We left off the last post having accommodated MODTRAN radiance to the Planck curves, and having promised to derive the temperatures of the individual lines from an inversion of the Planck radiance formula to give the Planck brightness temperature.
Unfortunately, we were unable to make this work. Several different formulae for brightness can be found that yield somewhat different results, but they all show conformance to the general shape of the CO2 radiance deviation.
We decided to use the following two because despite their different approaches, when scaled they become identical.
Below is what we get using the first equation above not scaled:
We really don’t know what to make of this. The systematic decrease in brightness from lower to higher wave numbers across the CO2 deviation in relation to radiance seemingly gives little hope that useful temperatures can be gained this way.
The good news is that a fresh look at the Stephan-Boltzmann approach to temperature using an emissivity slightly above .2 gives a far more satisfying result.
Staley and Jurica (1970) derived a full column emissivity for CO2 of .2. We are frankly astonished that by tweaking this slightly higher we were able to get such good agreement across the ~250 wave numbers of the CO2 deviation. Perhaps .225 is the correct column emissivity. At any rate, we will use this approach to get our temperatures henceforth.