When extreme heat gripped the Pacific northwest recently, people noticed. They noticed in Seattle, Washington, where they set an all-time record high of 104°F on June 27th, only to break it the next day at 108°F. They noticed in Portland, Oregon, where they set the all-time record high of 112°F on June 27th, only to break it the next day at 116°F. They noticed in Lytton, Canada, where they set the all-time record high for all of Canada at 121°F, only to burn to the ground the following day.

Naturally this has led to speculation about the relationship of this particular heat wave to man-made climate change (global warming). One of the reasons we *expect* global warming to increase extreme heat, is illustrated in this graph:

It compares two probability distributions, one (in yellow) with a lower average, another (in red) with a higher average. As the average (i.e. the *mean*) temperature increases, the amount of extreme heat increases dramatically — provided, of course, that the *shape* of the distribution remains the same.

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Does the shape of the distribution remain the same? Of course not — at least, not perfectly so. But it turns out that for summer in the Pacific Northwest, the change in shape is quite small, in fact so small it’s hard to confirm it exists at all, while change in the average sticks out like a sore thumb.

Let’s use data for daily high temperature averaged over the “Pacific Northwest” as defined by the *study region* of a recent attribution study of the connection between this event and climate change, shown by the box in this map:

It covers the region from latitude 45°N to 52°N, longitude 119°E to 123°E. The data are from the ERA5 re-analysis data set, which was also used in that attribution study. It’s provided in Kelvins, but I converted that to degrees Fahrenheit for a more familiar temperature scale (at least, for my American readers). And here it is, daily temperature for the study region from 1950 through the end of June 2021:

Here’s data for just temperatures over 85°F, to show where the extreme heat has happened:

Note that before this year, the average temperature throughout the region never exceeded 95°F, but this year it easily broke 100°F.

I’ll define *summer* as the months June, July, and August. Here’s the data for just the summer months:

It’s straightforward to average the temperature during each year, except we should leave out the year 2021 because it’s not over yet. Averages for each year (except 2021) look like this:

The red line is the linear trend (from linear regression), is strongly statistically significant (p-value < .0001), and indicates that the average summertime temperature increased 3.9°F from 1950 through 2020.

There’s no doubt that the *average* has changed over time — nobody in his right mind disputes that. But what about the distribution itself? Has its shape changed, and in particular, what has happened to the “extreme heat” end of the distribution?

Let’s look at the hottest temperature for each year, but just for fun let’s omit this year’s record-breaking value:

The trend line (in red) is estimated by linear regression, is strongly statistically significant (p-value 0.001), and suggests an increase in yearly maximum temperature of 4°F from 1950 through 2020. So, for summer at least, it’s not just the *average* that has gotten hotter, so too has the yearly maximum value.

Still, I’d like to get at the distribution itself for summertime. To that end, I’ll transform temperature into temperature *anomaly* by subtracting, for each value, the *average for that time of year*. Doing so, I remove the annual cycle from the data (which is still present because early and late summer are cooler than mid-summer). Here are the anomalies:

They aren’t too different from the temperature data itself.

Now let’s separate the anomaly data into two big intervals of time: “early” from 1950 to 1985, and “late” from 1985 to now. We can estimate the probability distribution for each by making a histogram, as well as a smooth estimate, and I’ll compare them with pre-2000 data in blue and post-2000 data in red (solid lines are smoothed estimates):

It’s rather striking how almost all of the negative anomalies are less likely lately than earlier, and almost all of the positive anomalies are more likely now than before. It certainly seems plausible that the “late” distribution has the same shape as the “early” distribution, but shifted to the right, toward higher temperatures.

Let’s focus on the extreme-heat end by looking at the *survival function* (which is 1 minus the cdf, or cumulative distribution function) for high temperature anomalies:

There’s no doubt, the Pacific northwest is getting more extreme heat. But is that just because the distribution shifted to the right, or is shape change part of the story?

Let’s compute probability distributions, not for the anomalies as we just did, but for the anomalies *offset by a constant* to make their average values the same (in particular, zero). I’ll call these “De-Trended” anomalies (a crude term). Here are the histograms and smooth estimates of the probability density function:

There may be some shape change, but perhaps not — perhaps the differences are just random fluctuations within the uncertainty range. We’ll get a better idea of what’s happening on the high end with the survival function:

Again, it’s plausible that they are the same because they are almost entirely within each other’s uncertainty range. It is *possible* that the distribution has changed at the high end with the appearance of de-trended anomalies not before see, but the statistics don’t confirm that yet.

Perhaps the “go-to” statistical test for whether two distributions are the same, is the Kolmogorov-Smirnov test. When I use it to compare the anomalies (not de-trended), it proves they are different (p-value 0.00000000000000022), which of course they are because they have a different mean value. But when I compare the de-trended anomalies, there is no significant evidence of a difference between the distributions. The p-value is 0.4974 — not even *close* to significant.

My bottom line: The evidence demonstrates that extreme heat has gotten more frequent and more severe, and we can expect it to continue, because the probability distribution has shifted to hotter values with no confirmable change in its shape. But it is possible (not yet confirmed) that the recent heat wave is so much hotter than we’ve seen before that there’s more going on — which means we can expect even worse.

Yet people continue to dispute even the simple idea, which is easily shown for the Pacific northwest, that the increase in the average has brought with it an increase in extreme heat. One such is Judith Curry (from whom I got the first graph shown), who gives references to the peer-reviewed literature which I went and read, only to make me wonder whether or not *she* read them. At least she delivers on the promise of her blog title: hot air.

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