Guest essay by Antero Ollila
I have written blogs here in WUWT and represented some models of mine, which describe certain physical relationships of climate change. Every time I have received comments that if there is more than one parameter in my model, it has about no value, because using more than four parameters, any model can be adjusted to give wanted results. I think that this opinion originates from the quote of a famous Hungarian-American mathematician, physicist, and computer scientist John von Neumann (1903-1957). I found his statement in the Wikiquote:
“With four parameters I can fit an elephant, and with five I can make him wiggle his trunk”.
I do not know what von Neumann meant with his statement, but I think that this statement can be understood easily in the wrong way. I show an example in which this statement cannot be applied. It is a case of curve fitting. My example is about creating a mathematical relationship between the CO2 concentration and the radiative forcing change (RF). Myhre et al. have published equation (1) in 1988
RF = 5.35 * ln(C/280) (1)
where C is the CO2 concentration in ppm. Somebody might think that clever scientists have proved that this simple equation can be deduced by a pen and paper, but it is impossible. The data points RF versus CO2 concentrations have been calculated using the BBM (Broad Band Model) climate model and thereafter Eq. (1) has been calculated using a curve fitting procedure. There are no data points in the referred paper but only this equation and a graphical presentation. The BBM analysis method is the most inaccurate method of the radiative transfer schemes. The most accurate is the LBL method (Line-B-Line), which I have used in my calculations. Anyway Myhre et al. have shown in their paper that BBM and LBL methods give very closely same results.
I have shown in my paper that I could not reproduce Eq. (1) is my research study: https://wattsupwiththat.com/2017/03/17/on-the-reproducibility-of-the-ipccs-climate-sensitivity/
I carried out my calculations using the temperature, pressure, humidity, and GH gas concentration profiles of the average global atmosphere and the surface temperature of 15 ⁰C utilizing the LBL method. The first step is to calculate the outgoing longwave radiation (OLR) at the top of the atmosphere (TOA) using the CO2 concentration of 280 ppm. Then I increased the CO2 concentration to 393 ppm and calculated the outgoing LW radiation value. It happens in two steps: 1) transmittance or a radiation emitted by the surface and transferred directly to space, and 2) radiation absorbed by the atmosphere and then reradiated to space. The sum of these components shows that the OLR has decreased 1.03 W/m2 due to the increased absorption of the higher CO2 concentration. The other points have been calculated in the same way.
The values of the four data pairs of CO2 (ppm) and RF (W/m2) are: 280/0; 393/1.03; 560/2.165; 1370/5.04. Using a simple curve fitting procedure between the CO2 and the term ln(C/280), I got Eq. (2):
(2) RF = 3.12 * ln(C/280)
The form of equation is the same as in Eq. (1) but the coefficient is different. For this blog I carried out another fitting procedure using the polynomial of the third degree. The result is Eq. (3)
(3) RF = -3.743699 + 0.01690259*C – 1.38886*10-5*C2 + 4.548057*10-9*C3
The results of these fittings are plotted in Fig. 1.
As one can see, there is practically no difference between these fittings. The polynomial fitting is perfect, and the logarithmic fitting gives the coefficient of correlation 0.999888, which means that also mathematically the difference is insignificant. What we learned about this? The number of parameters has no role in the curve fitting, if the fitting is mathematically accurate enough. The logarithmic curve is simple, and it shows the nature of the RF dependency on the CO2 concentration very well. The actual question is this: Are the data points pairs calculated scientifically in the right way. The fitting procedure does not make a physical relationship susceptible and the number of parameters has no role.
So, I challenge those who think that an elephant can be described by a model with four parameters. I think that it is impossible even in two-dimensional world.
All is already said in the comments above but a simple curve fitting and intrapolation is generally OK. When you go to extrapolation then you may go wrong. This is situation in the climate models. And there is no simple fitting but elephant fittings.
