Curve fitting and the number of parameters

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.

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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.

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168 thoughts on “Curve fitting and the number of parameters

  1. I don’t think the issue is necessarily that models with more parameters are wrong, just that the more parameters a model has, the more places it can be wrong.

    A classic example is RADIATION.F in the GISS GCM called ModelE. This one piece of spaghetti Fortran code has thousands of baked in floating point constants, many of which have absolutely no documentation, most notably, those related to atmospheric absorption coefficients.

    On the other hand there’s the SB equation for a gray body, which has only one parameter called the emissivity. For calculating the yearly average relationship between the surface temperature of a region of the surface and the emissions into space above that region, it fits almost perfectly using an effective emissivity of about 0.62 +/- 5% across all regions of the planet from pole to pole.

    • co@isnotevil

      Is that emissivity for a particular CO2 concentration?

      I agree with taking a grey body approach, mainly because the atmosphere is not a black body in the relevant wavelengths.

      How then does ‘darkening’ the atmosphere by adding CO2 affect the emissivity?

      Next point: There are two contributors to the atmospheric temperature just above the ground: convective heat (Hc) transfer from the hot surface and radiated or re-radiated infra-red energy.

      Regular CO2
      Hc + ir => Air temperature

      It is claimed that as the CO2 concentration increases, the temperature of the surface air will increase as the emissivity of the atmosphere increases in both directions: up and down.

      Hc + CO2 => HC

      High CO2
      HC + ir => Higher air temperature

      What is strange is that if we take the opposite tack, that of reducing the CO2 concentration and reducing the emissivity of the atmosphere to a low value, the equation for air temperature still has two components:

      Low CO2
      Hc + ir => Air temperature

      but the atmosphere will have lost its ability to cool radiatively (low emissivity).

      The reason ir thermometers have a scalable emissivity function is to be able to get the correct temperature from the IR being radiated. When the surface is like polished silver, it emits nearly no IR at all and the IR device reads Low. A correction is made to report a higher temperature (the real temp) by adjusting the emissivity button.

      Polishing metal to a mirror is not different from an atmosphere that loses its ability to cool by radiation. The Hc or HC contribution will continue to be made, of course, and the atmosphere will have lost the ability to radiate that energy. The air temperature will then rise inexorably until some gas starts to radiate or the air heats the surface sufficiently at night.

      If you do your calculations using an emissivity of 0.10 instead of 0.62, what air temperature must be reached in order to dispose of the same amount of energy as currently reaches the surface? It will be a heck of a lot more than 15 C at the surface.

      It is generally true that in order to cool a hot body more effectively, the radiating function – the emissivity – has to increase. Adding CO2 to the atmosphere increases the emissivity and is therefore bound to lower the temperature until it is again in net radiative equilibrium. It is very difficult to find CAGW in the physics of a radiating atmosphere (even without the clouds). Check Trenberth’s diagramme. The effect of Hc is missing and hidden among the radiative weeds.

      • Crispin,
        The emissivity changes slightly with increasing CO2 concentrations and it’s this slight change in emissivity that’s equivalent to the 3.7 W/m^2 of post reflection solar forcing said to arise from doubling CO2. Technically, only solar energy forces the system. Changing CO2 concentrations is a change to the system itself and not a change to the forcing, although any change to the system can be said to be EQUIVALENT to a specific change in the forcing keeping the system constant.

        Quantitatively, the effective emissivity can be calculated as 1 – A/2, where A is the fraction of NET surface emissions that are absorbed by atmospheric GHG’s and clouds, considering that half of this must eventually be emitted into space and the remaining half re-emitted back to the surface. The absorption coefficient A, can be calculated as the cloud fraction weighted absorption, where with no clouds, all absorption is by GHG’s and with clouds, absorption is dominated by the liquid and solid water in clouds, which like GHG’s eventually re-radiate that energy up and down in roughly equal parts.

        We should only be concerned with the AVERAGE temperature of the radiating surface, which is nearly coincident with what we consider the ‘surface’ to be. The temperature of the atmosphere doesn’t drive the surface temperature, but is a consequence of the surface temperature which is determined by its own specific radiative balance. Convection affecting air molecules in motion (i.e. their kinetic temperature) is irrelevant to the radiative balance as the translational motion of air molecules doesn’t participate in the RADIATIVE balance, but is the source of the energy required to obey the kinetic temperature lapse rate dictated by gravity. Note that whatever effects non radiative transports like latent heat and convection have on the surface temperature is already accounted for by that temperature and its corresponding BB emissions, thus has a zero sum contribution to the NET radiation of the surface whose temperature we care about. This is something else that Trenberth’s energy balance diagram is misleading about, especially since he refers to the return of non radiative energy as ‘back radiation’.

      • Crispin,
        One other point is that it’s not the emissivity of the atmosphere that’s being modelled, but the emissivity of a gray body model of a system whose temperature is that of the surface and whose modelled emissions are the fraction of emitted surface energy that’s eventually radiated into space. Any energy emitted by the atmosphere was emitted by the surface some time in the past and any energy emitted by the surface and absorbed by the atmosphere can be re-emitted into space or back the surface only once. The ultimate time or place where energy emitted by the surface ultimately leaves the planet or returns to the surface is irrelevant.

        Increasing CO2 concentrations decreases the effective emissivity and the energy that was not otherwise emitted must be returned to the surface, contributing to making it warmer than it would be otherwise.

      • Co2isnotevil said

        “Increasing CO2 concentrations decreases the effective emissivity and the energy that was not otherwise emitted must be returned to the surface, contributing to making it warmer than it would be otherwise.”

        Battisti and Donohoe say the exact opposite happens in the long run in their 2014 study that says after a period of time (decades)……”and the negative LW feedbacks that strongly increase OLR with warming” The name of the study is
        “SHORTWAVE AND LONGWAVE RADIATIVE CONTRIBUTIONS TO GLOBAL WARMING UNDER INCREASING CO2” They say it takes decades for the feedbacks to kick in but that eventually it is the SW radiation which causes global warming not the LWIR. The authors go on to say “Only if the SW cloud feedback is large and negative could the SW coefficient become small and the resulting energy accumulation be dominated by reduced OLR. That is all they say about clouds in the report but Battisti admits that he doesnt have a good handle on clouds.because the media summary copy which I assume was approved by the authors including Battisti before being released to the media said “clouds remain one of the big unknowns under climate change”. So if that is true and everybody seems to agree because noone can model clouds properly and no one has any equations for the clouds(in other words there is no theory that explains the clouds relationship to IR forcing then why in the hell havent all the climate scientists told the politicians that because clouds are too much of an unknown we should do nothing until we can figure it out.
        THEY HAVENT DONE THAT. INSTEAD THEY TOLD THE POLITICIANS THAT OUR MODELS PREDICT LARGE INCREASES IN GLOBAL WARMING THEREFORE YOU SHOULD ACT and we are all suffering financially because of it.
        I believe THAT ALL OF YOU ARE WRONG and that Willis is right about his convection with clouds theory having everything to do with it , assuming that changes in the sun cause only minor climatic changes (which noone has been able to prove or disprove). WHAT A MESS

      • Changing CO2 concentrations is a change to the system itself and not a change to the forcing, although any change to the system can be said to be EQUIVALENT to a specific change in the forcing keeping the system constant.

