Mathematics – Numerical Analysis
Scientific paper
2011-09-28
Mathematics
Numerical Analysis
submitted to Journal of Scientific Computing
Scientific paper
Differential optical absorption spectroscopy (DOAS) is a powerful tool for detecting and quantifying trace gases in atmospheric chemistry \cite{Platt_Stutz08}. 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 uses polynomial fits to remove a slowly varying background, and known reference spectra to retrieve the identity and concentrations of reference gases. An open problem is to identify unknown gases in the fitting residuals for complex atmospheric mixtures. In this work, we develop a novel three step semi-blind source separation method. The first step uses a multi-resolution analysis to remove the slow-varying and fast-varying components in the DOAS spectral data matrix $X$. The second step decomposes the preprocessed data $\hat{X}$ in the first step into a linear combination of the reference spectra plus a remainder, or $\hat{X} = A\,S + R$, where columns of matrix $A$ are known reference spectra, and the matrix $S$ contains the unknown non-negative coefficients that are proportional to concentration. The second step is realized by a convex minimization problem $S = \mathrm{arg} \min \mathrm{norm}\,(\hat{X} - A\,S)$, where the norm is a hybrid $\ell_1/\ell_2$ norm (Huber estimator) that helps to maintain the non-negativity of $S$. The third step performs a blind independent component analysis of the remainder matrix $R$ to extract remnant gas components. We first illustrate 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 multiple trace gases from the residuals.
Finlayson-Pitts Barbara. J.
Sun Yadong
Wingen L. M.
Xin Jack
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