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 $T$ and spectral emissivity $\epsilon (k)$. A standard argument^{3}^{,}^{5} gives the posterior probability in terms of a MAXENT estimator Display Formula
$P(I|T,\epsilon ,\sigma )=exp[\u2212(I\u2212IFM)22\sigma 2(Ta)]dI\sigma (Ta),$(12)
in terms of a forward model Display Formula$IFM=f[\u2211i=1m\lambda i\epsilon i(k)Bk(Tn)]\u21d4f(x),$(13)
that is some function of the $n$’th trial, in each spectral bin $k$. We note that while the equation of transfer is linear, the dependence of its solution $IFM$ upon $\epsilon i(k)Bk(Tn)$ need not be. The assumed noise variance $\sigma 2$ is shown as having a formal dependence upon a parameter, the “annealing temperature” $Ta$, which governs the annealing schedule for the search for an MAP solution. The joint posterior probability in $J$ spectral bands is proportional to Display Formula$P({Ik}|T,\epsilon ,\sigma )=\u220fk=1Jexp{\u2212[Ik\u2212IFM(k)]22\sigma 2(Ta)}dI\sigma (Ta).$(14)
If radiance $Ik$ in each of $J$ 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 $T$ given prior knowledge $K$ is given by Bayes’ theorem as Display Formula$P(T,\epsilon i(k)|{Ik},K)=P[T,\epsilon (k)|K]P[{Ik}|T,\epsilon i(k),K]P({Ik}|K).$(15)
The noise variance is assumed to be known and the functional dependence of probabilities upon $\sigma i$ is omitted. The prior probability $P({Ii}|K)$ for the radiances ${Ik}$ has no dependence upon $T$ 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 $T$ and has spectral emissivity $\epsilon (k)$, given available knowledge $K$^{5}Display Formula$P[T,\epsilon (k)|K]dT\u221d\u220fkd\epsilon (k)dTT,$(16)
where $P[T,\epsilon i(k)|{Ii},K]$ is the conditional probability for the hypothesis that the surface temperature is $T$, and the spectral emissivity $\epsilon k$, given observed radiances ${Ii}$ and prior knowledge $K$.