The bunny still has pen in paw (cont., cont., cont., cont.,.cont....)
Professor Keller, thank you for your patience. (this is becoming a bit of an obsession, but Ms. Rabett is out of town and the coneys want to play) Today, we would like to explore Essex, et al.'s claim of Contradictory Trends in Global Temperature Averages.
Recall that the authors have claimed that there is no physical basis for preferring one type of average above another, and has defined a series of averages, although this is clearly not as discussed earlier. To demonstrate their claim, they selected data from twelve stations from the GISSTEMP archive.
Phoenix, Arizona; Caragena, Columbia; Dublin, Ireland; Chiang Mai, Thialand; Jan Smuts, South Africa; Honolulu, Hawaii; Sioux Falls, South Dakota; Egedesminde, Greenland; Salehard, Russia; Ceduna, Australia; Halley Antarctica; Souda, IndiaWe have written you previously about the absence of a Souda, India climatology station in the GISS archive. For the purposes of this post, we will omit that station. As you will see this has no effect. Let us return to the definition that is given for the r averages. The s averages will be the same, only worse)
r-mean Average = [1/N (x1^r + x2^r+ ....... +xN^r)]^(1/r) for all rAlthough in principle r could be non-integer, we will assume r integer. If someone has a reason why we should consider r non-integer, he or she should be prepared to deal with roots of negative numbers. Essex, et al., computed monthly means across the stations in Celsius, and a linear trend for each of the monthly averages was fit by ordinary least squares. Trends were displayed for r ranging from -125 to +125 (see Figure 2 in their publication)
As academics, we have all had the experience of students blindly copying numbers from some instrument without questioning whether the apparatus was functioning, and handing in a pretty graph whose information content was nil. Such is Figure 2. While the definitions are clear, Essex, et al., have not considered the physical implications of averages for various values of r, especially in Celsius. This is rather curious in a paper which presumes to establish that there is no physical basis for global temperature anomalies (or even local ones). Even cursory examination shows that
- For even values of r, negative and positive values of temperature contribute the same, positive amount to the average. If one used even r to average a set of negative temperatures, the average would be positive. This is unphysical, and not an acceptable way of averaging Celsius temperatures.
- For values of r above 1, r-averaging for both positive and negative values of r overweights the measurement with largest absolute value. This is not a property one wishes in an statistic that represents the entire set of measurements.
- For negative values of r, r-averaging selects the measurement with the smallest absolute value.
One can use Kelvin (add 273.16 to all the measurements) in which case there is no possibility of a negative reading, and convergence to the absolute maximum and minimum values is slower as would be expected if one simply expanded the temperatures in Taylor series as T(K)=273.16 + T(C).
At this point the question occurs, did Essex, et al. use Kelvin or Celsius. While have not completed the calculation (the computer mice went off for a St. Patrick's day boozer), the shape of their trend line is indicative......
Tomorrow we will complete the calculation and explore further physical reasons why Essex, et al.'s treatment of their data set leads to unphysical conclusions.