Bishop Hill Is there such a thing as a global temperature?
Mar 14, 2007 Climate: McKitrick Climate: Surface This is the question asked in a paper by Essex, McKitrick and Andresen in a fascinating paper which can be found here. (Mathematics alert!). This is my understanding of it - I haven't done any maths since university days, so if I'm wrong I'm sure someone will put me right.
Some quantities, like weight, can be added and therefore averaged. If you take an 2oz mass and a 1 oz mass you can say with certainty that their total mass is 3oz. Because the sum of the two masses means something, you can calculate an avarage of 1.5oz and this figure has a useful meaning also. These kinds of measures are called extensive variables. Pressure, on the other hand, can't be treated in this way. If you add a system at 2 atmospheres to one at 1 atmosphere you don't get a system at 3 atmospheres. Because the sum of the two pressures has no meaning, the average likewise is meaningless. These are called intensive variables.
Temperature, as you might suspect, is an intensive measure. This means that when you add two temperatures together, the answer cannot be a temperature. It's meaningless. As the authors point out, dividing this meaningless sum by the number of components cannot give you an answer which has a meaning.
If the average of temperatures is not a temperature, then perhaps it's an index - a number which tracks whatever it is that drives the climate? If this is the case, then it is presumably necessary to describe how the average of temperatures - a statistic - is driven by the underlying climate driver, or at least to show some correlation between the two. They also need to demonstrate that the statistical measure they have chosen is better than any other measure they could have chosen. These alternative measures might well demonstrate a completely different trend to the average.
A third alternative is that the average is neither a temperature or an index, but a proxy for energy. But unfortunately there appear to be problems with this argument too. For a start, to do so is to use an intensive measure as a proxy for an extensive one. Secondly, the relationship between energy and climate is not understood. How then is it possible to know that the average of temperatures is a valid proxy?
It's not instantly obvious to the lay reader, but there are lots of different kinds of means. We're used to dealing with arithmetical means ("averages") but you can also have geometric means, harmonic means and any number of other means. For some systems, physics suggests which is the correct one to use. But, alas, this is not the case for global temperature.
As if to rub this point in, the paper demonstrates that there are in fact an infinite number of different means for global temperature. Which, they ask, is the correct one? Why has the scientific community alighted on the mean it has? They go on to show that, for the same set of data, different means can show a rising trend or a falling one. In other words, if a different averaging method to the one used in climate science had been chosen, we might now be having a crisis about global cooling... again.
It's a fascinating piece of work, some of which is beyond my understanding. If you are mathematically inclined, do take a look and tell me what you think.
Reader Comments (16)
http://timlambert.org/2005/04/gwsbingo/
See first two cells.
Global average temperatures are tricky even for the Hadley Centre; whose figures are used by the IPCC. The beginning of the 20th century is now 0.2C cooler. Why? How? They will not disclose how they calculate their figures.
See this post on Climate Audit:
http://www.climateaudit.org/?p=1106
I thought this would cheer you up?
What's the reason why not? You're meant to be setting me right.
Kit
I must say, I do think anything which is not verifiable should be thrown out on the spot.
"I always cringe when I see Tim Lambert's Bingo used to defend AGW"
I would cringe too if I was on the side of an argument whose proponents routinely used arguments so commonly advanced and rebutted - and in some cases so daft - that canned answers were available.
Bishop,
Did you read the link? I would have thought that the notion that global temperature can be anything you want plus can go up or down unconstrained by natural laws or empirical observations was absurd enough on its face. Most people would be inclined to check if they'd divided by zero somewhere or forgotten about a law of physics before going any further, surely?
But that is what this claim amounts to. Do you really think that whether the world is warming or cooling is a matter of definition/taste? Is it really meaningless to say that during ice ages the world was on average colder than it is now? Is it really plausible that under some other definition of 'average' I can say the last ice age was warmer than today?
From what I can see Lambert is not writing about the same paper that I have been reading.He refers to a book which I haven't seen.
Lambert's point seems to be that there are some errors in the data series worked on. I don't think that changes the point that different averages can produce different trends does it? The paper I've referred to does this theoretically as well as on a temperature series, so I think it's sound.
He also says that a weighted average is the correct measure because that will be the temperature when equilibrium is reached. What I don't understand is - weighted by what? If you had two masses and joined them with a conductor then, as Lambert says, the temperature at equilibrium would be the weighted average(by mass) of the two temperatures. But when you have two air temperatures what is it? Presumably it's something to do with air pressure, viscosity, specific heat capacity or something like that.
