In the last post I mentioned that a change in either the mean value or the variance of a probability distribution will greatly affect the probability of extreme events. I also mentioned that a change in variance has a more profound impact on the likelihood of extremes than a change in mean value.
One of the things which led me to believe that we have witnessed a change in variance is a paper by Hansen et al which discusses many aspects of temperature change, including variability in temperature. The basic kind of graph (of which there are many versions) from which they conclude increased variability is this one (left graph of figure 9):
The caption describes the graph thus:
Frequency of occurrence (y-axis) of local temperature anomalies divided by local standard deviation (x-axis) obtained by binning all local results for 11-year periods into 0.05 frequency intervals. Area under each curve is unity. Standard deviations are for the indicated base periods.
Note that the most recent 11-year period shows much greater spread than earliest ones. The “conclusions” section specifically mentions (my emphasis):
Seasonal-mean temperature anomalies have changed dramatically in the past three decades, especially in the summer. The shift of the probability distribution (Fig. 9, left) is more than one standard deviation. In addition, the probability distribution broadens, the warming shift being greater at the high temperature tail of the distribution than at the low temperature tail.
I used the temperature data from climate divisions of the U.S. mainland (USA48) to estimate the variability in temperature by a similar (but not the same) method and got a different result, which caused me to wonder why. I might have figured it out, but I could be wrong.
There are three panels in figure 9, the other two show their result when using a different baseline period than the 1951-1980 standard, here’s the third one:
Note that the most recent 11-year period no longer shows dramatically greater spread than earliest 11-year periods. They ascribe the different result to this:
The effect of alternative base periods on the frequency of temperature anomalies is shown in Fig. 9. Use of a recent base period alters the appearance of the probability distribution function for temperature anomalies, because the frequency of occurrence is expressed in units of the standard deviation. Because climate variability increased in recent decades, and thus the standard deviation increased, if we use the most recent decades as base period we “divide out” the increased variability. Thus the distribution function using 1981-2010 as the base period (right graph in Fig. 9) does not expose the change toward increased climate variability.
I believe they’re mistaken.
Using a different scale factor (standard deviation) would indeed reduce the variability, but it should do so for all time periods equally. Hence the relative variability would be preserved — both early and late decades would show less variability, but late would still show more than early.
Why then the difference? I think it’s because the distributions are estimated from temperature records for many small regions covering a large area (in this case, northern hemisphere land), and different regions within that area have warmed differently. This can cause the combined distribution to widen even if no single region has experienced increased variability.
An example may illustrate. Suppose our area is composed of only two regions. During the baseline period their distributions will necessarily have the same mean value, since we’re using anomalies relative to that baseline. Suppose also that anomalies in both regions follow the normal distribution, and both have the same standard deviation. Then during the basline period the distribution of the combined area will be that of either region.
Now suppose that one region warms by 1 standard deviation while the other does not (or more generally, that they simply warm by different amounts), but that there is no change in variability in either region. Due to the change in mean, their distributions are no longer the same — let me graph them like this:
Now let’s compare the distributions for the combined area before (in black) and after (in red) the change:
The combined-area distribution has widened considerably. But this isn’t because either region has increased temperature variability. It’s because for the baseline period both regions have the same mean value so their distributions coincide, but for the later time period they have different means from having warmed overall by different amounts. Combining the two identical distributions with different means yields a distribution with increased dispersion.
I think a better way to gauge the evolution of temperature variability (apart from average temperature, which we know has changed) would be to take each small-region record, and instead of just computing anomaly, actually de-trend the anomalies. I did this, using a modified lowess smooth for the de-trending so I could remove nonlinear trends. This truly isolates the variations from the trend. I then scaled the de-trended anomalies by the standard deviation for each month, since winter months show much higher standard deviation than summer months. This defines de-trended standardized anomaly, which I then studied to look for changes in variability over time.
Subjecting the de-trended standardized anomalies for all 344 climate divisions in USA48 to the same calculation described in Hansen et al. gives this
There’s no visible sign of any change in the amount of temperature variability from one 11-year period to any other. Also, there’s no issue about baseline period, since de-trending the divisional anomalies before standardizing and combining removes the influence of the choice of baseline period (but in case you’re interested, the baseline period I used was the entire time span 1895-2012.5).
That doesn’t mean there’s no change at all in the variability of temperature for this area (USA48). In fact I think there are much better ways to search for variability change in these data, than just estimating the probability functions for visual graphical comparison. Also, the graphs from Hansen et al. are for individual seasons whereas my graph is for all months of the year (seasonal patterns are a subject I have yet to investigate) But this much seems clear to me: any variability change which may be present is far smaller than indicated by the analysis of Hansen et al.
As I said earlier, it’s possible I might be mistaken. I could have misinterpreted the procedure they followed. There could be some flaw in my own method of using de-trended standardized anomalies, of which I’m not aware. I haven’t yet studied the data by individual seasons. I sure think there’s a much better way to scan the data for signs of variability change, and to test whether such change is significant. But when it comes to increased variability in month-to-month average temperature for the USA48 area — I’m skeptical.