include the temperature $T$ in the notation $\lambda_{E,T}$ to emphasize this dependence). Heuristically, $\lambda_{E,T}\left(\xi_{1},\xi_{2}\right)$ gives the probability density that $\sigma_{\varepsilon}\left(E',T_{0}\right)=\xi_{1}$ and $\sigma_{\varepsilon}\left(E',T\right)=\xi_{2}$ for $E'$ in a small neighborhood of $E-\delta\leq E'\leq E+\delta$, where $\delta$ is large relative to the characteristic line spacing $\varepsilon$ but small relative to the macroscopic variation of interest (i.e., $\varepsilon\ll\delta\ll1$). For small $\varepsilon$, we then have from (\[eq:conditional expected value\]) that the conditional expected value, $$\chi\left(x,E,T\right)\equiv\int_{0}^{\infty}\xi_{2}\lambda_{E,T}\left(x,d\xi_{2}\right),\label{eq: chi, conditional expected value}$$ approximately minimizes the average error, $$\mathbb{E}\left(\sigma_{\varepsilon}\left(E,T\right)-\chi\left(\sigma_{\varepsilon}\left(E,T_{0}\right),E,T\right)\right)^{2}\equiv\int_{0}^{\infty}\int_{0}^{\infty}\left(\xi_{2}-\chi\left(\xi_{1}\right)\right)^{2}\lambda_{E,T}\left(d\xi_{1},d\xi_{2}\right),$$ in the limit of small $\varepsilon$. Note that, if the assumption (\[eq:correlated assumption, relaxation section\]) exactly holds, then the joint probability measure $\lambda_{E}\left(\xi_{1},\xi_{2}\right)$ is supported on the curve $\xi_{2}=\chi\left(\xi_{1}\right)$.

To numerically approximate (\[eq: chi, conditional expected value\]), suppose that $\Delta E$ is chosen so that $0<\varepsilon\ll\Delta E$. Divide the range of $\sigma_{\varepsilon}\left(E',T_{0}\right)$ and $\sigma_{\varepsilon}\left(E',T\right)$, $E'\in\left[E-\Delta E/2,E+\Delta E/2\right]$, into temperature-dependant and energy-dependant “bands” $\sigma_{j}\left(E,T_{0}\right)$ and $\sigma_{j'}\left(E,T\right)$. For example, in Section \[sub:An-atmospheric-TRT\], we take logarithmically spaced bands between the minimum and maximum opacity values in $\left[E-\Delta E/2,E+\Delta E/2\right]$. Now define the discrete probabilities $$p_{j,k}\left(E,T\right)\approx\frac{\lambda\left(E'\in\left[E-\Delta E/2,E+\delta/2\right]\mid\sigma_{j}\left(E,T\right)\leq\sigma_{\varepsilon}\left(E',T\right)\leq\sigma_{j+1}\left(E,T\right),\sigma_{j'}\left(E,T_{0}\right)\leq\sigma_{\varepsilon}\left(E',T_{0}\right)\leq\sigma_{j'+1}\left(E,T_{0}\right)\right)}{\delta},\label{eq:p_=00007Bi,j=00007D(E,T)}$$ where $\lambda$ again denotes the Lebesgue measure. Then using the same reasoning as in Section \[sub:Discrete-approximation-of\], we approximate $\chi\left(x,E,T\right)$ as $$\chi\left(x,E,T\right)\approx\sum_{j'}p_{j,j'}\left(E,T\right)\sigma_{j'}\left(E,T\right),\,\,\,\,\text{if}\,\,\,\,\sigma_{j}\left(E,T_{0}\right)\leq x<\sigma_{j+1}\left(E,T_{0}\right).$$ In practice, we compute the 