Stochastic Time-Series Spectroscopy

Spectroscopically measuring low levels of non-equilibrium phenomena (e.g. emission in the presence of a large thermal background) can be problematic due to an unfavorable signal-to-noise ratio. An approach is presented to use time-series spectroscopy to separate non-equilibrium quantities from slowly varying equilibria. A stochastic process associated with the non-equilibrium part of the spectrum is characterized in terms of its central moments or cumulants, which may vary over time. This parameterization encodes information about the non-equilibrium behavior of the system. Stochastic time-series spectroscopy (STSS) can be implemented at very little expense in many settings since a series of scans are typically recorded in order to generate a low-noise averaged spectrum. Higher moments or cumulants may be readily calculated from this series, enabling the observation of quantities that would be difficult or impossible to determine from an average spectrum or from prinicipal components analysis (PCA). This method is more scalable than PCA, having linear time complexity, yet it can produce comparable or superior results, as shown in example applications. One example compares an STSS-derived CO$_2$ bending mode to a standard reference spectrum and the result of PCA. A second example shows that STSS can reveal conditions of stress in rocks, a scenario where traditional methods such as PCA are inadequate. This allows spectral lines and non-equilibrium behavior to be precisely resolved. A relationship between 2nd order STSS and a time-varying form of PCA is considered. Although the possible applications of STSS have not been fully explored, it promises to reveal information that previously could not be easily measured, possibly enabling new domains of spectroscopy and remote sensing.