The issue of what is the more appropriate initial value should be used for assumed linear extrapolation of surface temperature needs to be clarified.
From my perspective the greenhouse ratio would be 33K/(288-255)K. This produces a value of one equivalent with the initial value of the greenhouse or atmospheric effect. Optionally, 33K/(288-33) or 33/255 is proposed. This implies and initial ratio of 0.129 and changes to that ratio may provide a better approximation.
Comparing the two, a ten percent change in temperature would be 36.3/(288-255) or 10% increase. In the other case,36.3/255=0.142, then 0.142/0.129=10%, my method just reduces a step.
The question can this be used for determining a climate sensitivity to the change in CO2, it could not be. It would be an estimate for the change in climate due to a sensitivity of forcing at the surface only. In the atmosphere, the procedure would be used as a change in sensitivity for the atmosphere only.
A second question would be when using the ratio for combined fluxes if the ratio should include adjustments for the nonlinear relationships. For a small change the linear assumption should be fine, but at what point would the approximation produce significant error?
By assigning temperatures to the flux change in Era based on S-B, a 10% change in Era would result in 10.6% change in temperature. So there is error due to non-linearity, whether that error is acceptable for the situation would depend on the purpose of the estimation. Huh?
Update: Possibly, that error may be due to issues on the K&T cartoon other than the 235 at the TOA. 24+78+40+26=168 Where on average that should equal 390, the flux of a black body at 288K, at the surface.
I will get back to this, but it looks like K&T double dipped. The way I have drawn in the NASA budget should be closer to reality. The proportional method I was using may allow me to tweak it, though I still have some issues with dF/dT=4F/T at the surface. We will see.
Just a Note: The NASA cartoon is a little better because it shows radiant heat rising into the atmosphere, then splitting into absorbed and directly radiated to space. Water in the atmosphere is responsible for some percentage of the radiation in the direct window. So we have three basic heat fluxes with different rates of flow varying with conditions. The total flux from the surface will be approximately 390Wm-2, the minimum average downwelling will be 155Wm-2 and the maximum should be a percentage of the flux interacting with the atmosphere decreasing toward 155Wm-2.
The wind is blowing, so while I wait before going fishing, the little greenhouse effect thing on the left of the drawing is and initial estimate. The 216 and 72 radiative fluxes just happen to equal 288, not a mistake, just chance. That does change our initial heat fluxes to Er=288, El=79 and Et=24. Now a 3.7W/m^2 increase in Er gives us (288+3.7)/(390+3.7)=0.7409 from an initial 288/390 = 0.7385. A decrease equals (288-3.7)/(390-3.7)=0.736. The difference of the two from initial is the approximate ratio of the change in forcing which is both for flux and temperature as they happen to be equal. For radiative change, dF/dT=4F/T, is a good approximation. That does not necessarily hold for the latent, sensible or non-interactive radiative fluxes at the surface, but should be in the ballpark for such a small change. For an increase, 0.7409-0.7385=0.0024..., time 288 yields 0.707 For a decrease, 0.736-0.7385=-0.0025 times 288 yields -0.7214. Using dT(T)=dF/4F, I believe that results in a Planck parameter of approximately 0.18 :)
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