        How refreshing to see that stated.

      • We have a GH effect in our atmosphere and CO2 has a contribution of about 13 %. By increasing the CO2 concentration, the magnitude of the GH effect will increase, because the present absorption in the atmosphere is at the level of 90 %. The only way that CO2 concentration increase would have no effect, is the negative water feedback effect. According to the measurements, the amount of water vapor in the atmosphere is practically constant. Certainly it is not positive as calculated in computer models.

      • aveollila,

        The 90% absorption of surface emissions is incorrect. This comes from Trenberth’s balance and if you ask him where this came from, he has no good answer, My HITRAN based analysis tells me that the clear sky absorbs between 60% and 70% (avg 65%), is all by GHG’s and dependent on water vapor content. Nominally, water counts for more than 1/2 of this, CO2 is a bit more than 1/3, Ozone is most of the rest and a few percent arises from CH4 and other trace gases. Cloudy skies absorb on average about 80%, much of which is by liquid and solid water in clouds and would be absorbed independent of GHG concentrations between clouds and the surface. The planet is nominally 2/3 covered by clouds, so the total absorption fraction of surface emissions is 0.33*0.65 + 0.66*0.80 = 0.74, or about 74%.

        If 90% was correct, then 350 W/m^2 of the 390 W/m^2 of surface emissions is absorbed, leaving 40 W/m^2 passing through the transparent window. This requires 200 W/m^2 more emissions into space to be in balance, which can ONLY come from the 350 W/m^2 of absorption, leaving 150 W/m^2 to be returned to the surface, which added to the 240 W/m^2 of solar input adds up to the 390 W/m^2 of emissions. What this would mean is that only 43% of the energy absorbed by the atmosphere is returned to the surface while 57% of atmospheric absorption escapes out into space, even as the IPCC ambiguous definition of forcing requires 100% of what’s absorbed to be returned to the surface. Both violate the geometric constraint where half must be returned to the surface and the remaining half is emitted out into space.

        If 74% is absorbed, the total is about 290 W/m^2, leaving 100 W/m^2 passing through which requires additional emissions of 140 W/m^2 of the 290 W/m^2 absorbed to be in balance, leaving 150 W/m^2 returned to the surface which added to 240 W/m^2 offsets the 390 W/m^2 of emissions and is much closer to the required 50/50 split. A more precise analysis based on weather satellite imagery converges even closer to the 50/50 split dictated by geometric factors.

        Many are confused by Trenberth’s latent heat and thermals whose effect is already accounted for by the average temperature and its average emissions and offset by the return of latent heat and thermals back to the surface which Trenberth incorrectly lumps into his bogus ‘back radiation’ term. It’s crucially important to distinguish between energy transported by matter which is not part of the RADIATIVE balance and can not leave the planet (man made rockets not withstanding) and the energy transported by photons which exclusively comprises the RADIATIVE balance,

      • co2isnotevil,

        Two quick questions for you that have been vexing me.

        First, why 50/50 between surface and space for atmospheric radiation? The only place where the surface would equal 50% is right at the surface. That is, if we consider that any individual molecule can radiate spherically in any direction, the cross section of this sphere occluded by the earth’s surface diminishes from 50% as you ascend in altitude. Or, is this reduction so insignificant that it’s not worth refining? Just curious.

        Secondly, going back to a previous discussion in which, if I remember correctly, you argued that IR couldn’t thermalize through collisions since it was internal molecular energy …energy like stretching, or bending or whatever rather than kinetic (sorry, can’t remember the correct terms off the top of my head). I’m not sure I’ve understood your position correctly…but if I’m close, it leads me to ask why we accept that some types of EMR can thermalize, but not LWIR? Microwaves immediately come to mind. (Note, if I’m so far out in left field with my misunderstandings, then just tell me. I haven’t really had a lot time to dig into this in my thoughts to make sure I’m asking a sensible question, so I’m “emotionally prepared” for you to tell me to go pound sand.) :)

        rip

      • Rip,
        The 50/50 split is a consequence of geometry. The area of the top boundary with space is about the same as the area of the boundary with the surface where energy enters only from the surface boundary, but leaves through both boundaries comprising twice the total area upon which energy arrives. To be clear, the running global average is not exactly 50/50 and varies a couple percent on either side and locally, clouds can modulate the ratio over a wider range. It makes sense that with sufficient degrees of freedom, a system like this will self organize to be as close to ideal as possible, as this minimizes changes in entropy as the state changes. This is a consequence of deviations from ideal increasing entropy and the 50/50 split, as well as conformance to the SB Law describing a gray body emitter are both ideal behaviors.

        Regarding thermalization, the only possible way for CO2 to thermalize vibrational energy is to convert it to rotational energy that can be POTENTIALLY shared by a collision. These rotational bands are microwave energies and this energy represents the spacing of the fine structure on either side of the much higher energy vibrational modes. There’s no possible way to convert all of the energy of a vibrational mode into the energy of translational motion in one transaction since Quantum Mechanics requires this to be an all or nothing transfer, which can only happen by the collision with another GHG molecule of the same type where the energization state flips. Small amounts can be transferred back and forth in and out of rotational modes and is what spreads out the fine structure on either side of primary absorption lines. Rotational modes can theoretically be transferred to arbitrary molecules, but once more, this is an all or nothing conversion, although high N rotational modes are possible where less than N can be transferred. In general though, the transfer of state energy by collision is limited to collisions of like molecules where state is exchanged.

        The only thermalization mode that makes sense is when an energized water vapor molecule condenses upon a water droplet, warming that droplet.

    • Another point is how well known are the values of the parameters.
      If the values are well known and tightly constrained, there is not a problem.
      If the value is so poorly known that the modeler is free to put in whatever value makes his model work, then we have problem.

      • And a great many of those parameters are poorly known at best. The classic example is aerosols,. As you go back into time, we have very little knowledge how much aerosols have released in any given year. What the compositions of those aerosols were, and from where they were released.
        As a result you can plug a wide array of numbers into your model, with equal likelihood of entering the correct amount.
        If the model isn’t given you the result you want, just tweak the amount of aerosols until it does.

