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

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. &nbsp 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. &nbsp 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.