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OK, that is a good idea. However, there are some constraints:

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Tamino's temperature data start in 1950, but the early years are noisy and I'm reluctant to go back much more than 50 years (1967-ish)

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So let's split the 1967-2018 period in half, and have two non-overlapping, 25-year periods (1967-1992 and 1993-2018).

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That gives ECS of 2.3 (early) and 2.8 (late). Not too surprising, since the full period was 2.5.

Now for the million-dollar question: is the increase due to just random noise, or is it the non-stationarity that ASLR alluded to?

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This is back-of-the-napkin stuff, here, people. But that's what napkins are for, right?

Right, my napkins are practically unreadable by now.

Thanks for the new calculation and graph.

Here are my reactions:

A) The period from the late 1930's to the late 1950's is a rather unique, interesting, and potentially valuable time for consideration in this debate about ECS. Total anthro forcing paused and did not increase at all for 20ish years (1937 to 1958 for example).

B) The formal definition of TCS used in modeling climate requires a very fast rate of increase in CO2 forcing - 1% per year for 72 years to achieve a doubling.

Synthesizing these points, while the forcings paused for 20 years we were not at all observing anything related to TCS, but were slowly evolving towards equilibrium (ECS) for few decades. When forcings resumed their increase, the rate of change in CO2 forcing was far less than 1% per year at the beginning, and is still barely above 1/2% per year now.

So the temperature increased more than it would in a strict TCS model at the beginning of your time period - it was able to get slightly closer to equilibrium. For the second half, the forcing increased faster, and therefore closer to the TCS condition.

So the scatter plot graph of CO2 forcing vs temp has a bias towards higher temps at the left side than it would if TCS were calculated at the full rate of CO2 increase. This means your slope shows a bias towards flatter on the left, and steeper on the right.

Prediction:

If we are foolish enough to continue accelerating our CO2 emission, or if natural sinks break down, and thereby we see CO2 forcings increase at a rate close to 1% per year ...

... Then the ECS calculated by your method would be even higher on the next 25 year segment, as the true TCS conditions are felt.

Next calculation discussion:

Tamino's method of removing non-anthro effects is useful for shorter time spans, and a valuable refinement for your calculation, but over much longer time spans the natural forcings are mostly constant.

Therefore I think it would be interesting to repeat your method on longer time spans, and include pre-1950 data to get a longer term view.

The limitation I discussed above still apply... but it still may be useful to see how it has evolved over longer time spans.