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Cohen, Happer, Lindzen in WSJ and colors of noise

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Physicists Roger Cohen, William Happer, and Richard Lindzen wrote a calm and sensible text about weather extremes in the Wall Street Journal:

'Climate Consensus' Data Need a More Careful Look
They mention that the optimum CO2 concentration is probably significantly higher than the present one and enumerate some climate skeptics who are not conservative, among related sociological things.

But the bulk of the article is dedicated to clarifications of the fact that there's no unusual trend in the weather extremes.

Tornadoes, droughts, hurricanes, and wildfires are discussed as the most important examples of the fact that these largely unpredictable events are doing what they have always been doing. They randomly appear and disappear.

Despite the fact that these topics have been discussed for years and one could say decades, I think that most people – and maybe even most people who have science degrees – remain remarkably ignorant about the basic logic of all these things, especially the statistical properties of weather events. In particular, people don't distinguish "red noise" (or similar "pink noise") from the "white noise" (or quantities well approximated by it).

The global mean temperature has been changing during the glaciation cycles; the maximum and the minimum differ by nearly 10 °C which is significant. A cool world or a warm world persists for thousands of years. Smaller but qualitatively similar changes may be seen at shorter timescales such as several centuries (compare the medieval warm period and the little ice age). There are numerous drivers that influence the temperatures and allow them to remain "elevated" or "depressed". So it is surely incorrect to say that the annual global mean temperatures for each year are independent from those in the previous year or two. That would be needed for us to say that they behave as the "white noise".

Still, the white noise is the only "type of random data" that most people are familiar with. The white noise would look like this:

White noise. Good enough for tornadoes, hurricanes, or even earthquakes as functions of the year.

At every "moment", whatever its tiny length is, the quantity is distributed according to some fixed distribution and it is independent from the value at any previous moment. For that reason, the "jump" of the quantity from one moment to another is almost always infinite. In other words, the huge frequencies are heavily represented in the Fourier decomposition of the white noise.

We could say that tornadoes, hurricanes, wildfires, and other things are more or less described by the white noise. Every year is different, the previous year is forgotten and the "qualitative" weather events and natural catastrophes arrive randomly according to this most widely understood type of a random distribution. Each year, you throw dice and the number you get (reparameterized by a fixed function) gives you an estimate how many natural catastrophes of a certain type you get during the year.

Everything we know indicates that extreme weather events that depend on dramatic changes of temperature, pressure, or humidity as a function of space or time are nicely described as the white noise. That's why we don't observe any trend in those graphs. The graphs are highly variable but each year, the previous year (or two) are forgotten. It's always more sensible to build your expectation upon the ensemble of many (or thousands of) previous years, regardless of the fact that most of them are very far.

The climate alarmists often say that the number of extreme events such as tornadoes and hurricanes increases and they like to pretend that such claims are scientifically justified and that such claims are a part of the "greenhouse effect science" package. But these propositions are complete lies that are unfortunately often bought by a large fraction of the public. Once you look at extreme things driven by rapid change or large gradients such as tornadoes, there exists no empirical reason to think that anything is changing about them; and there exists no theoretical explanation why it should be happening. According to everything that science may say, such changes are simply not happening. Tornadoes are as random and as frequent as they were decades or centuries ago.

On the other hand, the temperature is more accurately modeled by the "red noise" or at least some kind of "pink noise" which is something in between the red noise and the white noise. While the "white noise" is a collection of random numbers that are independent from the previous ones, "red noise" is what you get by integrating (or partial summing of) the "white noise". In other words, "red noise" is the graph of a random walk. To emphasize the relationship between the two, let me also say that the derivative of the red noise is the white noise.

Graphs of red noise. Good enough model for temperatures or prices of stocks.

Each new year or new moment or new datapoint isn't independent from the previous one. Instead, it stands on the "shoulders" of the previous one and adds a random deviation away from it. Consequently, as you can see on the picture above, the graphs of red noise are much more continuous than the graphs of white noise we started with. In other words, the Fourier decomposition of the red noise suppresses the contribution from the very high-frequency oscillations and it adds strong low-frequency wiggles.

