Stefani's Paper Illustrates the Failure of MDPI Peer Review

A recent paper[1] published in the MDPI journal Climate by Frank Stefani provides a wonderful illustration of why we should never treat papers from MDPI journals as having any competent, let alone robust peer review. This paper argues that TCR = 1.1°C (0.6°C - 1.6°C) for doubling CO2. I'm not going to evaluate the entire paper here, since that would take too much time. The paper does make some counterfactual claims, like there's a "nearly perfect correlation of solar activity with temperatures over about 150 years." That's objectively false, but the correlation between CO2 forcings and GMST has an r^2 = 0.88. There's also some comical contrarian alarmism in this paper: "we fear that the huge Milankovitch drivers will—perhaps much too soon—massively interfere with the solar and anthropogenic factors that were considered in this paper." There's a lot we could say about this paper, but I want to focus here on some elementary math errors that would have been noticed by any competent reviewer. I can say this with confidence because I found these errors, and MDPI reviewers should have the capacity to find any of the errors I can find.

Blatant Math Errors

As a plausibility check for his (way too low) TCR estimate, Stefani compared his results to the results of other studies that have documented observable evidence of the greenhouse effect. When I looked at his numbers for two of these studies a couple things became apparent. The most obvious problem is that Stefani confused what we may call CO2-only (or pre-feedback) sensitivity (I'll refer to this as S) with TCR. TCR is the expected temperature response that would occur from doubling CO2 before an equilibrium temperature is reached and without considering the earth's energy imbalance. It can be approximated by the slope of the a plot of GMST on the y-axis and CO2 forcings on the x-axis (provided that all other forcings cancel out). However, S refers to the equilibrium temperature response to doubling CO2 before the effects of positive or negative feedbacks. These are two different values, with S almost certainly smaller than TCR.

Stefani then miscalculated S, and that's putting it generously. What he actually did is assume a value for S and then calculate a different value for S. Let's use his quote about Feldman's paper[2] to see what he's doing: 

Feldman et al. published results from two clear-sky surface radiative forcing measurements with the Atmospheric Emitted Radiance Interferometer (AERI) between 2000 and 2010, when the CO2 concentration increased (according to their estimate) by 22 ppm. They observed an increase of 0.2 W/m2 during this decade, which amounts to 2.4 W/m2 (per 2× CO2). With the usual zero-feedback sensitivity parameter of 0.27 K/(W/m2), this would translate into 0.65 K (per 2× CO2). If we were to use the modified value of 0.31 K/(W/m2) (which allows for variations with latitude [75]), we would obtain 0.75 K.

This is a convoluted mess. Here's what he did. He took the following equation to calculate radiative forcing (RF) from an increase in CO2:

RF= α*ln(C1/C0) 

In this equation, C0 and C1 are an initial and final CO2 concentration. In most papers the RF for 2xCO2 is ~3.71 W/m^2, so if you solve for He solved for α in this equation, you get α = 5.35. However, what Stefani did is take values from Feldman's paper: RF = 0.2 W/m^2, C0=370 ppm and C1=392 ppm from the Feldman paper and solve for α. So,

RF = 0.2 = α*ln(392/370) and α = 3.46

Then to calculate what would be the RF (2xCO2), he solved,

 

RF = 3.46*ln(2) = 2.40 W/m^2

Now there is no temperature value here, so you can't calculate S or TCR from this. So what he did is insert a "zero-feedback sensitivity parameter" of 0.27°C/W/m^12. This uses another standard equation of dT = λ*RF. Stefani is is just assuming a "CO2-only sensitivity" (S) of 1°C and an RF (2xCO2) = 3.71 W/m^2 where α = 5.35, not 3.46. Assuming α = 5.35, he calculated 1/3.71 = 0.27°C/W/m^2. If you use 1.15°C for S then you get his alternate of 0.31°C/W/m^2. With this assumption of  S = 1°C, he takes his calculation of an RF of 2.40 W/m^2 for 2xCO2 in Feldman's paper and calculated S:

S = 0.27*2.4 = 0.65°C.

Notice the trick here. He assumes S = 1°C in order to calculate S = 0.65°C. It cannot be that S can equal both to 0.65°C and 1°C. The contradiction here comes from the fact that he used two values for α and therefore two values for RF (2xCO2). If he had used 2.4 W/m^2 consistently, he of course would just end up with the same S value he assumed: 1/2.4 = 0.42°C/W/m^2 and S = 0.42*2.4 = 1°C. And this S value would simply be the pre-feedback sensitivity he assumed from the beginning. I could do the same calculations assuming an S of 3°C and arrive at an S of 3°C as well, provided that I keep the same value for α in all my calculations.

Stefani goes on to make the same mistake with the numbers he took from the Rentsch paper.[3] Using C0 = 373 ppm, C1 = 410 ppm, and RF = 0.358 W/m^2 = α*ln(410/373), he calculated α = 3.79, and he calculated RF (2xCO2) = 3.79*ln(2) = 2.63 W/m^2. Then he calculated S = 2.63*0.27 = 0.71°C. However, if he used consistent values for α and RF (2xCO2), he would have calculated S to be exactly what he assumed them to be (either 1°C or 1.15°C).

Conclusion  
What should be obvious here is that Stefani did not confirm anything about his calculations using values from either Feldman or Rentsch. What he did is assume a no-feedback sensitivity value S = 1°C (or 1.15°C) and then use two different values for α to lower the value he assumed. At no point did he actually calculate either TCR or S from the data in these papers. Even if he had done his calculations correctly, he would only have been able to calculate the values he assumed.

Competent reviewers would be able to see these errors if they looked at this paper carefully. I know this because I found them. These are not errors that can be explained away; these are obvious mathematical errors. The best explanation for this paper is that there was no competent peer review. And these types of errors are not uncommon in climate-related papers published in MDPI journals that I've examined before. This is par for the course. So now when people share with me papers from MDPI journals, I claim that the burden of proof is on the person promoting the paper to show that the paper isn't total garbage.

References:

[1] Stefani F. Solar and Anthropogenic Influences on Climate: Regression Analysis and Tentative Predictions. Climate. 2021; 9(11):163. https://doi.org/10.3390/cli9110163

[2] Feldman DR, Collins WD, Gero PJ, Torn MS, Mlawer EJ, Shippert TR. Observational determination of surface radiative forcing by CO2 from 2000 to 2010. Nature. 2015;519(7543):339‐343. doi:10.1038/nature14240

[3] Rentsch, C.P. Radiative forcing by CO2 observed at top of atmosphere from 2002 to 2019. arXiv 2019, arXiv:1911.10605