Lubos’s missing post

The required notch filter itself is the key disease showing that the particular solar model is almost certainly incorrect

More than two months ago, Jo Nova’s partner David Evans sent a group of people including your humble correspondent impressively looking and formally convincing documents about a new solar theory of the climate. I have spent many hours with reading them and thinking about them, exchanging e-mails with David, and so on. Because the documents were rather long, I needed an hour at the very beginning to see what the model really says, but that was followed by many other hours of reading.

Sometime on the second day, I became pretty much certain that the model is wrong. At that time, I should have stopped all interactions because they were unlikely to be constructive and I was at risk that I wouldn’t even be thanked for the intense hours even though David would tell me he was incorporating my feedback – and this worry seems to have materialized, indeed. Not that it’s too important! 😉 I did stop spending my time a few days later, anyway.

More importantly, I think that the climate cannot work like that and if you look how the theory works and what is used as evidence in favor of the theory, it’s very clear that there is no evidence at all. Now when the theory is no longer embargoed, see big news I and big news II on Jo Nova’s blog, let me summarize the model a little bit concisely.

David’s goal is to claim that the whole evolution of the global mean temperature – or a big portion of it, to say the least – and especially the 20th century global warming and its various intense episodes may be due to the Sun. Not just the Sun. He specifically means the total output of the Sun, the total solar irradiance (TSI). I will use the symbol P(t) for this total irradiance. After all, it’s the power of a sort.

P(t), the TSI, is expressed in watts per squared meter. Up to an additive shift, the graph of P(t) since the year 1600 AD or so looked something like this:

Clicking any graph will magnify it.

You see that the solar output is oscillating with an approximately 11-year periodicity. That’s the periodicity associated with the changing polarity of the Sun’s inner magnet. The magnet goes up and down and it takes 22 years or so to return to the original up position. The sign of the Sun’s magnetic field doesn’t matter much, at least not for the number of sunspots, so the effective periodicity of the observable quantities including P(t) is 11 years or so, not 22 years.

Aside from the 11-year cycle, the graph – largely extracted from the counting of the sunspots and the known correlations between sunspots and the TSI – shows longer periods of a relative silence of the Sun. They (the Maunder minimum and the Dalton minimum) apparently coincide with the little ice age and the cool early 19th century. I do find it conceivable if not likely that these correlations between the solar activity and the cold eras are more than coincidences and there may be some very slow causal, solar-driven mechanism changing the climate at the longer-than-decadal timescales. What I have a problem with is the “notch filter” part of David’s claim.

He says that the graph P(t) above drives the Earth’s global mean temperature which looked like this since 1600 AD:

By “driving”, he effectively means that this temperatureT(t) at some momentt is a linear combination of the solar outputs at all the previous moments:

I explicitly wrote the integral so that the delay of the response t′ is a positive number but you should really imagine that the integral goes from −∞ to +∞ and the response function – the function containing the coefficients R(t′) – is zero for negative values of t′.

David’s theory is then fully specified by the response function R(t′) – he calls it the “transfer function” but I am used to the term “response function” which is why I use it all the time. OK, what is the response function with which his model predicts correct global mean temperatures as a function of the total solar irradiance? Using various algorithms and choices to cut the data, its Fourier transform of the response function R~(f) as a function of the frequency f (the periodicity is 1/f) looks like this:

At the beginning, what impressed me is that different choices of the periods of time and different methodologies to calculate R~(f) give you pretty much the same result. The function has the sharp local minimum near 1/f=11years. This minimum is what justifies the adjective “notch” in front of the “filter” (“filter” means that the hypothesized relationship is expressed via a response function). But within a day, I came fully back to my senses. It’s obvious why the sharp minimum is there in all the versions of the graph. Why is it there?

