Anthony Watts has kindly offered the Talkshop the exclusive on Ned Nikolov and Karl Zeller’s reply to the article by Willis Eschenbach published at WUWT, which we accept, gladly.
Reply to: ‘The Mystery of Equation 8’ by Willis Eschenbach
Ned Nikolov, Ph.D. and Karl Zeller, Ph.D.
In a recent article entitled ‘The Mystery of Equation 8’ published at WUWT on January 23 2012, Mr. Willis Eschenbach claims to have uncovered serious mathematical and conceptual flaws with two principal equations in our paper ‘Unified Theory of Climate‘. In his ‘analysis’, Mr. Eschenbach makes several fundamental errors, the nature of which were so elementary that our initial reaction was to not respond. However, after 10 days of observing the online discussion, it became clear that a number of bloggers have fallen victim to the same confusion as Mr. Eschenbach. Hence, we decided to prepare this official reply in an effort to set the record straight. This will be the only time that we respond to such confused criticism, since we believe that the climate science community has much more serious issues to discuss.
Demystifying the Mysteries of Equations 7 and 8
We begin with the most amusing claim by Mr. Eschenbach, which he calls ‘the sting in the tale’. First, some background: in our original paper, we use 3 principal equations that form the backbone of our new ‘Greenhouse’ concept. For consistency, we use here the same formula numbering as adopted in the original paper. Equation (2) calculates the mean surface temperature (Tgb) of a standard Planetary Gray Body (PGB) with no atmosphere, i.e.
where So is the solar irradiance (W m-2), αgb = 0.12 is the PGB shortwave albedo, ϵ = 0.955 is PGB’s thermal emissivity, σ = 5.6704×10-8 W m-2 K-4 is the SB constant, and cs = 0.0001325 W m-2 is a small constant, the purpose of which is to ensure that Tgb = 2.725K when So = 0.0. The derivation and validation of this formula is discussed in more detail elsewhere. We redefine the ‘Greenhouse Effect’ as a near-surface Atmospheric Thermal Enhancement (ATE) measured by the non-dimensional ratio (NTE) of a planet’s actual mean near-surface temperature (Ts) to the temperature of an equivalent PGB at the same distance from the Sun, i.e. NTE = Ts / Tgb(where Tgb is computed by Eq. 2). We then use observed data on surface temperature and atmospheric pressure (Ps) for 8 celestial bodies to derive an empirical function relating NTE to Ps employing non-linear regression analysis. The result is our Eq. (7), which describes all planetary data points with a high degree of accuracy:
The key conceptual implication of Eq. (7) is that, across a broad range of atmospheric planetary conditions, the ATE factor is completely explained by variations in mean surface pressure. In Section 3.3 of our original paper, we specifically point out that NTE has no meaningful relationship with other variables such as total absorbed solar radiation by planets or the amount of greenhouse gases in their atmospheres. In other words, pressure is the only accurate predictor of NTE (i.e. ATE) we found. This fact appears to have completely escaped Mr. Eschenbach’s attention.
From Eq. (7) we derive our Equation 8 (the subject of Eschenbach’s analysis) in the following manner. First, we solve Eq. (7) for Ts, i.e
Secondly, we substitute Tgb for its actual expression from Eq. (2) to obtain:
Thirdly, we combine the fixed parameters 2/5, αgb, ϵ and σ in Eq. (7b) into a single constant, i.e.
Fourth, we use the newly computed constant along with the symbol NTE(Ps) representing the EXP term of Eq. (7b) to write our final Eq. (8):
Basically, Eq. (8) is Eq. (7b) expressed in a simplified and succinct form, where NTE(Ps) literally means the ATE factor as a function of pressure!
Let’s now look at how Mr. Eschenbach interprets Eq. (7) and its relationship to Eq. (8). He correctly identifies that Eq. (7) has 4 ‘tunable parameters’(the correct term is regression coefficients, but never mind this minor terminological inaccuracy for now). He then espouses:
Amusingly, the result of equation (7) is then used in another fitted (tuned) equation, number (8).
