Bounded Variable Least Squares -- Application of a Constrained Optimization Algorithm to the Analysis of TES Emissivity Spectra

Introduction: The objective of any linear spectral unmixing procedure is to determine the abundance at which the components represented in a predetermined end-member library are present in the observed target. This is done by modeling an observed spectrum as a linear combination of end-member spectra. Following the work of Ramsey and Christensen [1] and Feely and Christensen [2] linear unmixing has become a fundamental tool for analysis and interpretation of thermal infrared emissivity spectra. This technique was expanded upon by Smith et. al [3] to include inferred Martian atmospheric end-member spectra for the purpose of analyzing Mars Global Surveyor Thermal Emission Spectrometer (TES) data. The simultaneous modeling of atmospheric and surface contributions to the observed TES spectrum in a single linear system has become the most accessible means by which the surface emissivity spectrum and inferred surface mineralogy can be isolated from a given TES spectral observation [4]. In this work we examine the application of an advanced constrained optimization algorithm to the problem of linear spectral unmixing and evaluate its utility in the analysis of TES emissivity spectra. Constrained Linear Optimization: The fundamental problem to be solved in linear spectral unmixing analysis can be expressed as a matrix equation