Guest Post by Willis Eschenbach
After I published my previous post, “An Observational Estimate of Climate Sensitivity“, a number of people objected that I was just looking at the average annual cycle. On a time scale of decades, they said, things are very different, and the climate sensitivity is much larger. So I decided to repeat my analysis without using the annual averages that I used in my last post. Figure 1 shows that result for the Northern Hemisphere (NH) and the Southern Hemisphere (SH):
Figure 1. Temperatures calculated using solely the variations in solar input (net solar energy after albedo reflections). The observations are so well matched by the calculations that you cannot see the lines showing the observations, because they are hidden by the lines showing the calculations. The two hemispheres have different time constants (tau) and climate sensitivities (lambda). For the NH, the time constant is 1.9 months, and the climate sensitivity is 0.30°C for a doubling of CO2. The corresponding figures for the SH are 2.4 months and 0.14°C for a doubling of CO2.
I did this using the same lagged model as in my previous post, but applied to the actual data rather than the averages. Please see that post and the associated spreadsheet for the calculation details. Now, there are a number of interesting things about this graph.
First, despite the nay-sayers, the climate sensitivities I used in my previous post do an excellent job of calculating the temperature changes over a decade and a half. Over the period of record the NH temperature rose by 0.4°C, and the model calculated that quite exactly. In the SH, there was almost no rise at all, and the model calculated that very accurately as well.
Second, the sun plus the albedo were all that were necessary to make these calculations. I did not use aerosols, volcanic forcing, methane, CO2, black carbon, aerosol indirect effect, land use, snow and ice albedo, or any of the other things that the modelers claim to rule the temperature. Sunlight and albedo seem to be necessary and sufficient variables to explain the temperature changes over that time period.
Third, the greenhouse gases are generally considered to be “well-mixed”, so a variety of explanations have been put forward to explain the differences in hemispherical temperature trends … when in fact, the albedo and the sun explain the different trends very well.
Fourth, there is no statistically significant trend in the residuals (calculated minus observations) for either the NH or the SH.
Fifth, I have been saying for many years now that the climate responds to disturbances and changes in the forcing by counteracting them. For example, I have held that the effect of volcanoes on the climate is wildly overestimated in the climate models, because the albedo changes to balance things back out.
We are fortunate in that this dataset encompasses one of the largest volcanic eruptions in modern times, that of Pinatubo … can you pick it out in the record shown in Figure 1? I can’t, and I say that the reason is that the clouds respond immediately to such a disturbance in a thermostatic fashion.
Sixth, if there were actually a longer time constant (tau), or a larger climate sensitivity (lambda) over decade-long periods, then it would show up in the NH residuals but not the SH residuals. This is because there is a trend in the NH and basically no trend in the SH. But the calculations using the given time constants and sensitivities were able to capture both hemispheres very accurately. The RMS error of the residuals is only a couple tenths of a degree.
OK, folks, there it is, tear it apart … but please remember that this is science, and that the game is to attack the science, not the person doing the science.
Also, note that it is meaningless to say my results are a “joke” or are “nonsense”. The results fit the observations extremely well. If you don’t like that, well, you need to find, identify, and point out the errors in my data, my logic, or my mathematics.
All the best,
w.
PS—I’ve been told many times, as though it settled the argument, that nobody has ever produced a model that explains the temperature rise without including anthropogenic contributions from CO2 and the like … well, the model above explains a 0.5°C/decade rise in the ’80s and ’90s, the very rise people are worried about, without any anthropogenic contribution at all.
[UPDATE: My thanks to Stephen Rasey who alertly noted below that my calculation of the trend was being thrown off slightly by end-point effects. I have corrected the graphic and related references to the trend. It makes no difference to the calculations or my conclusions. -w.]
[UPDATE: My thanks to Paul_K, who pointed out that my formula was slightly wrong. I was using
∆T(k) = λ ∆F(k)/τ + ∆T(k-1) * exp(-1 / τ)
∆T(k) = λ ∆F(k)(1 – exp(-1/ τ)) + ∆T(k-1) * exp(-1 / τ)
The result of the error is that I have underestimated the sensitivity slightly, while everything else remains the same. Instead of the sensitivities for the SH and the NH being 0.04°C per W/m2 and 0.08°C per W/m2 respectively, the correct sensitivities should have been 0.05°C per W/m2 and 0.10°C per W/m2.