Still confused by this LWIR whereby half goes off to space and half back down, as if that is the end of the story. Surely that half gone down is similar to solar LWIR, it gets caught up in another cycle where half (of the original half) goes off to space, rinse and repeat over and over for quarter, then an eighth, until it has all gone to space. This is broadly the same as saying it all goes to space, so there is no need to detail downwelling IR physics and mechanisms. Geoff
Just one general comment here. Nobody has commented “the canonical form” as Schmidt et al. (2010) have called it – namely equation (1) of Myhre et al. What is the calculation basis of it: a) Can you find it, b) It is the correct average atmosphere, c) What are the data points, d) What are the validation evidences of the climate model? I can tell you that this equation just pops up in the original article and there are no validation data. But equation (1) is still the corner stone of IPCC approved climate models.
Myhre’s paper is here. But the log dependence was knows to Arrhenius, in his 1896 paper:
About validation of my model. The warming effects of CO2 are always based on calculating the total absorption caused by GH gases in the atmosphere. In the clear sky conditions, the total absorption should be the same as the downward total absorption at the surface according to the Kirchhoff’s radiation law. The synthesis analysis of Stephens et al. (2012) shows an average value of 314.2
Wm-2 of 13 independent observation-based studies. The value of the same flux by Ollila (2014) is 310.9 Wm-2 meaning the difference of 1.0 %. The error of CO2 absorption calculation code of HITRAN (2012) has been confirmed to be less than 1 % in the actual atmospheric conditions (Turner et al., 2012). The LW radiation flux at TOA in the clear sky conditions according to the spectral analysis calculations (Ollila, 2014) is 265.3 Wm-2. According to the NASA CERES (2017) satellite observations from 2000 to 2010 this flux has been 265.8 Wm-2. The difference is only 0.19 %. These results are not a direct validation of the correct warming value of CO2 but is rather convincing that the atmospheric model and the LBL-calculation give results so close to the observations.
And the main results of my model versus IPCC model: climate sensitivity parameter 0.27 / 0.5 and the RF value of 560 ppm versus 280 ppm 2.16 / 3,7 W/m2.
Here is a recent paper titled
“Why logarithmic? A note on the dependence of radiative forcing on gas concentration”
I derived the log equation from particle physics years ago and posted it in Judith Curry’s blog. The coefficient cannot be derived from theory but from curve fitting of model or empirical data
Below is what I posted a couple of years ago. I didn’t show the actual derivation of the “Strangelove Equation” I must have written it in a scratch paper and threw it away after posting this. LOL My equation uses change in temperature rather than radiative forcing but you can convert RF to dT by using the empirical relationship 3.3 W/m^2/K
Salby also misunderstood the role (or non-role) of atmospheric opacity to longwave IR in the greenhouse effect. Opacity is not important. What is important is what is called in particle physics as the mean free path. In Feynman’s famous lecture, he used the bullet analogy to explain the double-slit experiment in quantum mechanics. I will also use the bullet analogy to explain the particle interaction between CO2 molecules and IR photons.
Imagine the IR photons are bullets emanating from the ground going up. The atmosphere is a layer of foam. The CO2 molecules are steel balls impeded in the foam. The bullets can penetrate the foam so we say the foam is transparent to bullets. But when a bullet hits a steel ball, it stops the bullet so we say the steel ball is opaque to bullets. If there’s enough balls impeded in the foam, the bullets cannot penetrate the foam so we say the foam is now opaque to bullets.
But there’s a difference between bullets and IR photons. The photons are constantly being re-emitted and re-absorbed. Like a bullet after hitting a ball, it stops and it moves again hitting another ball and so on until it emerges on the top side of the foam. So the foam isn’t really opaque to bullets. The average distance the bullet moves before hitting a ball is the mean free path. The more balls in the foam, the shorter the mean free path.
Every time a bullet hits a ball, the bullet imparts its kinetic energy converting into thermal energy. If we are dealing with just one bullet, the number of hits doesn’t really matter since its kinetic energy is fixed. No increase in thermal energy. But we are dealing with a constant stream of bullets like a machine gun firing. If we take a snapshot (time interval dt = 0) of a foam with one ball vs. with 10 balls, we will see one hit vs. 10 hits. The number of hits matter since the thermal energy is 10 times greater.
