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## Wondering Willis and his canonical mathturbation

Sou | 12:16 AM

Update: See below - in case anyone thought that Wondering Willis Eschenbach had more grey matter between his ears than the average WUWT-er, his response to Nick Stokes should set you straight (updated archive here).

Wondering Willis Eschenbach seems to like my nickname for him.  Today on Anthony Watts anti-science blog, WUWT, Willis talks about his peregrinations. (Archived here, updated here.)

Willis' article is ostensibly about how he thinks he has found an equation that fits all climate models (with a bit of fudging).

Willis doesn't have a very good memory.  He puts up a link to what he says was his "last post", but it wasn't.  He'd published another one in between.  Today he goes back to his wonderings about climate models.  He really hasn't learnt a thing, despite lots of help from people who understand science to a greater or lesser degree.

Willis said he has plotted data from 19 climate models.  He wrote:
The inputs to the models are the annual forcings (the change in downwelling radiation at the top of the atmosphere) for the period 1860 to 2100. ...Each model is using its own personal forcing, presumably chosen because it produces the best results …

But that's not how climate models work.  Complex climate models don't have annual forcings as inputs. They don't have "personal forcings" in the way Willis describes. They are physical models.  They would be programmed with amounts of incoming solar radiation, volcanoes, the amount of greenhouse gases and land use changes.  It's not as simple as plugging in annual Watts/m2.  As Scott K Johnson describes at ArsTechnica.com:
Climate models are, at heart, giant bundles of equations—mathematical representations of everything we’ve learned about the climate system. Equations for the physics of absorbing energy from the Sun’s radiation. Equations for atmospheric and oceanic circulation. Equations for chemical cycles. Equations for the growth of vegetation. Some of these equations are simple physical laws, but some are empirical approximations of processes that occur at a scale too small to be simulated directly....

Willis refers once again to what he calls a "canonical equation".  It's only "canonical" in Willis' mind.  It's not even canonical in the mind of many of the denialati at WUWT.  Willis writes:
Now, the current climate paradigm is that over time, the changes in global surface air temperature evolve as a linear function of the changes in global top-of-atmosphere forcing.  The canonical equation expressing this relationship is:
∆T = lambda ∆F (Equation 1)
That equation can be viewed as a simple energy balance model of climate.  It's not a "canonical equation" but a simplification.  It's not used in the models that Willis wrote about.  It's used in very simple models only. Further down in his article, Willis writes:
Another implication of the mechanical nature of the models is that the models are working “properly”. By that, I mean that the programmers of the models firmly believe that Equation 1 rules the evolution of global temperatures … and the models reflect that exactly, as Figure 2 shows. The models are obeying Equation 1 slavishly, which means they have successfully implemented the ideas of the programmers.

Willis is wrong, of course.  Wrong in more than one respect.  People who work with complex climate models do not accept that there is a simple linear relationship between surface temperature and radiative forcing.  If they all thought that then they wouldn't be constructing hugely complex physical models of the earth system.  The relationship cannot be a simple linear relationship because different parts of the system operate on different time scales.  There are also positive and negative feedbacks, which also operated on different time scales.  For example, the effect of a climate forcing on the oceans operates over short, medium and very long time scales.  The effect of radiative forcing on ice sheets also operates on short, medium and very long time scales. The effect of a radiative forcing on the surface temperature can operate on a very short time frame as seen in surface temperature charts as well as longer time frames as the ocean, ice sheets and atmosphere equilibrate.

This next sentence of Willis' must be a mistake:
However, the models are built around the hypothesis that the change in temperature is a linear function of temperature
A change in temperature is a linear function of temperature?

There are more "oddities".  After deciding that ∆T = lambda ∆F or, to put it another way, ∆T/∆F = lambda, or to put it into words, the ratio of the change in temperature to the change in forcing equals lambda, Willis oddly writes he thinks it is odd:
Now, an oddity that I had noted in my prior investigations was that the transient climate response lambda was closely related to the trend ratio, which is the ratio of the trend of the temperature to the trend of the forcing associated with each model run.
Unless I've misunderstood what he wrote, he's now saying his "canonical equation" is an oddity?

Thing is, that I can't easily work out what Willis has done or what he thinks he's done.   Here is his equation, for what it's worth:

∆T = lambda * ∆F * ( 1-e^( -1/tau )) + ( T[n-1] – T[n-2] ) * e^(-1/tau)  [Eqn. 2]

Willis writes:
In Equation 2, T is temperature (°C), n is time (years), ∆T is T[n] – T[n-1], lambda is the sensitivity (°C / W/m^2), ∆F is the change in forcing F[n] – F[n-1] (W/m2), and tau is the time constant (years) for the lag in the system.

