Guest Post by Willis Eschenbach
[NOTE the update at the end of the post.] I’ve continued my peregrinations following the spoor of the global climate model data cited in my last post. This was data from 19 global climate models. There are two parts to the data, the inputs and the outputs. 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. The outputs of the models are the temperature hindcasts/forecasts for the same period, 1860 to 2100. Figure 1 shows an overview of the two datasets (model forcings and modeled temperatures) the nineteen models, for the historical period 1860-2000.
Figure 1. Forcing (red lines, W/m2) and modeled temperatures (blue lines, °C) from 19 global climate models for the period 1860-2000. Light vertical lines show the timing of the major volcanic eruptions. The value shown in upper part of each panel is the decadal trend in the temperatures. For comparison, the trend in the HadCRUT observational dataset is 0.04°C/decade, while the models range from 0.01 to 0.1°C/decade, a tenfold variation. The value in the lower part of each panel is the decadal trend in forcing. Click any graphic to enlarge.
The most surprising thing to me about this is the wide disparity in the amount, trend, and overall shape of the different forcings. Even the effects of the volcanic eruptions (sharp downwards excursions in the forcings [red line]), which I expected to be similar between the models, have large variations between the models. Look at the rightmost eruption in each panel, Pinatubo in 1991. The GFDL-ESM2M model shows a very large volcanic effect from Pinatubo, over 3 W/m2. Compare that to the effect of Pinatubo in the ACCESS1-0 model, only about 1 W/m2.
And the shapes of the forcings are all over the map. GISS-E2-R increases almost monotonically except for the volcanoes. On the other hand, the MIROC-ESM and HadGEM2-ES forcings have a big hump in the middle. (Note also how the temperatures from those models have a big hump in the middle as well.) Some historical forcings have little annual variability, while others are all over the map. Each model is using its own personal forcing, presumably chosen because it produces the best results …
Next, as you can see from even a superficial examination of the data, the output of the models is quite similar to the input. How similar? Well, as I’ve shown before, the input of the models (forcings) can be transformed into an accurate emulation of the output (temperature hindcasts/forecasts) through the use of a one-line iterative model.
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]
In this equation, “∆T” is the change in temperature from the previous year. It can also be written as T[n] – T[n-1], where n is the time of the observation. Similarly, “∆F” is the change in forcing from the previous year, which can be written as F[n] – F[n-1]. Finally, lambda is the transient climate response (°C / W/m^2). Because I don’t have their modeled ocean heat storage data, lambda does not represent the equilibrium climate sensitivity. Instead, lambda in all of my calculations represents the transient climate response, or TCR.
The way that I am modeling the models is to use a simple lagging of the effects of Equation 1. The equation used is:
∆T = lambda * ∆F * ( 1-e^( -1/tau )) + ( T[n-1] – T[n-2] ) * e^(-1/tau) [Eqn. 2]
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.
So … what does that all say? Well, it says two things.
First, it says that the world is slow to warm up and cool down. So when you have a sudden change in forcing, for example from a volcano, the temperature changes more slowly. The amount of lag in the system (in years) is given by the time constant tau.
Next, just as in Equation 1, Equation 2 scales the input by the transient climate response lambda.
So what Equation 2 does is to lag and scale the forcings. It lags them by tau, the time constant and it scales them by lambda, the transient climate response (TCR).
In this dataset, the TCR ranges from 0.36 to 0.88 depending on the model. It is the expected change in the temperature (in degrees C) from a 1 W/m2 change in forcing. The transient climate response (TCR) is the rapid response of the climate to a change in forcing. It does not include the amount of energy which has gone into the ocean. As a result, the equilibrium climate sensitivity (ECS) will always be larger than the TCR. The observations in the Otto study indicate that over the last 50 years, ECS has remained stable at about 30% larger than the TCR (lambda). I have used that estimate in Figure 2 below. See my comment here for a discussion of the derivation of this relationship between ECS and TCR.
Using the two free parameters lambda and tau to lag and scale the input, I fit the above equation to each model in turn. I used the full length (1860-2100) of the same dataset shown in Figure 1, the RCP 4.5 scenario. 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.