Combining the Stefan-Boltzmann law and the equation of mean free path, which I will not bore you with the details, this equation is derived:
dT = logx (N/No)
Where dT is the change in air temperature; N is the no. of CO2 molecules per unit volume; No is the initial value of N; x is a logarithmic base, which is an empirically derived constant. I call this the Strangelove Equation
It can be shown that the Strangelove Equation is mathematically equivalent to the generally accepted formula:
RF = 5.35 ln (C/Co)
Where RF is the TOA radiative forcing; C is the CO2 atmospheric concentration. I leave the mathematical proof as an exercise for interested readers
If the coefficient cannot be derived from the theory, is it a theory?
To Dr. Strangelove and to pagyfylec.
Firstly I agree with pagyfylec. The coefficeint 5.35 has not been deduced or derived by a theory and using onlu brains and a pen. It is a result of the curve fitting procedure. The curve fitting is not á problem itself. But the data points are. Therefore the value of the coefficient according to my calculations is 3.12. And I have much better validation to show than Myhre et al. have done.
Ave and paqy
I do not assume the value of the coefficient. My equation uses the log base as a free parameter. This is the simplest way to write the relationship between temperature and greenhouse gases that is consistent with physical laws (SB law and particle interaction) But the climate is more complicated because of heat transfer via fluid dynamics.
If a coefficient or constant is derived from experiments or observations, is it still a theory? Yes, this happens all the time. Newton did not determine the gravitational constant G from his theory. G was determined from astronomical observations and experiments by Cavendish. The Standard Model has many empirically derived constants.
You ask a lot. I answer a little in my blog:
https://roskasaitti.wordpress.com/2018/02/23/myhren-yhtalo-modtran-ja-ollila/
(language: very Finnish)
Some more information my be find here:
https://wattsupwiththat.com/2014/04/12/a-modtran-mystery/
Sorry to say but those are not complete answers to Your questions.
As a professional scientist (analytical chemist) I routinely used curve fitting in my work, mainly to construct calibration curves for analytical instruments. There have been many times I have had to caution the engineers I work with that the ONLY practical purpose of curve-fitting is interpolation of measurements. Any *extrapolation* performed with a fitted curve is an invitation to disaster, as David Middleton’s chart demonstrates.
From another retired analytical chemistry, full agreement with you. After decades of hindsight, it seems that analytical chemistry has particularly tight criteria for methods, not just curve fitting, but understanding accuracy and precision as well?
The success or failure of the analytical chemistry dependent greatly on good accuracy performance.
(A metallurgist colleague once described analytical chemistry as a mineral process with 100% recovery). Geoff.
I do fully agree.
https://publications.mpi-cbg.de/Mayer_2010_4314.pdf
Well, it can be argued that 4 or 5 complex parameters are actually 8 or 10 parameters, but, still…
Wow. I didn’t think this before but using even a single parameter technically may add an arbitrary amount of information! Let p be a ‘master’ real parameter, and let N be a number of mutually independent real parameters. Each parameter coefficient a(n) where 0 < n <= N, can be defined as the sum of every N'th bit in p starting from bit n. Each a(n) behaves as it was an independent real, but they're all defined by the chosen p. In physics, this kind of behaviour is not usual, but in a chaotic system, a parameter value may have more effect than what it looks like.
This is the process you use to make a bijection between R and R^n. Works in math, but not admissible in physics, where we have a stronger demand: each parameter must have its own unit and meaning. Obviously your master parameter is just a dimensionless index with no physical meaning, so it doesn’t qualify as a parameter for a physicist.
I’m not sure von Neuman was thinking of taking the same parameter to many powers.
@Antero Ollila
Your process seems fine to me, in that, it allows for prediction and test. It is just an abacus.
You can now do the calculation for any new point, and test if the value you get this way match the value you get out of your curve-fitting .
and you MUST do it.
A pretty standard test is: “what happen at point zero?”,
both equation (1) and (2) RF = K * ln(C/280) (where K is either 3.12 or 5.35) yield that if CO2 disappears from atmosphere, RF drops to negative infinity, and ground temperature drops to absolute zero
Do we need to qualify such claim?
