Simulated Annealing Approach to the Temperature-Emissivity Separation Problem in Thermal Remote Sensing Part One: Mathematical Background

The method of simulated annealing is adapted to the temperature-emissivity separation (TES) problem. A patch of surface at the bottom of the atmosphere is assumed to be a greybody emitter with spectral emissivity $\epsilon(k)$ describable by a mixture of spectral endmembers. We prove that a simulated annealing search conducted according to a suitable schedule converges to a solution maximizing the $\textit{A-Posteriori}$ probability that spectral radiance detected at the top of the atmosphere originates from a patch with stipulated $T$ and $\epsilon(k)$. Any such solution will be nonunique. The average of a large number of simulated annealing solutions, however, converges almost surely to a unique Maximum A-Posteriori solution for $T$ and $\epsilon(k)$. The limitation to a stipulated set of endmember emissivities may be relaxed by allowing the number of endmembers to grow without bound, and to be generic continuous functions of wavenumber with bounded first derivatives with respect to wavenumber.