# 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 and 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. [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. [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. [4] J.-F. Cardoso, High-order contrasts for independent component analysis, Neural Computation, 11 (1999), 157-192. doi: 10.1162/089976699300016863. [5] A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications, John Wiley and Sons, New York, 2005. [6] P. Comon and C. Jutten, Handbook of Blind Source Separation: Independent Component Analysis and Applications, Academic Press, 2010. [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. [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. [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. [10] T. Gomer, T. Brauers, T. Heintz, J. Stutz and U. Platt, MFC Version 1.99, 1995. [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. [12] J. Huber and E. Ronchetti, Robust Statistics, Wiley, 2009. [13] A. Hyvärinen, Fast and robust fixed-point algorithms for independent component analysis, IEEE Trans. on Neural Networks, 10 (1999), 626-634. [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. [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. [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. [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. [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. [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. [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. [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. [22] U. Platt and J. Stutz, Differential Optical Absorption Spectroscopy: Principles and Applications, Springer, 2008. [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. [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. [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. [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. [27] J. White, Long Optical Paths of Large Aperture, J. Opt. Soc. Amer., 32 (1942), 285-288. doi: 10.1364/JOSA.32.000285. [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.

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. [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. [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. [4] J.-F. Cardoso, High-order contrasts for independent component analysis, Neural Computation, 11 (1999), 157-192. doi: 10.1162/089976699300016863. [5] A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications, John Wiley and Sons, New York, 2005. [6] P. Comon and C. Jutten, Handbook of Blind Source Separation: Independent Component Analysis and Applications, Academic Press, 2010. [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. [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. [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. [10] T. Gomer, T. Brauers, T. Heintz, J. Stutz and U. Platt, MFC Version 1.99, 1995. [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. [12] J. Huber and E. Ronchetti, Robust Statistics, Wiley, 2009. [13] A. Hyvärinen, Fast and robust fixed-point algorithms for independent component analysis, IEEE Trans. on Neural Networks, 10 (1999), 626-634. [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. [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. [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. [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. [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. [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. [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. [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. [22] U. Platt and J. Stutz, Differential Optical Absorption Spectroscopy: Principles and Applications, Springer, 2008. [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. [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. [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. [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. [27] J. White, Long Optical Paths of Large Aperture, J. Opt. Soc. Amer., 32 (1942), 285-288. doi: 10.1364/JOSA.32.000285. [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.
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