So both eq (1) and (2) are WRONG, period. No matter how well they fit any number of points.
On the other hand, your eq. (3) means that, if CO2 drops to zero, RF = -3.743699. Or, said otherwise, contribution of 280 ppm CO2 to GHE is 3.74 W/m². It doesn’t seems totally absurd. But it is totally different from previous claim, and you have no right, just because it fits well, to claim that is true, or even just better than another curve fitted with another polynomial equation with 2 or 6 parameters. .
Van Neuman warning is not about the number of parameters per se, it is against the belief that the parameters you obtained are the mathematical translation of a real physical process, so it automatically has explicative power, and you can automatically take it for some law of Nature
Said otherwise:
do you claim that, in eq.(3) RF = -3.743699 + 0.01690259*C – 1.38886*10-5*C2 + 4.548057*10-9*C3,
the coefficient 0.01690259 is the effect of some physical process?
pagyfelec.
I did not showed this feature in my original text, because I could not think that somebody would use this fitting curve outside its limits. Those limits are the data points shown in text and the lower limit is 280 ppm and the upper limit is 1370 ppm. Very probably the concentrations up to 2000 ppm works stil fine, but there is no need for that, because we will never see those conditions.
Well, I miss the part where you issue the caveat that it can be used for interpolation, but certainly not for extrapolation, as Tadchem above warned. I stand corrected.
Then I just don’t understand the point of your post.
Obviously, mathematics says that you CAN perfectly fit a 4 point set with any 4 parameters (or more) function, like a polynomial of order 3 or more. And, if you allow some non-perfect fit, you can add any function with Taylor polynomial, which includes basically all function we are familiar with ( ln, exp, sin, etc. ) and some more
So you did. And there were infinite number of ways to do it. With exactly zero physical meaning (even if the four point were perfect measurement of the world).
So what?
There MAY BE something interesting when you succeed in fitting four points with LESS than four parameters, because this is unusual, mathematically speaking. If this happen, either you are very (un)lucky, or there is a law that constrains the four points, and this is exactly what you are looking for. The more points, the less parameters, and the more interesting the relationship, and the chance it efficiently describe real world.
But it looks like you don’t claim to have theory, just a curve fitting between 280 and 2000 ppm. What’s the use?
Curve fitting is certainly OK for lots of use, like, for instance, to calculate a single parameter, all other being known or constrained by mathematics or physics as described https://wattsupwiththat.com/2018/02/22/curve-fitting-and-the-number-of-parameters/#comment-2750362 . But here, none of them are known or constrained, and if I understand you correctly you don’t claim that any of them has physical meaning (that is, despite the eq 3 fit, you don’t claim that CO2 contribute 3.74 W/m² forcing when at 280ppm; (*) )
So, what was your point?
(*) obviously IPCC DOES claim that eq 1 has some physical meaning, and this make eq 1 very different from your eq 2, despite the similarity in form)
“And there were infinite number of ways to do it”
Sure. But you let your imagination run too much. A nice smooth approximation is probably OK in intrapolation and pretty OK for some extrapolation.
So how large an error follows from using a linear fit between 280ppm and 500 ppm (something like from 1900 to 2070) compared to an ln fit?
Anyway, your claim is interesting, and, if it was me, I were just thinking what I got wrong in my data points. But, every now and then, people get things more right than never before.
A good question. The maximum difference of the RF value is 0.131 W/m2 corresponding the temperature difference of 0.035 degrees Celsius only. Sometimes the eyes tells the truth quite well.
The polynomial fitting and the log fitting are a close match to each other only for a limited range of CO2 concentrations. Above this range, the logarithmic curve continues is decreasing slope which is what actually happens with CO2, while the polynomial curve will take off upwards due to the positive coefficient of it’s highest order term, a cubic one.
You can always improve a fit by adding more parameters, but the added parameters don’t necessarily mean anything. That’s what Occam’s Razor is all about. To check that a new term is worth adding, check it’s t score or partial-F value.