(If the graphs depict pressure as a function of time, you will hear a lower-pitch noisy sound than in the case of the white noise. If the graphs depict the electromagnetic field, the resulting noisy "light" will look redder because the lower-frequency red light will be overrepresented relatively to the white noise which is considered "color neutral" here; that's why it's called "red noise".)

These low-frequency wiggles inevitably look like "trends" over arbitrarily long timeframes.

Another concept in physics that is almost synonymous with "red noise" or "random walk" is the Brownian motion. Albert Einstein's 1905 paper – and an independent Polish mountaineer Marian Smoluchowski's 1906 paper – derived some basic mathematical patterns about the Brownian motion. The typical distance \(s(t)\) that the particle moves after time \(t\) goes like\[

\overline{s(t)} \sim C\cdot \sqrt{t}.

\] So it increases with \(t\) but it increases less quickly than the direct proportionality would predict. The graph of \(s(t)\) may be drawn and you may calculate the apparent trend between \(t=0\) and \(t=t\):\[

\overline {|v(t)|} = \frac{s(t)}{t} \sim \frac{C}{\sqrt{t}}.

\] You see that the "apparent velocity" or "apparent rate" goes to zero as you increase the period of time over which you look at the quantity. But it goes to zero very slowly. You need to increase the timescale 9 times for the apparent rate to decrease 3 times.

The global mean temperature is at least morally closer to the red noise, much like the prices of stocks. The price of a stock is pretty random and if you look at its evolution over a few years, it's often twice as high at one moment than another moment. But it's very unlikely for the stock to jump 100 percent or drop 50 percent overnight. Instead, it's the change of the price from the previous day to the new day that is random so to see large enough changes, you usually have to wait for a longer time. (That wasn't the case of the white noise.)

For this reason, it's totally normal for the temperature in a year (or few years) after a warm year to be comparably warm, too. The year 1998 was warm and due to the basic properties of the red noise or pink noise, it's very unlikely for nearby years to among the coolest ones. That's simply prohibited by the continuity of the temperature graph. And the temperature graph is continuous (although not smooth) because the Earth has a finite heat capacity and an infinitely abrupt change of the temperature would require an infinite amount of energy.

That's why all the comments of the type "most of the years in the last decade are among the top 10 warmest years" are not surprising at all. That's exactly what the natural theory of the noisy graphs – even without any human or other "special" influences – predicts. It predicts such a clustering of warm years simply because the temperature is a continuous function of time; the noise apparent in the temperature graphs is closer to the red noise than the white noise.

Being excited about this clustering is exactly as silly as saying "Look, 4 out of the history's 5 best days for the Apple stock price occurred in April 2012 (the only exception is yesterday, isn't it a stunning proof of something?". No, it's not shocking at all. There simply had to be a maximum near $640 and the moments near the maximum are likely to have similarly high values. (I am not claiming that the Apple stock won't get above $632 in the future; I am not claimiing that it will, either. Instead, we just talk about the known history.)

On the other hand, there are many other quantities that depend on time and that have no reason to be continuous – e.g. the number of tornadoes in the year Y – and those things don't have to remember and don't remember the previous year or two. So you will find no trends in them.

Things related to the drought are somewhere in between. The precipitation itself is more or less "white noise" (especially above the ocean) because whether it rains today is mostly independent on whether it was raining a month or a year ago. However, the soil and other entities on the surface of the globe have some inertia. They store water or they may run out of water (and this also influences the evaporation rate and precipitation) and if they do so, chances are that such conditions will continue for the following year or two or three, too. This inertia means that the dry years will tend to be clumped more than what you would expect from the white noise, i.e. from the complete independence. But the inertia diminishes if you look at timescales longer than a decade. If there are some unlucky coincidences that create drought in some region, chances are high that in 10 years, the situation returns to the "normal" (calculable from recent centuries, for example). That was the case of the Dust Bowl in the 1930s.

