unknown solution $\psi\left(\mathbf{x},\boldsymbol{\Omega},E\right)$. Like the multigroup method, this requires one to make some approximation in order to close the system. We note that the group averaged solution is recovered from (\[eq:multiband average\]) via summing over bands $b$ within each group $g$.

In the context of atmospheric TRT calculations, the correlated k-Distribution method [@ARK-GRO:1972] and the multigroup k-Distribution method [@MOD-ZHA:2001] have been shown to drastically reduce the computational cost of direct line-by-line calculations, and have similarities to the current homogenization approach. Another method that is related to the current homogenization approach is the so-called Opacity Distribution Function (ODF) method [@STR-KUR:1966], which also takes a statistical approach toward coarse-graining the TRT equation in energy. See [@MODEST:2013] for a clear overview of these, and related, methods.

Finally, let us mention that the homogenization in energy of transport equations in the absence of scattering can be analyzed using techniques developed by Tartar [@TARTAR:1990]. Unlike when the opacity has rapid spatial variation (see [@DUM-GOL:2000]), the homogenized version of the transport equation *is not* simply another transport equation with a homogenized opacity. In fact, the homogenized equation is an integro-differential equation, where the integral equation is nonlocal in the spatial variable. An alternative homogenization approach for 