derive so-called “grey” or frequency independent approximations, thereby reducing the complexity of solving the full TRT problem. Many opacity homogenization techniques have been developed over the years. Here we mention only those most closely related with the method developed in this paper.

A commonly used averaging technique (the multigroup or picket fence) starts by integrating the transport equation over the energy interval $\left[E_{g},E_{g+1}\right]$. This formerly results in a transport equation for the group averaged intensity, $$\psi_{g}\left(\mathbf{x},\boldsymbol{\Omega}\right)=\int_{E_{g}}^{E_{g+1}}\psi\left(\mathbf{x},\boldsymbol{\Omega},E\right)dE.$$ However, the group averaged opacities $\sigma_{g}$ in the resulting multigroup equations depend on the unknown solution $\psi\left(\mathbf{x},\boldsymbol{\Omega},E\right)$, and an approximation is therefore needed to close the system. Typically, either a Rosseland or Plank mean opacity is used; generally, these closures are only accurate when $\left[E_{g},E_{g+1}\right]$ is small relative to the variation of $\sigma\left(E\right)$ or in certain limiting physical regimes (e.g. the optically thick limit). See, e.g., [@POMRAN:1971] for details. We remark that the averaging method developed in this paper does not require one to postulate such a closure relationship.

The multiband method [@CUL-POM:1980] is another approach for averaging the transport equation. The formulation results in a multigroup-type system for $$\psi_{g,b}\left(\mathbf{x},\boldsymbol{\Omega}\right)=\int_{\sigma_{g,b}}^{\sigma_{g,b+1}}\int_{E_{g}}^{E_{g+1}}\delta\left(\sigma\left(E\right)-\xi\right)\psi\left(\mathbf{x},\boldsymbol{\Omega},E\right)dEd\xi.\label{eq:multiband average}$$ The multiband equations depend on an averaged opacity $\sigma_{g,b}$ that again involves the 