Since this is ONLY a curve fit based not on physics but on observations-to-date, here’s an additional observation:
The curve fit can have no better uncertainty than the observations it is fit to.
In other words:
– The surface temperature observations to date have an uncertainty. Just for grins I’ll pick a number not quite out of thin air: +/- 1 degree C.**
– This wonderful curve fit has the exact same uncertainty, even for interpolation purposes.
(**My son in law works with the biggest source of national standards calibration equipment in the world, and has pointed me to some very interesting libraries on the subject. I claim almost-ignorance as a result 🙂 )
(Oops. Should have said “has no better uncertainty” rather than “has the exact same uncertainty.”
The problem is that if you have as many as four or more free variables, such as a1, a2, a3 and a4, and put them in some function like f(x) = a1*x*x*x + a2 x*x + a3*x* + a4, it is usually possible to find values that match the graph quite exactly to any historical data.
This has often misled researchers, because they think the good match on historical data means that they have found a governing function for the data, and therefore use that function to predict future values.
Dr. Pielke used to include a polynom like this on his monthly temperature series, but he warned correctly that the graph was for entertainment purposes only
/Jan
A general comment concerning what was the meaning of my blog.
a). As I wrote, there are always comments, if there are more than one parameter in an equation that this equation has about no value, because using several parameters, anybody can fit a model as they like without any physical meaning. My example about the curve fitting shows that the number of parameters have no role in the curve fitting as long as the fitting is good enough. In this case a logarithmic fitting with one parameter is as good as a polynomial fitting with four parameters.
b) The curve fitting does not destroy the physical relationship like in this case of the dependency of RF on CO2 concentration. The equations (1) and (2) have that relationship and they are very handy to use. I find this claim in many comments and for me it means that those readets do not understand the physical and mathematical basis of these calculations.
c) I have reproduced the study of Myhre et al. in which way eq. (1) has been produced. I did not get the same result; the formula is same, but the coefficient is different. It was as not my invention to try logarithmic fitting, but I copied it right away from Myhre et al. and it works just fine. It looks like that some readers have no problems in accepting equation (1) even though they seem to have no idea, in which way it was created. But when I show the calculations and even validation information for my eq. (2), they think there is no physical basis at all. I have studied this issue pretty carefully and I can say that the only way to find out, if eq. (1) is correct or is eq. (2) is correct, is to carry out the same calculations. Then you will know.
“As one can see, there is practically no difference between these fittings. The polynomial fitting is perfect, and the logarithmic fitting gives the coefficient of correlation 0.999888, which means that also mathematically the difference is insignificant. What we learned about this? The number of parameters has no role in the curve fitting”
I don’t think your conclusion follows from your example curve-fitting exercise. All your example does is alternately fit two equations, each using a different number of parameters, to an extremely simple, slowly changing curve, and all you have demonstrated is that this simple curve can be reasonably “fit” by selecting coefficients in front of one independent parameter, and that adding parameters doesn’t get you much marginal benefit in this simple exercise.
But if the curve you’re trying to fit with an equation has many inflection points, i.e. goes up for a while, then down, etc., and at varying slopes each way – like, say. annually averaged temperatures over the last century – then the number of free parameters is certainly going to make a difference in your ability to fit an equation to the complex curve.
Kurt. Certainly so.
This is why I am interested in the understanding the most basic testable physics first . The mean temperature of an asymmetrically irradiate ball is TOTALLY determined by the power spectra of its sources and sinks and its absorptivity=emissivity spectrum . Calculating the change in that spectrum with changes in CO2 or any other surface or atmospheric component should be not impossible to quantitatively analyze .
Yet I have never seen this fundamental physical analysis laid out .
The problem is , once you do that , and have dissected the spectral components that go into the radiative equilibrium , apparently in the neighborhood of 20 degrees below that of a gray ball in our orbit , you are even further away from explaining the gap between that and our bottom of atmosphere temperature . And NO GHG quantitative testable theory for that has ever been presented , nor given the violations of math and thermodynamics will it ever be .
