Four temperature indices, 1960-2014 (WoodForTrees)

Climate sensitivity calculations

Climate Sensitivity” is a measure of the (in)stability of the Earth's temp­er­a­tures, most commonly defined as the globally averaged temperature increase to be expected from a doubling of atmospheric carbon dioxide (e.g., an increase from 300 ppmv to 600 ppmv). (See also TCR and ECS.)

The most straightforward and obvious way of estimating climate sensitivity to a doubling of CO2 is by examining the result of the “experiment” which we've performed on the Earth's climate, by raising the atmospheric CO2 level from about 311 ppmv in 1950 (or 285 ppmv in 1850) to about 408 ppmv in 2018. We simply examine what happened to temperatures when the atmospheric CO2 level was raised by 31% (or 43%), and extrapolate from those observations.

However, there are a few pitfalls with that approach. For one thing, natural global temperatures variations due to ENSO can be larger than the “signal” we're looking for, so it is important that we choose an analysis interval which avoids those distortions. For another, it would be a mistake to assume that all of the warming which the Earth has experienced since pre-industrial conditions was due to anthropogenic CO2, because much of that warming occurred when CO2 levels were still very low, and because we know of other factors which must have contributed to warming, such as rising levels of other GHGs, and probably aerosol/particulate pollution abatement.

So the key question is, how much of the warming can be attributed to rising CO2 level? In the calculations below, the assumed answer to that question is an explicit parameter, “A” (for “Attribution”).

You can calculate an estimate of TCR sensitivity, using the time period and temperature index of your choice, as follows:

A = attribution to anthropogenic CO2, e.g., 0.5 = 50% attribution
T1 = initial global average temperature (or temperature anomaly) for your chosen time period
T2 = final global average temperature (or temperature anomaly)
C1 = initial CO2 (or CO2e) value
C2 = final CO2 (or CO2e) value
S = sensitivity in °C / doubling of CO2

The formula is very simple:

S = A × (T2-T1) / ((log(C2)-log(C1))/log(2))
S = A × (T2-T1) / (log2(C2/C1))

For example, to capture most of the period of rapid CO2 level increases, while avoiding distortions from major ENSO spikes, we could use the period 1960-2014:

CO2, log-scale ENSO index

Over that period, CO2 level rose from about 317 ppmv in 1960 to about 399 in 2015. Depending on which temperature index you trust, temperatures rose by about 0.5 °C (HADCRUT3) or about 0.75 °C (GISS), or somewhere in-between; let's use the midpoint, 0.625 °C:

Four temperature indices, 1960-2014 (WoodForTrees)

If T1 is 0.00, T2 is 0.625, C1 is 317 (in 1960), C2 is 399 (in 2015), and A is 50%, then:

S = 0.5 × (0.625-0) / ((log(399)-log(317))/log(2))
      We can use Google as a calculator to find:
S = 0.94 °C / doubling

Note #1: ECS is usually estimated to be about 1½ × TCR.

Note #2: the above discussion doesn't mention minor GHGs like O3, CH4, N2O & CFCs. To take them into account, there are two simple approaches you can use. One is to substitute estimates of “CO2e” (CO2 equivalent) for C1 and C2. The other is to adjust A to account for the fact that some portion of the warming (perhaps one-fourth) is due to other GHGs.

Other than that, the attribution factor, A, is really just an educated guess, but it is based on expert opinion. The AMS frequently surveys meteorologists and asks them what percentage of the last 50 years' warming they attribute to “human activity” (presumably mostly GHGs). This bar chart is from their 2017 survey report:

2017 AMS Meteorologist Survey Summary

As you can see, the “average” or “midpoint opinion” of American broadcast meteorologists is that a little over half of the warming was caused by human activity (presumably mostly by CO2):
   (.905×15/92)+(.7×34/92)+(.5×21)+(.3×13/92)+(.085×08/92) = 57%

So if we attribute 57% of the warming to anthropogenic causes, and 75% of that to CO2, the attribution factor, A, should be 0.75 × 0.57 = 0.43 (43%), resulting in a calculated TCR sensitivity estimate of 0.81 °C per doubling of CO2.

Note that our calculation includes the effects of both positive and negative temperature feedbacks.

For ECS, multiply that result by 1.5, yielding 1.21 °C per doubling of CO2.

The ECS/TCR ratio is sometimes estimated as high as 1.65:1. If we use that multiplier we could get the ECS estimate up to 1.34 °C per doubling of CO2, which is still slightly below the IPCC's “low end” estimate of 1.5 °C per doubling.

On the other hand, if the ECS/TCR ratio is only 1.25:1, then ECS = 1.25 × TCR = 1.01 °C per doubling.

Even if 100% (rather than 57%) of the warming since 1960 is attributed to anthropogenic causes (and 75% of that anthropogenic warming is attributed to CO2), TCR still comes out to only 1.41°C per doubling, and ECS = 1.5 × TCR = 2.12°C.

It is very difficult to approach the IPCC's “midrange” estimate of 3°C per doubling, using this sort of analysis.

Another approach

For a different approach to estimating climate sensitivity, which results in an estimate of ECS rather than TCR, I can recommend this blog post by “SteveF” at The Blackboard climate blog: 
…and this follow-up: