MSimon. I thought you were better than this - but maybe you have just not looked in detail at the paper. Robert Brown actually provides a better critique than me - I just said the experimental data was overfitted with a short explanation - he provides two long comments in which he points out why, and answers the author's replies to his first comment. The quality of Robert's critique, and the reply, is telling.
In general "if the curve fits it will be useful" is just not true in physics because correlation and causation are two different things. This curve has no physical basis for the arbitrary coefficients used in it to fit the data. It has a lot of arbitrary coefficients. It fits only a small number of data items (the multiple airless planets/moons are obviously trivial).
Any random function can be made to fit data if you allow enough fudge factors.
The Ideal Gas Law is not the point here, because to use it he needs to make assumptions which embody arbitrary coefficients, and these are cherry-picked to match the data.
Finally, when you take into account variations between his planets not considered by him but obviously physically relevant, his results clearly do not match reality. The appearance of a match is a result of fudged post hoc curve fitting.
In fact every criticism levelled at GCMs by the anti-AGW lobby, though in that case not obviously true because the models are dynamical and all parameters have either physical basis or validating physical simulations, are true 100% of this fudged model.
Now, here is Robert Brown's criticism written at some length which I happen mostly to agree with: and will happily explain for you furtehr should you wish:
Before I or anyone can consider the goodness, or uniqueness, of your fit to your data, surely one needs to have the probable or possible sources of error accounted for and error bars included in the numbers for use in the regression program. One can get truly horrible errors fitting a set of noisy data with a single one size fits all error bar (especially one that is too small so it places too much weight in the fit on data that is actually not known particularly accurately, even more so when one is fitting a small set of data with a large set of parameters). In the meantime, as I said, fit the data with a cubic spline — it is just as meaningful. What you’ve done is no different from Roy Spencer’s “cubic fit” presented on his lower troposphere temperature curve — presented a curve that smoothly interpolates the data, sure, but that is physically unmotivated and hence meaningless except as a guide to the eye. Spencer openly acknowledge it. You’ve written a paper on it, claiming that your arbitrary fit is “derived” by virtue of roughly interpolating the data.
Now let’s talk about Equation 7 itself. You yourself in figure 6 plot “potential temperature”. Potential temperature is a dimensionless quantity like the one you hope to understand in the form of N_{TE} — I get it. Note well that in the case of potential temperature, because it is based on and indeed actually derived from some fundamental physics, the two numbers that appear: P_0 = 1 atm and the exponent $0.285$ are both entirely physical!. The one is a reference pressure that not only is relevant but sets the scale of pressure-temperature relationships for the entire atmosphere, the other is related to \gamma and the atmosphere’s actual molecular composition. This is characteristic of “good physics”, or at least of plausible physics. The quantities make physical sense even before one digs into and learns to understand where they come from.
For some reason you presented Equation 7, the result of your nonlinear regression fit, in a form that was not as manifestly dimensionless as potential temperature in figure 6, after claiming it as inspiration. I have helped you out there by filling in the characteristic pressures that go with your choice of exponents. These pressures are clearly absurd, are they not? Unlike P_0 in potential temperature, 54,000 atmospheres is a pressure that appears nowhere in the physics describing ideal gases, in physical processes that might possibly be relevant on the surface of Europa or Triton or Mars or Venus.
I’ve played the “fitting nonlinear functions” game myself, for years, as part of finding critical exponents from scaling computations, and in the process I learned a thing or two. One thing I learned is that it is often possible to get more than one fit that “works”, and that the fit that works best may not be the one you are seeking, the one that makes physical sense. Often this is a matter of the error bars or lack thereof. Too small error bars will often “constrain” the best fit away from the true trend hidden in the data. The problem is compounded when one is fitting data with multiple independent trends, such as a fast decay mixed with a slow decay (multiple exponential).
Your data clearly has such multiple trends with completely distinct physics — you misrepresent it as a single fit, but presenting it in dimensionless form clearly shows that you are really proposing two different physical processes occurring at the same time with completely different characteristic dimensions. I think this is as clear a signal as you will ever see that you are overfitting the information content of the data, and would do far better to just fit the larger planets on your list with a single dimensionless form, preferrably after putting error estimates into all of the data in your table 1 and using the correct bond albedo for the planets in question, and adding references.