I appreciate the comment about all sorts of parameters being buried in thousands of lines of Fortran , That’s why I am only interested in APL level executable explanations which are as or more succinct than the explication of the physics in any textbook — leaving no place to hide .
To CO2isnot evi about the LW absorption
I write this answer here, because I could not find a proper place among the comments.
Your comment is correct, but I may disagree with some numerical values. I am sorry for using inaccurate language. I meant that the LW flux emitted by the surface will be absorbed almost totally (90 %) – the accurate number being 88 %. This means that only 12 % of this emitted LW flux can pass through the atmosphere without absorption. The cloudy sky conditions prevail about 66 % of the time and the cloudy sky absorbs totally LW radiation.
Another side of this phenomenon is that the absorbed LW flux does not disappear (transform into heat) but it reradiates into all directions. The overall figures are quite simple: The emitted LW flux is 396 W/m2, the outgoing LW flux at the TOA is 240 W/m2 and it means that the total energy flux absorbed is 156 W/m2. This radiation energy is transformed into heat and it is the main source in maintaining the temperature profile of the atmosphere.
This “flux” business is very misleading, because it can be “net” or “gross”, and here it is gross. Gross figure don’t add up, because other heat source/sink come into play and can be (and are ) converted from/to LW.
More proper calculation would be (per your figures)
The emitted LW flux is 396 W/m2, 88% of it absorbed, that is, 348, and 12% = 48 directly get through. So atmosphere need to provide 192 outside, to add up to 240, and has 348-192=156 (+ other sources: directly from the sun , convection, …) to contribute to “back radiation”
Temperature profile of the atmosphere only depends on lapse rate, which curiously only depends on some convection heat being provided at the bottom, to power the buoyancy stratification machine. Radiation energy doesn’t matter, as evidenced by “inversion” condition, occuring when surface radiation is at max and convection at zero
I disagree in one point. You write that “atmosphere need to provide 192 outside, to add up to 240, and has 348-192=156 (+ other sources:directly from the sun , convection, …) to contribute to “back radiation””. I understand this statement that the absorbed radiation flux 156 is used for back radiation. I do not buy this idea. The total absorbed radiation flux is this 156 W/m2 and it is transformed into heat (higher temperature of the air). In the same way the other sources of energy fluxes are transformed into heat: SW absorption by the air (56) and the clouds (14), LW absorption by the clouds (55), thermals (25) and latent heat (90). The LW flux into space is the sum of three flux: cloud top radiation (40), the atmosphere (170) and the transmitted radiation (28) – totally 238 in this energy balance scheme.
The absorption by the atmosphere starts from the surface and it continues through the atmosphere but it is almost done at the tropopause (98 %).
The original study is here in the list: http://www.jcbsc.org/issuephy.php?volume=4&issue=1
well, I tried to avoid this misinterpration “the absorbed radiation flux 156 is used for back radiation”, by using “+ other sources” statement, and obviously failed. I should had been more explicit (but it meant longer and sidewalked)
Of course atmosphere just pool all energy, whereever it comes from, into heat, and then tap from the heat pool for all uses. There is no process directly transforming a part of absorbed LW directly into back-radiation.
Besides, I just hate this average budget thing, basically equivalent to a flat homogenous non rotating Earth close to equilibrium.
Earth is NOT any of these thing. Not flat, it has poles and tropics. Not homogenous, it has sea and land. Rotating, it has days and night. It has heat reservoirs that alternatively fill and dump, with hugely different size and in and out velocities. It as LIFE, for god sake, that tap on sun energy and is the very reason atmosphere has and keeps O2 instead of CO2 and CH4. Out of equilibrium, everywhere and everytime. In fact, the whole point of climate science is to figure out all the process struggling to bring back the equilibrium, and … failing.
It blows my mind that some people dare call themselves scientists, when working with such crude conceptions. These people would be useless just to understand a basic home electric circuitry, or financial operations of their round the corner shop, and they claim to cope with climate. And they are believed. Amazing.
To CO2isnot evil about the impacts of clouds.