The actual things that have changed are things such as the global mean temperature that are expected to change and "oscillate around the new values" because their behavior is closer to the red noise (random walk), much like the behavior of the stock prices or the position of a particle undergoing the Brownian motion. But the global mean temperature – which may only be calculated with the desired accuracy if you have accurate thermometers everywhere and at all times and you're extremely careful about the statistical procedures used to compute the average – has only changed by something like 0.7 °C per century, something a human would probably not be able to detect at all even if the surrounding temperature was stable (the real one has additional oscillations by dozens of degrees at all time so the human detection of the underlying "global trend" is completely impossible).

If one focuses on the precise values of the temperature, it is totally inevitable – a trivial consequence of the red noise – that he gets some "apparent trends". That's what the red noise always does. But the actual temperature changes coming from this stuff are tiny and they affect neither humans nor tornadoes. All things such as tornadoes depend on properties of the climate system that are not changing. And if they're changing, the changes are negligible.

You know, one could naturally expect that there is some residual dependence. For example, the number of tornadoes could be proportional to a power of the absolute temperature, \(T^k\). I am talking about the absolute temperature because it is the more natural quantity describing temperature in physics (for example, the volume of the ideal gas at fixed pressure is proportional to the absolute temperature).

Now, the global mean temperature has changed from 288 kelvins to 288.7 kelvins in a century (or some numbers of this kind, no one knows because there is not even any precise universal definition of the "global mean temperature"). That's a change by 0.25 percent. Consequently, \(T^k\) has changed by \(k\) times 0.25 percent per century. Unless \(k\) is much greater than one, and it would be unnatural if it were much greater (so it is unlikely), we see that change in the predicted number of tornadoes per year will be smaller than one percent or so (relatively to the average predicted for the year 1912). That's obviously undetectable even if such an effect existed. And by the way, if it existed, it's likely that a warmer world will lead to smaller, and not greater, amount of similar activity because this activity depends on temperature gradients and they're predicted to decrease, and not increase, in a warming world (because the equator-to-pole temperature difference drops as the poles are claimed to be warming faster).

There's an immense layer of irrationality underlying the whole climate panic. Every rational person knows that a 0.7 °C increase of some carefully-averaged temperature is undetectable by humans and irrelevant for all other phenomena that matter. It's just a tiny change of the temperature. Nature routinely deals with much larger and harsher changes every day – larger changes per day than the centennial change someone claims to be dangerous. For the same reason, things like tornadoes and hurricanes have no good reason to behave as anything else than the "white noise", so there won't be any trend and there isn't any statistically significant trend in any of these quantities. So it's trivial to see that given the estimates for the rates that we have measured, there can't exist absolutely any reason for concern when it comes to the climate change.

Saying that people talking about a dangerous climate change are good scientists is preposterous beyond imagination. They're dimwits incapable of understanding (or unwilling to understand) basic concepts of mathematics, statistics, and natural sciences.

And that's the memo.

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reader Smoking Frog said...

I think that most people – and maybe even most people who have science
degrees – remain remarkably ignorant about the basic logic of all these
things, especially the statistical properties of weather events. In
particular, people don't distinguish "red noise" (or similar "pink
noise") from the "white noise" (or quantities well approximated by it).

It's remarkable that you find it remarkable that most people (not most people with science degrees) are ignorant of the basic logic etc. The things that people don't know, in every area, not only science and math - you have no idea! It's so pronounced in young people that you'd think you were living in the movie "Idiocracy."

reader Gene said...

I think the terms "white noise" and "red noise" were originated by radio engineers and electrical engineers, not physicists. When you are trying to detect an electromagnetic signal you really need to understand the role of noise, which is always present. If your receiver or transducer (think headphones) bandwidth cuts off below your own upper audio limit, the noise sounds lower in frequency. White noise is much more irritating.

SETI's attempts to pick up signals from other civilizations out there are noise-limited, of course. Our early TV broadcasts of "I Love Lucy" programs may reach out 60 light years from earth but our friends out there, if they exist, can't detect them. CMB noise is the big problem; it is everywhere.

reader Gene said...

