A Puzzler, where did Loehle go too far?

what's too far said he
where you are said she - ee cummings

The bunnies have been asking what's wrong with the rather strange roughly cylindrical brown object that Craig Loehle left lying on the sidewalk at Atmospheric Environment. The constant nattering has woken Eli from his retirement nap so he might as well have a go.

This is a case, where when you see the trick you want to go punch the referees, decapitate the editor and shoot the author into space with a one way ticket. It starts with a paper by Hofmann, Butler and Tans in the same journal which looked at how fast the CO2 mixing ratio has been growing since ~1800. They find a close correlation between population growth and the mixing ratio, indeed, in 1987 (note added in proof) Newell and Marcus had suggested that monitoring CO2 mixing ratio growth was a reasonable way of monitoring global population.

HBT show that the CO2 mixing ratio has been growing very close to exponentially since ~ 1960 when monitoring at Mauna Loa started, although going back further than that on the basis of ice cores shows slower growth in earlier times. They then try to tie everything together to suggest

Besides showing the insight gained by removing pre-industrial CO2 in explaining the curvature in the Mauna Loa CO2 record, and organizing the confusion on growth rates, does this new analysis of the CO2 record have any further use for the future? One possibility is to use it to watch for the expected and necessary break of the exponential nature of CO2 and its growth rate. Fig. 4 shows an exponential fit to the past 14 years of Mauna Loa anthropogenic CO2 data and the residual between the data and the exponential function. The residual has varied from zero less than 1% over this time period. A sustained downward trend of the residual by more than 1% (in the absence of any known major volcanic event) would be a clear sign of a change in the nature of anthropogenic atmospheric carbon dioxide, and a possible gauge of progress in the inevitable need to limit atmospheric CO2. Similarly, a break in the close relation between anthropogenic CO2 and population would signal that a change in ‘‘business as usual’’ had occurred.

For anyone who needs a shorter HBT, they said watch for deviations from exponential growth on short to medium term intervals to spot changes in emissions. They did not predict future emissions but they did discuss the entire record. Let Eli see what Loehle did according to Loehle

A paper by Hofmann et al. (2009, this journal) is critiqued. It is shown that their exponential model for characterizing CO2 trajectories for historical data is not estimated properly. An exponential model is properly estimated and is shown to fit over the entire 51 year period of available data. Further, the entire problem of estimating models for the CO2 historical data is shown to be ill-posed because alternate model forms fit the data equally well. To illustrate this point the past 51 years of CO2 data were analyzed using three different time-dependent models that capture the historical pattern of CO2 increase. All three fit with R2 > 0.98, are visually indistinguishable when overlaid, and match each other during the calibration period with R2 > 0.999. Projecting the models forward to 2100, the exponential model comes quite close to the Intergovernmental Panel on Climate Change (IPCC) best estimate of 836 ppmv. The other two models project values far below the IPCC low estimates. The problem of characterizing historical CO2 levels is thus indeterminate, because multiple models fit the data equally well but forecast
very different future trajectories.

Watch the pea. The IPCC estimate is for A PARTICULAR EMISSION SECENARIO, it is not a projection of the last fifty years of measuring CO2 mixing ratios, nor is it the only emission scenario. Fast work there folks. Still, Eli is a trusting bunny so let us go on. The three fitting forms and Loehle's values for the parameters are

• the exponential, a + b exp(ct),
  a=259.4, b=2.978 x 10^-13, and c= 0.01677
  
  
• the quadratic, a + bt + ct²
  a= 45 318.5, b= -46.7684, and c = 0.0121
  (Gives a really bad fit. a is adjusted to 45504 for a decent fit)
  
  
• and the saturated, c + a(1-exp[b(t-d)])²
  a= 616.694, b= -0.0067, c= 311, and d= 1945.7
  

and this is what Craig shows,
but Eli made a somewhat fuller picture, using the Loehle's values (click on the graphs for better views)

The exponential model, although not perfect, does match the CO2 mixing ratio much better at earlier times than 1959, so, contrary to Loehle, the three models are not equally plausible. But there's more. Eli also did his own fit.

The saturating curve here is much closer to the data and the exponential curve. Still, young bunnies, there is still something like junior high school wrong with what Craigie poo did. Flip to the continuation for the answer


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Loehle extrapolated all of his fits from the year dot. Little ones can spot this in his value for b in the exponential fit and for a in the quadratic and d in the saturated one. If the time is replaced the three fitting forms by [t-1985] you get much much improved saturated fits. The quadratic fit still sucks at early times (and late ones) but even it comes closer to the other two..

The extrapolations are shown below.

	Model	Year 0 Loehle	Year 0 Eli	Year 1985 Eli
2000	Exp	370	369	369
	Quad	367	367	370
	Saturated	368	370	370
2050	Exp	513	513	510
	Quad	479	479	490
	Saturated	467	500	512
2100	Exp	846	846	819
	Quad	651	651	668
	Saturated	567	729	834

The referees and the editor missed the slight of hand, they also did not read Hoffman, et al. The graphs demonstrate that quadratic fits are not industrial grade for this problem, and the table and graphs show that Loehle's fits are fishy. Doesn't anyone check the programming? Where are the auditors when you need them.

Eli has placed an Excel spreadsheet at the memory hole for those who want to play around with the numbers.