The absorption of CO2 in the atmosphere happens below 1 km altitude. The cloud ceiling (the bottom of the cloud cover) is globally in 1.5 km altitude. In this sense clouds have no effect in the absorption process of CO2. When we talk about the clouds, we should always remember that the clouds have two roles: they reduce incoming SW radiation and increase the absorption of the outgoing LW radiation. (By the way the SW absorption by the atmosphere is about constant regardless of the degree of the cloudiness (70 W/m2)). The cloudiness varies all the time regionally. The global temperature measurements show that globally the surface temperature is 0.3 C higher than during the clear sky. Because the overall GH effect is about 34 C, the net contribution of clouds in GH effect is only about 1 %. Schmidt et al. (2010) calculated the LW absorption effects of the clouds but they left deliberately off the SW effects.
You cannot use surface temperature being 0.3 C higher average than during the clear sky as a 0.3C net cooling effect of clouds. Only an effect of the condition that result in clouds (or not), including whether you must use dry or moist lapse rate, and very different balance at day or night.
No surprise that clouds correlate with cooler condition, since they exist when it is cool enough for water to be water or ice instead of gas.
Correlation is not causation; and causation must not be reversed.
About the emissivity of CO2.
I do not know this issue at all. I have used the LBL-method, which is based on the spectral analysis calculations. I have not seen the emissivity of CO2 anywhere. The spectral calculations are based on the absorption and emission properties of GH gas molecules and the properties of the real atmospheric conditions: composition, temperature, pressure, altitude. For me it looks like that CO2 emissivity has no role in these calculations but I may be wrong, because I am not an expert in molecular physics.
You are not wrong, mkelly is, if this is what you are refering to. You have not seen emissivity of CO2 anywhere because, as co2isnotevil pointed out, gas “essentially emit everything that they absorb, so in an absorption band, the effective emissivity is 1.0 and outside the absorption bands, the effective emissivity is 0. ”
Said otherwise, you did see emissivity of CO2, that’s nothing other than spectra of CO2, but you don’t find a single number that would be called “emissivity of CO2” that would be good for every use. I guess you did a calculation that could be called emissivity of CO2 in the situation you examined.
I can buy easily your explanation
It’s worse than you thought…
Dr. Ollila’s taut article (2/22/2018) of just 700 words conceals two huge icebergs with peaks poking up through the blanket, one mathematical and one scientific.
Von Neumann’s remark refers to the fact from mathematics that given enough independent variables, an exact fit exists to any finite number of data points. Ollila demonstrates that himself by fitting a third degree polynomial (4 independent terms) to 5 data points. Two points determine, not just fit, a straight line; 3, a quadratic; 4, a cubic; etc. Curve fitting to data, which includes noise, is statistics, which has little predictive power. It is the failing species of science — Popperian academic science, not Baconian real science. Predict with a statistical model requires the assumption that whatever Cause & Effect relationships existed when acquiring the data remain in effect for the analysis.
Curve fitting is vital in science for postulating Cause & Effect relationships. It should be evaluated on the basis of its power to represent data, measured by R-squared, which is directly convertible to signal to noise power ratio, (SNR).
Ollila’s equation (1), above, RF = 5.35 * ln(C/280), is for Earth’s Outgoing Longwave Radiation (OLR). He attributes it to Myhre, et al., 1988 but the correct date might be 1998. Myhre (1998) provides the equation, comparing it in Figure 1, p. 2717, to an earlier estimate by Hansen, et al., (1988). Myhre, et al., reports the equation in Table 3, p. 2718, which IPCC modifies slightly for TAR Table 6.2, p. 358, including Hansen’s ln-polynomial, but in every case under the qualification simplified expressions. However, none of the sources provides an earlier, more complex expression. It is a biased approximation from modeling.
Equation (1) and Hansen’s ln-polynomial are already fits to data – data from simulations. Myhre fitted to radiative transfer (RT) models, and Hansen to temperature models. For temperature models, the relationship is logarithmic by virtue of the climatologists’ modeling assumption that a scaling of C, the concentration of CO2, produces an increment in temperature, ΔT, which IPCC named the temperature sensitivity. Myhre’s logarithm is simply a free choice. A logarithm is readily fitted to any concave down data set, and is exact in rare circumstances.
