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

Yuanchang Sun; Lisa M. Wingen; Barbara J. Finlayson-Pitts; Jack Xin

doi:10.3934/ipi.2014.8.587

Article Contents

2014, Volume 8, Issue 2: 587-610. Doi: 10.3934/ipi.2014.8.587

This issue Previous Article Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography Next Article

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
2.
Department of Chemistry/Mathematics, University of California at Irvine, Irvine, CA 92697
3.
Department of Mathematics/Mathematics, University of California at Irvine, Irvine, CA 92697

Received: September 2012

Revised: December 2013

Published: May 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 65F99, 62J05; Secondary: 92J99.

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