    • CO2isnotevil, I posted this before.

      In 1954, Hoyt C. Hottel conducted an experiment to determine the total emissivity/absorptivity of carbon dioxide and water vapor11. From his experiments, he found that the carbon dioxide has a total emissivity of almost zero below a temperature of 33 °C (306 K) in combination with a partial pressure of the carbon dioxide of 0.6096 atm cm. 17 year later, B. Leckner repeated Hottel’s experiment and corrected the graphs12 plotted by Hottel. However, the results of Hottel were verified and Leckner found the same extremely insignificant emissivity of the carbon dioxide below 33 °C (306 K) of temperature and 0.6096 atm cm of partial pressure. Hottel’s and Leckner’s graphs show a total emissivity of the carbon dioxide of zero under those conditions.
      http://www.biocab.org/Overlapping_Absorption_Bands.pdf

      • mkelly,
        Clouds are complicated because they affect both the input side by modulating the albedo and the output side by modulating the transparency of the atmosphere which also warms the surface as the transparency decreases.

        Your hypothesis can’t explain why we only see about a 3 db (50%) attentuation in those absorption bands where the HITRAN line data tells us that the probability of absorbing a photon emitted by the surface is nearly 100%. If the emissivity of the GHG molecules is as low as you say, what’s the origin of all the energy we see at TOA in those absorption bands?

        Also, the emissivity of the atmosphere is insignificant, relative to the emissivity of the surface itself, at least without accounting for energized GHG molecules and the water in clouds. GHG molecules act more like a mismatched spectrally specific transmission line between the surface and space.

      • I read that study there were significant questions with it. Until someone else does an actual experiment this is all conjecture the study author only did elementary calculations. and even those were confusing. . The study is a joke compared to a real scientific report. We need better than that to expose the hoax.

      • CO2, It is not my hypothesis but Hottel’s. He wrote books on radiative heat transfer and his charts are used in design calculations for combustion chambers.

        If you don’t like Hottel, then please tell me what is the emissivity of CO2 in our atmosphere at present concentrations.?

      • I agree with the comments of CO2isnotevil and Alan Tomalty. The study referred by mkelly is not convincing and its theory is not generally accepted by other scientists as far as I know. I rely on the Spectral Calculator LBL-method and the HITRAN database. The overall results of the outgoing LW radiation as well as the downward overall radiation matches with observations. For me it is the best evidence about the calculation basis.

      • @ mkelly
        what’s this strange “atm cm of partial pressure” unit?
        does this means you have to multiply by the actual thickness (kilometers for the atmosphere) relative to a cm standard (km/cm = 10^5)?
        In which case, well, you could easily have both a “total emissivity of almost zero” in the standard case, and a significant emissivity in the real world.

      • mkelly,

        “If you don’t like Hottel, then please tell me what is the emissivity of CO2 in our atmosphere at present concentrations.?”

        Emissivity is a bulk property and isolating the emissivity of a trace gas in the atmosphere is a meaningless exercise. GHG’s 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. It’s also crucial to understand that when I talk about the emissivity, I’m talking about the bulk emissivity of a system comprised of a surface emitting energy corresponding to its temperature and an atmosphere between the emitting surface and space where the NET emitted energy is observed.

        The idea that there is any NET thermalization of CO2 is absolutely wrong. While there’s conversion between the vibrational states and rotational states of CO2, this goes both ways. The fine structure of CO2 absorption lines is very clear about this having lines on both sides of the primary resonance. A line at a lower frequency means that energy was taken from a rotational state to make up the difference, while a line at a higher frequency means that the excess energy was put into a rotational state. The fine structure is relatively symmetric around the primary resonance which means that energy is added to and removed from rotational states in nearly equal and opposite amounts.

        Note that there is some NET thermalization from water vapor, which happens when an energized water vapor molecule condenses upon a liquid water droplet and the state energy warms the resulting droplet.

      • @mkelly
        the graph doesn’t say what this unit is. Just hint at being bar*cm, not bar/cm.
        And you are not supposed to use bar nor cm, anyway, SI units are pascal (= newtons/m²) and meters. bar*cm would some sort of newton/meters, that’s the unit for a spring stiffness, which also apply to a air spring. Is that the context of Hottel work? I don’t know. Do you?

      • “If you don’t like Hottel”
        There is nothing wrong with Hottel. The problem is Nasif Nahle’s elementary misreading of the graph which mkelly seems determine to blindly propagate. The units of bar cm aren’t intended to suggest work done. The emissivity of a gas basically depends on the mass (volume*density) per unit area. 1 bar cm on that graph means the emissivity of a layer 1 cm thick at 1 atm pressure. So the bottom curve, .06 bar cm, is the emissivity of a 0.6 mm thick layer of pure CO2 at 1 atm. Nasif Nahle has no idea how to read this plot, and neither, it seems, does mkelly. The relevant curve for the atmosphere would be right at the top of that plot.

      • @mkelly @Nick Stockes
        never thought i would have to thanks Nick, but he fact is, I do.
        400ppm of a few km atmosphere is in the meter range, so
        “The relevant curve for the atmosphere would be right at the top of that plot”, indeed. That is, close to 0.2 at usual temperature ; and that’s NOT almost zero, for emissivity

  2. It is well known quote. Fermi asked Neumann to evaluate mathematical model. Neumann asked how many FUDGE parameters are there; when Fermi said “5”, Neumann tossed out model and said that. The key is the word “Fudge”, i.e. not determined by experiment.

      • It is interesting to see the enormous range in energy fluxes in the CMIP5 models (CMIP5 =Climate Model Incomparison Project) used by IPCC in their Assessment Report 5.
        The energy balance over land and oceans: an assessment based on direct observations and CMIP5 climate models – Wild et al 2014

        Here are some examples of the range of energy fluxes that is spanned out by the models (See Table 2: Simulated energy balance components averaged over land, oceans and the entire globe from 43 CMIP5/IPCC AR5 models at the TOA, atmosphere, and surface)

        Surface (All units: W/m2):
Solar down: 18.6
        Solar up: 10.5
        Solar net: 17.2
        Thermal down: 18.5
        Thermal up: 11.8
        Thermal net: 15.7
        Net radiation: 17.2
        Latent heat: 13.9
        Sensible heat: 13.1
        (Averages are taken over the period 2000–2004)

        Taking into account that the current energy accumulation on earth is estimated from observation of ocean warming to be approximately 0.6 W/m2 (Ref.: “considering a global heat storage of 0.6 W m–2» ref.: IPCC;AR5;WGI;page 181; 2.3.1 Global Mean Radiation Budget), I think it is fair to assume that the models would have been all over the place withou heavy parameterization.