This is the first demonstration of misunderstanding in his analysis (with far reaching consequences as discussed below), where he fails to grasp that Eq. (8) follows simply and directly from Eq. (7) after a few straightforward algebraic rearrangements, and that it contains no additional tunable parameters! Instead, Mr. Eschenbach smugly informs our fellow bloggers that the constant 25.3966 is yet another tunable parameter, which he labels t5 (his Eq. 8sym)?! We point out that the fixed parameters used to produce this constant have been defined and set prior to carrying out the regression analysis that yielded Eq. (7). Indeed, it could not have been any other way, because these parameters are required to estimate the PGB temperatures (Tgb) used in the calculation of NTE values, which are subsequently regressed against observed pressure data. Thus, Eschenbach now leads the readers astray telling them that we use 5 tunable parameters instead of 4. Fascinating! Next, in a state of total confusion, he makes the following stunning proposition:
We can also substitute equation (7) into equation (8) in a
slightly different way, using the middle term in equation 7. This
Ts = t5 * Solar^0.25* Ts / Tgb (eqn 10
What middle term? This twisted line of reasoning is astounding, because it reveals an utter misunderstanding of basic algebra compounded with an inability to follow content, thus leaving the reader literally speechless! This error leads Eschenbach to his central false claim that our Eq. (8) simply meant Ts = Tgb * Ts / Tgb, and therefore reduces to Ts = Ts!? One can only stand in disbelief before such nonsense! This is what Mr. Eschenbach jubilantly calls ‘the sting in the tale‘. It is a big sting, alright, but in his tail, not ours! He proudly reiterates this ‘finding’ once again in the Conclusion section of his article leaving no doubt in reader’s mind about his analytical ‘skills’.
Blinded by a profound misunderstanding, Mr. Eschenbach pompously concludes in regard to the constant 25.3966 that what we have done is “estimate the Stefan-Boltzmann constant by a bizarre curve fitting method”. He further states: “And they did a decent job of that. Actually, pretty impressive considering the number of steps and parameters involved”. Wow! Hands down, such a conclusion could easily qualify for the Guinness Book of Records on Miscomprehension!
The rest of Eschenbach’s ‘revelations’ in regard to our Equations (7) and (8) are less flamboyant but equally amusing. He argues that the small constant cs in Eq. (2) is pointless while failing to understand the physical realism it brings to the new model (Eq. 8). Since the goal of our research was not just to derive a regression equation, but to develop a new physically viable model of the ‘Greenhouse Effect’, this constant is important in two ways: (a) it does not allow the PGB temperature to fall below 2.725K, the irreducible temperature of Deep Space, when So approaches zero; and (b) it enables Eq. (8) to predict increasing temperatures with rising pressure even in the absence of solar radiation. Indeed, if we set cs = 0.0, then Eq. (8) would always predict Ts = 0.0 when So = 0.0 regardless of pressure, which is physically unrealistic due to the presence of cosmic background radiation.
A major portion of Eschenbach’s criticism focuses on the ‘accusation’ that all we had done is just ‘curve fitting’ devoid of any physical meaning. In an Update to his article, Eschenbach attempts to prove that he can do a better job in fitting a curve through our planetary NTE values using an equation with fewer free parameters. His simplified version of our Eq. (8) has 3 regression parameters (instead of 4) and reads:
Figure 1. Absolute errors of predicted planetary mean surface temperatures by Eschenbach’s simplified equation and by N&Z’s Equation (8). Errors are assessed against the observed mean surface temperatures listed in Table 1 of Nikolov & Zeller’s original paper.
Note that his expression is in a sense more empirical than our Eq. (8), because the coefficient in front of So has been erroneously treated as a tunable (regression) parameter, hence distorting our PGB Eq. (2). Figure 1 compares the absolute deviations of predicted planetary surface temperatures from their true values (listed in Table 1 of our original paper) using Eschenbach’s regression equation and our Eq. (8). It is obvious to a naked eye that Eschenbach’s formula produces far less accurate results than our Eq. (8). This was also recently quantified statistically by Dan Hunt in an article published at the Tallbloke’s Talkshop. For example, Eschenbach’s equation predicts Earth’s mean temperature to be 295.2K, which is 7.9K higher that observed. This is not a small error, because the last time our Planet was 7.9K warmer than present some 40M years ago the earth surface was ice-free, and Antarctica was covered by subtropical vegetation! Of course, being a construction manager, Mr. Eschenbach likely has a limited understanding of Earth’s climate history and what a 7.9K warmer surface actually means. However, the fact that he claims aloud a superior accuracy of his simplified equation over ours is puzzling to say the least. His exact words were:
Curiously, my simplified version actually has a slightly lower RMS
error than the N&Z version, so I did indeed beat them at their own game. My
equation is not only simpler, it is more accurate
This statement blatantly contradicts the evidence. Mr. Eschenbach does not know that we have extensively experimented with exponential functions containing various numbers of free parameters many months before he became aware of our theory, and we have found that it takes a minimum of 4 parameters to accurately describe the highly non-linear relationship between NTE and surface pressure (Eq. 7). The basic implication of Eschenbach’s analysis is that one could indeed use a 3 parameter exponential function to predict planetary temperatures from solar irradiance and surface pressure but with far less accuracy. Truly enlightening!
