Global Warming Basics: Signal and Noise — and Trend

Even when climate is constant, unchanging, the weather is not.

Temperature is one aspect of weather (and therefore of climate), so it’s in constant flux, whether we’re looking at a single location or an average over the whole globe. The changes we observe, whether of ever-changing temperature or any other weather variable, can be divided into two broad categories: signal and noise.

To understand how they differ, let’s begin with an imperfect but insightful definition of the difference between climate and weather:

Climate is what on an average we may expect,
weather is what we actually get.
— Andrew John Herbertson

By this definition, climate is the signal; it’s what we expect to happen on average. The fluctuations that distinguish weather from climate make up the noise. The weather itself is the combination of “what on an average we may expect” (the signal) and those fluctuations (the noise), so it’s “what we actually get.”

If the long-term average for a given day’s high temperature is 60F (degrees Fahrenheit), that’s the signal. If the actual temperature on that day turns out to be 65F, that’s the weather. The difference — an extra 5F of warmth — is the noise.

The expectation we’re talking about is “over the long term.” If we say the climate is for today’s high to be 60F, that’s what we expect for today’s date over the long haul. But the weather forecast for today might be for a high of 65F, because weather forecasts try to anticipate the noise.

Noise is essentially the random part of the weather. You might be surprised to learn that something random may still be, at least in part, somewhat predictable. It depends on what kind of noise it is, which leads to another fact which can be quite a surprise — that there’s more than one kind of noise. But that is a story for another day.

For a clearer idea of how signal and noise differ, I’ll rely on the old adage that a picture is worth a thousand words.

KREMSMUENSTER

Kremsmuenster, Austria has been a center of learning and of science since it was founded around Kremsmuenster Abbey in the year 777. It’s also the location of a particularly long and high-quality temperature record. Being in Europe, their temperature data are in C (degrees Celsius) rather than F (degrees Fahrenheit).

Here’s the daily high temperature at Kremsmuenster for each day of the two-year period from January 1st 1968 through December 31st 1969:

It certainly fluctuates a lot from day to day, giving the distinct impression of randomness (noise) being part of the temperature. But in addition, there seems to be a pattern, one which repeats consistently: a cycle of seasons, with temperature colder in winter, hotter in summer.

That pattern isn’t followed perfectly; one year is not simply a repeat of the preceding. This is clear if we compare two different years, say plotting temperture by day of the year with 1968 in blue and 1969 in red:

Although they seem to be following the same overall pattern, the differences can be substantial, as much as 15C (which is 27F) from one year to the next for the same date.

While the pattern isn’t followed perfectly, it is followed consistently. This is clear if we look at more than just two years. Let’s look at the 30-year period from 1940 through 1969:

The cycle of the seasons is evident throughout. That it follows the one-year cycle consistently is obvious by plotting all thirty years by day of the year:

Yes, colder in winter and hotter in summer is not an accident, nor is it random — it’s predictable. It’s what enables us to have at least some idea what the temperature will be on a given date, years in advance.

The cycle of the seasons is the signal. The fluctuations about that cycle are the noise. That last graph gives us a pretty good picture of both, showing what we expect on average for each day of the year, as well as some idea of how much temperature fluctuates about that expectation. And that’s what climate really is; a better definition is this:

Climate is the mean and variation of weather over long periods of time (typically, at least 30 years).

The mean is the mathematical term for “what on an average we expect.” The variation is the essential nature of the fluctuations about that mean (and is the part of climate left out of Herbertson’s definition). To specify the climate you have to describe the mean (the signal), saying what it is you expect it to be, and you have to describe the variation (the noise), for instance how big the fluctuations are likely to be. To specify climate completely, you have to do so not just for temperature, but a lot of other things too, such as pressure, humidity, precipitation, wind, etc.

ANOMALY

The signal we’ve described so far is “merely” the seasonal cycle. We can quantify it in many ways; I won’t burden you with complex mathematics, I’ll just show you the result of one (very good) way of doing so:

The red line shows “what on an average we may expect” based solely on the cycle of the seasons. If, for each day, we subtract that from “what we actually get,” we’ll have transformed temperature into something called temperature anomaly, which just might give us a picture of the noise alone, because we just might have removed the signal. It looks like this:

There certainly is plenty of fluctuation. But overall, it doesn’t seem to be going anywhere — just fluctuating. It’s at least plausible that the temperature anomaly is only noise, that there’s no signal left, in which case that seasonal cycle really is the climate.

SLOWING IT DOWN

Let’s compute the average temperature anomaly for each month of each of the 30 years:

The monthly averages also wiggle around a lot. That’s no suprise, because when you average noise you get — noise.

The noise you get from averaging tends to be smaller than the noise you started with, something that’s clear from the fact that the monthly average anomaly values never get as high as +10 or as low as -10, while the daily anomaly values ranged from well above +10 down to below -15.