The logarithmic relationship is essential to the RF climate modeling. It has the effect of being invariant to a scaling of C, in particular to a doubling of CO2 concentration with respect to the chosen baseline, e.g. 28 ppmv, 28000 ppmv, or anything else. That means that the modeler needn’t bother to figure out an operating point, the reference point for CO2 concentration.
The logarithmic equation is accurate, only over the region to which it was fitted. This RF application derives from OLR absorption, which varies between 0 and 1 (100%). The logarithm, though, ranges between minus and plus infinity. So the fit at best is an assumption, an approximation, and a modeling convenience.
IPCC reported that the RT error was not just large, but the largest source of error in its RF modeling of climate. One reason for this error is that radiation absorption is nonlinear, which means that the average RT over Earth is not equal to the RT over the average atmosphere. No method exists for objectively determining an effective atmosphere, including cloudiness and greenhouse gas concentrations (esp. water vapor and CO2), that will cause RT to produce an objective and realistic average OLR.
Consequently, the RF model for climate, relying directly or indirectly on Equation (1) and implemented in the GCMs, does not work, which should not surprise anyone. Worse, the GCMs predict an Equilibrium Climate Sensitivity (ECS) for a doubling of CO2 of 3C, estimated from measurements at 0.7C, has a confidence level of about 2.5% per IPCC, where anything less than, say, 80% would be deemed unsuccessful. Much, much worse, the measured ECS assumes that the CO2 is the Cause and the Temperature rise, T, is the Effect, requiring that CO2 lead T. Physics (Henry’s Law) says CO2 will lag T. Climatologists have yet to report which leads and which lags. The Vostok record confirms the physics. Not only that does CO2 lag T, but that it lags by as much as a millennium.
A vital role exists for curve fitting and maybe counting the number of parameters. OLR radiative forcing just isn’t it.
Unfortunately. the author has misunderstood von Neumann. And he is wrong in saying about his graphic, that “The fitting procedure does not make a physical relationship susceptible and the number of parameters has no role.”
Von Neumann was making the point that a polynomial with enough terms can be fitted even if you have no knowledge of the process that generated the data. The path of a short-range missile conforms to a parabola, for which the physical process is known. If you had no knowledge of the process, you could fit the path of the missile with a polynomial.
In the case of the author’s graphic, the logarithm fits because the generating process is multiplicative. In the case of the relationship between CO2 and its radiative effect, the multiplier is less than unity. You would choose the logarithm because you already have a theory.
Well you would if you had been in the UK during the bombing when you had those semi-transparent blackout curtains. Each layer of curtain blocked out some light emitted from your window, but with each layer you got diminishing value for money. Or you might have been involved in using thin layers of lead to act as X-ray shielding and you may have observed the same thing as with the blackout curtains.
Von Neumann was making the point that a purely empirical approach (using polynomials) does not tell us anything about the physical process that generated the data. We cannot go from pure empiricism to theory.
Which is why Richard Feynman said, “In general, we look for a new law by the following process. First, we guess it…”. This is how Feynman got around the problem faced by the logical positivists. It’s also a simple exposition of Karl Popper’s basic idea about theory in science.
I looked at the data for delay at a port and I saw a hyperbola. Then I fitted a rectangular hyperbola to the data.
In a recent paper the Viscount Monckton analyzed the fit of a rectangular hyperbola to data for estimating the water vapour feedback parameter and its impact on climate projections.
Household income distribution can be calculated from Household Income and Expenditure surveys. I see lognormal distributions because I am interested in the incomes of working people up to about the 80th percentile. Others focus upon the Gini Coefficient or a Pareto distribution to describe income distribution. They seem to be more interested in income inequality and the higher income groups above the 80th percentile.
These examples show what Richard Feynman meant when he said we guess at theories.
Our guesses about theories depend a lot on the perspective we bring to the table.