How do we allocate partial CO2 doublings for Empirical Climate Sensitivity?

Question

How do we allocate partial CO2 doublings for Empirical Climate Sensitivity? I applied partial doublings on a linear basis, is this correct?

Climate Threat to the Planet:Implications for Energy Policy and Intergenerational Justice Jim Hansen December 17, 2008 http://www.columbia.edu/~jeh1/2008/AGUBjerknes20081217.pdf

Empirical Climate Sensitivity 3 ± 0.5C for 2 x CO2
(Hanson said it could be 6C in the longer term)
http://www.columbia.edu/~jeh1/2008/TargetCO2_20080407.pdf

If we assume that the formula above still holds and
assume the current rate of CO2 increasing at 1.9% per year
and assume 280 PPM as the CO2 base.
and assume that partial doublings can be allocated linearly.

Years.......CO2
From.......Per.......CO2
Now........Year......Total
....0..........3.00......410
....4..........3.17......420.....1.5C
...17.........4.05......467.....2.0C
...27.........4.89......513.....2.5C
...36.........5.80......560.....3.0C

Keith McClary provided this link, which provided the data below it:
https://en.wikipedia.org/wiki/Climate_sensitivity#Estimating_climate_sensitivity

IPCC Fifth Assessment Report 2014
66% likely: Empirical Climate Sensitivity 3 ± 1.5C for 2 x CO2
Possible: Empirical Climate Sensitivity 1C to 6C for 2 x CO2

The best information available provided above shows the climate sensitivity. The next thing that we need is the procedure for allocating fractional doubling.

https://www.ipcc.ch/site/assets/uploads/sites/2/2019/05/SR15_Chapter1_Low_Res.pdf SPECIAL REPORT: GLOBAL WARMING OF 1.5 ºC
CH 01
Framing and Context The following is on page 66
Expert judgement based on the available evidence (including model simulations, radiative forcing and climate sensitivity) suggests that if all anthropogenic emissions were reduced to zero immediately, any further warming beyond the 1°C already experienced would likely be less than 0.5°C over the next two to three decades, and also likely less than 0.5°C on a century time scale.

It seems we already missed the 1.5C target:
3 * log2(410/280) degree increase, or about 1.65°C

Answer 1

To get a rough estimate you should be using a logarithmic interpolation rather than a linear interpolation. The response to added $\text{CO}_2$ would be linear if the atmosphere had barely any $\text{CO}_2$ in it. Even 280 ppm does not qualify as "very little". "Very little" would mean a handful of parts per million, at most.

The issue is that the atmosphere is rather opaque at those frequencies where $\text{CO}_2$ absorbs and emits thermal infrared radiation. Instead of a linear response, the response in terms of radiative forcing is roughly logarithmic. This is why climatologists talk about the effect of a doubling the $\text{CO}_2$ levels. The radiative forcing of an increase from 140 ppm $\text{CO}_2$ to 280 ppm is more or less the same as the response to an increase from 280 ppm to 560 ppm, or to an increase from 560 ppm to 1120 ppm.

Answer 2

You can use the formula ΔT=λ·α·㏑(C/C₀), (natural logarithm) where C is the later concentration of CO₂ and C₀ the earlier concentration. λ and α is constants. This formula is valid for a solid sphere but since 70 % of the surface of Earth is water, which warms up much deeper and accumulate about 90 % of the accumulated energy, it isn't totally clear how to use the formula correct.

One way could be to use the formula on global land temperatures and transform to global temperature for sea and land afterwards, somehow.

Otherwise, the calculations won't be exact.