Friday, July 18, 2008

How Well Does a Sensitivity of 3.46C Hindcast?

[Note: Revised 08/02/2008]

In the last post we estimated the most likely climate sensitivity to CO2 doubling by means of an analysis of temperature change rates. The result (3.46C) is in the high end of the range of sensitivities considered plausible by the scientific community. A hindcast should not only tell us if the estimate is in fact too high, but it should also test some of the other results from the analysis. And to make it interesting, we will do a hindcast of the last 150 years. Sound crazy? See Figure 1.

global warming hindcast co2

This turned out much better than I expected. In fact, I suspect the chart might beg disbelief among some readers, so I'm making the spreadsheet available here (XLS). Formulas can be verified to match those of the analysis.

The only inputs to the hindcast are (1) CO2 atmospheric concentrations from 1853 to 2004 (estimated in ppmv as described at the end of this post), and (2) observed temperatures from 1853 to 1856. The observed temperatures used (Column D) are actually central moving averages of period 7.

My expectation for the hindcast was that error would accumulate, and in the end we would have a deviation from the observed temperature trend, but hopefully not a big one. That's because the way temperature for year Y is predicted in the hindcast is by adding the temperature in Y-2 plus the predicted temperature change rate in Y-1 times 2. Intuitively, it doesn't seem like this technique would tend to maintain accuracy over a time series this long.

There is a good reason why the model hindcasts this well, nevertheless. First, it helps that formulas were derived in part from the data we're hindcasting. But more importantly, what we're looking at is a self-correcting system. Local variability cannot make the system resolve its imbalance any faster or slower. If temperature becomes higher than it should be, for whatever reason, the temperature change rate will drop. Similarly, temperatures lower than they should be will be corrected by a positive change in the rate. Sooner or later, the observed trend will rejoin the predicted trend.

This speculative observation is testable in the hindcast. We can break the chain of predicted temperatures, insert artificial values, and see if the model resolves. This can be done in the spreadsheet by modifying one of the predicted temperature columns (e.g. column K, any row greater than 9). What I did is introduce an artificial warming between 1910 and 1913 so it ended up at 0.1C. The results can be seen in Figure 2.

global warming hindcast

I think that's interesting, and I'm sure there's some insight about what's been occurring since 1998 somewhere in there.

For those who are interested in the details, the following is a recap of the results from the analysis that are used to produce the hindcast.

  1. T' = 11.494 log C - 28.768

  2. R = (T' - T) * 0.0915

  3. An unexplained lag of 3 years for imbalance to take effect on the rate of temperature change.


  • C = The atmospheric concentration of CO2 given in ppmv.

  • T' = The equilibrium temperature, given in degrees Celsius anomalies as defined in CRUTEM3v data set.

  • T = The observed temperature. In the hindcast, this is actually the predicted temperature, except for 4 years we use as inputs.

  • R = The rate of temperature change, given in degrees Celsius per year.

The high and low hindcast predictions are based on the confidence interval given in the formula for R.

As an example, the following is how the predicted temperature for 1857 is calculated.

T(1857) = T(1855) + 2 * R(1856)

R(1856) = 0.0915 * (T'(1853)-T(1853))

That's all the hindcast is.

Next up: We'll attempt a forecast.


Hank Roberts said...

Why do you say 3.46C is at the high end of the range? I thought it was close to the best estimate albeit out of a fairly skewed curve.

Joseph said...

Because I understand the IPCC sensitivity range is 1.5 to 4.5C. I get the impression it's a bit on the high end.

Also, I found an annual reconstruction of CO2 concentrations from ice cores that goes from 1830 to 1975. If I use this instead of my estimates based on emissions, the result goes down to 3.13C. The range of CO2 concentrations is larger though. The 150-year hindcast looks better too. I might write about that sometime.

Since I'm only considering CO2 in these analyses, there's no doubt some confounding with other anthropogenic forcings. I'd actually expect sensitivity to CO2 doubling to be a bit lower than 3.13C if no other forcings changed when CO2 concentrations change.

In fact, I tried a 1000-year hindcast, and the model right here is undercasting by about 0.3-0.4C.

I can probably figure it out by doing a fairly complicated Monte Carlo-ish multivariate analysis with more forcings.