Andrew N. Edmonds, PhD.

Copyright 2011 © Andrew Edmonds, all rights reserved.

My recent article “The Chaos theoretic argument that undermines Climate Change modelling” that appeared on my own blog and the “Watts up with that” site created a large number of responses.

Most of them were surprisingly complimentary, but the key things for me were that the audience were quite capable of taking in a complicated explanation, and that it flushed out some of the arguments for the defence.

I’d like to expand on some of the themes in the first article, since I know now that my audience can take it, and to meet those counterarguments head on.

Firstly, I hope you noticed that my criticisms apply to all models that might be used to predict long term climate, and that I haven’t got drawn into a debate about one model or another.

In order to answer the most sophisticated counterargument I have to dip a bit deeper into the forms such a model might take.

Put simply, the counterarguments all really filter down to one which goes like this:

*Chaotic systems are constrained by attractors. Attractors have averages and standard deviations, so they can be treated as bounded and subtracted from the underlying signals by filtering. *

This implies that a general climate change model could be formalised as:

Modelable things + chaos = observed values

They go on to argue that this makes their models valid, since once you get rid of the chaos you have nice reliable values.

I didn’t in my last post explain what an attractor is. Because chaotic systems can be produced by relatively simple equations, there are methods available for processing them to get insight into these equations.

The Dutch mathematician Floris Takens worked out how to do this using what’s known as the Takens embedding theorem.

The idea of this is surprisingly simple. You take samples of the time series, at a rate to be determined, and then create a set of coordinates by using these samples. For instance, if you were selecting 3 dimensions, and your samples were X_{1}, X_{2}, X_{3}, etc. You would create coordinates by taking these in threes, so:

(X_{1}, X_{2}, X_{3 }),(_{ }X_{2}, X_{3}, X_{4})_{ },(_{ }X_{3}, X_{4}, X_{5})…( X_{n}, X_{n+1}, X_{n+2})

When you plot these points as a chart, something wonderful can happen:

This is a view of the Lorenz equations plotted in three dimensions, and then, of course flattened for a 2D page.

There are two variables in the above process, the sampling rate, or the *separation *of the samples, i.e. do you take every sample, every second sample, every third, and the *embedding dimension*. In our example this is three, but it might be any number from 2 up.

If you go back to our spread sheet from the last posting, take one of the series we generated, and embed this with a dimension of two and a separation of one you get:

To do this, just make two copies of the series and paste one of them one cell down on the other, and then plot the pairs of values in a scatter graph.

You can see that at the heart of well-behaved series like these there is order.

In practice with real-world time series the process of selecting the correct embedding and separation values is fraught with difficulties. This is discussed in my PhD thesis if you are really interested.

Most real world chaotic systems have another aspect. When it was discovered that a simulation of a column of air behaved chaotically, this did not mean that the whole of the earth would be one chaotic system.

Rather, There would be many chaotic systems, all perhaps obeying the same equations, but with different initial conditions and interacting with each other. This is known as *multi-chaos*.

When you add many such systems together the results are no longer likely to produce such pretty pictures.

The attractor for the post 1500 temperature data is shown below:

Interestingly, our current location is in the main mass to the top right, but by no means outside the historic range of the attractor.

So, now you know about attractors. How about the original criticism? Can we average out the effects of chaos? After all, these attractors do seem to be nice safe and bounded.

There are several reasons why not, each of which is in itself conclusive.

1) Averaging is a process of low pass filtering. Chaotic systems are not band limited; they have low frequency components too. You cannot say at some point that you have filtered out all the chaos to leave only the fundamentals.

2) Even if you could, you would then rely on the chaos’s only effect being additive, and the underlying system being linear. We know that no aspect of weather is linear over its entire range. If weather is non-linear then the effect of chaos will have more than additive effect.

3) The whole global warming scare is based on the premise that at least some changes are cumulative; that if you increase the level of CO2 you drive processes that again increase the level of CO2 and that cause a temperature runaway. If the effects of chaos are also cumulative anywhere in the model, then the chaos cannot be subsequently removed from the results.

4) You don’t get chaos in linear systems!

Because, of course, the chaotic element of weather is unpredictable it cannot just be subtracted: You’d have to know its exact value to do this. Instead Climate modellers try to filter it out.

Back in the days of vinyl records and cassette tapes, most of the noise was in the higher frequencies. You could remove noise by turning down the treble. Averaging data does exactly the same thing. The problem is that chaos creates low frequencies too. This is rather like the turntable rumble you used to get with vinyl records, which because it was low frequency could only be removed by turning down the bass, which is the opposite of what the critics propose.

If the chaos in weather were small compared to everything else, and band limited in frequency (both of which are not the case) you might be able to remove it by low pass filtering if the process the chaos was affecting was linear. So again using the analogy of a hi-fi system, you expect your amplifier to be linear, and if you had lots of high frequency noise you would expect to be able to filter it out with the treble filter. What would happen if you had so much noise that your pre-amplifier started to clip – i.e. to move into the non-linear regions? No amount of filtering would get rid of that.

It’s the same with weather. Simple things like whether it rains, sleets or snows are determined by minute changes in temperature when you are around 0^{o}C, yet at higher temperatures it’s always rain. Once there is snow on the ground subsequent weather will be affected. This is just like your pre-amplifier clipping. Once you get this, chaos can no longer be additive, it’s multiplicative or worse, it creates harmonics, and it can’t be filtered out.

In the first article I graphed a simple series created by accumulating the values of our really simple chaotic series. The fact that this series wanders all over the place is evidence of the presence of very low frequencies in chaos, but also illustrates that if any aspect of the weather model tends to accumulate chaotic effects, then the chaos cannot be filtered out.

Finally, the clincher: I’ve tried to explain that, in order to be able to filter out chaos its effects must be purely additive to be reversible, to be able to take it out again. This means the processes in those computer models must be linear. If nonlinear, then the effects of chaos will spread and contaminate everything, and it will be impossible to isolate the chaos from the rest. The study of chaos is part of the study of nonlinear dynamics. **Chaos cannot occur in linear systems**.

I rest my case.