He's regurgitated his curve-fitting equation from older articles of his.   He's said that his lambda (TCR) "ranges from 0.36 to 0.88 depending on the model".  Then he says that "using the two free parameters lambda and tau to lag and scale the input, I fit the above equation to each model in turn".  So I take that to mean that he fudged his lambda and tau until he got his equation to fit each model.

Willis wrote:
Note that the same equation is applied to the different forcings in all instances, and only the two parameters are varied. The results are shown in Figure 2. In all cases, the use of Equation 2 on the model forcings and temperatures results in a very accurate, faithful match to the model temperature output.

So assuming he got his T (temperature) from the models and fudged his lambda and tau, the only question is from where did he get his F (forcing).  There are clues in the comments.  Nick Stokes speculates Willis got the forcings from a paper by Forster et al (2013).  If that's the case then it's no surprise that Willis' equation 1 fits because that's what they used to estimate the forcings from the models.  Nick Stokes writes:
December 2, 2013 at 2:47 am
“The inputs to the models are the annual forcings (the change in downwelling radiation at the top of the atmosphere) for the period 1860 to 2100.”
I agree with Rhoda here. The forcings are often expressed as radiative equivalents, but they aren’t the actual input to models. Those inputs are the direct physical quantities, such as GHG concentrations, or for some modern AOGCM’s, the actual emissions (from scenarios). The radiative forcings in W/m2 are back-calculated for comparison. Hansen describes that here:
“We compute Fi, Fa, Fs and Fs* for most forcing mechanisms to aid understanding and to allow other researchers easy comparison with our results.”
I believe the forcings quoted here are from the paper by Forster et al. They are explicitly computed by those authors; they call them adjusted forcings (AF). They were not model inputs. They say:
“Forster and Taylor [2006], hereinafter FT06, developed a methodology to diagnose 60 globally averaged AF in Coupled Model Intercomparison Project phase 3 (CMIP3) models and we use the same approach here within CMIP5 models, taking advantage of their improved diagnostics and additional integrations to improve the methodology.”
In fact, the close association with the “canonical equation” is not surprising. F et al say:
“The FT06 method makes use of a global linearized energy budget approach where the top of atmosphere (TOA) change in energy imbalance (N) is split between a climate forcing component (F) and a component associated with climate feedbacks that is proportional to globally averaged surface temperature change (ΔT), such that:
N = F – α ΔT (1)
where α is the climate feedback parameter in units of W m-2 K-1 and is the reciprocal of the climate sensitivity parameter.”
IOW, they have used that equation to derive the adjusted forcings. It’s not surprising that if you use the thus calculated AFs to back derive the temperatures, you’ll get a good correspondence.

Or, to put it simply - Nick Stokes again, first quoting Willis:
December 2, 2013 at 3:39 am
“To remind folks, the canonical equation, the equation around which the models are built, is Equation 1 above, ΔT = lambda ΔF, where ΔT is the change in temperature (°C), lambda is the sensitivity (°C per W/m2), and ΔF is the change in forcing (W/m2)”
It doesn’t have anything to do with the way the models are built. It’s not their canonical equation. What it is is Forster’s equation (1), which he used to infer the adjusted forcings that you are using from the model output. The math is circular. You are feeding his Eq (1) derived AFs into your analysis and coming up with Eq (1).

Assuming Willis plots his charts using a time interval n=1 year, then it seems to me quite possible that he is just doing a short walk each time, which is self correcting.  Each next step in his temperature/time series could be taking a temperature reading from the model.  But I could be wrong.  If I'm wrong and he seeds the start of his series with the model then he should be able to make projections.  But I've never seen him do that.

Willis offered what he said was a file in R and an excel spreadsheet, for people who want to play with his numbers.  I have never used R and I don't have any current plans to do so but you can download it from here if you want to.  Not being familiar with R I wasn't able to figure out what he was doing.

Willis also provided a text file (which he erroneously called an excel file), with what I presume was the output from each of the models he used.  I've uploaded his text file to Google docs here and his R file here in case any HW reader wants to have a peep.

As usual, I don't think Willis is proving what he thinks he is proving.  Maybe someone more familiar with maths and models could comment.  I expect I've not got it quite right.  But then I very much doubt that Willis has it quite right either.