        The spread demonstratest that these models cannot all be be right. To the extent that some of these models may seem right, they may also seem right for wrong reasons – tuning and parametrization to get a desired result.

    • AGW specializes in Fudge. They (the climate scientists) are the biggest candy makers of all time. However we will eventually take their candy away from them.

  3. “I do not know what von Neumann meant with his statement”

    It was just a metaphor, but to put in in tangible terms:

    The 4 parameters are height, width, length and mass. The fifth is the wiggle rate of its trunk.

      • “So would the LXWXH and mass of a mouse allow you to recreate an elephant?”

        Yes, if you can set those four parameters as you wish rather than as determined by experiment.

      • If you replace the mouse model with an elephant model, the same 4 parameters can produce a really tiny elephant. Keep in mind that I’m talking about the model of an average elephant (or mouse), not the model of a specific instance of a real animal which would require far more parameters.

        This illustrates the problems with GCM’s as they try to project the EXACT weather going forward in order to predict trends in the average climate. It’s much more accurate and far easier to use a simpler model that predicts the average based on macroscopic properties.

      • This would be a pointless exercise as there are so many elephants in the room already that there is little wiggle room remaining.

  4. Adding parameters, or changing the function to fit is an exercise in what my advisor called bumpology. The key is without some physical reason why a curve is quadratic or logarithmic, it is all just a fitting exercise.

    First comes the physical understanding, then the fitting.
    The problem in complex systems like climate, we often don’t know the whole physics.

    • We know quite well that the dependency of RF on the CO2 concentration is not linear. By applying the method what I described we can calculate the RF value per a CO2 concentration. In order to utilize these results, we need to carry out a mathematical fitting. And right here these comments pop up that it is just a fitting exercise, because there is no physical reason or connection. That is not so. There is a physical connection between each RF per CO2 concentration values. The curve fitting procedure does not destroy this physical connection. Utilizing the curve fitting we can find out a simple equation between the CO2 concentration and RF value. And this equation is very useful and in this case it is perhaps the most important equation or formula in the climate change science.

      Antero Ollila

      • It that case, the physical relationship is known, the curve fit is just a way of representing that relationship in a compact manner.

        It is a completely different issue when climate models, with many parameters, are tuned to match historical data and claimed to have precision in prediction based largely on that explicit and implicit tuning.

      • Re aveollila
        **Utilizing the curve fitting we can find out a simple equation between the CO2 concentration and RF value. And this equation is very useful and in this case it is perhaps the most important equation or formula in the climate change science.**
        This curve relates CO2 levels to Radiative Forcing. Correct?
        However, these are calculations? Correct?
        Now I noticed that these “forcings” are used to calculate future additional warming by CO2 as the CO2 increases to the end of the century.
        Are there any measurements of the warming by the changes in CO2?
        Or are we assuming that the warming increases as the “calculated” forcing increases?

      • Can you describe an in-situ experiment whose data matches the model curve? Are there published papers with results?

      • This may work, but physically if we don’t have the mechanism we can not eliminate the possibility that the CO2 and T aren’t both the effect of an unknown and unmeasured separate cause.

      • Antero,

        One thing that neither the LBL or BBM calculation of RF handles is clouds. The influence of clouds reduces the incremental effect of increasing GHG concentrations since the clouds would be absorbing that energy anyway and the net effect from a cloud or GHG absorbing energy is roughly the same relative to the NET energy balance. This reduces the global RF as a function of CO2 concentrations by as much as a factor of 2.

    • J said “we often don’t know the whole physics”.

      i would conjecture that we know very LITTLE of the physics.

      • To Jeanparisot.
        I have not seen any simple study showing the experimental relationship between the temperature and the CO2. The reason is that temperature measurements are too noisy and the other forcing factors like the Sun are too strong and therefore the small CO2 effect cannot be easily noticed. I have read a quite fresh study of Abbot & Marohasy and they have used the method of machine learning. Their result for the climate sensitivity is the same what I have found: 0.6 C.
        https://www.sciencedirect.com/science/article/pii/S2214242817300426

  5. I find the focus on these complicated, vast models strange. You can model what might happen to temperature with a three line model:

    1. Change in CO2 concentration
    2. Sensitivity of temperature to change in CO2
    3. Effect on temperature change of feedbacks.(using say the percentage increase or decrease in sensitivity)

    The trouble then is that 2 and 3 are unknown. and the big models don’t make them more known, because they can’t – the starting assumptions are not well enough known and the climate is non-linear and possibly chaotic, with far too many parameters for models to give you the knowledge claimed.

    Almost all (perhaps all?) complicated, large models are a waste of time and effort – all economic models are a bust, as are the models used in things like traffic forecasting.

    • There should be a wholescale review of any science that uses computer modeling. Computer modeling is leading us down the wrong path in many situations in science.

      • Alan, in a nutshell my slight knowledge of climate science and associated ‘computer modeling’ tells me that ‘computer modeling’ is not science, and therefore should not lead any proper scientist anywhere. I use a simple computer model to show the possible value of my retirement fund in a decade’s time given three or four assumptions, but I sure can’t take that value to the bank! A more apt comparison for climate is between computerised weather forecasting which at best can predict the regional weather around here for the next week, and climate models which purport to be meaningful on a global scale (yeah, right) decades or even centuries into the future. The former is science, the latter is astrology. I generally ignore any report or ‘research’ that includes the phrase ‘climate models’, period. Perhaps you should too.

    • Phenix44. I do agree with your approach because it has been my approach, too. I have written a blog about my model including the real contributor, which is the Sun:
      https://wattsupwiththat.com/2017/11/21/new-study-tries-to-link-climate-models-and-climate-data-together-in-a-semi-empirical-climate-model/

      The warming effects of CO2 can be summarized like this:
      1. RF = 3.12 * ln(CO2/280)
      2. dT = 0.27 * RF
      3. The residence time of the total CO2 is 55 years and the anthropogenic CO2 is 16 years.
      4. The contribution of CO2 in the GH effect is about 13 %.
      That’s it.

  6. von Neumann was criticizing economists and mathematicians who tried to model relationships by starting with a large number of variables and then dropped the variables that were not statistically significant from their models. This practice is now become common in data analytics (big data). Unfortunately, from a statistical theory perspective, the identified relationships cannot be distinguished from pure coincidence in a particular data set.

  7. I first heard that Carbon Dioxide’s ability to create heat was a Logarithmic Curve about ten years ago.
    Interesting the article above.