Figure 2. Temperatures (hindcast & forecast) from 19 models for the period 1860 to 2100 (light blue), and emulations using the simple lagged model shown in Equation 1 (dark blue). The value for “tau” is the time constant for the lag in the system. The ECS is the equilibrium climate sensitivity (in degrees C) to a doubling of CO2 (“2xCO2”). Following the work of Otto, the ECS is estimated in all cases as being 30% larger than “lambda”, which is the transient climate response (TCR). See the end note regarding units. Click to enlarge.
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. Note that the worst r^2 of the group is 0.94, and the median r^2 is 0.99. In other words, no matter what each of the models is actually doing internally, functionally they are all just lagging and resizing the inputs.
Other than the accuracy and fidelity of the emulation of every single one of the model outputs, there are some issues I want to discuss. One is the meaning of this type of “black box” analysis. Another are the implications of the fact that all
of these modeled temperatures are so accurately represented by this simplistic formula. And finally, I’ll talk about the elusive “equilibrium climate sensitivity”.
Black Box Analyses
A “black box” analysis is an attempt to determine what is going on inside a “black box”, such as a climate model. In Figure 3, I repeat a drawing I did for an earlier discussion of these issues. I see that it used an earlier version of the CCSM model than the one used in the new data above, which is CCSM4.
Figure 3. My depiction of the global climate model CCSM3 as a black box, where only the inputs and outputs are known.
In a “black box” analysis, all that we know are the inputs (forcings) and the outputs (global average surface air temperatures). We don’t know what’s inside the box. The game is to figure out what a set of possible rules might be that would reliably transform the given input (forcings) into the output (temperatures). Figure 2 demonstrates that functionally, the output temperatures of every one of the climate models shown above in Figure 2 can be accurately and faithfully emulated by simply lagging and scaling the input forcings.
Note that a black box analysis is much like the historical development of the calculations for the location of the planets. The same conditions applied to that situation, in that no one knew the rules governing the movements of the planets. The first successful solution to that black box problem utilized an intricate method called “epicycles”. It worked fine, in that it was able to predict the planetary locations, but it was hugely complex. It was replaced by a sun-centered method of calculation that gave the same results but was much simpler.
I bring that up to highlight the fact that in a “black box” puzzle as shown in Figure 3, you want to find not just a solution, but the simplest solution you can find. Equation 2 certainly qualifies as simple, it is a one-line equation.
Finally, be clear that I am not saying that the models are actually scaling and lagging the forcings. A black box analysis just finds the simplest equation that can transform the input into the output, but that equation says nothing about what actually might be going on inside the black box. Instead, the equation functions the same as whatever might be going on inside the box—given a set of inputs, the equation gives the same outputs as the black box. Thus we can say that they are functionally very similar.
Implications
The finding that functionally all the climate models do is to merely lag and rescale the inputs has some interesting implications. The first one that comes to mind is that regarding the models, as the forcings go, so goes the temperature. If the forcings have a hump in the middle, the hindcast temperatures will have a hump in the middle. That’s why I titled this post “Mechanical Models”. They are mechanistic slaves to the forcings.
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.
Climate Sensitivity
Finally, to the question of the elusive “climate sensitivity”. Me, I hold that in a system such as the climate which contains emergent thermostatic mechanisms, the concept of “climate sensitivity” has no real meaning. In part this is because the climate sensitivity varies depending on the temperature. In part this is because the temperature regulation is done by emergent, local phenomena.
However, the models are built around the hypothesis that the change in temperature is a linear function of forcing. To remind folks, the canonical equation, the equation around which the models are built, is Equation 1 above, ∆T = lambda ∆F, where ∆T is the change in temperature (°C), lambda is the sensitivity (°C per W/m2), and ∆F is the change in forcing (W/m2)
In Equation 1, lambda is the climate sensitivity. If the ∆F calculations include the ocean heat gains and losses, then lambda is the equilibrium climate sensitivity or ECS. If (as in my calculations above) ∆F does not include the ocean heat gains and losses, then lambda is the short-term climate sensitivity, called the “transient climate response” or TCR.