### Update

Willis has written a response to Nick Stokes, arguing that his equation: ∆T = lambda ∆F is different to this equation from Forster et al (2013) N = F - αΔT or αΔT = F-N, where:
• N = top of atmosphere energy imbalance
• F = climate forcing and
• T = temperature
• α  = the climate feedback parameter that is proportional to the change in temperature ΔT in Wm-2K-1 .

Although the two equations are not identical, they effectively resolve to the same thing.  Willis still doesn't seem to appreciate the models do not input forcings in the manner he thinks.  Willis wrote:
PS—It’s not “Forster’s equation” either, it’s the reported forcing from the models as shown in the CMIP5.

While I was writing the above update, Willis added a second comment but I'm not sure that he realises yet that his computations are circular.  Willis Eschenbach says (excerpt):
December 2, 2013 at 10:54 am
Thanks, Nick. I see I spoke prematurely above. Dang, I hadn’t realized that they had done that. I was under the incorrect impression that they’d used the TOA imbalance as the forcing … always more to learn.
So we have a couple of choices here.
The first choice is that Forster et al have accurately calculated the forcings. If that is the case, then the models are merely mechanistic, as I’ve said. And as you said, in that case it’s not surprising that the forcings and the temperatures are intimately linked. And if that is the case, all of my conclusions above still stand.
The second choice is that Forster et al have NOT accurately calculated the forcings. In that case, we have no idea what is happening, because we don’t know what the forcings are that resulted in the modeled temperatures.
w.
Willis doesn't appear to understand what Nick Stokes wrote.  It's not that the forcings and temperature are intimately linked (of course they are).  The point is that Willis' workings are circular.  He used Forster13 forcings in his equation, not model forcings.  Therefore his charts should come out the same as the models, just as they did.

Willis' conclusions were a bit off track.  He says the models are "simply incapable" of a main task they have been asked to do.  I don't agree.  The sensitivities reported in Forster13 don't vary hugely.  The ECS range is 2.08 to 4.67 for ECS and 1.1 to 2.5 for TCR.  The analysis of forcings in Forster13 throws light on some of the differences.  Willis knows better than all those scientists though.  He wrote his conclusions as:
Conclusions? Well, the most obvious conclusion is that the models are simply incapable of a main task they have been asked to do. This is the determination of the climate sensitivity. All of these models do a passable job of emulating the historical temperatures, but since they use different forcings they have very different sensitivities, and there is no way to pick between them.
Another conclusion is that the sensitivity lambda of a given model is well estimated by the trend ratio of the temperatures and forcings. This means that if your model is trying to replicate the historical trend, the only variable is the trend of the forcings. This means that the sensitivity lambda is a function of your particular idiosyncratic choice of forcings.

Willis still has a few fans, but he also has people querying his assumptions and workings. (Archived here, updated here.)

John B. Lomax eats it all up and swallows it whole:
December 2, 2013 at 12:09 am
You should really send this to the OMB. Your one-liner equation could save our government (and we taxpayers) a few \$B by replacing all of the complex computer models.

Rhoda Klapp quizzes Willis on his "forcings" and says:
December 2, 2013 at 12:22 am
Are the inputs really in units of watts/m2? It was my impression that the modellers used CO2 levels and modelled radiative physics in terms of local conditions. Taking some average figure for a supposed forcing, no matter how accurate the average is, can never be a satisfactory model input. This is true of any forcing, not just radiative.

cd says in part what I was thinking about his fudging lambda and tau to get a fit:
December 2, 2013 at 3:14 am
Willis
All your points follow. However, if I understand your method, you’re essentially fitting a function and playing about with lamda and tau and until you get a reasonable fit with the models. While this gives you an “adaptive model”, it does sound like a statistical model of the models and therefore is one of many possible solutions – although in truth you now have your own climate model that was designed to mimic the ones your testing.
This brings me on to your point on model being a Black Box. They aren’t, there are a number of online articles/lectures as well as journal papers that explain what types of algorithms they use right down to up-scaling methods and even what type of programming paradigms are chosen. So I think this is unfair, you’re almost suggesting that we should somehow be suspicious of a model because their unfathomable complexity hides a simple, and limited, algorithm – and such commentary, implicitly suggests stealth by design. If they seem like Black Boxes then that’s because you haven’t made the effort to find out what makes them tick.

#### 1 comment:

1. "However, the models are built around the hypothesis that the change in temperature is a linear function of temperature."

He might be right. I always panicked at the Hockeystick being an exponential or even combinatorial growing curve but parabolic might cover the thing well. It would also bring back the Medieval Warm Period, which as we know was even hotter than the present :)

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