A new improved estimate of climate sensitivity!?

Someone (thanks Sven) recently drew my attention to Forster and Gregory, 2006: The Climate Sensitivity and Its Components Diagnosed from Earth Radiation Budget Data.

Using results from the ERBE project, they have estimated climate sensitivity to lie in the range 1.0-4.1C (95% confidence interval). This looks like a very impressive headline result, but on closer examination, it's not quite what it seems (the authors are very up-front about this, I'm not for a minute accusing them of trying to mislead anyone). In fact their paper is a rather technical one that presents a method to investigate the different contributions to overall sensitivity, and it does not make a big noise about this overall result.

So, why is this 1.0-4.1C range not an exciting new result? A recent paper by Frame et al (including Gregory) gives a nice explanation and illustration of the point, and I'll give a simple reprise of it here. In order to use Bayes' Theorem to generate a probabilistic estimate from observational data, we need to start from some prior distribution - typically, we choose this to be something fairly ignorant (like a uniform distribution) in order to minimise the risk that this will strongly distort our results.

However, climate sensitivity S is related to something called the feedback parameter L via the equation

S=3.7/L

(where 3.7 is the radiative forcing of doubled CO₂, in Wm^-2). S and L are both unknown, of course. This inverse relationship means that a uniform prior in S is a strongly biased prior in L, and vice-versa. Whichever prior we choose has strong implications for our result, one way or the other. There really is no way round this problem, short of a whole new treatment of probabilistic inference such as imprecise probability (which IMO is not really justified in this case, although it's an interesting idea). Frame et al recommend that if you want to estimate S, you should choose the prior to be uniform in S. I disagree with some aspects of their presentation (about which more later) but agree that this is probably a sensible convention. However, F&G chose their prior to be uniform in L, for reasons which they explain and justify - basically, they are primarily interested in L and its components, and do not wish to bias those results.

I've attempted a simple reanalysis of their results to illustrate the importance of their decision. Again, this is rather similar to (but simpler than) the example that Frame et al present. I'm not sure that all details of my analysis are strictly valid because I'm not starting from F&G's original data but merely working backwards from their result, but I think it shouldn't be far off.

Case A, the top row, are the results presented by F&G (click the pic for a larger version). The blue lines give their prior distributions and the red is the posterior. You can see that the top left (feedback) has a nice flat prior, which gives a perfectly Gaussian posterior with 95% CI of 0.9-3.7 (cos that's what I assumed as an input based on F&G). The top right, however, has a strongly skewed prior, which favours low values. The 95% CI of the posterior here is 1.0-4.1C, as F&G state.


The lower row (Case B) are what they would have obtained if they had chosen a uniform prior in S. In this case, the left plot has a strongly skewed prior and the posterior is shifted to lower values compared to Case A - in fact it has a bimodal distribution. The LH peak would have shot off to infinity but I truncated the calculation. The bottom right graph shows the effect on climate sensitivity - the posterior has a long tail that just keeps on going (it's clearer on the larger version below). In this calculation, the 95% confidence interval of the posterior runs from 1-15C, but that's only because I artificially truncated the calculation at 20C. Without that limit, I think it would not have had a finite upper bound at all.

So, that's why F&G didn't trumpet their results as having solved the climate sensitivity estimation problem.

However, their analysis does clearly favour a moderate value for S over an extreme one, and it appears to be just about entirely independent of all the constraints that we used in our recent paper (hmm...I wonder if anyone will argue that the overlap in obs with the recent temperature record makes this not strictly valid?). Therefore, it can (in fact should) be added into the mix in order to generate an improved overall estimate. I've done this below:


For context, I've left on the original 3 constraints (blue) that we used to generate the main result (red). F&G's constraint (technically a likelihood function, the same shape as the posterior in the lower right plot above) is the green dashed line. You can see how it stubbornly refuses to drop to zero as it leaves the right hand edge of the plot, but that doesn't matter as we already know those values are implausible.

Multiplying the three blue curves and the green dashed one, we get the green solid shape. This is now a new improved estimate of climate sensitivity, which takes account of F&G's work. The upper limit has dropped to about 3.9C. The median is lowered to about 2.5C. You can also see how the peak is slightly higher, indicating the lower uncertainty that arises as a natural consequence of adding more information.

I don't intend to write another paper presenting this result :-) I think it is reasonable to err on the side of overestimating uncertainty, to make some allowance for the "unknown unknowns". And we'd already made it as clear as we could in the paper that we didn't really think there was as much as a 5% chance of S exceeding 4.5C. But in any case F&G's work does, I think, add some significant extra support and credibility to what we've already presented. I await the forthcoming IPCC draft with interest...