In summary, the tight exponential relationship between NTE and pressure is real, and the fact that it is described by a function, which coefficients cannot be easily interpreted in terms of known physical quantities, does not invalidate that relationship! This is because it is a higher-order emergent relationship, which summarizes the net effect of countless atmospheric processes including the formation of clouds and cloud albedo. This relationship might not be precisely reproducible in a lab, simply because it may require a planetary scale to manifest. However, a lab experiment should be able to validate the overall shape of the curve defining the thermal enhancement effect of pressure over an airless surface. BTY, this shape is already supported by the response function of relative adiabatic heating defined by Poisson’s formula (Fig. 6 in our paper).
Actually, as I’ve pointed out very precisely above, equation 8 is just as algebraic restatement of your definition of N_{TE}. You’ve simply inserted an empirical heuristic fit to your data to replace the data itself. This isn’t a derivation of anything at all, it is curve fitting, which is a game with rules. Mann, Bradley and Hughes tried to play this game and broke the rules when they built the infamous Hockey Stick. Mckittrick and McIntyre called them on it.
I’m trying to keep you from making the same sort of mistake. You fit the data with the product of two exponentials of ratios of the surface pressure to arbitrary powers. Why? Well, exponentials are functions that are 1 when their argument is zero, so you fit two of your data points (badly if you leave out error bars or use the actual data in your table) for free without using a fit parameter, and come darn close to a third, close enough that — lacking error bars and given a monotonic relationship — you can count it as “well fit” whatever the error really is.
You then are really “fitting” five data points with four free parameters. Skeptics often quite rightly mock the warmist crowd for their global climate models with highly nonlinear behavior and enough free parameters that they can be tuned to fit past temperature data, accurate or not, as nicely as you please, and we are not surprised when those fits of past data turn out to be poor predictors of either future trends or even earlier past data (hindcast). We mock them because it is well known in the model building business that with enough free parameters and the right choice of functional shapes you can fit anything, but unless you treat error in the data with the respect it deserves and include some actual physics in the choice of functions being fit, the result is unlikely to actually capture the physics.
Listen, in fact, to your own argument. There is a dazzling amount of physics involved in the processes that establish the surface temperatures on the planets in your list. One can split the planets up into completely distinct groups — two airless planets near the sun with no surface ice, two nearly airless planets that are completely coated in high-albedo ice, one water ice, one frozen N_2, one of which is heated by a tidal process that still isn’t well understood, the other of which is hypothesized to have a greenhouse trapping of heat by the semi-transparent N_2 ice that replenishes its atmosphere. Of the four planets with substantial atmospheres all of them have an optically thick greenhouse gas content and all of them therefore have tropospheres and stratospheres and lapse rates driven by vertical convection across the temperature differential between the surface and the tropopause.
Yet somehow none of this matters? Calling it an “high-order emergent relationship” is just fancy talk for “we found a fit and have no idea what it means”, but it isn’t surprising that you can fit the data with an arbitrary form with four free parameters, especially without error bars or any criterion for judging goodness of fit.
How is your fit more informative than fitting the data with a spline, or with a polynomial, or with anything else one might imagine? I’ve already pointed out that your figure 6 is precisely why one should not believe your result. In it, P_0 means something, and so does the exponent. There is nothing “emergent” about it, it is really a derived result, and when it turns out to approximately describe actual atmospheres we gain understanding from it.
What does the 54,000 bar in your fit mean? What does the 202 bar in your fit mean? What does the exponent 0.065 mean? You cannot answer any of these questions because you have no idea. How could you? They are all completely irrelevant to the pressures present on the planets in question. They have precisely as much meaning as the arbitrary coefficients of a cubic spline or any other interpolating function or approximate fit function that could be used to approximate the data, quite possibly as well or better than the fit that you found if you actually add in error bars
[Reply] In fact Ned has addressed your concerns regarding your oft repeated assertion, please revue his recent reply to you again. I’d also like you to answer my question which I’ll repost here:
“Please could Robert explain the physical basis of the imaginary number ‘i’ (or ‘j’ in engineering) the product of which when multiplied by itself is minus 1, which is used extensively in electronics design and control engineering? Presumably any competent Duke physicist at the time of the invention of this imaginary quantity which defies the laws of mathematics would have rejected it out of hand for being “absurd nonsense” and therefore of no possible use? – Thanks – TB. .
Note this reply from Nikolov is telling. He does not address the substantial criticism. He counter-attacks by asking a question unrelated to the criticism which does in fcat not make any point at all, since i is NOT and arbitrary parameter
So far the total information content of your paper is:
* We do a better job of defining/computing a baseline greybody temperature T_{gb} for the planets.
