-1964. doi: 10.3934/dcds.2012.32.1939 [19] Siyao Zhu, Jinliang Wang. Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1999-2019. doi: 10.3934/dcdsb.2020013 [20] Xuan Tian, Shangjiang Guo, Zhisu Liu. Qualitative analysis of a diffusive SEIR epidemic model with linear external source and asymptomatic infection in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021173

2020 Impact Factor: 1.639