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Saturday, August 1, 2009

Denialists and Secondary School Maths

Recently a paper appeared in the Journal of Geophysical Research, written by MClean, de Frietas and Carter.

The concluding sentence of the paper claims:

Finally, this study has shown that natural climate forcing associated with ENSO is a major contributor to variability and perhaps recent trends in global temperature, a relationship that is not included in current global climate models.

Carter made an even stronger point in a press release quoted here:

The close relationship between ENSO and global temperature, as described in the paper, leaves little room for any warming driven by human carbon dioxide emissions. The available data indicate that future global temperatures will continue to change primarily in response to ENSO cycling, volcanic activity and solar changes

The definitive response to this piece is given by Tamino in this post. Other comments can be found at: Real Climate, Michael Tobis, Grumbine, and Deep Climate.

You might expect that the authors attempted to correlate ENSO with global temperature, but this is not what they did (because the correlation is very weak). Instead they developed a new temperature time series by by taking the difference between 12-month running averages, 12 months apart. For ease of expression I will call this the differencing method. Tamino made the comment that this process effectively produces "time derivatives of temperature and SOI".

I can't add much to the comments made in the links above, but what struck me was Tamino's comment that the differencing method constituted differentiation. I decided to investigate this point.

My first approach was to play with the numbers in Excel. I entered a linear time series (y = 2x) and differenced it (=B3-B2). The result was 2, ie the derivative of y = 2x. I did the same for y = x2, and got data representing the derivative, 2x and for x3 getting 3x2. This confirmed that the differencing method produced derivatives, but what was the theoretical explanation.

A mathematical way of writing the differencing method used by the authors would be: f(x + 1) - f(x). The number 1 refers to the one year time difference they use for the calculation. I thought back to introductory calculus and remembered that derivatives can be calculated by evaluating the limit as h -> 0 of ( f(x + h) - f(x) ) / h . If you substitute 1 for h (ie the one year time difference) you get the following: ( f(x + 1) - f(x) ) / 1 , which evaluates to f(x + 1) - f(x), which is the differencing method the authors used.

A fun, and quick, introduction to calculus is given in the following video. The relevant formula can be found at the 3 minute 58 second mark. The only difference is that the presenter uses delta x rather than h.

So, what implication does this have for the McLean et al paper? Remember that the authors claim that ENSO might explain " ... recent trends in global temperature ... ". Now the trend in temperatures over the last 30 years has been linear. If you take the derivative of a linear function you will get a constant - ie a number. For example, dy/dx (2x) = 2 - the trend disappears.

So the analysis used by the authors removes the trend, consequently there is now way the final sentence is supported by the analysis of the paper.

What stuns me is the maths used in this post to invalidate the main conclusion of the paper could be understood by a year 11 Maths Methods student within a week of so of starting the study of calculus. This is not high powered maths! One wonders how the authors could have made such a mess.

Horatio Algernon, who writes poetry about climate issues as well as discussing the details in prose wrote the following poetic commentary on McLean et al :

The denialosphere is all abuzz,
Noisier than it ever was.
A paper by Carter and McClean,
(Careening down the passing lane)
Claims that temperature's rise and fall,
Hugs el Nino, as flies hug the wall,
With little room for greenhouse gases,
To explain the melting of glacier masses?

But alas, the overconfident tone,
Is wishful thinking, so hold the phone.
The authors have greatly oversold,
The "correlation", truth be told,
Between ENSO and the global temp,
As pot proponents do with hemp,
And most of the recent temperature rise,
Is not explained as they surmise.

In fact, the results are nothing new
The climate scientists knew this too:
ENSO causes "downs and ups",
In global temp like brief "hiccups",
While over decades, the temperature rises,
From greenhouse gases, with few surprises.

Tamino has done some more damning analysis here .

UPDATE 2: Deep Climate discussed the paper here .

The comment submitted to the JOURNAL OF GEOPHYSICAL RESEARCH, is found here .

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