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. I speculated at that time (based on only the few models for which I had data back then) that lambda would be equal to the trend ratio. With access now to the nineteen models shown above, I can give a more nuanced view of the situation. As Figure 4 shows, it turns out to be slightly different from what I speculated.
Figure 4. Transient climate response “lambda” compared to the trend ratio (temperature trend / forcing trend) for the 19 models shown in the above figures. Red line shows where lambda equals the trend ratio. Blue line is the linear fit of the actual data. The equation of the blue line is lambda = trend ratio * 1.03 – 0.05 °C / W/m-2.
Figure 4 shows that if we know the input and output of a given climate model, we can closely estimate the transient climate response lambda of the model. The internal workings of the various models don’t seem to matter—in all cases, lambda turns out to be about equal to the trend ratio.
The final curiosity occurs because all of the models need to emulate the historical temperature trend 1860-2000. Not that they do it at all well, as Figure 1 shows. But since they all have different forcings, and they are at least attempting to emulate the historical record, that means that at least to a first order, the difference in the reported climate sensitivities of the models is the result of their differing choices of forcings.
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.
Are there more conclusions? Sure … but I’ve worked on this dang post long enough. I’m just going to publish it as it is. Comments, suggestions, and expansions welcome.
Best regards to everyone,
w.
A NOTE ON THE UNITS
The “climate sensitivity” is commonly expressed in two different units. One is the c
hange in temperature (in °C) corresponding to a 1 W/m2 change in forcing. The second is the change in temperature corresponding to a 3.7 W/m2 change in forcing. Since 3.7 W/m2 is the amount of additional forcing expected from a change in CO2, this is referred to as the climate sensitivity (in degrees C) to a doubling of CO2. This is often abbreviated as “°C / 2xCO2
DATA AND CODE: As usual, my R code is a snarl, but for what it’s worth it’s here, and the data is in an Excel spreadsheet here.
[UPDATE]. From the comments:
Nick Stokes says:
December 2, 2013 at 2:47 am
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.
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 Nick 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.
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Nick
All of those criticisms apply to the similar programs used for numerical weather forecasting.
That’s a bit naughty. The spatial resolution of short-term weather models is far, far finer than climate models because of the shorter time period. The higher resolution also means that short-term atmospheric models can deal with sensitivity to starting conditions in a far more detailed and accurate manner.
cd says: December 4, 2013 at 1:41 am
“That’s a bit naughty. The spatial resolution of short-term weather models is far, far finer than climate models because of the shorter time period.”
Not really. The stated criticisms referred to the variable behaviour of water vapor/clouds, and the failure to achieve 1000ft resolution to do water “properly”. That applies to both climate models and GWP. The NCDC Global forecasting system (GFS) uses 28 km cells. GCM’s tend to use 100 km or so for cost reasons, but that doesn’t make them worthless.
Nick
The up-scaling required in order to simulate energy transfer is obviously proportional to the size of your cells. It has huge implications, to say that it doesn’t would make one wonder why they bother using evermore powerful computers for the expressed purpose of increasing descretised voluminous resolution.
Can I add however, that they may be physics based, but the implementation of the physics is very poor. Another point, as Willis has mentioned are not total deterministic. They have stochastic components – their projections are highly dependent on starting conditions and starting assumptions. In short, each run is a non-unique solution.
Sure, but that’s a feature, not a bug. Weather is chaotic, and these things work by literally integrating the weather forward in time (in e.g. five minute timesteps). A butterfly-wing-flap change in the initial conditions creates an entirely different pattern of both weather and climate five or ten years into the future. The idea is that probable climate is determined by averaging over a sampled distribution of possible weather futures. Sadly, this is not actually the case — what is determined by the sampled distribution of the possible weather futures is precisely that — the distribution of possible weather futures conditioned on the assumptions built into the model program.
