Over on WUWT (http://wattsupwiththat.com/2010/07/13/calculating-global-temperature/), Zeke Hausfarther and Steven Mosher have been discussing the calculation of global temperature from station data. They list several methods of combining records, noting that most of the major indices use the Common Anomalies Method (CAM). They mention, but do not discuss, the First Differences Method (FDM).
In fact, FDM is far superior to any method based on averaging anomalies. Simply averaging anomalies relative to each station’s mean (as in the much-discussed Steig et al 2009 study of Antarctic temperatures — see below) greatly understates any trend there may be in the data, while using a common period unnecessarily restricts the available data. At the same time, FDM eliminates the need for opaquely complex adjustments for TOBS or MMTS, and automatically takes care of station moves.
Suppose, to take a purely hypothetical example, that four stations, A, B, C and D, have partial data on years 1-5, as indicated in the following table:
Table 1: Temperatures (°C)
| Station | Yr 1 | Yr 2 | Yr 3 | Yr 4 | Yr 5 |
| A | 13 | 14 | – | – | – |
| B | – | 10 | 11 | – | – |
| C | – | – | 22 | 23 | – |
| D | – | – | – | 15 | 16 |
Clearly all the stations point to a +1°C/yr uptrend in temperature.
If we compute anomalies for each station, as was done by Steig et al (2009) (to the best of my knowledge), and then average over available stations (S09 did something much more elaborate), we obtain the following:
Table 2: Anomalies
| Station | Yr 1 | Yr 2 | Yr 3 | Yr 4 | Yr 5 |
| A | -.5 | +.5 | – | – | – |
| B | – | -.5 | +.5 | – | – |
| C | – | – | -.5 | +.5 | – |
| D | – | – | – | -.5 | +.5 |
| Avg. | -.5 | 0 | 0 | 0 | +.5 |
The Least Squares trend in the average anomalies is only 1/6 1/5 °C/yr. [(-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 10, not 12 as I computed last night!]
The First Difference Method, on the other hand, first computes annual differences as available for each station and then averages the first differences as available across stations. The average first differences are then cumulated from an arbitrary starting point (Year 1 in the following table), and then may be adjusted as anomalies relative to any desired reference period (Years 1-5 in the last row of the table):
. Neither of us had a good answer, but I had to know. 
