 Solution

Use the optical-depth weighting stipulated in the paper. A jump in the Planck function at ${\tau }_1$ gives $dB/d\tau =(B_1-B_0)\delta (\tau -{\tau }_1)$, soThe upper isothermal layer emits $\pi B_0(1-e^{-{\tau }_1})$ and transmits the lower region's blackbody spectral irradiance with factor $e^{-{\tau }_1}$. The lower layer and black surface have the same $B_1$, so their combined radiation entering the upper layer is $\pi B_1$, independent of how its emission is divided between the lower gas and surface.

More generally, integration by parts rewrites the supplied expression asThe terms are attenuated surface emission and the atmosphere's own emission. In the original boundary-plus-gradient form, $\pi B_0$ is the isothermal reference flux and the integral corrects for temperature variation below the top. Optical depths of order unity contribute most strongly. These formulas follow the paper's specified exponential weighting; a full angular radiative transfer calculation uses ray-angle-dependent optical paths rather than identifying every ray with a vertical path.