The real world, on the other hand, follows a single trajectory that (as we can see by comparing the actual data to the actuam model outputs) may not resemble any of the sampled trajectories and may not even be within the envelope of the distribution of possible weather futures for any given model. Indeed, I’m pretty sure that for many of the CMIP5 models, the actual weather/climate trajectory for the last 15-20 years is not within the envelope of possible weather futures for the models, which is why I advocate rejecting those models on the basis of hypothesis testing.
As to whether or not the physics implementation is “poor” — that’s very difficult to judge. Again the issue is in part chaos — if one writes nonlinear ODEs for a chaotic oscillator, the implementation of the physics can be identical on two different systems as far as the definition of the ODEs themselves are concerned, but if you either feed the exact same initial conditions into two different differential equation solvers or infinitesimally different initial conditions into a single differential equation solver or the same initial conditions into a single differential equation solver but with slightly different settings for the tolerance and error, you may well find that the solution goes to completely different places quite rapidly. The same is true for the entire class of “stiff” differential systems, or systems where the derivatives are highly sensitive to small numerical errors (e.g. one’s involving subtracting two big numbers to get a small number to get the slopes for the next time step). In some cases there IS NO “good” physics implementation, at least not one that we can (yet) compute.
Ordinarily one would judge the quality of the implementation of a large, multi-component problem by seeing how it works, by comparing its output to the real world. But failure doesn’t really tell you whether it is the physics per se that is or isn’t well done, or if the failure is numerical, or if it is produced by simple bugs in the code, or if the model omits important physics, or if the model simply cannot be run at the requisite granularity to get the right answer. If you try integrating ANY complex system with a non-adaptive ODE solver, you basically are betting that the errors produced by e.g. Euler’s method (or any more sophisticated scheme) at the granularity you can afford to reach can be somehow regularized or renormalized so that they don’t accumulate systematically. There are plenty of problems for which this assumption is simply incorrect — plenty of very SIMPLE problems for which it is incorrect. Try integrating a planetary orbit with Euler’s method with “large” timesteps, or simply use two different small timesteps. You rapidly get into trouble — the very kind of troubles that the “mass fixer” and “energy fixer” components of CAM are trying to ameliorate. But “fixing” the mass and energy (or energy and angular momentum) ex-post-facto in a numerical integration — which might work adequately for a “simple” two body planetary orbit — is unlikely to work well in a nonlinear Navier-Stokes problem because the system can easily have completely distinct solution classes — different attractors, as it were — that a true integration will jump between while a constrained integration may find itself locked to a single attractor. It is more or less impossible to prove that any given renormalization like this or truncation of an adaptive method will preserve the important gross FEATURES of the actual trajectory.
rgb
Nick Stokes says:
December 4, 2013 at 5:41 am
This is OT, but I believe you said some time ago that you had published a method for estimation of the Laplace transform of a system given I/O data. I would like to learn more about your formulation, and would appreciate a citation/link of some sort. Thanks.
rgb:
Use of “prediction” in the manner that you suggest serves to obscure pathological features of global warming climatology that I prefer to expose. You can get an understanding of the mechanism by which these features are obscured by reading my peer-reviewed article at http://wmbriggs.com/blog/?p=7923 . In brief, this mechanism is application of the equivocation fallacy.
Dear Terry,
As far as I’ve been able to tell, over our several exchanges on this subject, nobody really cares if you think that the equivocation fallacy is relevant to the problems with GCMs. Whether they predict, project, prophesy, or pretend, they solve physics-based model systems of equations that are tuned to the past, initialized in the present, and compute a putative system state into the future. Name that future state any way that you prefer, but do not pretend that the name you use has anything whatsoever to do with the single relevant question — is that future state — be it a prediction, a projection, a prophesy, or a complete fiction invented to sell snake oil to gullible natives — a reliable representation of the actual future of the system being modelled.