Yes and no. Yes to the integral, no to ignoring the bond albedo, especially in the case of Europa and Triton where there is no conceivable justification for doing so.
* We define a dimensionless ratio between empirical T_s and T_{gb}. We tabulate this computed ratio for the data, forming an empirical N_{TE} dataset with eight objects.
Sure.
* We heuristically fit a four parameter functional form. The fit works. It is unique. It must be meaningful.
Lacking error bars on your data, you cannot possibly assert that it is unique. There could be dozens of functional forms, some of them with fewer free parameters, that produce comparable Pearson’s \chi^2 for the fit once you add in error bars. I rather expect that there will be, especially if you correctly treat the bond albedo for planets with almost no atmosphere and no exposed regolith that reflect away over half their incident insolation without being heated by it.
The fit you obtain is not meaningful. If you disagree, give me a physical argument for the 54000 bar, the 202 bar, and the exponent of 0.065. The only parameter of your four parameter fit that is plausible is the 0.385, although even that number would need to be connected to some actual physics in order to obtain meaning.
* The real meaning is that only surface pressure explains surface temperature, because we were able to fit a functional form to T_s(P_s).
Excuse me? I can fit any set of data pairs with any sufficiently large basis. If the data is monotonic I can almost certainly fit it with fewer free parameters than there are data points, especially if I completely ignore the error estimates for the data points! Lacking the error bars, you cannot even compute R^2 and plausibly reject no trended correlation at all! I’m not suggesting that this is reasonable for your particular data set, only that you are far away from presenting a plausible argument for uniqueness or correlation that implies causality. In two of the four planets in your list, it’s rather likely the case that surface temperature implies surface pressure, not the other way around! The chemical equilibrium pressure of N_2 over a thick layer of N_2 ice or O_2 over water ice is far more likely to be the self-consistent result of surface temperature, not its cause.
In the end, you are left where you started — that there is a monotonic trend to the data that you cannot explain or derive, and because of flaws in your statistical analysis you cannot even resolve difference between competing explanations including the simplest one that the last four planets have surface temperatures dominated by the greenhouse effect and their albedo, the first two are greybodys to a decent approximation (that somehow turned into 1.000 to four presumed significant digits in your Table 1), and two are special cases described by a completely different physics than the others (dominated by the incorrectly used albedo), and to some extend different even from each other.
Nothing in your analysis rejects this as a null hypothesis. You cannot even assert that it does without including an error analysis in your data and fit.
To conclude, you have two choices. You can ignore my objections above and plow ahead with your paper as is. You might get it past a referee, although I somewhat doubt it. You can in the process continue to get all sorts of uncritical positive feedback on it on the pages of this blog and have it trumpeted as “proof” that there is what, no actual GHE? That gravity alone heats atmospheres? I’ve heard all sorts of absurd punchlines bandied about, and your result can be used to support any or all of them if one ignores the statistical and methodological flaws.
Or, you can fix your paper. Include references, for example. Use the correct bond albedos. Here’s a small challenge for you. Apply your formula to Callisto, to Ganymede, to other planetary bodies. Callisto is an excellent case in point. It has an albedo almost twice that of the moon, It is the warmest of Jupiter’s moons — warmer in particular than Europa, for good reason given the difference in their albedos . It has an atmosphere with a surface pressure around 0.75 microPa, it will fit right in there on your table. It puts the immediate lie to any assertion that your fit is either predictive or universal, as its surface pressure is lower than Europas and its surface temperature is higher than Europas and if you use your “universal” T_{gb} formula for it the lower albedo will further raise N_{TE} for it relative to Europa. Your nice monotonic curve won’t be monotonic any more, and you can see some of the consequences of ignoring albedo, atmospheric composition (Callisto’s is mostly CO_2, hmmm), error estimates, and using cherrypicked data to increase the “miraculous” impact of your result.
I honestly hope that you fix your paper. There may well be something worth reporting in there in the end, once you stop trying to prove a specific thing and start letting the data speak. I actually rather like what you are trying to do with T_{gb}, but if you want to actually improve this you can’t just leave physics out at will, especially not when looking only at the temperature of moons tells you that your assumptions are incorrect even before you get to actual planets with actual atmospheres. Also, if you do indeed do your statistical fits correctly, you might find something useful — a less “miraculous” fit that is still good given the error bars and that has characteristic pressures and exponents with some meaning,
Best regards,
rgb