It might look as though there’s some pattern to the monthly averages, maybe even some signal present (besides the cycle of the seasons, which we’ve removed). But that’s not the case. This isn’t the simplest type of noise, instead it’s a kind that can easily give the false impression of a signal. To distinguish signal from noise requires some (sometimes subtle) mathematical analysis. I won’t burden you with that either, but I’ve done the analysis and during this time span, from 1940 through 1969, there’s no real evidence that temperature anomaly in Kremsmuenster is anything but noise, so the temperature itself is a seasonal cycle plus noise.

The conclusion is that during this 30-year span, the climate in Kremsmuenster was stable — at least as far as temperature is concerned.

A LONGER VIEW

Let’s average temperature anomaly over each month, but instead of doing so only from 1940 through 1969, let’s do it for the entire Kremsmuenster record, from 1876 through 2015:

Now there’s an even stronger impression of some pattern. It really looks like the monthly anomalies have done more than just fluctuate due to noise. In fact the anomalies seem to be higher recently — at least, on an average — than they used to be.

If we average temperature anomaly over each full year rather than each month, that impression is even stronger:

The visual evidence of some pattern, some signal, is vivid. But we have to be careful before drawing conclusions; it’s far too easy for the mind to imagine patterns when they’re not real, to see “pictures in the clouds” that look like everything from the Sistine Chapel to the Mona Lisa when we’re really just looking at noise.

That’s where the fancy math comes in. In this case, the pattern turns out to be real; this isn’t just noise, there’s signal there that we haven’t removed, more than just the seasonal cycle.

A better estimate of what that signal is, can be obtained by some other fancy mathematical methods. Here’s the result (shown as a red line) from one of my favorites:

Now we can plainly see a distinct rise in temperature recently. It’s not a trivial change either; Kremsmuenster temperature has increased by about 3C (5.4F) since right around 1970. Nor is it just noise, not even one of the unusual kinds of noise that gives false impressions, it passes the statistical tests. This is a real change in “what on an average we expect,” i.e. it’s climate change.

Climate-wise, 3C is a lot. The difference in global average temperature between a full-on glacial period and now, between mile-thick ice sheets covering Chicago and warm summers there, is “only” about 5C. The change Kremsmuenster has already seen is more than half that.

And it has happened in less than 50 years. That’s fast.

Averaged over the whole globe, temperature hasn’t increased as much as it has at Kremsmuenster. The global warming so far has only been about 1C, compared to the 3C in this small Austrian town. But in both cases, for Kremsmuenster and for the globe as a whole, the change already has been substantial.

TREND

The upshot is that the temperature signal at Kremsmuenster is more than just the seasonal cycle. That cycle continues, but in addition there has been a rise over the last decades. Both the cycle and that upswing are part of the signal.

There’s a name for the part of the signal that isn’t cyclical; it’s called the trend.

Sometimes when people say “trend” they’re referring to a linear trend. That’s a pattern which follows a straight line. But another, and often more useful, definition is whatever part of the signal isn’t cyclic, and that can often be (as in the case of Kremsmuenster temperature) different from a straight line.

We began by decomposing temperature data into two components: signal and noise. We have now further decomposed the signal into two components: cycles and trends. For temperature, the cycle consists of the coming and going of the seasons. The trend is … global warming.

TREND AND NOISE

There’s a saying, that “just because climate changes, that doesn’t mean we won’t still have weather.” The point is that just because there’s a signal, in particular a trend, that doesn’t mean we won’t still have noise.

The noise makes it harder to see what the trend really is. If one day is hotter or colder than the day before, it’s easy to believe that could just be noise, one of those unpredictable fluctuations from day to day. But many people aren’t aware that such fluctuations exist on longer time scales: from month to month, from year to year, even from decade to decade. The impact of noise is to bring about change in temperature for no apparent reason.

If one year is hotter or colder than the year before, that too might just be noise. But it could also be a combination of year-to-year noise and a trend. And if the time span we’re looking at is short, chances are the trend (if there is one) hasn’t had enough time to accumulate enough change to be noticeable compared to the noise. It takes time for the signal to “rise above the noise.”

That’s one of the tricks global warming deniers use to claim the absence of a trend, or a change in (or even a halt to) global warming. They limit what they show you to a brief span of time. That way, the jumping up and down due to noise (which is still there, even when climate changes) can hide the trend.

They usually pick an especially hot time to start, one which is extra hot due to the noise. That often means 1997 or 1998, because that’s when a giant el Niño made temperature soar. But el Niño is one of those noise factors, not part of the trend, so the years that followed weren’t as hot, and that made it look like the trend was going down, when really it continued steadily up.

Any time someone shows you a temperature graph starting in 1997 or 1998, you can rest assured that they’re trying to take advantage of the noise to mislead you about the trend. Don’t trust ’em.

As for noise — the fluctuations — making the trend harder to see, Neil DeGrasse Tyson illustrates that in this video:

It’s an excellent description. It’s especially good at revealing the difference between changes that are persistent and give us an idea where we’re heading, and those which are random and really don’t tell us anything about what to expect in the future. That’s what climate is about: what we expect to happen, on average.

My advice is twofold. First, don’t trust those who start their temperature graphs in 1997 or 1998. Second, keep your eye on the man, not the dog.

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