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Temperature emissivity separation: Estimation with a parameter affecting both the mean and variance of the observation

We consider a model for temperature-emissivity separation (TES) in hyperspectral image processing. The emissivity is modulated by both the black body function and the atmospheric downwelling. The interaction has made it difficult to extract both temperature and emissivity, since offsets in one can be compensated by the other. Working with only a single wavelength component, we propose here a model in which the downwelling is considered as a random variable (or vector). The emissivity thus contributes to both the variance and mean of the observations. This leads to a maximum likelihood estimator for the emissivity. We compute an expression for the bias of this estimator, and show how it can be used to produce an unbiased estimator. An estimator for the temperature is also given. These two estimators can be used iteratively, providing separation of the temperature and emissivity components.