Predictions of hurricane incidents/incidence after Katrina have not panned out, as everyone knows (although the true believers ignore this fact). More important is the probability that a tropical depression will develop into a hurricane. This phenomenon seems to be driven by a strong negative deviation from the adiabatic lapse rate (not warm seas) and, since Katrina, it is about half of its pre-Katrina average. I don't know whether this is statistically significant.

reader Shannon said...

I find the article on the Wall Street Journal quite unusual. It's like the voice of wisdom gently blowing through some crazy hysteria... and hopefully soothing it. Refreshing.

reader Oxo said...

About six months ago, there was a link to a video of an American prof debunking a lot of the panic science. Can some one post the link again for me

reader Werdna said...

A more mathematical (and thus physics orient approach) would be to describe the shapes of the power density functions of the frequency. White Noise is basically flat, "pink noise" has power density proportional to 1/f, "Brownian" noise proportional to 1/f^2, "blue" to f, "violet" to f^2.

So instead of "pink noise" I have heard the term "1/f noise" used.

reader Gordon Wilson said...

There was a remarkably stupid poll in Canada that was reported in the media---the conclusion was that "only 2% of Canadians believe that the climate is not changing".
Well, not bad then. Only 2% of Canadians are morons.
Of course the climate is changing. It always has---it is not time independent.
The conclusion of the pollsters was obviously that 98% of Canadians believe in AGW catastrophic climate change, which is not true at all....dishonest poll question.
Angela Merkel is visiting Canada and talking to Harper about a free trade agreement between Canada and the EU. I would be really wary----Canada could be pressured into the PIGS bailout, and EU environmental laws could be foisted on Canadian companies driving up prices. Merkel also in her visit was going to Halifax to "visit climate scientists". This was not part of the agenda set by the Canadian government. She was not invited to do this---it is obvious pressure on Harper to support IPCC generated agendas supported by most European countries. ie, it is rude meddling by a former scientist who should know better and be ashamed of herself.
Note---I am all for good environmental protections, but not for throwing money away on climate hype.

reader Luboš Motl said...

I just wanted to congratulate you to being in the top 2% smartest people of Canada, Gordon. It doesn't follow from the stupid poll but the result is probably true, anyway. ;-)

reader Daniel said...

I agree that linking extreme events to global warming is non-scientific, it takes much longer to see the trend. To be fair most climate scientists (I read) seem to say the same thing.

As someone far more knowledgeable about physics than myself you know that greenhouse gasses trap heat. There are more greenhouse gasses than before the industrial revolution. So the earth must be heating, all things being equal.

What is your problem with the theory?
1) The greenhouse effect doesn't exist
2) We aren't causing the rise in greenhouse gasses
3) Climate change isn't something to worry about
4) Climate change is happening but its not worth the cost to try to change it.

Or some combination.

reader Luboš Motl said...

Dear Daniel, my problem with the doctrine is described by your points 3,4 as well as by much more important points 5-200 that you haven't even noticed. Is that some political correctness that one is only allowed to have one of four prescribed problems with the most irrational pseudoscientific movement of the present era?

reader Daniel said...

I wasn't trying to limit you to my options.
Feel free to list any I have omitted.

I was curious to know what your specific objections are.

If im interpreting correctly you think most change in climate is part of some longer cycle that we aren't contributing to?

reader Daniel said...

Regarding the point I replied to.
I think you edited / were editing the comment after I replied or maybe I read the email alert and it is truncated.

The idea that someone can limit your options is not valid for anyone who reads your blog :)

Im not trying to burn anyone at the stake. My assessment of the consensus of climatologists is the "left wing" view (some how its aligned to politics). But I dont have an emotional stake in it.

I'm in no position to deliberate on the evidence.

Do you agree that the majority of climatologists subscribe to AWG?

Or are you talking about your reading of scientists in general?

My main point was just to find out what your position is. There are many different ones. I just wanted to separate the political part of your view from the science.

