that the absorption opacity $\sigma_{\varepsilon}^{a}\left(E,T\right)$ is of the form $$\sigma_{\varepsilon}^{a}\left(E,T\right)=\chi\left(\sigma_{\varepsilon}\left(E\right),E,T\right),\label{eq:basic opacity assumption}$$ where $\chi\left(\sigma,E,T\right)$ is continuous in its first argument and $\sigma_{\varepsilon}\left(E\right)$ is an appropriate function (that may or may not be directly related to the original opacity $\sigma_{\varepsilon}^{a}\left(E,T\right)$). This assumption, in particular, is a generalization of the commonly assumed assumption in atmospheric TRT calculations that $\sigma_{\varepsilon}^{a}\left(E,T\right)=\chi\left(\sigma_{\varepsilon}\left(E,T_{0}\right),T\right)$, where $T_{0}$ is a fixed reference temperature (see [@QIA-LIO:1992] for its use in the correlated k-Distribution method). In Section \[sub:Relaxation-of-the\], we relax the assumption in (\[eq:basic opacity assumption\]) of an exact equality, and discuss how to numerically compute an approximation $\sigma_{\varepsilon}^{a}\left(E,T\right)\approx\chi\left(\sigma_{\varepsilon}\left(E,T_{0}\right),E,T\right)$, for a given reference temperature $T_{0}$, that is optimal in a certain sense.

Assume that $\sigma_{\varepsilon}^{a}\left(E,T\right)=\chi\left(\sigma_{\varepsilon}\left(E\right),E,T\right)$. Then from the solution of the transport equation for $\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)$, parameterized by the real number $\xi\geq0$, $$\begin{aligned}
\boldsymbol{\Omega}\cdot\nabla_{\mathbf{x}}\Psi+\left[\chi\left(\xi,E,T\left(\mathbf{x}\right)\right)+\sigma^{s}\left(E,T\left(\mathbf{x}\right)\right)\right]\Psi & = & \chi\left(\xi,E,T\left(\mathbf{x}\right)\right)S\left(T\left(\mathbf{x}\right),E\right)+Q_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right)+\nonumber \\
 &  & \int_{\mathbb{S}^{2}}\int_{0}^{\infty}\int_{0}^{\infty}\Sigma^{s}\left(\mathbf{x},E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)\Psi\,\lambda_{E'}\left(d\xi\right)dE'd\boldsymbol{\Omega}',\label{eq:transport equation on extended phase space, intro}\end{aligned}$$ we can exactly recover the homogenized solution by weighting against the Young measure of $\sigma_{\varepsilon}\left(E\right)$, $$\psi_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right)=\int_{0}^{\infty}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)\lambda_{E}\left(d\xi\right).\label{eq:psi_0 as weighting against Young measure, intro}$$ Heuristically, $\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)d\lambda_{E}\left(\xi\right)$ weights the solution of the transport equation (\[eq:transport equation on extended phase space, intro\]) by the probability that $\sigma_{\varepsilon}\left(E\right)=\xi$ when $\varepsilon\ll1$, and integration over all possible values $\sigma$ yields the 