  8. It isn’t clear that there is anything to be learnt from this. Apart from the obvious fact that there is a unique nth order polynomial passing through any set of n+1 points which is all the author seems to have shown. A more useful graph would be to look at the difference between the polynomial and the log curve for CO2 values greater than 1400. It is then clear that they diverge very quickly. For example if C is 2500 then 3.12*ln(C/280) =6.8
    while the polynomial fit gives 22.8. It is then clear which curve is a better fit to the physics.

    • There is no need for the time being in the planet like Earth to create these kind of relationship for the CO2 concentrations. We simply have not enough fossil fuel to make it happen.

      • Who said that fossil fuel is the source of the CO2?

        Look back in history… it’s pretty clear that climate changes result in massive CO2 changes. :)

  9. “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.”

    So your calculations are for this specific assumption and nothing else?

    • Using 15 ⁰C, an arithmetical average is bogus except in the case where temps are actually all equal. Any body with hot bits and balancing cold bits will radiate a given amount at a LOWER average temp than if temps were uniform. Rotation makes a difference too.

    • Tom in Florida. I do not know what you have expected me to do. I have shown the results of the reproduction of the equation, which shows the RF value dependency on the CO2 concentration. If you have any knowledge about the climate science, it is the most important physical relationship between the global warming and the increasing CO2 concentration. You need only the value of a climate sensitivity parameter and oops, you can calculate by yourself, what is the climate sensitivity value.

  10. curve fitting is guessing … not theory –> data —> proof … i.e. its not science its a circle jerk …

  11. Here’s the problem with polynomial fitting…

    If I extend the trend line to the next doubling of CO2, the polynomial skyrockets. There’s no predictive value.

    • Yup. A polynomial can be good for constructing a calibration curve. But this requires a) data and b) that it is only used within the range of available data. Neither of these apply to the prognostications of climate science catastrophe.

      • There is no data here at all (or “prognostications”). He calculates forcing using BBM on a standard atmosphere. So there is no issue of “range of available data”. The only reason why he is fitting to four points is that he didn’t want to calculate more.

      • “There is no data here at all”

        Sort of like most of the planet when it comes to inventing the mythical ‘average global temperature.

      • Yup. I’ve used instruments with calibration curves that are 7th order polynomials. Just need very precise measurements of calibration standards and high resolution readings from the instrument being calibrated. But in most cases the coefficients of 4th and higher order terms are 10^-5 or smaller and of no practical importance relative to other uncertainties of measurement. In almost any lab, we’d be very happy to be able to confidently claim an uncertainty of +/- 1% or better for measuring anything. Where much lower uncertainty is necessary, it becomes a matter of whether the cost is justified. e.g. A NIST certified primary reference standard can cost 10 to 100 times a commercial secondary standard reference.

      • Au contraire, Mr Stokes. This is the sort of thing that global warming catastrophists, and environmentalists at large, indulge in all the time. It is the bread and butter of their existence. Demonstrate, say, a clear relationship between human mortality and increased temperature over the range 40 to 50 Celsius and then claim rising temperatures are bad for humans at some other temperature outside the range.

        Sure, that is an example that few people will fall for, but it sums up much of the thinking. They just don’t usually couch their most outrageous assertions in mathematics or a form simple enough for the unwary to immediately spot it and laugh it off. But a mathematical curve fit still expresses a thought or hypothesis.

      • To Nick Stokes.
        This is interesting. You write: “He calculates forcing using BBM on a standard atmosphere. So there is no issue of “range of available data”.”
        I guess you did read carefully enough the content of the blog. It was not me using BBM method. It was Myhre et all. I used the line-by-line (LBL) method.

      • Dr Ollila
        “I used the line-by-line (LBL) method.”
        Apologies for that. I read the sentence “The data points RF versus CO2 concentrations have been calculated using the BBM (Broad Band Model) climate model” when looking up what you had used, and didn’t read on.

    • And you fully deserve what you get.
      Here the third order polynomial is almost surely overfit. That is to say a second order fit would provide an adequate representation of the data. Adding the third term to the fit results in an improvement, but the improvement is not statistically significant. {wee p-values again!} Therefor, the data is said to be overfit. It is well known that extrapolating (projection, in ClimateScience) overfit polynomials leads to disaster at a geometrically increasing rate.

      Everybody maintains that polynomials are no good for making predictions.
      In fact, polynomials are great for making scary predictions.

      {As the blue line shows, choosing a mathematical function which properly represents the underlying physical phenomena, gets you closer to the truth, faster. But that is science, and we do not do things like that in ClimateScience!}.

      • @ Nick Stokes:
        Interesting point.
        I am an empirical type since way back. Whenever I see data graphed, I instantly think measurement data. That is really all I have ever worked with. Back in the day, people would construct theoretical curves to explore some phenomena, of course. Often these curves would often be calculated point by point. But we never generated experimental data sets via calculation alone.

        This whole modern idea of calculating a data set and then treating it as measurement data sometimes gets confusing to me. I perhaps sometimes lose sight of which is which.

        On the other hand:
        1) You have a set of data points. It does not matter how the data was obtained; calculated, measurement, tooth fairy, whatever.
        2) You select a model; linear, polynomial, logarithmic, and so on.
        3) A statistical test informs how well your model represents the data set. complete with wee p-values.

      • “A statistical test informs how well your model represents the data set”
        If there were a dataset. But it actually tells you only about the points “you chose to calculate”. There is no underlying randomness. You might as well to a p-test for however well the log function is approximated by a truncated Taylor series.

      • Polynomials are great for making “spectacular” predictions in either or both directions… ;)

      • “A statistical test informs how well your model represents the data set”
        All that is being tested here is the difference between 2 deterministic curves. The test is just some measure of difference eg sum squares. This head post is really just showing show well the stated polynomial approximates the log. That is just ordinary stuff about functions. Here is the continuous plot:

    • “There’s no predictive value.”
      There is no prediction here. It is just a relation between calculated values and a simplifying curve. 1400 ppm is not an observed value. If you want to know the value at 2500 ppm, you don’t have to wait; you can calculate it in the same way.

      • Nick
        OT.
        You showed a graph comparing GHCNM adjusted/unadjusted the other day. Can I ask if you calculated individual anomaly baselines for the two sets or if you used one common? If one common, was QCU or QCA used as the baseline?

        Thanks

      • MrZ,
        I have described the process here. It is a least squares method, which doesn’t require data in any particular interval. BEST subsequently adopted a similar method. Once the anomalies have yielded a spatial average, one can subtract the global average for any particular interval so the mean will be zero there. I do that with 1961-90.

        This is done separately for adjusted and unadjusted. But both are then normalised to 1961-90.

