We score trial mixtures by that we most wish to maximize; the posterior probability for the observed spectral radiance to originate from a surface patch with temperature and spectral emissivity . A standard argument3,5 gives the posterior probability in terms of a MAXENT estimator Display Formula
(12)in terms of a forward model Display Formula
(13)that is some function of the ’th trial, in each spectral bin . We note that while the equation of transfer is linear, the dependence of its solution upon need not be. The assumed noise variance is shown as having a formal dependence upon a parameter, the “annealing temperature” , which governs the annealing schedule for the search for an MAP solution. The joint posterior probability in spectral bands is proportional to Display Formula
(14)If radiance in each of bands originating from a patch on the Earth’s surface has been detected at the top of the atmosphere, the posterior probability that the surface patch is at a temperature given prior knowledge is given by Bayes’ theorem as Display Formula
(15)The noise variance is assumed to be known and the functional dependence of probabilities upon is omitted. The prior probability for the radiances has no dependence upon and for our purposes may be absorbed into an overall normalization.24 Equation (15) is evaluated with the aid of the prior probability for the surface to be at temperature and has spectral emissivity , given available knowledge 5Display Formula
(16)where is the conditional probability for the hypothesis that the surface temperature is , and the spectral emissivity , given observed radiances and prior knowledge .