27. Scaling of temperature anomalies for Australia

In my analysis of temperature data from both New Zealand (Posts 6-10) and Australia (Posts 18-21) I have speculated about the nature of the temperature fluctuations. These fluctuations have not behaved like normal white noise. For a start, when data from multiple stations are averaged, the standard deviation of the fluctuations barely changes. If the fluctuations behaved like white noise then the standard deviation would be expected to decrease by a factor of √N, where N is the number of stations in the average. This does not happen.

Then there is the scaling behaviour. If the data is smoothed with a moving average of length N points, again you would normally expect to see the standard deviation of the smoothed data decrease by a factor of √N compared to the unsmoothed data. Again this does not happen. The standard deviation decreases, but generally only by about a factor of N ^0.25. This is important because it essentially predicts how big the fluctuations will be for longer temperature records where each data points represents a longer time interval, such as yearly or decadal averages, rather than the monthly averages that form the basic station records I have been discussing here.

So what do we see for the mean temperature for Australia that I presented in Fig. 26.1 in the last post? In the following discussion I have restricted the analysis to data over a 140 year period from 1874 to 2013. This is because data earlier than this is derived by averaging over much less than 7 states, and will also be based on a much smaller number of individual stations.

Well, firstly the standard deviation of the data in Fig. 26.1 is only 0.627 °C. This is significantly less than that seen in individual temperature records, and in individual states like New South Wales where the value was 1.07 °C. The reason for this decrease is that the temperature record for Australia involves combining records from across the country, and the country is very big. It is, for example over 3600 km from Perth to Brisbane. Yet in Post 11 I showed that the correlation between monthly temperature records for stations that are spatially separated decreases almost linearly with the distance of separation (see Eq. 11.1). In fact, once you get to separations of more than 3000 km, the data in Fig. 11.2(a) suggest station records become completely uncorrelated. This is what appears to be happening here.

The monthly temperature data from Western Australia and Queensland will be almost totally uncorrelated as most of their respective stations are over 3000 km apart. So averaging the two will reduce the standard deviation by a factor of almost √2. Including the other states in the average could increase the reduction factor to √7, but this will be mitigated by the fact that the distances between those states are much less and so their correlations will be greater. The net result is a compromise with the standard deviation appearing to decrease by about 40% compared to the individual state values.

The second point to note is that the fluctuations in the mean temperature for Australia in Fig. 26.1 appear to conform to a Gaussian noise spectrum. This is confirmed below in Fig. 27.1.

Fig. 27.1: The distribution of anomaly values for the mean temperature record for Australia (blue curve), together with the predicted Gaussian distribution for a standard deviation of 0.627 °C.

However, these fluctuations are not white noise. If we smooth the data with different moving averages of size N months, and then recalculate the standard deviation, we get the data shown in Fig. 27.2 below. This is the scaling behaviour I described in the second paragraph above. For a more detailed explanation of its methodology, see Post 17.

Fig. 27.2: The change in standard deviation of the mean temperature anomaly for Australia (see Fig. 26.1) after smoothing with a moving average of size N. The gradient of the best fit line is -0.256 ± 0.003 and R² = 0.9993

The two significant features of the data in Fig. 27.2 are the gradient of the linear regression best fit line and the magnitude of the residual for the data relative to that line. The gradient is -0.256 ± 0.003 while the rms (root mean square) residual is 2.7% of the standard deviation of the y-values, the residuals being the vertical distance from the data point to the best fit line. This gives a value for R² of 0.9993, indicating a very good fit to the data.

The rms residual of 2.7% is incredibly low compared to previously determined values for individual states, while the gradient is incredibly close to -1/4. These two results may not be coincidental, and hint at the existence of a fundamental truth within the data, that these temperature records are fractal in nature, with the patterns of fluctuations over different timescales exhibiting strong self-similarity. If so, then this implies even 100-year mean temperatures would exhibit fluctuations with a standard deviation of over 0.1 °C and a range of over 0.4 °C. It would also suggest that changes of more than 0.5 °C in the average temperature of different centuries would be commonplace over the course of millenia. It should be noted that the data in Fig. 26.1 in the last post indicates that the average temperature in Australia over the whole of the 20th century was actually 0.063 °C less than that for the last 50 years of the 19th century.

So, are all the instrumental temperature records and their averages fractals? Do they exhibit self-similarity with a fractal dimension of 0.25 as I have suggested previously? Or is the data just white noise on an oscillating background as the data from South Australia might suggest? The answer is, it is too early to tell. We need more data and longer datasets. What we can say is that the temperature trend for Australia since 1853 in Fig. 26.1 is not a simple hockey stick.