-w.]
joeldshore says:
June 2, 2012 at 8:05 am
Joel, you’re disagreeing with the citations you provided earlier. One of them says:
So it’s claiming that observations show a constant increase in relative humidity, and it says that the models agree with it. But you say the models show stable or decreasing relative humidity …
Finally, FWIW (which may not be much), the model results from the NCEP Reanalysis Model are here … they show variations but no overall trend in either relative or specific humidity over the last decades.
So I’d say that despite your certainty, the question is still wide open …
Is there water vapor feedback? Me, I’d say yes, but it is overwhelmed by the clouds. Note in my analysis, for example, that it spans a fairly wide temperature range. During the summer we must get much more moisture in the air than in the winter … but despite that, the resulting temperature is well modeled using a constant climate sensitivity. This observational evidence strongly supports the secondary role of water vapor in determining the temperature.
You still haven’t grasped the nettle, Joel. I’ve hindcasted the temperature, with shocking accuracy, using nothing but the clouds and the sun. This means that whatever other feedbacks and mechanisms might be involved, there is very, very little left for them to explain … and thus it indicates that the other mechanisms play only a very small role in determining temperature.
w.
lgl says:
June 2, 2012 at 4:44 am
Thanks, lgl. You are right, but remember that these are Watts per square metre, not total amounts. As a result, they should be “2* Net Sun Global”, because you cannot simply add them, you need to average them instead.
w.
Overall this is a great article. The astounding simplicity and the very small residuals suggests you are on the right track.
=====
Agreed. The accuracy of the result suggests either Willis has discovered a truly remarkable correlation, or there is a nonsense in the math, or that we have a statistical fluke of low probability.
Many have argued that the complexity of the climate system prevents accurate prediction from first principles as is being attempted in the climate models. The problem is computer run times for many (most) problems grows exponentially with problem size, making them impractical top solve on computers.
In fact,much of computer science is concerned with discovering efficient algorithms. Computer methods that do not grow exponentially with run times. Otherwise, you have no option but to reduce resolution, as is done in the climate models. However, this leads to problems of reliability in the answers, as has been seen.
Another method is to tackle the problem from a completely different direction. To bypass first principles and look for a “rule of thumb”. Something that gives a quick, reliable, but less precise answer. In other words, an answer that it close to correct all the time. As compared to an answer that it very precise, but sometimes spectacularly wrong.
This is what I see in the work Willis has done. He has found what appears to be a very good rule of thumb to predict temperature, that appears to be more reliable than the method of first principles used by climate models.
What would I believe be an interesting exercise to validate the model would be for Willis to run the model forward, to provide a range of prediction of future temperatures based on IPCC scenarios for CO2, solar activity and albedo and see where that leads.
Have the prediction down on paper and plot it as we go forward, with its own page on WUWT to track performance for the whole world to see. A contest between the “Willis Method” or “Eschenbach Technique” and the best of the climate models. Keep a running score on the accuracy of the various techniques.
I agree with others that you have made a reasonable case that albedo is the response variable which has the greatest control over temperature, but you have not provided evidence showing that CO2 contribution to albedo change is negligible. Given the narrow bands in which CO2 affects the albedo, this should be straightforward in and of itself, but then you hit the warmist claim of positive feedback with water vapor, and since water vapor contributes significantly to albedo, we are right back to the fundamental argument over whether there is a positive feedback or not.
So, all in all, it appears to me that you have put forward another way of looking at the problem, but have not really resolved anything in a way which would force the opposing side out of their bunkers.
Bart says:
June 2, 2012 at 10:33 am
So, all in all, it appears to me that you have put forward another way of looking at the problem, but have not really resolved anything in a way which would force the opposing side out of their bunkers.
=====
Agreed, because everyone is concentrating on “why” without simply getting on with the science. Everyone agrees that the sun, albedo and CO2 are in some fashion related to temperature. All that is really being argued is how much and in what direction. And folks are pulling out pieces of the puzzle to support their arguments.
However, what Willis has done is to draw a line on the table that says “it doesn’t matter how many pieces there are to the puzzle, it doesn’t matter how big the pieces are, it doesn’t matter their color, size,or shape. No matter what, when assembled correctly, the puzzle will fit snugly within the area drawn, with very little error”
What Willis has really shown is that understanding the mechanism by which the various forcings determine temperature is not required to accurately predict temperature. We know this to be true for the tides. Willis has now demonstrated this to be true for global temperature.