The only point in correcting language is when there is a serious misunderstanding associated with the usage and when the general population of language users agree that the correction is relevant. Both are necessary — lie and lay and good and well are both excellent examples of cases where literal application of terms could lead to serious misunderstandings — “I’m doing good” as colloquially used does not, in fact, literally mean “I’m doing well”, in spite of the fact that all of us who do indeed understand the difference between a noun and an adverb still understand those who use the former phrase perfectly — errm – good — as intending the latter phrase.:-) We rarely correct adults who use it incorrectly (especially in the south, where the incorrect usage might even be the norm, and hence according to the true rules of language, no longer be incorrect usage), however, as they might well be offended and knock out our teeth in response and besides, communication is well-established even with the malapropism.
Just a thought. It might help if you stop acting as the WUWT grammar/usage nazi and instead concentrate on the relevant issues, which are not linguistic, they are numerical and statistical. Not that this suggestion will have any impact on your future behavior — if you were capable of responding to it you would have already done so.
rgb
This is OT, but I believe you said some time ago that you had published a method for estimation of the Laplace transform of a system given I/O data. I would like to learn more about your formulation, and would appreciate a citation/link of some sort. Thanks.
What are you looking for, Bart? Matlab and Mathematica have laplace transforms built in. The GSL and many other numerical packages have numerical integration tools galore. And the transform itself is basically a real integral evaluated on a grid — there are two or three methods for solving it efficiently that you can google up pretty easily, and you can probably find C/C++ source (at least) in the open source world. For data on a fixed grid, you can probably get by with a simple summation of f(s_j) = f(x_i) exp(s_j x_i) (for s_j in a list and x_i in some normalized interval). If the data is on a variable grid, you’ll have to weight the sum with the variable interval. If you want to get fancier, you can take the data and e.g. spline it and then run a laplace transform with an actual quadrature routine on the spline, but I doubt that is going to improve your result and it makes assumptions about the smoothness of the data. Finally, you can probably use a FFT to do an LT as the two convolutions are basically the same with the LT a special case of the FFT, although I haven’t ever done it.
But matlab, and maybe octave too (haven’t looked) can do it out of the box, if not for a data list for a function that interpolates a data list..
rgb
rgbatduke says:
December 4, 2013 at 9:42 am
Thanks, RGB. Mainly, I wanted to see if Nick had a particularly robust and efficient (in both senses) formulation.
When dealing with real world stochastic data, the processing is not as straightforward as you might imagine. For example, using the FFT directly for PSD estimation is not efficient in a statistical sense. The variance does not decrease with longer record length, and you must perform additional processing to trade off bias and variance to obtain a good result. Choosing how to balance bias and variance is the classic conundrum in estimation theory.
Also, the convolution of an LT (or Z-transform, as it would become for discrete time data) is significantly different from an FFT. The FFT makes use of periodicites to drastically reduce the number of mathematical operations. When your kernel is non-unitary, you lose that advantage.
Bart says: December 4, 2013 at 8:46 am
Bart,
Yes, the paper is here. Unfortunately I can’t find a non-paywall version.
rgbatduke says: December 4, 2013 at 9:42 am
“What are you looking for, Bart? Matlab and Mathematica have laplace transforms built in.”
It’s an inverse Laplace Transform method. But is is in Matlab.
I’ve put a copy of the inverse Laplace paper here. There’s a recent review article here which explains and compares various methods, including ours. It has now been published in Numerical Algorithms.
Nick Stokes says:
December 4, 2013 at 1:29 pm
Thanks, Nick. Not exactly what I was expecting, but it might be useful.
Which is to say, I am sure it is useful, but it might be useful to me.
rgb:
Thank you for taking the time to reply. By your speculation that “nobody really cares if you think that the equivocation fallacy is relevant to the problems with GCMs,” you make an ad hominem argument. As most of us know, an ad hominem argument is illogical.
A fact of global warming climatology, which you steadfastly refuse to address, is that no statistical population underlies the IPCC climate models. What say you about the absence of this population?
Willis Eschenbach:
I’m sorry to have missed the excellent comments that you left on my testimony to an EPA hearing. I learned that Anthony had published my submission only yesterday and after the opportunity to respond had passed. If you’d like and are still tuning in, I’ll respond in this thread.