From the paper's explanation of the weighting process used, this is done to give equal influence to isolated sampling points as to those more tightly bunched. As far as I can see this can't underpin Lambert's explanation of the theoretical justification for the use of an average. He needs to explain why if he were to join two sample points with a pipe, the air would obtain an equilibrium temperature equal to the average of the two.
Am I making sense?
Anyway. Time for bed.
I am sure there are subtleties like that to be explored but the idea that this makes average temperature essentially so arbitrary that you can turn a rising trend into a falling one still seems like nonsense, surely?
Plus, if you follow this to its conclusion then all the denier's claims about temperature fall with it. After all when they were claiming that temperature wasn't warming, or talking about medieval temperature, or CO2 lagging temperature, they must have been talking about something, right? Are all those statements incoherent also?
Apparently these guys have got their paper published now since Lambert wrote that rebuttal. Tim et al are all over that here:
http://scienceblogs.com/deltoid/2007/03/party_at_rabetts.php
Also re the GGWS, there is some interesting stuff emerging here:
http://scienceblogs.com/stoat/
Have you read the paper? I think they demonstrate pretty effectively that you can get a different trend from a different average, because they demonstrate it to be true in theory as well as on some real data.
The argument against seems to be that the they are measuring temperature anomalies rather than absolute temperatures.
I'm scratching my head at the moment and trying to work out whether this remains valid. I'm struggling to see that a trend in an anomoly in an average (where the anomoly is against another average) somehow eliminates the "average has no meaning" problem.
(Just to clarify that last bit, as I understand it the weather stations report the max and min temperatures for the day which are averaged. This is then compared to the 30 year moving average.)
Weighted by the heat capacities (= specific heat capacity * mass) of the different components, provided that the heat capacities are not themselves functions of temperature, otherwise the mean temperature is the solution of a more complicated equation.
Tim Lambert is correct. There is nothing inherently problematic about defining a mean temperature.
But from what I understand the weighting that is used is done in order to move from the actual coverage of sampling points on the surface of the earth to a uniform one.
If this is going to be valid, we have to be measuring identical things at each point. If this is air temperature, then we are assuming that the composition of the air is identical in each place, and, I guess, that the air pressure is identical too. Can these really be valid assumptions?
But with regards to your other points, no, you don't have to make any assumptions about uniform composition and pressure of the atmosphere. The concept of a mean temperature is still perfectly well defined.
But as it happens, these are reasonable assumptions. The atmosphere is roughly 80% N2, 20% O2 + traces of other gases everywhere. Variations in composition only have a small effect on the heat capacity of the air. And at atmospheric conditions, the heat capacity of a gas is independent of the pressure, so that assumption is simply irrelevent.
The wikipedia article on SHC suggests that it is dependent on pressure. Is this wrong?
My assertion is qualified by the phrase "at atmospheric conditions". At very high pressures, where an ideal gas law doesn't apply, such is in a high-explosive, then the inter-molecular forces will affect the heat capacity.
But at atmospheric conditions, the inter-molecular forces can be reasonably approximated as instantaneous collisions, which have no effect on the heat capacity. Then the molar heat capacity, at constant volume, is
(no. of degrees of freedom/molecule) * (gas constant)/2, provided that the temperature is low enough that the internal structure of the molecules is unimportant, and high enough that other quantum mechanical effects can be ignored.
My naive understanding of the issue is driven by a comment on Real Climate which justified their weighted average approach by saying that if you have two masses (presumably of the same material) and join them with a conductor then they will eventually reach the weighted average of the two starting temperatures. Fine. But for this to work for two gases you have to know how much of the gas you have at each point at which you measure T. They are trying to get from a series of temperature measurements the majority of which are land-based to one where they are spread in a uniform fashion over the globe, you recall.
I had (again possibly naively) imagined a layer of unit cubes of atmosphere over the surface of the earth, which could be weighted so as to give an average T.
I understand your point about molar heat capacities being largely unaffected by pressure, but there is a different number of moles of air (or its constituents) in each of my unit cubes because the air pressure is different. Therefore a weighted average can't apply.
Or at least that's my theory. ;-)
They don't need to be of the same material. The argument is quite general.
"Therefore a weighted average can't apply."
Yes, it can.
The weight's are the heat capacities of the different components. So, in your example, that would mean the heat capacities of each of your cubes, which is just the molar heat capacity (approximately constant, since the composition doesn't vary very much) multiplied by the number of moles in the cube.
I should also add that the air pressure (and density) fluctuates on a much shorter time-scale (of the order of a day or so) than the changes of interest, so any errors introduced by unmeasured density fluctuations will be reduced by filtering the data.
It may be possible to argue that a particular calculation hasn't weighted or filtered the data properly. However, it's simply nonsense to claim that the weights are arbitrary and there is no definition of mean temperature that should be preferred over any other.