What Willis has shown is that it is possible to accurately predict future temperatures as we currently do now with the tides. From very basic observational evidence using simple, low-cost computer models.
What Willis has shown is that the hundreds of millions of dollars spent on complex solutions was money spent going in the wrong direction. The solution lay not in mastering complexity, but in mastering simplicity.
Bart:
Your post at June 2, 2012 at 10:33 am concludes by saying to Willis:
I strongly agree. I remind that I concluded my above post June 1, 2012 at 12:52 am by saying to Willis:
Richard
Another analogy comes to mind. Everyone is arguing how many angels (forcings) are dancing on the head of a pin (temperature). Some say the pin is large because there are lots of angels. Some say the pin is small because there are few angels. Some say there are many angels but the angels are small. Others say there are few angels but the angels are large.
Willis has shown that no mater how many angels, the pin is “this big”. Since the real question we are trying to answer is the size of the pin, not the number of angels, it is time to stop arguing over the number of angels as a means of measuring the size of the pin.
The number of angels and their size is not needed. We know the size of the pin, from the song the angels are singing (solar, albedo, co2).
joeldshore says: June 2, 2012 at 8:05 am
Steve Keohane says:
Atmospheric RH% is going down. http://i38.tinypic.com/30bedtg.jpg
(1) Relative humidity going down is not incompatible with absolute humidity going up. (Most climate models predict relative humidity to stay about constant or decrease a bit overall as the climate warms.)
(2) You give no source or other information for the data set you show but I believe you have cherry-picked a particular re-analysis of radiosonde data with known severe problems. This data does not agree with the much better satellite data available (and I think it even disagrees with other re-analyses of the radiosonde data).
Sorry to disrupt your fantasy, I did not cherry pick anything. Here is an updated version of that graph, I simply stored these two graphs when I came across them in 2008 and 2012, I did not generate them. http://i48.tinypic.com/2qlfnzn.jpg
I understood them to be US gov’t data.
Here is another albeit shorter time frame and different pressures from NOAA
http://i48.tinypic.com/14mwa5y.jpg
ferd berple:
With respect, you are misunderstanding the situation created by Willis analysis.
For example, at June 2, 2012 at 10:55 am you say;
But Willis’ analysis denies that.
As Willis says himself
So, he specifically states that
“Sunlight and albedo seem to be necessary and sufficient variables to explain the temperature changes over that time period.”
And he specifically lists “CO2” was something he “did not use” in his analysis.
This implies that CO2 is NOT a significant variable which affects global temperature.
But, as I said in my post at June 1, 2012 at 12:52 am
And, importantly, as I said in my post at June 1, 2012 at 4:10 am
So, Willis analysis implies the effect of atmospheric CO2 concentration is not significant to recent global temperature change but does not prove it.
Furthermore, at June 2, 2012 at 10:55 am you suggest
No, Willis has NOT shown that. Indeed, he refutes that saying at June 1, 2012 at 1:31 am
Willis’ analysis is VERY important. So, understanding what his analysis does and what it does not do are also important.
Richard
An expanded version of this should be in Nature Climate Change,
instead of this month’s sprawling pile of modelling articles:
Quantifying Future Climate Change (7 pages)
Evaluation of Climate Models Using Paleoclimate Data (8 pages)
Multistability and Critical Thresholds of the Greenland Ice Sheet (4)
Overestimation of Mediterranean summer temperature projections due to MODEL DEFICIENCIES (imagine seeing that phrase at all, in a title yet!)
This is atop the usual pile of Doomsday-Soon articles!
Though you need a subscription if you actually want to bother reading these,
the incestuous list of references suffices to spell out the closed-circle aspect of this field.
How much longer before the rank odor of sanctimonious fraud is so overwhelming that actual science, such as this article, penetrates into that rotten field to bring long-overdue sanity?
When will the supercomputers they egregiously misuse be re-purposed for useful work?
Willis,
As usual, a very thought provoking post. What you touch on here I see as an extension of your post on ‘The cold equations’ (Jan 28 2011). Together these seem to point to a very good path for growing a model from the simple to the complex by adding terms to eliminate discrepancies. What I like about your approach is that it can be directly related to physical quantities. With an electrical background I kind of see it as building up an ‘analog computer and solving the equations digitally. The climate system is obviously a system of many modes, each having its own time constant. But it is still frequently found informative to explore simple single time constant approximations and then extend them.
