The notion of "back radiation" can be traced back to the Schwarzschild equations for radiative transfer from 1906, which formally mathematically decompose net radiative heat energy transfer into the difference of opposite two-way energy transfers, in order to faciliate symbolic solution by analytical mathematics. This method is also used by Chandrasekhar in his book on radiative transfer, and there correctly referred to as "formal solution".
Both Schwarzschild and Chandrasekhar thus use a formal mathematical decomposition similar to formally rewriting a physical Stefan-Boltzmann law
- $Q=\sigma (T_1^4-T_2^4)$ (1)
- $Q =\sigma T_1^4 - \sigma T_2^4$, (2)
Compare with Fourier's law of heat conduction $Q=\sigma (T_1-T_2)$ (or in differential form $Q=\sigma dT/dx$), which nobody would even think of splitting into $Q=\sigma T_1- \sigma T_2$ (or in differential form $Q=\sigma T_1/dx-\sigma T_2/dx$), not even a first year student, since a system acting like that with "back conduction", would be unstable and thus could not exist. "Back radiation" is as un-physical and unstable as "back conduction". Fourier would turn in his grave at the mere thought of such a horrendous concept.
For an illuminating comparison of one-way and two-way equations for radiative heat transfer, see this article by Joseph Reynen.
Notice that another way of formally rewriting (1) is:
- $Q =\sigma T_1^4+GHOST_1 - (\sigma T_2^4 +GHOST_2)$,
This is the way a pyrgeometer measuring DLR as atmospheric "back radiation" functions.
PS1 David Andrews states on p 84 of An Introduction to Atmospheric Physics, about two-way radaitive heat transfer:
- We find F-up and F-down by a sequence of tricks.
PS2 The fundamental error is clearly exposed in the above book:
- If the Earth is assumed to emit as a black body: $Q=\sigma T^4$...(page 5).
- The ground is assumed to emit as a black body (page 6).