I don't really want to get into the evidence because as you rightly say its many megabytes of information and its not going to be meaningfully sorted out here.

I agree with you that people are terrible at spotting real patterns. The climate is a superposition of many epicycles. Which makes the year by year analysis pointless. Its only useful over timescales long enough to be statistically significant.

The one thing about the medieval warm period and little ice age is that they were regional and not global (from the latest I have read) so its hard to know what to think of it. Ive read they might have been caused by solar output variation. But thats a side point.

Regarding the "climate change might be good". That's true, It might be. Which is another huge whole topic :) Again I'm not in a position to make a judgement on my own reading. Its horribly complicated with many carbon cycles interacting.

I also agree people should separate the science from the politics. Nature is always right but what we choose to do or not do is a value judgement.

reader Luboš Motl said...

Do you agree that the majority of climatologists subscribe to AWG?

It's called AGW. Again, the answer depends on how one defines a climatologist. If a climatologist is a person who studies the climate using the scientific method, then I disagree. No genuine climatologist really endorses AGW. If you define a climatologist as someone who is labeled in this way by a political organization, then I probably agree even though their "subscription" still doesn't mean that they actually believe that the framework of ideas is right. Many of them just "take the left wing view", if I use your candid admission, or they've determined that the "subscription" is helpful for their income which is what those people *really* care about.

At any rate, I agree that in the real world today, most people described by "climatologists" by various official bodies "subscribe" to AGW. I also agree that most people described as "climatologists" are dishonest jerks and many but not most of them should get many years in prison.

I don't really want to get into the evidence because as you rightly say its many megabytes of information and its not going to be meaningfully sorted out here.

That's not what I wrote. I wrote that there are megabytes of data and propositions one may say about the climate. But that doesn't mean that to demonstrate every point, one needs megabytes. I have made several important claims that have actually be demonstrated, or it takes a minute to verify that my claims are correct and the fearmongers' claims are not, and if you just refuse to verify that my statements are correct, then you are not approaching the problem fairly.

reader Daniel said...

I dont think I am being idelogical, I know that there is that element. If you accept the notion that we are causing climate change and you want to enthuse the public, public figures tend to latch on to any extreme event to hook people.

So I can understand why politicians do this but it makes me cringe because its just as likely to backfire.

Regardless of the issue, humans aren't good at planning for or dealing with non-immediate threats.

Most climatologists I see or read about tend to be far less alarmist than the ones you talk about. Even if you don't think climate change is bad, you can't be certain so some consideration of all outcomes seems reasonable. Though its hard to have a reasonable public debate on anything.

It seems there are two points we are down to..

a) Significant AGW is happening or not

If you dismiss all the experts in the field then there isn't much to go on :)

The main data that makes this look man made, for me anyhow (again, im deferring to the consensus, but if I have to pick something) is that the rate of temp increase over the past 100 years is way faster than in any time since the end of the last ice age.

Climate reconstruction from ice cores and the like show a strong correlation between CO2 and temperature.

Of course both of these things have problems on the surface. But you yourself brought up the fact that there are cycles in cycles. Things like solar output do effect warming, el ninos, aerosols in past had a cooling effect etc.

b) Climate change is neutral, good, bad

That is of course the biggest question. Clearly sea level rise has a direct impact on modern life. Most of the population lives on the coast lines. This is where a value judgement comes, the cost of doing something compared to the economic damage.. or if nothing happens. But I think we both agree on the point that the earth is in a warming phase whatever the reason. The Dutch have dealt with being mostly under the sea level but Im not sure how practical that is on a large scale. If the rise is not dramatic then a slow transition might not be that bad. At least for the rich countries :p

The effects on weather are harder to predict. Of course if agricultural land turns to desert and desert to grassland that would be a big problem in the short to medium term, but this is much more speculative.

Somewhere in there people can have a reasonable discussion on what they want to do. Of course thats nothing, so you are safe :)

If AGW is the main forcing Im not sure cutting emissions would even work. Once you push a chaotic system out of balance you cant just turn it off.

reader Gene said...

