I retrieved data myself, temperature at the 850 hPa level (about 1.5 km altitude) from 20th-century reanalysis, from ECMWF. I’ve looked at two locations on the equator, at longitude 45E and 60E, selected because they’re two places for which Sardeshmukh shows an increase in mean temperature but not in the probability of extreme temperature, defining “extreme” as more than 2 standard deviations above the mean. Here’s his graph of how the mean temperature has changed (in units of standard deviations):
When I analyzed the data, I got a different result than he got. A very different result. Somethin’ ain’t right, and I really want to know what’s goin’ on.
Most of us have seen graphs like this (e.g. the top panel here):
The thick black line shows the pdf (probability density function) for the normal distribution when the mean is zero and the standard deviation is one, while the thick red line shows the same thing when the mean value is increased by 1, a full standard deviation. The colored regions show the area under the pdf for values higher than 2, which is the probability of getting values that big or larger — we might even call them “extreme” values. For the black curve the chance of exceeding 2 (the extreme-high probability) is only 0.02275, but after the mean value gets bigger (all other things being equal) that probability increases to 0.1587, just about 7 times higher.
Let’s look at this in a different way. Instead of plotting the pdf (probability density function) which has the characteristic “bell curve” shape, let’s plot the cdf (cumulative distribution function) instead. Actually let’s use a different form of that, often called the survival function, which is just 1 minus the cdf, because the survival function is nothing more nor less than the probability of exceeding a given value.
Note that at x-value 2 the survival function is just 0.02275, but after increasing the mean by 1 it increases to 0.1587.
So that’s what happens when we increase the mean, all other things being equal: the chance of exceeding some “extreme” cutoff threshold also increases. In this case, dramatically (by a factor of 7) because the mean increased by an entire standard deviation.
What if we increase the standard deviation (say, by 50%), but leave the mean alone? That looks like this in terms of the pdf (like the second panel here):
This time we haven’t shifted the distribution left or right, but we have widened it. Doing so also increases the probability of values above 2, in this case from 0.02275 to 0.09121, a four-fold increase. If we look at the same thing in terms of the survival function, we get this:
Having introduced this, let’s look at daily temperature data from longitude 45E along the equator, at the 850 hPa level, from reanalysis data:
Note that temperature is in Kelvin, which is why the numbers are so high. Visually it doesn’t seem to have changed much, but if we transform from raw temperature to temperature anomaly (in order to remove the seasonal cycle) we get this:
According to Sardeshmukh’s presentation (which Judith Curry was kind enough to provide), here’s my reading of what he did. First he isolated the anomaly data for Jan-Feb-Mar (winter in the north, summer in the south). Next he isolated two 25-year long time spans: 1901-1925 and 1981-2005. Then he estimated, for each, the probability distribution in order to estimate the probability of extreme heat, defined as more than 2 standard deviations above the mean. He doesn’t say which time span defines this cutoff limit, but he did say explicity that the same absolute temperature cutoff was used for both sections. Finally, he computes how that probability has changed from the first time span to the second, and the graph he shows indicates that the probability didn’t change by much, not (according to my reading of the color scale) by more than 0.001.
I did the same thing. And here’s what I got:
The dashed line shows the cutoff limit of 2 standard deviations when we define the mean and standard deviation using the initial time span. The fact that it’s so close to the numerical value 2 is just a coincidence.
To get a better idea of how things changed, let’s zoom in on the upper range:
Note that from the first to the second time span the probability of exceeding that cutoff limit increased, and not by a tiny amount. It increased by about 0.014 which, from what I can see, is at least 14 times more than Sardeshmukh reports. Somethin’ ain’t right.
And there’s something else I find quite interesting. In his presentation Sardeshmukh includes what he refers to as a “math slide” in which he defines a “probability shift index” thus:
He mentions it as applying to the Gaussian (normal) distribution, but in fact it can be defined for any distribution, Gaussian or not. It can also be modified to indicate, not just the sign of the change, but by how much the probability of exceeding some limit will change. And here it is, in different notation (my own) which I’ll define for you:
In this equation, is the change in the survival function when the mean and standard deviation change but the shape of the probability distribution doesn’t change, i.e. the change in probability of exceeding the limit. is the pdf for a normalized version of the temperature limit, where the normalized version is simply
is the change in mean, is the change in standard deviation.
Plugging in the numbers, this suggests that for the observed change in mean and standard deviation the “exceedance probability” will increase by 0.013. The observed increase was 0.014. That contradicts Sardeshmukh’s reported value of less than 0.001, and contradicts his assertion (at least for this location) that the change in exceedance probability “looks nothing like the mean warming pattern.” One cannot claim, on the basis of this data, that the change in exceedance probability is much affected by change in the shape of the distribution.
If instead of defining our cutoff by the mean and standard deviation of the first interval, we had done so using the second interval, then the cutoff would be different. It would look like this:
The exceedance probability still increases, but now only by 0.007. That’s still at least 7 times more than Sardeshmukh shows.
Of course that’s only one location. What about the other one I’ve looked at, for longitude 60E? This:
Again I’ve set the cutoff limit (the dashed line) at the 2-sigma level for the first time span. Now the exceedance probability has increased by about 0.06, whereas Sardeshmukh’s graph of changes in exceedance probability shows it to be negative.
Somethin’ ain’t right. Maybe I’ve made a mistake. But maybe Sardeshmukh has, or maybe there’s more to what he did than meets the eye. I am genuinely curious to know, what’s going on?
I’ve also looked at 20th-century reanalysis data for these locations, not at the 850 hPa level, but at the surface (often called “T2m” for temperature 2 meters above the surface). They give the same results. I do have to wonder, if the subject is the increase in extreme temperatures due to global warming, why would you study the temperature at 850 hPa? It doesn’t make sense to me.
I’ve also looked at actual thermometer data for daily temperature, from a number of locations (mostly courtesy of ECA&D, the European Climate Assessment & Dataset network). Results: the same.
I was fully prepared to study these data, confirm Sardeshmukh’s results, then post an admission of error. Prominently, unambiguously. I still am — but thus far my attempts to confirm Sardeshmukh’s results have only contradicted them. Somethin’ ain’t right.