Ok. So I took your single time constant model which seems to show encouraging results and looked at it as an energy storage unit of capacity C=mass times heat capacity., a current source (in electrical terms , which is the solar energy input. And a coupling out of the system. This output coupling is the parameter which determines your tau value. This coupling in the present case is a function of Co2, moisture and all the rest. Your model assumes this constant at some value (a place to start). When I use The sigma*T^4 formula for energy out. I find that I have to adjust (epsilon if you like) to a value of ~ 0.6 to get the temperature to rise to the measured level.
An extension here could be this as adjusting the temperature to the TOA level. A resistance to TOA and a small atmosphere storage reservoir could be added to incorporate a more physical model. Also the ocean could be broken into shallow /deep storage with resistance between.
In the ultra simple model above The heat capacity adjusted to roughly the equivalent of 70 M depth of water matches the temperature phase and amplitude.
This is all pretty quick, so I hope there are not too many mistakes.
I see Bart is on the thread. Help us out here Bart, I think this is along the lines of the systems analysis approach that you advocate.
The Matlab/Octave code for a crude step integration solver is attached. The result is the figure below
load alb_temp;%load year,albedo,solar flux,avg temp,for globe,nh,sh.
% from W. Eschenbach post may 31,2012.
%These data are in a 10 column array named at;
%C=1e4;%Guess 1 atm air columnheat capacity
C=2.9e9/15;%700M depth of water(adjusted by divisor)
sigma=5.67e-8;%Boltzmann constant
%B=0.3;%Bond albedo
Tn=260;%Start the calculation at this temperature
%otherwise build up ts long due to T^4 (not a lineat eq).
delt=1;%time increment (mo)
t=at(:,1);
B=at(:,3);
nsam=length(t);
%T=at(8,:);%global average temperature (C)
T=zeros(size(t));
I=(1-B).*at(:,6);%average global solar flux (W/M^2)
Tm=at(:,9);%global temp
%hold
for n=1:nsam;
Tn=Tn+3600*24*30*delt*(I(n)-0.615*sigma*Tn^4)/C;
T(n)=Tn;
endfor
plot(t,T-273,’r’)
hold
plot(t,Tm,’b’);
plot(t,(I-mean(I))/10+mean(Tm),’g’)
richardscourtney says:
June 2, 2012 at 11:13 am
In a sane and rational world, the shoe would be on the other foot: the alarmists would have to prove other factors were not responsible, not the realists having to prove CO2 is not.
I conclude from this evidence that we do not live in a sane and rational world.
“• I have been showing that the recent rise in atmospheric CO2 concentration can be attributed to factors other than anthropogenic CO2 (and have been vilified for it).”
And, I have recently shown that it must be attributable to other factors, though you disagree, and I’m sure neither one of us wants to revisit that issue at this time. Interested parties can review the debate at this thread.
Hi Willis,
As argues before, with the annual cycle you are just calculating the frequency response of a simgle frequency, , which doesn’t have an effect on multidecadal periods. Other people also arrived at 0.1 K/wm-2 for the annual cycle so your value isn’t new either
http://members.casema.nl/errenwijlens/co2/Climate_sensitivity_period.gif
I suggest you read the recent paper by J. H. van Hateren who considers the complete spectral response one of the conclusions reads
” The transient climate response (response after 70 years of 1 % yearly rise of CO2 concentration) is 1.5 ± 0.2 °C.”
http://rankexploits.com/musings/2012/empirically-based-estimate-of-climate-sensitivity/
the online paper
J. H. van Hateren, 2012, A fractal climate response function can simulate global average temperature trends of the modern era and the past millennium, Climate Dynamics, 10.1007/s00382-012-1375-3
http://www.springerlink.com/content/348g07361627360x/fulltext.html
Willis,
”
Second, the sun plus the albedo were all that were necessary to make these calculations. I did not use aerosols, volcanic forcing, methane, CO2, black carbon, aerosol indirect effect, land use, snow and ice albedo, or any of the other things that the modelers claim to rule the temperature. Sunlight and albedo seem to be necessary and sufficient variables to explain the temperature changes over that time period.