Simply because the integral equation above says that the temperature T(t) is the convolutionof the response functionR(t′) and the total solar outputP(t). If you know some basic calculus of Fourier transforms and convolutions, or if you’re ready to check it using some verifiable identities for integrals, you will agree that the Fourier transform of the convolution is the product of Fourier transforms. So translating my previous integral equation which may be schematically rewritten (using the star as a symbol of the convolution) as

to the frequency representation gives

up to some irrelevant, convention-dependent normalization factors, minus signs in frequencies, and/or complex conjugations if needed. That’s great because the Evans response function in the frequency representation may be simply calculatedas

This frequency-based Evans response function is simply the ratio of the Fourier-transformed global mean temperature and the Fourier-transformed solar output! Because the Fourier-transformed solar output has a peak (maximum) near the period 1/f of 11 years and because this Fourier transform appears in the denominator, R(f) will obviously have a similar minimum over there. The response function R~(f) will go close to zero for 1/f close to 11 years. It has to go to zero because it’s needed to suppress the effect of the 11-year cycle that is seen in the sunspots and the TSI but that isn’t seen in the global mean temperature!

I think that many of you will agree that the marketing point used as the title on Jo’s blog

For the first time – a mysterious notch filter found in the climate

is pure demagogy. What the near-vanishing of R~(f) for 1/f close to 11 years really means is that the most obvious possible proof of the direct effect of the total solar irradiance doesn’t exist – the 11-year cycle isn’t present in the temperature data. This is a problem – potentially a huge problem – for any theory that tries to present the solar output as the primary driver even at the decadal scale and faster scales. It’s surely nothing to boast about. It makes the solar theory of the climate much less likely, not more likely. Suggesting otherwise is a case of demagogy.

Still, you might be impressed that the response functions that I repeat for your convenience

always come out very similar, very close to the relatively narrow grey strip. At least I was intrigued by this observation. However, you should realize that the vertical axis is logarithmic and the strip goes from 1k to 3k for some k∈R: the response function for the given frequency is nearly tripled if you go from the grey strip’s lower end to its upper end. The numbers in the interval (1k,3k) may be written as 2k±1k, so the relative standard deviation is about 50 percent. No wonder that for most frequencies in most reconstructions, the response function may be written as the averaged one plus minus fifty percent. The accuracy with which you hit the grey strip is completely unspectacular.

In other words, there is certainly no precision evidence – and probably no significant evidence at all – that there exists any universal function R~(f) that would describe the behavior of the hypothetical solar-driven climate. The right function R~(f) needed to “predict” the right data is directly calculated from the data and its detailed values will depend on which intervals of time and which data you will try to use (TSI in some period) and “predict” (temperature in some period). R~(f) is the ratio of two functions that have pretty much nothing to do with one another. The only universal feature is that the denominator is amplified near 1/f equal to 11 years, so the ratio will be suppressed for the same frequencies and for the same reason.

This is the basic sketch explaining why I don’t believe that there is anything correct about the notch-filter theory. There are climate skeptics who will endorse any claim or idea that goes against the “consensus”. I am surely not one of them and I think that this theory is approximately as unjustifiable as the theory that we will face a climate catastrophe before 2100 which is why I am critical to both.

Natural mechanisms on Earth just won’t produce a response function that happens to vanish exactly for the 11-year periodicity!

That’s especially the case because the value 11 years for the periodicity isn’t even quite constant. If the “notch filter” on the Earth were programmed to kill the 11-year cycle, it would still fail to kill the 10-year cycle comparably accurately and because the cycles sometimes take 10 years only and the Fourier transform of TSI is still rather high for 1/f=10years, the temperature record should have a peak near that value, anyway. It doesn’t.

The sub-decadal and similarly fast oscillations of the climate don’t have anything to do with the TSI.

David is making lots of other claims that go beyond the simple model of the response function. For example, the warming between the late 1970s and late 1990s is explained by “his theory”, too. I don’t see any evidence for that at all. The warming is predicted if the response function and the delay is appropriately fine-tuned so that the prediction appears. But this is not an explanation. It’s just a replacement of one fact by another, equivalent fact in an unnatural parameterization of the same variables. There should exist a simplified explanation why such a solar model produced warming since the late 1970s and I don’t think that he has one.

