= Solution
The <Beer-Lambert law> solves the source-free attenuation term $dI/ds=-\beta_{\rm ext}I$ of the <radiative transfer equation>. Extinction includes absorption and scattering \b[out of that ray]. Missing terms are thermal emission and scattering into the ray from other directions, including repeated scattering and surface reflection.
A cloud can remove nearly all direct sunlight while still supplying substantial diffuse downward radiation. Consequently the total surface solar flux can greatly exceed the direct-beam answer, although reflection sends some energy back to space and absorption can heat matter. At $550\,\mathrm{nm}$ ordinary atmospheric thermal emission is negligible. No unique total flux follows from extinction alone; the single-scattering albedo, angular redistribution and boundary conditions are needed.<Radiative transfer in the atmosphere by Met Office>[Absorption, scattering and thermal-emission processes]{ref}
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