in each layer come from the 1976 US standard atmosphere), and consider the absorption opacity corresponding to water vapor. For all three examples, we demonstrate a small number of energy discretization parameters can capture the solution with between $0.1$ and $1$ percent accuracy, and using orders of magnitude fewer parameters than the standard multigroup formulation with Planck-weighted opacities for comparable accuracies. We note that, for these examples, we expect that the correlated k-Distribution method can yield the same accuracy with a comparable number of parameters.

The homogenized transport equation and its discrete approximation\[sec:The-homogenized-transport\]
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In this Section \[sub:Derivation-of-homogenized\], we first derive the homogenized equations (\[eq:transport equation on extended phase space, intro\]) and (\[eq:psi\_0 as weighting against Young measure, intro\]). We then discuss the discrete approximation of the Young measure in Section \[sub:Discrete-approximation-of\], and derive the discrete approximation (\[eq:transport multiband-multigroup, intro\]) and (\[eq:formula for psi\_o, discretized Young measures\]) in Section \[sub:Derivation-of-discrete\]. Finally, we conclude this section with a heuristic derivation of the homogenized system for the time-dependent TRT equations.

Derivation of the homogenized system \[sub:Derivation-of-homogenized\]
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To derive (\[eq:transport equation on extended phase space, intro\]) and (\[eq:psi\_0 as weighting against Young measure, intro\]), first assume that the scattering kernel only depends 