      • Nick,

        Impressive!!!
        Will keep me busy for a while 8-)

        Simple comparison AVG(tavg.qca – tavg.qcu) using data with OK qcflag AND values>-99.99C gives -0.3C during 1930s and close to -0C since 2000.

        I have also noticed that tavg.qcu != (tmin.qcu+tmax.qcu)/2 for 5% of the OK records

        We talked about anomaly as linear… Interesting to group by station’s monthly baseline temp to see how temps deviates from that baseline depending on warmer or colder years. Dramatic for cold baselines and almost flat for warm baselines.

        Don’t want to hijack this thread so I’ll post comments on the site you linked.

      • Value to be predicted for 1400 or 2500 PPM? I see a strawman (polynomial) being constructed as a comparison to logarithmic, with the strawman getting drawn by some as an elephant, or an elephant lifting its trunk. And, I see this as a distraction from another elephant, the one in the room where we are. I suspect a game of distraction from seeing elephants in the room, by someone who I have seen as finding less effect of global atmospheric CO2 before – by claiming in a comment in a previous WUWT article that outgoing longwave IR gets completely absorbed by CO2 before getting past 1 km above the surface, even though in an earlier comment in that same WUWT article someone with his name published spectral absorption curves for the total thickness of the average global atmosphere for some wavelengths that matter having absorption by CO2 being notably between 0 and 100 % and all other named greenhouse gases having absorption well below 100%.

      • The purpose of trend line is to demonstrate the mathematical relationship between two variables for the purpose of interpolating and/or extrapolating values. A polynomial function can’t be used to extrapolate values.

        With climate data, the primary purpose of trend line functions is extrapolation.

      • I just repeat it once again. For the next 200 years at least, it is good enough to know the RF and warming effects of the CO2 concentrations up to about 1400 ppm. Going further than that is purely academic exercise. We should be able to know what is going to happen during this century. And it is not a problem to calculate the effects up to 10 000 ppm. Actually I have calcualted but I have not shown the results, because they have no meaning.

      • aveollila February 22, 2018 at 11:13 pm:

        “I refer to the reply before. There is no need for the concentrations above 1300 ppm. Simple like that.”

        The past history of the Earth had atmospheric CO2 concentration higher than 1300 PPM and somewhat known with global temperature somewhat known, although the accuracy and precision of global temperature and global atmospheric CO2 concentration get a little lower as one explores into the past? These are known enough to have been raised as argument points in WUWT for climate sensitivity to CO2 change being low and for atmospheric CO2 to be known to lag global temperature from 400,000-plus years ago (as a positive feedback mechanism) until humans started bigtime mining and burning of fossil fuels, that were mostly formed more than 100 million years ago. From at least 400,000 years ago to 150 years ago, the sum of carbon (including in carbon compounds) in the sum of the atmosphere, hydrosphere and biosphere was fairly constant. Global temperature change initiated by a cause other than atmospheric CO2 change often got reinforced by positive feedbacks from surface albedo change (especially in the middle-upper northern latitudes in the past few million years, more so when northern continental ice sheets were at least intermittently unstable while getting as far south as 40 degrees north), secondarily due to balance between oceanic and atmopspheric CO2 being a lesser positive feedback.

        Nowadays, the effect of increasing atmospheric CO2 has different effects than it did in about 398,500 of the past 400,000 years, because most of the atmospheric CO2 increase is not from global warming since the Little Ice Age, not even according to the NASA GISS version of how much global temperature increased since the Industrial Revolution. From at least 400,000 to around 150 years ago, global temperature change (initiated by something other than CO2 change, such as one or two of the Milankovitch cycles) was reinforced by a lagging transfer of CO2 to/from the atmosphere from/to the oceans. Nowadays, we have CO2 increasing in the atmosphere much more than according to the global temperature increase typically associated with such an amount of CO2 increase. Also nowadays, carbon in the carbon cycle(s) is increasing in all of the atmosphere, hydrosphere and biosphere, and usual from 400,000 to 150 years ago was for carbon in these to get shifted around in response to and to reinforce a temperature change, as opposed to the sum of carbon in the atmsophere, hydrosphere and biosphesre getting a sudden upward trend after the Industrial Revolution due to transfer of carbon from the lithosphere to the sum of the atmosphere, hydrosphere and biosphere.

      • Actually I have calculated but I have not shown the results, because they have no meaning.

        Prof Ollila,

        Your calculations may be of interest to geologists studying the issue of Snowball Earth.

        Snowball Winter in Finland?

        The recent satellite picture of a snow covered Finland is spectacular.

        I see that the forecast for Porvoo continues to be for cold weather this week.

        Please stop sending us your global warming!

    • David. I simply repeat the point given already in my comments. The maximum value of 1370 ppm is good enough. We have not enough fossil fuel reserves to increase the CO2 concentration above 1000 ppm. So in this case this kind of extrapolation is a theoretical exercise only.

      • Fossil fuel “reserves” are irrelevant to how high the atmospheric CO2 concentration could be. Without burning any fossil fuels, CO2 levels have routinely exceeded 1,000 ppm over the Phanerozoic Eon. Less than 1,000 ppm has been the exception rather than the rule.

        The purpose of developing a mathematical relationship between CO2 and temperature is to be able to quantify and predict how one variable will affect the other.

      • In fact, one long term test of the models is which parameter will go wildly out of bounds first: temperature or CO2.

      • Mr. Pete: CO2 is increasing with global temperature increasing less than according to the “historical” (usual within the last 400,000 years) correlation. I see two factors for this:

        1) Global climate sensitivity varies with global temperature, being greater when northern hemisphere continents have higher mobility (advancing or retreating) of year-round sunshine-getting ice/snowpack coverage. I see this as having gotten to positive feedback occasionally to the extent of instability, at times when global temperature was colder than it got during the Little Ice Age. And when global temperature got warmer than that ofd the Medieval Warm Period, global climate sensitivity was lower, due to less surface albedo positive feedback and greater (negative) lapse rate feedback, the negative one on global temperature (a convective one) as opposed to the one on the global lapse rate itself.

        2) Historically within the past 400,000 years, atmospheric CO2 concentration generally lagged global (or icy polar continental region) temperature. During the first 399,800 of those years, the sum of carbon/CO2, as in carbon involved in the carbon cycle that I learned from library books and got taught about in school before “global warming” or “greenhouse effect” became political catchphrases, was fairly constant. There was not much formation or burning of “fossil fuels” during that time. And global temperature changes were usually caused by something other than atmospheric concentration of greenhouse gases, such as the Milankovitch Cycles affecting insolation of a range of Arctic and near-Arctic high latitudes and upper-middle latitudes, and arguably northern latitudes as low as around sometimes even a little below 40 degrees north often mattered, for the surface albedo positive feedback. More when the extratropical and “subtropical” (nearly and barely tropical, and “subtropical” gets up to about 40 maybe 41 degrees north according to some definitions as old as around 1980) gets intermittent or occasional snow coverage. Please note that Atlanta GA, Birmington AL, Nashville TN, Dallas and San Antonio and El Paso TX have gotten some notable wintry weather by more than one definition in the past few decades, mostly because historically the weather “has a temper” in the northern temperate (as opposed to temperant) zone.