”
I fully agree with your first sentence and third sentence. but you are not quite right on the second. When you used a real albedo measurement, you included the effects of aerosols, volcanic forcing, black carbon, land use, snow and ice albedo and EVERY other forcing that has scattering and reflection as its operational effect. I agree too that methane and co2 are not part of your model, except for what, if any, measureable effects upon temperature exist which are actually caused by them. Considering that your model describes or tracks the real world temperatures to the extent that around 90% or better is covered, it would seem there is extremely little contribution left for ghgs to be a part of after the albedo factor. For those detractors, that 90% means that 90% of the temperature variation, trends and fluctuations, are described by Willis’ model which means that this vast majority of temperature variation is not caused by co2.
As I recall, our beloved CAGW fanatics (Hansen et al ???) have managed to create a model that explains only around 30% of the variation which is usually not considered to be statistically significant – except in catastrophic climate disaster research papers. Of course, there is also the factor of causality which is not determined – which means the CAGW fanatics don’t actually know if co2 level increase changes cause warming or whether warming causes co2 level increases. Chemists on the other hand know what happens when you warm a liquid that has dissolved gases present which tends to support the warming causing co2 gas increases. The causality in Willis’ model is a bit more straight forward and actually makes sense.
Hans Erren:
Thankyou for your providing a post at June 2, 2012 at 1:17 pm which includes evidence to support your statement that “with the annual cycle you are just calculating the frequency response of a simgle frequency”. However, you ignore the fact that Willis Eschenbach has proved by demonstration that his method emulates global and hemispheric changes for the period from 1984 to 1998. Therefore, he has demonstrated that “a single frequency” is a sufficient model.
Importantly, your post misses the main point. Willis Eschenbach has shown that
“Sunlight and albedo seem to be necessary and sufficient variables to explain the temperature changes over that time period” (i.e. from 1984 to 1998).
The effect of rising CO2 is not a necessary input to his model.
Hence, those who want to claim e.g.
” The transient climate response (response after 70 years of 1 % yearly rise of CO2 concentration) is 1.5 ± 0.2 °C.”
need to explain why the analysis of Willis Eschenbach holds for the period of 1984 to 1998 but does not apply “after 70 years”.
Richard
cba:
re your post at June 2, 2012 at 2:17 pm, please read my post at June 2, 2012 at 12:03 pm.
Richard
Steve Keohane says:
In other words, you came across some data that agrees with what you want to believe. So, despite the fact that it disagrees with lots of data better suited for the purpose of looking at long-term trends and you have no idea where it is from or what the caveats and issues associated with it are, you conclude things from it that are in contradiction both with what the scientific community has concluded and basic physical principles.
If that is not a cherry pick, I don’t know what is!
Richard, 14 years is way too short to satisfy the Nyquist theorem for a centenial signal response.
Willis Eschenbach says:
Actually, I think the quote that you took from Soden is just poorly worded. If you read it in the context of the rest of the paper, it is clear that he is not saying they show a constant increase in the relative humidity. He is saying that they show an increase in upper tropospheric moisture that is equivalent to what you get if you assume that the relative humidity remains constant as things warm. He certainly should have worded that better, but if you read the rest of the paper, it will be obvious that this is what he was saying. [I talked about even slightly decreasing relative humidity as being compatible with the models because Dessler and coauthors wrote a paper in which he found a small decrease in relative humidity with time from the satellite data and thought that this was perhaps a bit incompatible with what the climate models predicted. However, in another paper a few years later, they actually checked what various models predicted and found that overall on average they did predict a slight decrease in relative humidity (although I think, within the variability of the models, it also included a completely stable relative humidity).]
Yes, which is why this NCEP reanalysis has been so popular with AGW skeptics. But, there are apparently known severe issues with the long-term trends in that reanalysis and it is contradicted by lots of other evidence from satellites (and other reanalyses too, I think) that show the moistening trend.
Statements like this drive me batty. If you think the seasonal cycle is in contradiction with the role of water vapor, then you should be able to demonstrate this, for example, by showing that climate models exaggerate the seasonal cycle as a result of the important role that the water vapor feedback plays in them. But, of course, that isn’t true since, as I noted, the analysis that I know of that looked at seasonal cycles that I linked to previously concluded that it supports climate sensitivity right in the IPCC range.
It may shock you but it doesn’t shock me. Yes, a simple model with a single time scale can do a good job at modeling the seasonal cycle, which is undoubtably dominated by solar effects. As to how well you did with the overall trend, I already told me the reasons for being skeptical about that; I doubt if the albedo data is even good enough to get that without large error bars.