By the way, curve fitting is an integral part of the classic science method. When dealing with an unknown process or phenomenon, taking measurements and using the data to fit curves is the only feasible approach to understand and develop a theory about the phenomenon. This method was extensively used throughout the 18th and 19th Century and a good part of the 20th Century to extract the so-called first principles in physics we currently employ to describe the World. However, arguing about curve fitting really misses the main point of our study.
Focusing on the Big Picture
What Mr. Eschenbach and a number of others have totally failed to grasp is the highly significant fact that the enhancement factor NTE (i.e. the Ts / Tgb ratio) is indeed closely related to pressure, and that no other variable can explain the interplanetary variation of NTE so completely. As Dr. Zeller pointed out in a recent blog post, given the simplicity of Eq. (8), it is a ‘miracle’ how accurately it predicts surface temperatures of planets spanning a vast range of environmental and atmospheric conditions throughout the solar system! This cannot be a coincidence! Rather it suggests the presence of a real physical mechanism behind the regression Equation (7) related to the thermal enhancement effect of pressure. This effect is physically similar (although different in magnitude) to the relative adiabatic heating observed in the atmosphere and described by the well-known Poisson formula derived from the Gas Law (see discussion in Section 3.3. and Fig. 6 in our original paper).
Even the mistaken analysis of Mr. Eschenbach could not manage to negate the above truth. He vigorously criticized our Eq. (8) using all sorts of faulty technical arguments only to arrive himself at a similar (albeit less accurate) equation that predicts planetary temperatures as a faction of the same two variables – solar insolation and pressure! His argument that one could arbitrarily use air density instead of pressure is groundless, because pressure as a force is the primary independent variable in the isobaric thermodynamic process of planetary atmospheres. Ground pressure depends solely on the mass of air column above a unit surface area and gravity, while air density is a function of temperature and pressure. In other words, density cannot exist without pressure. For a given pressure, the near-surface air density varies on a planetary scale in a fixed proportion with temperature, so that the product Density*Temperature = const. on average, i.e. higher temperature causes lower density while lower temperature brings about higher density according to the Charles/Gay-Lussac Law for an isobaric process.
We now draw attention to a key logical contradiction in Mr. Eschenbach’s approach. In the main text of his article, he makes the central claim that our Eq. (8) represented a mathematical nonsense, since according to his logic, it reduces to Ts = Ts (the TA-DA! moment). Yet, in the Update section, he uses data from Table 1 in our original paper to derive a very similar equation, which he calls a ‘simplified version’ of Eq. (8). So, according to Mr. Eschenbach, our Eq. (8) is numerically meaningless, while his equation based on the same data is mathematically sound. This raises the question, how poor do one’s reasoning skills have to be in order for one to contradict himself in such a ridiculous manner? We will let you be the judge …
We have shown in this reply that all criticism of our Equations (7) and (8) by Mr. Eschenbach is without merit. We emphasized the need for better understanding of and focusing on the big picture that our theory conveys. We propose to shift the discussion from meaningless argumentations about number of regression coefficients or number of significant digits of constants used, to how pressure as a force controls temperature and climate. In this regard, we would like to issue an appeal to all of you, who are capable of carrying out an intelligent discussion at a decent academic level to stop engaging in pseudoscientific, besides-the-point fruitless debates. We are here to discuss and offer a resolution to the current climate science debacle and welcome everyone who shares that goal. We are not here to promote or engage in endless circular talks or teach laymen ‘skeptics’ basic math and high-school level physics. Hence, we will no longer participate in dialogs of the kind that prompted this reply. We urge all sound thinking readers to do the same.