        Back to atmospheric concentration CO2, that was mostly a positive feedback

      • But we do have access to a vast quantity of limestone.
        All that is needed to raise the CO2 concentration above 1000 ppm is to manufacture a great deal of cement!

    • It sounds to me that the curve fitting argument here is a strawman one. The logarithmic effect of variation of CO2 is not perfectly logarithmic (it must change to linear, probably in a gradual manner, around some point as concentration decreases somewhere below 180 PPMV and I suspect it flattens slightly more than logarithmic does due to saturation effects with increase to past 1000-1200 PPMV), but it is known as better than a strawman (to knock down) for having its behavior being described by a logarithm function.

  12. Perhaps I misunderstand the thrust of what you are saying.
    If you are just talking about “models and the number of parameters”, apologizes in advance.
    But if you are in someway defending “Climate Models”, there are an “H” of a lot more parameters and interactions between those parameters than any “Climate Model” could ever hope to project.

    I said this some time ago. What “Climate Model” accounts for all of the possibilities?

    https://wattsupwiththat.com/2012/05/12/tisdale-an-unsent-memo-to-james-hansen/#comment-985181

    Gunga Din says:
    May 14, 2012 at 1:21 pm

    joeldshore says:
    May 13, 2012 at 6:10 pm
    Gunga Din: The point is that there is a very specific reason involving the type of mathematical problem it is as to why weather forecasts diverge from reality. And, the same does not apply to predicting the future climate in response to changes in forcings. It does not mean such predictions are easy or not without significant uncertainties, but the uncertainties are of a different and less severe type than you face in the weather case.
    As for me, I would rather hedge my bets on the idea that most of the scientists are right than make a bet that most of the scientists are wrong and a very few scientists plus lots of the ideologues at Heartland and other think-tanks are right…But, then, that is because I trust the scientific process more than I trust right-wing ideological extremism to provide the best scientific information.

    =========================================================
    What will the price of tea in China be each year for the next 100 years? If Chinese farmers plant less tea, will the replacement crop use more or less CO2? What values would represent those variables? Does salt water sequester or release more or less CO2 than freshwater? If the icecaps melt and increase the volume of saltwater, what effect will that have year by year on CO2? If nations build more dams for drinking water and hydropower, how will that impact CO2? What about the loss of dry land? What values do you give to those variables? If a tree falls in the woods allowing more growth on the forest floor, do the ground plants have a greater or lesser impact on CO2? How many trees will fall in the next 100 years? Values, please. Will the UK continue to pour milk down the drain? How much milk do other countries pour down the drain? What if they pour it on the ground instead? Does it make a difference if we’re talking cow milk or goat milk? Does putting scraps of cheese down the garbage disposal have a greater or lesser impact than putting in the trash or composting it? Will Iran try to nuke Israel? Pakistan India? India Pakistan? North Korea South Korea? In the next 100 years what other nations might obtain nukes and launch? Your formula will need values. How many volcanoes will erupt? How large will those eruptions be? How many new ones will develop and erupt? Undersea vents? What effect will they all have year by year? We need numbers for all these things. Will the predicted “extreme weather” events kill many people? What impact will the erasure of those carbon footprints have year by year? Of course there’s this little thing called the Sun and its variability. Year by year numbers, please. If a butterfly flaps its wings in China, will forcings cause a tornado in Kansas? Of course, the formula all these numbers are plugged into will have to accurately reflect each ones impact on all of the other values and numbers mentioned so far plus lots, lots more. That amounts to lots and lots and lots of circular references. (And of course the single most important question, will Gilligan get off the island before the next Super Moon? Sorry. 8-)
    There have been many short range and long range climate predictions made over the years. Some of them are 10, 20 and 30 years down range now from when the trigger was pulled. How many have been on target? How many are way off target?
    Bet your own money on them if want, not mine or my kids or their kids or their kids etc.


    • As for me, I would rather hedge my bets on the idea that most of the scientists are right than make a bet that most of the scientists are wrong and a very few scientists plus lots of the ideologues at Heartland and other think-tanks are right…But, then, that is because I trust the scientific process more than I trust right-wing ideological extremism to provide the best scientific information.

      Even though the vast majority of scientists agreeing with it cannot duplicate it, have never tried to duplicate it, and don’t have access to the the full descriptions of what was done to verify it using validated procedures?
      But the comparison is also largely miss-stated, as for the most part, it is one side saying we have successfully created a functioning accurate model and it has been “settled science” for over 25 years (despite being wrong when forecasting anything long enough to show statistical significance) vs. a side that says they don’t know enough to make the claims they are making and being able to show basic flaws.

  13. https://doi.org/10.1119/1.3254017
    Drawing an elephant with four complex parameters
    American Journal of Physics 78, 648 (2010); https://doi.org/10.1119/1.3254017
    Jürgen Mayer
    Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstr. 108, 01307 Dresden, Germany
    Khaled Khairy

    Log(x+a) is given as a Taylor series:
    = ln(a) + (x-a) / a – (x-a)2 / 2a2 + (x-a)3 / 3a3 – (x-a)4 / 4a4 + …
    The terms rapidly diminish as their power increases. That is why a polynomial can be fitted to a log curve.

    • “The terms rapidly diminish as their power increases. “
      Not very rapidly. The series converges only to x=2*a, and of course very slowly as you get close.

  14. Someday I will again come across Benoit Mandelbrot’s (quoted?) statement that reality is fractally complex..

    I learned that shortly after David. M. Raup (‘Nemesis’ ’Extinction’) encouraged me to pursue an assertion in discussion that extrapolating points on an epistemological map was freighted, not least by the low order of the model.

  15. First, parametres need to have a physical explanation for why the formulae might be right. For example, Grant Foster and Rhamstorf 2011 fitted 4 parametres to the annual temperature value but they had the solar cycle being a negative value, The higher solar irradiance got, the lower temperature that resulted. Obviously, the wrong result and an example of overfitting and or using the wrong input data (they actually used an incorrect solar irradiance dataset on purpose because it made the CO2 parametre look higher).