And, your model doesn’t speak at all to what feedbacks are or are not present.
Hans Erren says:
June 2, 2012 at 1:17 pm
“I suggest you read the recent paper by J. H. van Hateren who considers the complete spectral response one of the conclusions reads
” The transient climate response (response after 70 years of 1 % yearly rise of CO2 concentration) is 1.5 ± 0.2 °C.”
”
Hans, why would you even bother suggesting reading a paper that treats albedo, the most important variable in the whole climate discussion, as a constant. Granted, there are no albedo records that date back 70 years so van Hateren could not properly reconstruct a long term sensitivity at all. In the short term where we actually have some albedo measurements, we know that albedo varies by at least 10%. Anyone trying to determine sensitivity without considering this factor is doing nothing but generating random numbers. It’s like ascribing a 4% variation in TSI just for the heck of it with no data to back it up.
It is just like every other sensitivity calculation that has been done which ignores albedo as a significant variable. The results are worth absolutely nothing at all. It’s like trying to figure the gas mileage of a hybrid car without considering it gets plugged into the power charger every night and the engine only runs when the car batteries get low. Drive it only 10 miles a day and your gas mileage becomes almost infinite but your electric bill goes up significantly.
joeldshore
“Actually, I think the quote that you took from Soden is just poorly worded. If you read it in the context of the rest of the paper, it is clear that he is not saying they show a constant increase in the relative humidity. He is saying that they show an increase in upper tropospheric moisture that is equivalent to what you get if you assume that the relative humidity remains constant as things warm. He certainly should have worded that better, but if you read the rest of the paper, it will be obvious that this is what he was saying. [I talked about even slightly decreasing relative humidity as being compatible with the models because Dessler and coauthors wrote a paper in which he found a small decrease in relative humidity with…”
OOPS! someone better tell lacis and hansen. They assume something a bit different, where by rising T causes rising humidity and dwindling cloud cover in order to have an unstable positive feedback sufficiently strong to explain how a 1 deg C rise in co2 forcing will cause a 3-6 deg C total temperature rise. It seems that if RH stays constant, the water vapor feedback from a 5 deg C rise over the entire atmospheric column only adds about 30% more h2o vapor which is far less than a doubling and is in fact less than the effect of the co2 doubling – so much for the largest official ipcc feedback.
Hey, your co2 doubling + h2o vapor at constant RH(assuming a 5 deg C T rise) gives you a good 5 or 6 W/m^2 and you only need an additional 20 W/m^2 or so to get to the 5 deg C rise.
And, oh, you also need to explain why there would be a cloud decrease with a substantial increase in h2o vapor in the atmosphere and a slight rise in T. That means since we’re currently at around 62% cloud cover that this is as much as we can get. Drop the T and you start to lose cloud cover. Raise the T (accordning to lacis and hansen) and you start to lose cloud cover. Evidently, the only way Earth can get more than this 60% cloud cover is to have something other than h2o clouds – like venus. SARC OFF/
Hans Erren:
Your post at June 2, 2012 at 2:58 pm is an evasion of – and not an answer to – either of the points in my post addressed to you at June 2, 2012 at 2:41 pm.
Please answer the points if you can.
Richard
Thanks Willis
Here is my result for NH
http://virakkraft.com/Sun-Temp-NH.png
Net Sun: +1 W/m2 (june-june)
Temp: +0.25 C (july-july)
or close to 1 C pr CO2 doubling
Richard, apparently you don’t understand why a decade of observations is not sufficient to calculate the amplification factor for a century or a millenium.
Others: I introduced the van Hateren paper because he uses an elegant frequency domain approach, he also uses R scripts. But the most important is that even using Mann reconstructions, van Hateren’s climate sensitivity doesn’t exeed 2 degrees for CO2 doubling. Which is quite promising when using reconstructions that aren’t hockeysticks.
richardscourtney says:
June 2, 2012 at 12:03 pm
With respect, you are misunderstanding the situation created by Willis analysis.
Second, the sun plus the albedo were all that were necessary to make these calculations.
======
Thanks Richard, I haven’t had the time to check the math. If no CO2 was required in this model, then this implies the CO2 sensitivity is 0 C, given the size of the residuals. But this wasn’t the result, so I’ve missed something.