I just received yesterday a pair of tiny Bang and Olufsen earphones (earbuds) as a gift from Audi because I recently purchased an Audi automobile. They were accompanied by a booklet that states that the council of the European Union has issued a Directive that requires me to recycle them so that the materials and energy can be recovered and that the manufacturer (in China) provide markings on the product (if possible) and accompanying documentation showing this requirement. I suppose that the European Union is prepared to come to California and arrest me if I just throw them in with the garbage.

Christ almighty! I discard at least a thousand time this much trash every day and it is insane to think that it is economic to separate and recycle a gram or two of aluminum and plastic.

Beware the EU, Gordon; those folks are batshit crazy.

reader Luboš Motl said...

The main data that makes this look man made, for me anyhow (again, im deferring to the consensus, but if I have to pick something) is that the rate of temp increase over the past 100 years is way faster than in any time since the end of the last ice age.

There doesn't exist any valid scientific evidence supporting this proposition of yours. Quite on the contrary, there's diverse evidence – chronicles, temperature reconstructions etc. etc. – that the temperature changes in the 20th century were of the same order as the temperature changes in previous centuries or dozens of centuries. Also, there isn't any "problem" with sea levels that are changing by a rather constant rate, 2-3 mm/year, which is about 10 times slower than what the Earth experienced for 10,000 years about 5-15 thousand years ago.

But I think we both agree on the point that the earth is in a warming phase whatever the reason.

No, I don't agree with that. The Earth has experienced slight warming since the end 1980s, slight cooling from 1998, 10-degree warming during the last 20,000 or hundreds of thousands of years, cooling in the last millions of years that left us in the present ice age (in the sense of the multi-million-year periods; an ice age is a period in which there are permanent ice sheets in the Greenland and Antarctica). In the same way, the temperatures will go up and down between 2012 and year ABCD depending on ABCD and no one really knows what will happen with the temperature in the next 10, 20, or 50 years.

Climate reconstruction from ice cores and the like show a strong correlation between CO2 and temperature.

Except that it's trivial to prove that the CO2 changes are the consequences, and not the cause, of the temperature changes. I don't believe that you haven't understood the proofs yet. You're just deliberately spreading a scientific misconception. The reasons why we're sure that the temperature was driver are numerous and obvious: there is a lag in hundreds of years, CO2 follows with a delay; the temperature curves can be derived from purely astronomical cycles, so the truly primary driver is outside the Earth; we know the mechanisms that keep the temperature and concentrations of gases (not only CO2) correlated - outgasing by the oceans. You're shamelessly spreading your lies because you think it helps your "left wing view" but you know that all these things are lies, don't you?

If you dismiss all the experts in the field then there isn't much to go on :)

Wow, so they're suddenly "all experts" again. At any rate, it's good that we concluded that there isn't much to go on.

reader Luboš Motl said...

Dear Gene, wow, imagine how the VW-Audi group must be intrinsically strong (and how skillful the Germans are in general) if it's still able to do excellent business despite similar "mandatory perpetual motion machine" directives of the type you mentioned. ;-)

reader AJ said...

Thanks for the heads up Gordon. She's due here in Halifax at 5pm local time, just when I'll be trying to get across the bridge for a ballgame. Maybe she'll take the ferry across the harbour? Hopefully she won't get in my way :)

Looks like she's visiting Dalhousie University's oceanography faculty. Maybe she'll meet with Dan Kelly, the maintainer of R's oceanography package (oce). Maybe she wants a new function added... saveeuro(). The only reason I think that this should be in the oce package is that the Euro appears to be a floundering whale :)

reader AA said...

Ah yes, since the climatological peer-reviewed literature has past its credibility long-time ago (actually. since the birth of the term "global warming" in 1975 I think) - WSJ is the best source of climate change information ;-)

reader Luboš Motl said...

Well, it's surely at least 1,000% better than the journals where people like you and your pals from Czech Globe contribute. ;-)

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