    Secondly, “time” needs to be a factor in any energy flow equation (such as surface temperature). They use emissivity sometime to substitute for this factor but it needs to be way more in depth to accurately represent the trillions upon trillions of molecules of air, surface and water. Time needs to factored in at the pico-second level and hourly level. The “Ocean Lag” is a perfect example of how this can go wrong. Many formula say the Ocean Lag is 100s of years long, others say 30 years and then the climate models actually only use 7 years for the ln(Co2) formula. Complete garbage in other words..

    Third, don’t they think they need to actually test this ln(CO2) formula against what has actually happened on Earth, If you try to do this, you find there is ZERO correlation. It doesn’t mean the formula is wrong, it just means that many other factors such as Albedo, must play an even larger part.

    Fourth, what about the 99% of other atmospheric molecules. The theory treats them as having no impact at all. What if the atmosphere has 30% Oxygen rather than 21%? What if the atmosphere is 70% CH4 as in 3.5 billion years ago. Is Nitrogen actually an inert gas in a warm atmosphere? Not a chance. Nitrogen is absorbing energy and giving up energy just like every other molecule in the universe.

    The situation is just far too complicated for a ln(CO2) formula to be accurate in any way.

  16. Antero Ollila
    Thanks for your excellent discussion. The LBL log function fits the physics better.
    For how to curve fit an elephant’s trunk see:
    Drawing an elephant with four parameters

    Introduction
    Mayer et al. [1] relate the following anecdote:
    “A turning point in Freeman Dyson’s life occurred during a meeting in the SPring of 1953 when Enrico Fermi criticized the complexity of Dyson’s model by quoting Johnny von Neumann [1] `With four parameters I can fit an elephant, and with five I can make him wiggle his trunk'”, Mayer et al. [2].

    They then go on to provide the first successful implementation of von Neumann’s claim, based on trunk-ated Fourier series. Reproducibility is key to good science, so this page provides a Javascript implementation of the full five-parameter elephant, including trunk-wiggling. Mayer et al. require some censure for the lack of detail of their trunk-wiggling algorithm. The approach taken here was to form a second elephant by reflection of the distal portion of the proboscis of the primary pachyderm, parallel to the x-axis. The wiggling was then implemented by linear interpolation between the primary and secondary elephants, using a sinusoidally varying mixing coefficient. While it has not been possible to reproduce their results quantitatively, with complete accuracy, a qualitative reproduction of the main result proved entirely feasible.
    Conclusions
    Von Neumann was right; you can fit an elephant with four (complex) parameters, and with a fifth, you can make his trunk wiggle.
    References:
    [1] Jügen Mayer, Khaled Khairy and Jonathon Howard, Drawing an elephant with four complex parameters, American Journal of Physics, volume 78, pages 648-9 (2010). (doi:10.1119/1.3254017)
    [2] Freeman Dyson, A meeting with Enrico Fermi, Nature (London), 427(6972), 297 (2004).

    How to fit an elephant
    Source: “Drawing an elephant with four complex parameters” by Jurgen Mayer, Khaled Khairy, and Jonathon Howard, Am. J. Phys. 78, 648 (2010), DOI:10.1119/1.3254017.
    Posted: Freeman Dyson, A meeting with Enrico Fermi,

    • Noble Laureate Freeman Dyson wrote:

      . . .By the spring of 1953, after heroic efforts, we had plotted theoretical graphs of meson–proton scattering.We joyfully observed that our calculated numbers agreed pretty well with Fermi’s measured numbers. So I made an appointment to meet with Fermi and show him our results.. . .In a few minutes, Fermi politely but ruthlessly demolished a programme of research that my students and I had been pursuing for several years. He probably saved us from several more years of fruitless wandering along a road that was leading nowhere. I am eternally grateful to him for destroying our illusions and telling us the bitter truth. . . .
      Then he delivered his verdict in a quiet, even voice. “There are two ways of doing calculations in theoretical physics”, he said. “One way, and this is the way I prefer, is to have a clear physical picture of the process that you are calculating. The other way is to have a precise and self-consistent mathematical formalism. You have neither.” . . .
      physical picture, and the forces are so strong that nothing converges. To reach your calculated results, you had to introduce arbitrary cut-off procedures that are not based either on solid physics or on solid mathematics.”
      His physical intuition told him that the pseudoscalar meson theory could not be right. And so it was Fermi’s intuition, and not any discrepancy between theory and experiment, that saved me and my students from getting stuck in a blind alley. . .
      In desperation I asked Fermi whether he was not impressed by the agreement between our calculated numbers and his measured numbers. He replied, “How many arbitrary parameters did you use for your calculations?” I thought for a moment about our cut-off procedures and said, “Four.” He said, “I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” With that, the conversation was over. I thanked Fermi for his time and trouble, and sadly took the next bus back to Ithaca to tell the bad news to the students.

      Global climate models
      1) have far more than five fitted parameters.
      2) They try to model multiple weakly coupled non-linear chaotic systems
      3) with inadequate data
      4) without exhaustive boundary conditions, AND
      5) with widely varying economic and energy use scenarios.
      Better to recognize the impossibility of making accurate projections and try stochastic modeling, realistic resilience planning etc.

      • To David L Hagen.
        And a funny thing is that the same warming values can be calculated by both the most complicated GCMs and the simplest IPCC’s climate model as I have shown in my earlier blog:
        https://wattsupwiththat.com/2017/12/11/on-the-ipcc-climate-model/

        The IPCC’s simple model can be used for calculating both the transient and equilibrium climate sensitivity values and all the IPCC’s scenario values for different RCPs. How it is possible? Maybe they do carry out a lot of parameter fittings in GCMs in order to get the same results as by the simple IPCC model.

  17. It must be fun building hypothetical planets with hypothetical climates by modeling. I fail to see any relationship to reality. I can probably tell you more about the climate by stepping outside my door. It is often cold, cloudy and snowy in Alberta in winter. What does your model tell you? Building a model appears like a waste of good minds.

  18. Antero Ollila has transformed the logarithmic data such that all that is required is a linear regression. Then, upon solving this linear equation, we know, a priori, that a four degree polynomial will fit the data, as well.

  19. Your calc: RF = 3.12 * ln(C/280)

    Why not just use pi. 3.1415926537…? an irrational number.
    In nature, we find (pi) popping up all over the place.

    Besides, rationality left the climate modellers decades ago.

  20. 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.

  21. 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

  22. 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.

      • 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.

    • 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

      • 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.

  23. 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.

    • 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.

  24. @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.

  25. 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.

  26. 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.

  27. 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.

  28. 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.

  29. 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 :) )

  30. 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

  31. 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.

  32. “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.

  33. 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 .

  34. 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.

  35. 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.

  36. 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.

  37. 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.

  38. 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.

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