Eli writes a letter (cont.)

The following argument in Essex, McKitrick and Andresen proceeds from their flawed (FUBAR occurs as a kind description, FUMTU could suffice, although MOAFU might be better were it not needed for other parts of this paper ) discussion of the cup of coffee and the cold water. Clearly the linear average is preferred on physical grounds. A mass weighted linear average :

• reproduces the cooling curve one would obtain if the two liquids were first mixed.
• is independent of the zero of the temperature scale, and depends only on the magnitude of the degree as a multiplicative constant.
• is proportional to total energy for homogeneous systems using the Kelvin scale.

None of the other proposed averaging methods have these qualities. The only other proposed averaging method which had any physical justification was T^4, but in that case, instead of using the mandatory Kelvin scale, Celsius was used. Having misinterpreted the implications of their own example, the authors use their error to motivate (either that or they really want an IgNoble and the cash prize that goes with it) a detailed discussion of averaging, defining two types of averages:

r-mean Average = [1/N (x1^r + x2^r+ ....... +xN^r)]^(1/r)

s-mean Average =(1/s) [ln{ 1/N (exp[sx1] + exp[sx2]+ ....... +exp[sxN])}]

The r-mean for r=0 is unity. The s-mean is undefined. (Wait young bunnies.) While for other r and s these are mathematically well defined their utility varies. For example, for even powers of r, negative and positive temperatures will contribute equally and the average will always be positive. If r=2 and T=-2, -2, -2, the average would be +2. Applying the same principal to a balance sheet would make three years of losses an average gain (reality is a harsh Ms. Rabett). For negative values of r, if there were a zero in the data, the average would blow up. In those cases one might try a temperature scale where the zero was always lower than the lowest data value. (Kelvin anyone?)

For r>1 the averaging overweights large values, the same is true for the s averages which are used in economics. For the series 3, 4, 2, 5, 3, 5, 345 the average for r=1 is 52, r=3 is 180, r=5 is 234 and for s=1 the average is 343. While there may be reasons why one wants to do this, measuring the average temperature of a system is not as was shown in the first part of this letter.

In Section 4 Essex, et al. describe methods used to calculate global temperature anomalies. While the description is correct, Essex, et al., make a series of ex cathedra claims about these methods which are not justified in the text other than as tenets of belief, and for which there is a considerable refereed literature justifying the methods and discussing the magnitude of uncertainty or error that would be introduced. Essex, et al. draw broad and confident conclusions without justification or reference to other studies, refereed or not, and base further arguments on their assertions. One wonders why the referees did not object to this bootstrapping. Certainly a more sophisticated (adult, grown up.....) discussion of these issues is needed in this article. Essex, McKitrick and Anresen use this to claim that

So far we have shown that different averages exist (true), they are used (sometimes appropriately and sometimes not) and that contradictory trends can emerge between them (if you use them inappropriately). We have shown that the conditions exist in the atmosphere where such paradoxical behavior can be expected to be found. (not done, asserted, but probably true if you use an inappropriate average)

More tomorrow......