Similarly, he postulates that the most intense cooling in the last 4 centuries is imminent. I don’t see how it can follow from any arguments related to this theory. Again, for some choices of R(t) including the most represented delay, one might get a significant cooling. For most others, one doesn’t get it. If one gets such a prediction regardless of the data, it’s clearly a numerical artifact resulting from the fact that the functions P(t) and T(t) are not known for all t∈R but only in an interval. This inevitably leads to some unphysical behavior near the edges. Experts in signal processing would talk about ringing artifacts. Ironically enough, I think that David’s prediction of a dramatic imminent cooling is the result of the same “ringing artifact” as some of the failed numerical model-based predictions of a dramatic imminent global warming. Non-engineers and more formally trained people will call this particular thing the Gibbs phenomenon. One must be very careful whether he believes what he predicts near the edges if his model were obtained from the data that were truncated at these edges!

Thanks to his background, David talks about the “notch filter” and electric circuits that would emulate the same response function that suppresses the given frequencies. But he doesn’t actually even have the electric circuit that behaves in this way (although you may surely design a sufficiently complicated one, involving transistors as well as capacitors and resistors and coils, that would behave like that). But even if he had one, that would be very far from having evidence that the Earth’s climate is mathematically analogous to that circuit.

So he’s really not just one level but two levels from having anything that could count as a physical justification of the model. Not only the physical mechanisms based on well-known physical phenomena are unknown. He can’t even write down the differential equations for functions of time and their derivatives that would produce such a strange response function.

I think that such a complicated response function is unlikely to follow from any physical mechanisms you may imagine at all. Note that the TSI is supposed to affect the climate for all values of the delay simultaneously. It sort of requires Nature to remember all the previous values somewhere, in a register, and then combine their effects. Realistic response functions may be approximated by functions with clear peaks near a single frequency – or several frequencies. But this function R~(f) whose values for all f seem to matter seem immensely non-resonance-like, and virtually impossible to get from any underlying mechanism.

Maybe I could even prepare a rigorous proof that R~(f) shown on the graph cannot result from a Fourier transform of any causal function R(t′) that vanishes for t′<0. Maybe some extra conditions would be needed.

Again, I am ready to believe that the Sun has a significant impact on the Earth’s climate. But it must be either something else than the TSI, or the effect must be such that all the wiggles shorter than 20 years or so must be universally suppressed. A model that takes these frequency components seriously and removes the unwanted prediction of a 11-year temperature cycle by hand is hugely contrived and the probability that it’s zero or that it’s a leading driver of the climate is nearly zero. There’s no reason for response functions to have such sharp minima near the frequencies that are supposed to matter most and even if there were such minima, it’s hugely unlikely that their corresponding frequency (which should only depend on the dynamics of the Earth) will happen to agree with the characteristic frequency of the solar processes (solar magnetic fields). The Earth’s processes and the Sun’s processes are uncorrelated.

So it’s a piece of nicely done work which looks great but from a physics viewpoint, the required mechanisms seem impossible and from a statistical viewpoint, all the things presented as the virtues or the evidence are either tautologies – the Fourier transformation of a function back and forth – or they are vices that dramatically reduce the probability of the whole paradigm instead of increasing it.

If you ask about my guess, I am with Richard Lindzen and I do think that all the wiggles of the climate that are comparable to the changes in a decade or 20 years or to the changes by something modest of the order of 0.3 °C are due to the internal variability. These changes at these timescales are effectively “weather” which means that they’re largely chaotic and you won’t find any simple, demonstrably causal “external driver” that is responsible for all of them. Some people are obsessed with determinism so even if they abandon the dominantly CO2-driven theory of the climate, one which clearly disagrees with the empirical data, they still believe that there must be another equally clear driver of all the changes, by every tenth of a degree. But this isn’t necessarily so. It’s probably not the case. Many things just don’t have deterministic causes and the weather isn’t something that only affects the 1-day timescale. The significant effects of variable weather survive at much longer – and possibly all – timescales.

Only when the temperature changes become comparable to 8 °C, the difference between the interglacials’ and ice ages’ temperature, it’s clear that most of the changes are externally driven, by the Milankovitch cycles rooted in astronomy. The evidence that the Milankovitch theory works, as clarified by Roe, is overwhelming. But in principle, all temperature changes that are substantially smaller than 8 °C may totally nicely be due to the “weather”, some chaotic internal behavior of the Earth’s atmosphere and the world ocean that admit no sharp predictive and no clearcut attribution.